Geometry of logarithmic strain measures in solid mechanics
GGeometry of logarithmic strain measures insolid mechanics
Patrizio Neff , Bernhard Eidel and Robert J. Martin Published in Arch. Rational Mech. Anal., vol. 222 (2016), 507–572.DOI: 10.1007/s00205-016-1007-xIn memory of Giuseppe Grioli (*10.4.1912 – † November 1, 2016
Abstract
We consider the two logarithmic strain measures ω iso = (cid:107) dev n log U (cid:107) = (cid:107) dev n log √ F T F (cid:107) and ω vol = | tr(log U ) | = | tr(log √ F T F ) | = | log(det U ) | , which are isotropic invariants of the Hencky strain tensor log U , and show that they can be uniquelycharacterized by purely geometric methods based on the geodesic distance on the general linear groupGL( n ). Here, F is the deformation gradient, U = √ F T F is the right Biot-stretch tensor, log denotesthe principal matrix logarithm, (cid:107) . (cid:107) is the Frobenius matrix norm, tr is the trace operator and dev n X = X − n tr( X ) · is the n -dimensional deviator of X ∈ R n × n . This characterization identifies the Hencky (ortrue) strain tensor as the natural nonlinear extension of the linear (infinitesimal) strain tensor ε = sym ∇ u ,which is the symmetric part of the displacement gradient ∇ u , and reveals a close geometric relationbetween the classical quadratic isotropic energy potential µ (cid:107) dev n sym ∇ u (cid:107) + κ ∇ u )] = µ (cid:107) dev n ε (cid:107) + κ ε )] in linear elasticity and the geometrically nonlinear quadratic isotropic Hencky energy µ (cid:107) dev n log U (cid:107) + κ U )] = µ ω + κ ω , where µ is the shear modulus and κ denotes the bulk modulus. Our deduction involves a new fundamentallogarithmic minimization property of the orthogonal polar factor R , where F = R U is the polar decom-position of F . We also contrast our approach with prior attempts to establish the logarithmic Henckystrain tensor directly as the preferred strain tensor in nonlinear isotropic elasticity. Key words: nonlinear elasticity, finite isotropic elasticity, Hencky strain, logarithmic strain, Hencky energy,differential geometry, Riemannian manifold, Riemannian metric, geodesic distance, Lie group, Lie algebra,strain tensors, strain measures, rigidity
AMS 2010 subject classification: 74B20, 74A20, 74D10, 53A99, 53Z05, 74A05 Corresponding author: Patrizio Neff, Head of Chair for Nonlinear Analysis and Modelling, Fakult¨at f¨ur Mathematik,Universit¨at Duisburg-Essen, Campus Essen, Thea-Leymann Straße 9, 45141 Essen, Germany, email: patrizio.neff@uni-due.de,Tel.:+49-201-183-4243 Bernhard Eidel, Chair of Computational Mechanics, Universit¨at Siegen, Paul-Bonatz-Straße 9-11, 57068 Siegen, Germany,email: [email protected] Robert J. Martin, Chair for Nonlinear Analysis and Modelling, Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-Essen,Campus Essen, Thea-Leymann Straße 9, 45141 Essen, Germany, email: [email protected] a r X i v : . [ m a t h . DG ] O c t ontents + ( n ) as a Riemannian manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103.2 The geodesic distance to SO( n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 + ( n ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Further mechanical motivations for the quadratic isotropic Hencky model based on logarithmicstrain tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2.1 From Truesdell’s hypoelasticity to Hencky’s hyperelastic model . . . . . . . . . . . . . 284.2.2 Advantageous properties of the quadratic Hencky energy . . . . . . . . . . . . . . . . . 30 A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.2 Linear stress-strain relations in nonlinear elasticity . . . . . . . . . . . . . . . . . . . . . . . . 49A.3 Tensors and tangent spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49A.4 Additional computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51A.5 The principal matrix logarithm on Sym + ( n ) and the matrix exponential . . . . . . . . . . . . 51A.6 A short biography of Heinrich Hencky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 The concept of strain is of fundamental importance in elasticity theory. In linearized elasticity, one assumesthat the Cauchy stress tensor σ is a linear function of the symmetric infinitesimal strain tensor ε = sym ∇ u = sym( ∇ ϕ − ) = sym( F − ) , where ϕ : Ω → R n is the deformation of an elastic body with a given reference configuration Ω ⊂ R n , ϕ ( x ) = x + u ( x ) with the displacement u , F = ∇ ϕ is the deformation gradient , sym ∇ u = ( ∇ u + ( ∇ u ) T )is the symmetric part of the displacement gradient ∇ u and ∈ GL + ( n ) is the identity tensor in the groupof invertible tensors with positive determinant. In geometrically nonlinear elasticity models, it is no longernecessary to postulate a linear connection between some stress and some strain. However, nonlinear strain Although F is widely known as the deformation “gradient”, F = ∇ ϕ = Dϕ actually denotes the first derivative (or the Jacobian matrix ) of the deformation ϕ . [15, 16, 24] (cf. Appendix A.2 forexamples).There are many different definitions of what exactly the term “strain” encompasses: while Truesdell andToupin [204, p. 268] consider “ any uniquely invertible isotropic second order tensor function of [ the rightCauchy-Green deformation tensor C = F T F ]” to be a strain tensor, it is commonly assumed [106, p. 230](cf. [107, 108, 23, 159]) that a (material or Lagrangian ) strain takes the form of a primary matrix function of the right Biot-stretch tensor U = √ F T F of the deformation gradient F ∈ GL + ( n ), i.e. an isotropic tensorfunction E : Sym + ( n ) → Sym( n ) from the set of positive definite tensors to the set of symmetric tensors ofthe form E ( U ) = n (cid:88) i =1 e( λ i ) · e i ⊗ e i for U = n (cid:88) i =1 λ i · e i ⊗ e i (1.1)with a scale function e : (0 , ∞ ) → R , where ⊗ denotes the tensor product, λ i are the eigenvalues and e i arethe corresponding eigenvectors of U . However, there is no consensus on the exact conditions for the scalefunction e; Hill (cf. [107, p. 459] and [108, p. 14]) requires e to be “suitably smooth” and monotone withe(1) = 0 and e (cid:48) (1) = 1, whereas Ogden [162, p. 118] also requires e to be infinitely differentiable and e (cid:48) > , ∞ ).The general idea underlying these definitions is clear: strain is a measure of deformation (i.e. the changein form and size) of a body with respect to a chosen (arbitrary) reference configuration. Furthermore, thestrain of the deformation gradient F ∈ GL + ( n ) should correspond only to the non-rotational part of F . Inparticular, the strain must vanish if and only if F is a pure rotation, i.e. if and only if F ∈ SO( n ), whereSO( n ) = { Q ∈ GL( n ) | Q T Q = , det Q = 1 } denotes the special orthogonal group. This ensures that theonly strain-free deformations are rigid body movements: F T F ≡ = ⇒ ∇ ϕ ( x ) = F ( x ) = R ( x ) ∈ SO( n ) (1.2)= ⇒ ϕ ( x ) = Q x + b for some fixed Q ∈ SO( n ) , b ∈ R n ,where the last implication is due to the rigidity [174] inequality (cid:107) Curl R (cid:107) ≥ c + (cid:107)∇ R (cid:107) for R ∈ SO( n ) (witha constant c + > ε ≡ ε = sym ∇ u , then u is an infinitesimal rigid displacement of the form u ( x ) = A x + b with fixed A ∈ so ( n ) , b ∈ R n ,where so ( n ) = { A ∈ R n × n : A T = − A } denotes the space of skew symmetric matrices. This is due to theinequality (cid:107) Curl A (cid:107) ≥ c + (cid:107)∇ A (cid:107) for A ∈ so ( n ), cf. [151].In the following, we will use the term strain tensor (or, more precisely, material strain tensor ) to referto an injective isotropic tensor function U (cid:55)→ E ( U ) of the right Biot-stretch tensor U mapping Sym + ( n ) toSym( n ) with E ( Q T U Q ) = Q T E ( U ) Q for all Q ∈ O( n ) (isotropy)and E ( U ) = 0 ⇐⇒ U = ;where O( n ) = { Q ∈ GL( n ) | Q T Q = } is the orthogonal group and denotes the identity tensor. Inparticular, these conditions ensure that = U = √ F T F if and only if F ∈ SO( n ). Note that we do notrequire the mapping to be of the form (1.1). In a short note [32], R. Brannon observes that “ usually, a researcher will select the strain measure for which the stress-straincurve is most linear ”. In the same spirit, Bruhns [33, p. 147] states that “ we should [ . . . ] always use the logarithmic Henckystrain measure in the description of finite deformations. ”. Truesdell and Noll [203, p. 347] explain: “ Various authors [ . . . ] havesuggested that we should select the strain [ tensor ] afresh for each material in order to get a simple form of constitutive equation. [ . . . ] Every invertible stress relation T = f ( B ) for an isotropic elastic material is linear, trivially, in an appropriately defined,particular strain [ tensor f ( B )] . ” Similarly, a spatial or Eulerian strain tensor (cid:98) E ( V ) depends on the left Biot-stretch tensor V = √ F F T (cf. [74]). Seth-Hillfamily [190] E r ( U ) = (cid:40) r ( U r − ) : r ∈ R \ { } log U : r = 0 (1.3) λ e e e / e − e ˜e / Figure 1: Scale functions e r , ˜e r associated with the straintensors E r and (cid:101) E r = ( E r − E − r ) via eigenvalue λ . of material strain tensors , which includes the Biot straintensor E / ( U ) = U − , the Green-Lagrangian strain ten-sor E ( U ) = ( C − ) = ( U − ), where C = F T F = U is the right Cauchy-Green deformation tensor, the (ma-terial) Almansi strain tensor [2] E − ( U ) = ( − C − )and the (material) Hencky strain tensor E ( U ) = log U ,where log : Sym + ( n ) → Sym( n ) is the principal matrixlogarithm [105, p. 20] on the set Sym + ( n ) of positive def-inite symmetric matrices. The Hencky (or logarithmic)strain tensor has often been considered the natural or true strain in nonlinear elasticity [198, 197, 75, 88]. It is alsoof great importance to so-called hypoelastic models, as isdiscussed in [210, 76] (cf. Section 4.2.1). A very usefulapproximation of the material Hencky strain tensor wasgiven by Baˇzant [17] (cf. [165, 1, 49]): (cid:101) E / ( U ) := [ E / ( U ) + E − / ( U )] = ( U − U − ) . (1.4)Additional motivations of the logarithmic strain tensor were also given by Vall´ee [205, 206], Roug´ee [182,p. 302] and Murphy [142]. An extensive overview of the properties of the logarithmic strain tensor and itsapplications can be found in [209] and [154].All strain tensors, by the definition employed here, can be seen as equivalent : since the mapping U (cid:55)→ E ( U )is injective, for every pair E, E (cid:48) of strain tensors there exists a mapping ψ : Sym( n ) → Sym( n ) such that E (cid:48) ( U ) = ψ ( E ( U )) for all U ∈ Sym + ( n ). Therefore, every constitutive law of elasticity can – in principle – beexpressed in terms of any strain tensor and no strain tensor can be inherently superior to any other straintensor. Note that this invertibility property also holds if the definition by Hill or Ogden is used: if the strain Note that log U = lim r → r ( U r − ). Many more examples of strain tensors used throughout history can be found in [47]and [58], cf. [27, p. 132]. The corresponding family of spatial strain tensors (cid:98) E r ( V ) = (cid:40) r ( V r − ) : r (cid:54) = 0log V : r = 0includes the Almansi-Hamel strain tensor (cid:98) E / ( V ) = V − as well as the Euler-Almansi strain tensor (cid:98) E − ( V ) = ( − B − ),where B = F F T = V is the Finger tensor [69]. Bruhns [37, p. 41–42] emphasizes the advantages of the Hencky strain tensor over the other Seth-Hill strain tensors in theone-dimensional case: “
The significant advantage of this logarithmic (Hencky) measure lies in the fact that it tends to infinityas F tends to zero, thus in a very natural way bounding the regime of applicability to the case F > . This behavior can alsobe observed for strain [ tensors ] with negative exponent n . Compared with the latter, however, the logarithmic measure also goesto infinity as F does, whereas it is evident that for negative values of n the strain [ n ( F n − is bound to the limit − n .All measures with positive values of n including the Green strain share the reasonable property of the logarithmic strain for F going to infinity. For F going to zero, however, these measures arrive at finite values for the specific strains, e.g. at − for n = 2 , which would mean that interpreted from physics a total compression of the rod (to zero length) is related to a finitevalue of the strain. This awkward result would not agree with our observation - at least what concerns the behavior of solidmaterials. ” According to Truesdell and Toupin [204, p. 268], “ . . . any [ tensor ] sufficient to determine the directions of the principal axesof strain and the magnitude of the principal stretches may be employed and is fully general ”. Truesdell and Noll [203, p. 348]argue that there “ is no basis in experiment or logic for supposing nature prefers one strain [ tensor ] to another ”. Nevertheless, “[ in ] spite of this equivalence, one strain [ tensor ] may present definite technical advantages over another one ”
4s given via a scale function e, the strict monotonicity of e implies that the mapping U (cid:55)→ E ( U ) is strictlymonotone [130], i.e. (cid:104) E ( U ) − E ( U ) , U − U (cid:105) > U , U ∈ Sym + ( n ) with U (cid:54) = U , where (cid:104) X, Y (cid:105) = tr( X T Y ) denotes the Frobenius inner product onSym( n ) and tr( X ) = (cid:80) ni =1 X i,i is the trace of X ∈ R n × n . This monotonicity in turn ensures that the mapping U (cid:55)→ E ( U ) is injective.In contrast to strain or strain tensor , we use the term strain measure to refer to a nonnegative real-valued function ω : GL + ( n ) → [0 , ∞ ) depending on the deformation gradient which vanishes if and only if F is a pure rotation, i.e. ω ( F ) = 0 if and only if F ∈ SO( n ).Note that the terms “strain tensor” and “strain measure” are sometimes used interchangeably in theliterature (e.g. [108, 159]). A simple example of a strain measure in the above sense is the mapping F (cid:55)→(cid:107) E ( √ F T F ) (cid:107) of F to an orthogonally invariant norm of any strain tensor E .There is a close connection between strain measures and energy functions in isotropic hyperelasticity: anisotropic energy potential [84] is a function W depending on the deformation gradient F such that W ( F ) ≥ , (normalization) W ( QF ) = W ( F ) , (frame-indifference) W ( F Q ) = W ( F ) (material symmetry: isotropy)for all F ∈ GL + ( n ) , Q ∈ SO( n ) and W ( F ) = 0 if and only if F ∈ SO( n ) . (stress-free reference configuration)While every such energy function can be taken as a strain measure, many additional conditions for “proper”energy functions are discussed in the literature, such as constitutive inequalities [202, 106, 107, 11, 44, 127],generalized convexity conditions [10, 13] or monotonicity conditions to ensure that “stress increases withstrain” [154, Section 2.2]. Apart from that, the main difference between strain measures and energy functionsis that the former are purely mathematical expressions used to quantitatively assess the extent of strain ina deformation, whereas the latter postulate some physical behaviour of materials in a condensed form: anelastic energy potential, interpreted as the elastic energy per unit volume in the undeformed configuration,induces a specific stress response function , and therefore completely determines the physical behaviour ofthe modelled hyperelastic material. The connection between “natural” strain measures and energy functionswill be further discussed later on.In particular, we will be interested in energy potentials which can be expressed in terms of certainstrain measures. Note carefully that, in contrast to strain tensors, strain measures cannot simply be usedinterchangeably: for two different strain measures (as defined above) ω , ω , there is generally no function f : R + → R + such that ω ( F ) = f ( ω ( F )) for all F ∈ GL + ( n ). Compared to “full” strain tensors, this canbe interpreted as an unavoidable loss of information for strain measures (which are only scalar quantities).Sometimes a strain measure is employed only for a particular kind of deformation. For example, on thegroup of simple shear deformations (in a fixed plane) consisting of all F γ ∈ GL + (3) of the form F γ = (cid:16) γ
00 1 00 0 1 (cid:17) , γ ∈ R , we could consider the mappings F γ (cid:55)→ γ , F γ (cid:55)→ √ | γ | or F γ (cid:55)→ √ (cid:32) γ (cid:114) γ (cid:33) ; [47, p. 467]. For example, there is one and only one spatial strain tensor (cid:98) E together with a unique objective and corotationalrate dd t (cid:3) such that dd t (cid:3) (cid:98) E = sym( ˙ F F − ) = D . Here, dd t (cid:3) = dd t log is the logarithmic rate, D is the unique rate of stretching and (cid:98) E is the spatial Hencky strain tensor (cid:98) E = log V ; cf. Section 4.2.1 and [36, 210, 158, 216, 86]. The specific elasticity tensor further depends on the particular choice of a strain and a stress tensor in which to express theconstitutive law.
5e will come back to these partial strain measures in Section 3.2.In the following we consider the question of what strain measures are appropriate for the theory ofnonlinear isotropic elasticity. Since, by our definition, a strain measure attains zero if and only if F ∈ SO( n ),a simple geometric approach is to consider a distance function on the group GL + ( n ) of admissible deformationgradients, i.e. a function dist : GL + ( n ) × GL + ( n ) → [0 , ∞ ) with dist( A, B ) = dist(
B, A ) which satisfies thetriangle inequality and vanishes if and only if its arguments are identical. Such a distance function inducesa “natural” strain measure on GL + ( n ) by means of the distance to the special orthogonal group SO( n ): ω ( F ) := dist( F, SO( n )) := inf Q ∈ SO( n ) dist( F, Q ) . (1.5)In this way, the search for an appropriate strain measure reduces to the task of finding a natural, intrinsicdistance function on GL + ( n ). The remainder of this article is dedicated to this task: after some simple (Euclidean) examples in Section 2,we consider the geodesic distance on GL + ( n ) in Section 3. Our main result is stated in Theorem 3.3: if thedistance on GL + ( n ) is induced by a left-GL( n )-invariant, right-O( n )-invariant Riemannian metric on GL( n ),then the distance of F ∈ GL + ( n ) to SO( n ) is given bydist ( F, SO( n )) = dist ( F, R ) = µ (cid:107) dev n log U (cid:107) + κ U )] , where F = R U with U = √ F T F ∈ Sym + ( n ) and R ∈ SO( n ) is the polar decomposition of F . Section 3 alsocontains some additional remarks and corollaries which further expand upon this Riemannian strain measure.In Section 4, we discuss a number of different approaches towards motivating the use of logarithmic strainmeasures and strain tensors, whereas applications of our results and further research topics are indicated inSection 5.Our main result (Theorem 3.3) has previously been announced in a Comptes Rendus M´ecanique article[147] as well as in Proceedings in Applied Mathematics and Mechanics [148].The idea for this paper was conceived in late 2006. However, a number of technical difficulties had to beovercome (cf. [29, 156, 118, 129, 145]) in order to prove our results. The completion of this article might havetaken more time than was originally foreseen, but we adhere to the old German saying: Gut Ding will Weilehaben.
An approach similar to the definition of strain measures via distance functions on GL + ( n ), as stated inequation (1.5), can be employed in linearized elasticity theory: let ϕ ( x ) = x + u ( x ) with the displacement u .Then the infinitesimal strain measure may be obtained by taking the distance of the displacement gradient ∇ u ∈ R n × n to the set of linearized rotations so ( n ) = { A ∈ R n × n : A T = − A } , which is the vector space of skew symmetric matrices. An obvious choice for a distance measure on the linear space R n × n ∼ = R n of n × n -matrices is the Euclidean distance induced by the canonical Frobenius norm (cid:107) X (cid:107) = (cid:113) tr( X T X ) = (cid:114) n (cid:88) i,j =1 X ij . A distance function is more commonly known as a metric of a metric space. The term “distance” is used here and throughoutthe article in order to avoid confusion with the Riemannian metric introduced later on. Note that so ( n ) also corresponds to the Lie algebra of the special orthogonal group SO( n ).
6e use the more general weighted norm defined by (cid:107) X (cid:107) µ,µ c ,κ = µ (cid:107) dev n sym X (cid:107) + µ c (cid:107) skew X (cid:107) + κ X )] , µ, µ c , κ > , (2.1)which separately weights the deviatoric (or trace free ) symmetric part dev n sym X = sym X − n tr(sym X ) · ,the spherical part n tr( X ) · , and the skew symmetric part skew X = ( X − X T ) of X ; note that (cid:107) X (cid:107) µ,µ c ,κ = (cid:107) X (cid:107) for µ = µ c = 1 , κ = n , and that (cid:107) . (cid:107) µ,µ c ,κ is induced by the inner product (cid:104) X, Y (cid:105) µ,µ c ,κ = µ (cid:104) dev n sym X, dev n sym Y (cid:105) + µ c (cid:104) skew X, skew Y (cid:105) + κ tr( X ) tr( Y ) (2.2)on R n × n , where (cid:104) X, Y (cid:105) = tr( X T Y ) denotes the canonical inner product. In fact, every isotropic inner producton R n × n , i.e. every inner product (cid:104)· , ·(cid:105) iso with (cid:104) Q T X Q, Q T Y Q (cid:105) iso = (cid:104) X, Y (cid:105) iso for all
X, Y ∈ R n × n and all Q ∈ O( n ), is of the form (2.2), cf. [50]. The suggestive choice of variables µ and κ , which represent the shear modulus and the bulk modulus , respectively, will prove to be justified later on.The remaining parameter µ c will be called the spin modulus .Of course, the element of best approximation in so ( n ) to ∇ u with respect to the weighted Euclideandistance dist Euclid ,µ,µ c ,κ ( X, Y ) = (cid:107) X − Y (cid:107) µ,µ c ,κ is given by the associated orthogonal projection of ∇ u to so ( n ), cf. Figure 2. Since so ( n ) and the space Sym( n ) of symmetric matrices are orthogonal with respectto (cid:104)· , ·(cid:105) µ,µ c ,κ , this projection is given by the continuum rotation , i.e. the skew symmetric part skew ∇ u = ( ∇ u − ( ∇ u ) T ) of ∇ u , the axial vector of which is curl u . Thus the distance is dist Euclid ,µ,µ c ,κ ( ∇ u, so ( n )) : = inf A ∈ so ( n ) (cid:107)∇ u − A (cid:107) µ,µ c ,κ = (cid:107)∇ u − skew ∇ u (cid:107) µ,µ c ,κ = (cid:107) sym ∇ u (cid:107) µ,µ c ,κ . (2.3)We therefore find dist ,µ,µ c ,κ ( ∇ u, so ( n )) = (cid:107) sym ∇ u (cid:107) µ,µ c ,κ = µ (cid:107) dev n sym ∇ u (cid:107) + κ ∇ u )] = µ (cid:107) dev n ε (cid:107) + κ ε )] = W lin ( ∇ u )for the linear strain tensor ε = sym ∇ u , which is the quadratic isotropic elastic energy, i.e. the canonicalmodel of isotropic linear elasticity with σ = D ∇ u W lin ( ∇ u ) = 2 µ dev n ε + κ tr( ε ) · . (2.4) The family (2.2) of inner products on R n × n is based on the Cartan-orthogonal decomposition gl ( n ) = (cid:16) sl ( n ) ∩ Sym( n ) (cid:17) ⊕ so ( n ) ⊕ R · of the Lie algebra gl ( n ) = R n × n . Here, sl ( n ) = { X ∈ gl ( n ) | tr X = 0 } denotes the Lie algebra corresponding to the speciallinear group SL( n ) = { A ∈ GL( n ) | det A = 1 } . The distance can also be computed directly: since (cid:107)∇ u − A (cid:107) µ,µ c ,κ = µ (cid:107) dev n sym( ∇ u − A ) (cid:107) + µ c (cid:107) skew( ∇ u − A ) (cid:107) + κ ∇ u − A )] = µ (cid:107) dev n sym ∇ u (cid:107) + µ c (cid:107) (skew ∇ u ) − A (cid:107) + κ ∇ u )] for all A ∈ so ( n ), the infimum inf A ∈ so ( n ) (cid:107)∇ u − A (cid:107) µ,µ c ,κ = µ (cid:107) dev n sym ∇ u (cid:107) + κ [tr( ∇ u )] is obviously uniquely attained at A = skew ∇ u . o ( n ) R n × n ε = sym ∇ u skew ∇ u ∇ u Figure 2: The Euclidean distance dist ,µ,µ c ,κ ( ∇ u, so ( n )) = µ (cid:107) dev n ε (cid:107) + κ [tr( ε )] of ∇ u to so ( n ) in R n × n in the infinites-imal strain setting. The strain tensor ε = sym ∇ u is orthogonal to the infinitesimal continuum rotation skew ∇ u . This shows the aforementioned close connection of the energy potential to geometrically motivated measuresof strain. Note also that the so computed distance to so ( n ) is independent of the parameter µ c , the spinmodulus , weighting the skew-symmetric part in the quadratic form (2.1). We will encounter the (lack of)influence of the parameter µ c subsequently again.Furthermore, this approach motivates the symmetric part ε = sym ∇ u of the displacement gradient asthe strain tensor in the linear case: instead of postulating that our strain measure should depend only on ε ,the above computations deductively characterize ε as the infinitesimal strain tensor from simple geometricassumptions alone. In order to obtain a strain measure in the geometrically nonlinear case, we must compute the distancedist( ∇ ϕ, SO( n )) = dist( F, SO( n )) = inf Q ∈ SO( n ) dist( F, Q )of the deformation gradient F = ∇ ϕ ∈ GL + ( n ) to the actual set of pure rotations SO( n ) ⊂ GL + ( n ). Itis therefore necessary to choose a distance function on GL + ( n ); an obvious choice is the restriction of theEuclidean distance on R n × n to GL + ( n ). For the canonical Frobenius norm (cid:107) . (cid:107) , the Euclidean distancebetween F, P ∈ GL + ( n ) is dist Euclid ( F, P ) = (cid:107) F − P (cid:107) = (cid:113) tr[( F − P ) T ( F − P )] . Now let Q ∈ SO( n ). Since (cid:107) . (cid:107) is orthogonally invariant, i.e. (cid:107) (cid:98) QX (cid:107) = (cid:107) X (cid:98) Q (cid:107) = (cid:107) X (cid:107) for all X ∈ R n × n , (cid:98) Q ∈ O( n ), we find dist Euclid ( F, Q ) = (cid:107) F − Q (cid:107) = (cid:107) Q T ( F − Q ) (cid:107) = (cid:107) Q T F − (cid:107) . (2.5)Thus the computation of the strain measure induced by the Euclidean distance on GL + ( n ) reduces to the matrix nearness problem [104]dist Euclid ( F, SO( n )) = inf Q ∈ SO( n ) (cid:107) F − Q (cid:107) = min Q ∈ SO( n ) (cid:107) Q T F − (cid:107) . By a well-known optimality result discovered by Giuseppe Grioli [82] (cf. [150, 83, 131, 31]), also called“Grioli’s Theorem” by Truesdell and Toupin [204, p. 290], this minimum is attained for the orthogonal polarfactor R . 8 heorem 2.1 (Grioli’s Theorem [82, 150, 204]) . Let F ∈ GL + ( n ) . Then min Q ∈ SO( n ) (cid:107) Q T F − (cid:107) = (cid:107) R T F − (cid:107) = (cid:107)√ F T F − (cid:107) = (cid:107) U − (cid:107) , where F = R U is the polar decomposition of F with R = polar( F ) ∈ SO( n ) and U = √ F T F ∈ Sym + ( n ) .The minimum is uniquely attained at the orthogonal polar factor R . Remark 2.2.
The minimization property stated in Theorem 2.1 is equivalent to [132]max Q ∈ SO( n ) tr( Q T F ) = max Q ∈ SO( n ) (cid:104) Q T F, (cid:105) = (cid:104) R T F, (cid:105) = (cid:104) U, (cid:105) . (cid:3) Thus for nonlinear elasticity, the restriction of the Euclidean distance to GL + ( n ) yields the strain measuredist Euclid ( F, SO( n )) = (cid:107) U − (cid:107) . In analogy to the linear case, we obtain G L + ( n ) SO( n ) F − R U − = R T F − R F = R U U
Figure 3: The “flat” interpretation of GL + ( n ) ⊂ R n × n endowed with the Euclidean distance. Note that (cid:107) F − R (cid:107) = (cid:107) R ( U − ) (cid:107) = (cid:107) U − (cid:107) by orthogonal invariance of the Frobenius norm, where F = R U is the polar decomposition of F . dist ( F, SO( n )) = (cid:107) U − (cid:107) = (cid:107) E / (cid:107) , (2.6)where E / = U − is the Biot strain tensor. Note the similarity between this expression and the Saint-Venant-Kirchhoff energy [117] (cid:107) E (cid:107) µ,µ c ,κ = µ (cid:107) dev E (cid:107) + κ E )] , (2.7)where E = ( C − ) = ( U − ) is the Green-Lagrangian strain.The squared Euclidean distance of F to SO( n ) is often used as a lower bound for more general elasticenergy potentials. Friesecke, James and M¨uller [78], for example, show that if there exists a constant C > W ( F ) ≥ C · dist ( F, SO(3)) (2.8)for all F ∈ GL + (3) in a large neighbourhood of , then the elastic energy W shows some desirable propertieswhich do not otherwise depend on the specific form of W . As a starting point for nonlinear theories ofbending plates, Friesecke et al. also use the weighted squared norm (cid:107)√ F T F − (cid:107) µ,µ c ,κ = µ (cid:107) dev ( U − ) (cid:107) + κ U − )] = µ (cid:107) U − (cid:107) + λ U − )] , λ is the first Lam´e parameter, as an energy function satisfying (2.8). The same energy, also called the Biot energy [149], has been recently motivated by applications in digital geometry processing [43].However, the resulting strain measure ω ( U ) = dist Euclid ( F, SO( n )) = (cid:107) U − (cid:107) does not truly seem appro-priate for finite elasticity theory: for U → (cid:107) U − (cid:107) → (cid:107) (cid:107) = √ n < ∞ , thus singular deformations donot necessarily correspond to an infinite measure ω . Furthermore, the above computations are not compatiblewith the weighted norm introduced in Section 2.1: in general [149, 70, 71],min Q ∈ SO( n ) (cid:107) F − Q (cid:107) µ,µ c ,κ (cid:54) = min Q ∈ SO( n ) (cid:107) Q T F − (cid:107) µ,µ c ,κ (cid:54) = (cid:107)√ F T F − (cid:107) µ,µ c ,κ , (2.9)thus the Euclidean distance of F to SO( n ) with respect to (cid:107) . (cid:107) µ,µ c ,κ does not equal (cid:107)√ F T F − (cid:107) µ,µ c ,κ ingeneral. In these cases, the element of best approximation is not the orthogonal polar factor R = polar( F ).In fact, the expression on the left-hand side of (2.9) is not even well defined in terms of linear mappings F and Q [149]: the deformation gradient F = ∇ ϕ at a point x ∈ Ω is a two-point tensor and hence, inparticular, a linear mapping between the tangent spaces T x Ω and T ϕ ( x ) ϕ (Ω). Since taking the norm (cid:107) X (cid:107) µ,µ c ,κ = µ (cid:107) dev n sym X (cid:107) + µ c (cid:107) skew X (cid:107) + κ X )] of X requires the decomposition of X into its symmetric and its skew symmetric part, it is only well definedif X is an endomorphism on a single linear space. Therefore (cid:107) F − Q (cid:107) µ,µ c ,κ , while being a valid expressionfor arbitrary matrices F, Q ∈ R n × n , is not an admissible term in the setting of finite elasticity.We also observe that the Euclidean distance is not an intrinsic distance measure on GL + ( n ): in general, A − B / ∈ GL + ( n ) for A, B ∈ GL + ( n ), hence the term (cid:107) A − B (cid:107) depends on the underlying linear structureof R n × n . Since it is not a closed subset of R n × n , GL + ( n ) is also not complete with respect to dist Euclid ; forexample, the sequence (cid:0) n · (cid:1) n ∈ N is a Cauchy sequence which does not converge.Most importantly, because GL + ( n ) is not convex, the straight line { A + t ( B − A ) | t ∈ [0 , } connecting A and B is not necessarily contained in GL + ( n ), which shows that the characterization of the Euclideandistance as the length of a shortest connecting curve is also not possible in a way intrinsic to GL + ( n ), as theintuitive sketches in Figures 4 and 5 indicate.These issues amply demonstrate that the Euclidean distance can only be regarded as an extrinsic distancemeasure on the general linear group. We therefore need to expand our view to allow for a more appropriate,truly intrinsic distance measure on GL + ( n ). + ( n ) as a Riemannian manifold In order to find an intrinsic distance function on GL + ( n ) that alleviates the drawbacks of the Euclideandistance, we endow GL( n ) with a Riemannian metric . Such a metric g is defined by an inner product g A : T A GL( n ) × T A GL( n ) → R If X : V → V is a mapping between two different linear spaces V , V , then X T is a mapping from V to V , hencesym X = ( X + X T ) is not well-defined. The straight line connecting F ∈ GL + ( n ) to its orthogonal polar factor R (i.e. the shortest connecting line from F toSO( n )), however, lies in GL + ( n ), which easily follows from the convexity of Sym + ( n ): for all t ∈ [0 , t U + (1 − t ) ∈ Sym + ( n )and thus R + t ( F − R ) = R ( t U + (1 − t ) ) ∈ R · Sym + ( n ) ⊂ GL + ( n ) . Note that the representation of GL + ( n ) as a sphere only serves to visualize the curved nature of the manifold and thatfurther geometric properties of GL + ( n ) should not be inferred from the figures. In particular, GL + ( n ) is not compact and thegeodesics are generally not closed. For technical reasons, we define g on all of GL( n ) instead of its connected component GL + ( n ); for more details, we refer to[129], where a more thorough introduction to geodesics on GL( n ) can be found. Of course, our strain measure depends only onthe restriction of g to GL + ( n ). n ) R = polar( F )GL + ( n ) F dist ( F, SO( n ))= k U − k = k√ F T F − k Figure 4: The Euclidean distance as an extrinsic measure on GL + ( n ). on each tangent space T A GL( n ), A ∈ GL( n ). Then the length of a sufficiently smooth curve γ : [0 , → GL( n )is given by L ( γ ) = (cid:90) (cid:113) g γ ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) d t , where ˙ γ ( t ) = dd t γ ( t ), and the geodesic distance (cf. Figure 5) between A, B ∈ GL + ( n ) is defined as theinfimum over the lengths of all (twice continuously differentiable) curves connecting A to B :dist geod ( A, B ) = inf { L ( γ ) | γ ∈ C ([0 , + ( n )) , γ (0) = A, γ (1) = B } . Our search for an appropriate strain measure is thereby reduced to the task of finding an appropriate A GL + ( n ) B dist ( A, B ) = k A − B k dist ( A, B )Figure 5: The geodesic (intrinsic) distance compared to the Euclidean (extrinsic) distance.
Riemannian metric on GL( n ). Although it might appear as an obvious choice, the metric ˇ g withˇ g A ( X, Y ) := (cid:104)
X, Y (cid:105) for all A ∈ GL + ( n ) , X, Y ∈ R n × n (3.1)provides no improvement over the already discussed Euclidean distance on GL + ( n ): since the length of acurve γ with respect to ˇ g is the classical (Euclidean) length L ( γ ) = (cid:90) (cid:113) ˇ g γ ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) d t = (cid:90) (cid:107) ˙ γ ( t ) (cid:107) d t , the shortest connecting curves with respect to ˇ g are straight lines of the form t (cid:55)→ A + t ( B − A ) with A, B ∈ GL + ( n ). Locally, the geodesic distance induced by ˇ g is therefore equal to the Euclidean distance.However, as discussed in the previous section, not all straight lines connecting arbitrary A, B ∈ GL + ( n ) arecontained within GL + ( n ), thus length minimizing curves with respect to ˇ g do not necessarily exist (cf. Figure6). Many of the shortcomings of the Euclidean distance therefore apply to the geodesic distance induced byˇ g as well.In order to find a more viable Riemannian metric g on GL( n ), we consider the mechanical interpretationof the induced geodesic distance dist geod : while our focus lies on the strain measure induced by g , that is11 L + ( n ) b γ ( t ) = A + t ( C − A ) b γ ( t ) / ∈ G L + ( n ) γA CB Figure 6: The shortest connecting ( geodesic ) curves in GL + ( n ) with respect to the Euclidean metric are straight lines, thus notevery pair A, B ∈ GL + ( n ) can be connected by curves of minimal length. The length of the straight line γ : t (cid:55)→ A + t ( B − A )connecting A to B is given by (cid:82) (cid:113) ˇ g γ ( t ) ( ˙ γ ( t ) , ˙ γ ( t )) d t = (cid:107) B − A (cid:107) , whereas the curve (cid:98) γ connecting A to C is not containedin GL + ( n ); its length is therefore not well defined. the geodesic distance of the deformation gradient F to the special orthogonal group SO( n ), the distancedist geod ( F , F ) between two deformation gradients F , F can also be motivated directly as a measure ofdifference between two linear (or homogeneous ) deformations F , F of the same body Ω. More generally, wecan define a difference measure between two inhomogeneous deformations ϕ , ϕ : Ω ⊂ R n → R n viadist( ϕ , ϕ ) := (cid:90) Ω dist geod ( ∇ ϕ ( x ) , ∇ ϕ ( x )) d x (3.2)under suitable regularity conditions for ϕ , ϕ (e.g. if ϕ , ϕ are sufficiently smooth with det ∇ ϕ i > g on GL( n ), we must discuss the required propertiesof this “difference measure”. First, the requirements of objectivity (left-invariance) and isotropy (right-invariance) suggest that the metric g should be bi- O( n ) -invariant , i.e. satisfy g QA ( QX, QY ) = g A ( X, Y ) (cid:124) (cid:123)(cid:122) (cid:125) objectivity isotropy (cid:122) (cid:125)(cid:124) (cid:123) = g AQ ( XQ, Y Q ) (3.3)for all Q ∈ O( n ), A ∈ GL( n ) and X, Y ∈ T A GL( n ), to ensure that dist geod ( A, B ) = dist geod ( Q A, Q B ) =dist geod ( A Q, B Q ).However, these requirements do not sufficiently determine a specific Riemannian metric. For example,(3.3) is satisfied by the metric ˇ g defined in (3.1) as well as by the metric ˇˇ g with ˇˇ g A ( X, Y ) = (cid:104) A T X, A T Y (cid:105) . Inorder to rule out unsuitable metrics, we need to impose further restrictions on g . If we consider the distancemeasure dist( ϕ , ϕ ) between two deformations ϕ , ϕ introduced in (3.2), a number of further invariancescan be motivated: if we require that the distance is not changed by the superposition of a homogeneousdeformation, i.e. that dist( B · ϕ , B · ϕ ) = dist( ϕ , ϕ )for all constant B ∈ GL( n ), then g must be left- GL( n ) -invariant , i.e. g BA ( B X, B Y ) = g A ( X, Y ) (3.4)for all
A, B ∈ GL( n ) and X, Y ∈ T A GL( n ). The physical interpretation of this invariance requirement isreadily visualized in Figure 8.It can easily be shown [129] that a Riemannian metric g is left-GL( n )-invariant as well as right-O( n )-invariant if and only if g is of the form g A ( X, Y ) = (cid:104) A − X, A − Y (cid:105) µ,µ c ,κ , (3.5) Of course, the left-GL( n )-invariance of a metric also implies the left-O( n )-invariance. ϕ ϕ ∇ ϕ ( x ) ∇ ϕ ( x )dist geod ( ∇ ϕ ( x ) , ∇ ϕ ( x ))dist( ϕ , ϕ ) := R Ω dist geod ( ∇ ϕ ( x ) , ∇ ϕ ( x )) d x ΩΩ Figure 7: The distance dist( ϕ , ϕ ) := (cid:82) Ω dist geod ( ∇ ϕ ( x ) , ∇ ϕ ( x )) d x measures how much two deformations ϕ , ϕ of a bodyΩ differ from each other via integration over the pointwise geodesic distances between ∇ ϕ ( x ) and ∇ ϕ ( x ). where (cid:104)· , ·(cid:105) µ,µ c ,κ is the fixed inner product on the tangent space gl ( n ) = T GL( n ) = R n × n at the identitywith (cid:104) X, Y (cid:105) µ,µ c ,κ = µ (cid:104) dev n sym X, dev n sym Y (cid:105) + µ c (cid:104) skew X, skew Y (cid:105) + κ tr( X ) tr( Y ) (3.6)for constant positive parameters µ, µ c , κ >
0, and where (cid:104)
X, Y (cid:105) = tr( X T Y ) denotes the canonical innerproduct on gl ( n ) = R n × n . A Riemannian metric g defined in this way behaves in the same way on alltangent spaces: for every A ∈ GL + ( n ), g transforms the tangent space T A GL + ( n ) at A to the tangent space T GL + ( n ) = gl ( n ) at the identity via the left-hand multiplication with A − and applies the fixed innerproduct (cid:104)· , ·(cid:105) µ,µ c ,κ on gl ( n ) to the transformed tangents, cf. Figure 9.In the following, we will always assume that GL( n ) is endowed with a Riemannian metric of the form(3.5) unless indicated otherwise.In order to find the geodesic distancedist geod ( F, SO( n )) = inf Q ∈ SO( n ) dist geod ( F, Q )of F ∈ GL + ( n ) to SO( n ), we need to consider the geodesic curves on GL + ( n ). It has been shown [129, 134,87, 5] that every geodesic on GL + ( n ) with respect to the left-GL( n )-invariant Riemannian metric induced bythe inner product (3.6) is of the form γ ξF ( t ) = F exp( t (sym ξ − µ c µ skew ξ )) exp( t (1 + µ c µ ) skew ξ ) (3.7) If µ = µ c = 1 and κ = n , then the inner product (cid:104)· , ·(cid:105) µ,µ c ,κ is the canonical inner product, and the corresponding metric g is the canonical left-invariant metric on GL( n ) with g A ( X, Y ) = (cid:104) A − X, A − Y (cid:105) = tr( X T A − T A − Y ). Note that this metricdiffers from the trace metric (cid:101) g A ( X, Y ) = tr( A − XA − Y ), cf. [55]. Bϕ (Ω) ϕ (Ω) B · ϕ (Ω) B · ϕ (Ω) dist( ϕ , ϕ )dist( B · ϕ , B · ϕ ) = Figure 8: The distance between two deformations should not be changed by the composition with an additional homogeneoustransformation B : dist( ϕ , ϕ ) = dist( B · ϕ , B · ϕ ). with F ∈ GL + ( n ) and some ξ ∈ gl ( n ), where exp denotes the matrix exponential. These curves arecharacterized by the geodesic equation ˙ ζ = µ + µ c µ ( ζ T ζ − ζζ T ) , ζ := γ − ˙ γ . (3.8)Since the geodesic curves are defined globally, GL + ( n ) is geodesically complete with respect to the metric g .We can therefore apply the Hopf-Rinow theorem [111, 129] to find that for all F, P ∈ GL + ( n ) there existsa length minimizing geodesic γ ξF connecting F and P . Without loss of generality, we can assume that γ ξF isdefined on the interval [0 , γ ξF are γ ξF (0) = F and P = γ ξF (1) = F exp(sym ξ − µ c µ skew ξ ) exp((1 + µ c µ ) skew ξ ) , and the length of the geodesic γ ξF starting in F with initial tangent F ξ ∈ T F GL + ( n ) (cf. (3.7) and Figure11) is given by [129] L ( γ ξF ) = (cid:107) ξ (cid:107) µ,µ c ,κ . The geodesic distance between F and P can therefore be characterized asdist geod ( F, P ) = min {(cid:107) ξ (cid:107) µ,µ c ,κ | ξ ∈ gl ( n ) : γ ξF (1) = P } , that is the minimum of (cid:107) ξ (cid:107) µ,µ c ,κ over all ξ ∈ gl ( n ) which connect F and P , i.e. satisfyexp(sym ξ − µ c µ skew ξ ) exp((1 + µ c µ ) skew ξ ) = F − P . (3.9)Although some numerical computations have been employed [215] to approximate the geodesic distancein the special case of the canonical left-GL( n )-invariant metric, i.e. for µ = µ c = 1, κ = n , there is no knownclosed form solution to the highly nonlinear system (3.9) in terms of ξ for given F, P ∈ GL + ( n ) and thus noknown method of directly computing dist geod ( F, P ) in the general case exists. However, this parametrizationof the geodesic curves will still allow us to obtain a lower bound on the distance of F to SO( n ). The mapping ξ (cid:55)→ exp geod ( ξ ) := γ ξF (1) = F exp(sym ξ − µ c µ skew ξ ) exp((1 + µ c µ ) skew ξ ) is also known as the geodesicexponential function at F . Note that in general exp geod ( ξ ) (cid:54) = F · exp( ξ ) if ξ is not normal (i.e. if ξξ T (cid:54) = ξ T ξ ), thus the geodesiccurves are generally not one-parameter groups of the form t (cid:55)→ F exp( t ξ ), in contrast to bi-invariant metrics on Lie groups (e.g.SO( n ) with the canonical bi-invariant metric [136]). + ( n ) T A GL + ( n )= A · gl ( n ) T GL + ( n ) = gl ( n ) A X Y A − X A − Y A − g A ( X, Y ) = h A − X, A − Y i µ,µ c ,κ Figure 9: A left-GL( n )-invariant Riemannian metric on GL( n ) transforms the tangent space at A ∈ GL + ( n ) to the tangentspace T GL + ( n ) = gl ( n ) at the identity and applies a fixed inner product on gl ( n ) to the transformed tangents. Thus notangent space is treated preferentially. n ) Having defined the geodesic distance on GL + ( n ), we can now consider the geodesic strain measure, which isthe geodesic distance of the deformation gradient F to SO( n ):dist geod ( F, SO( n )) = inf Q ∈ SO( n ) dist geod ( F, Q ) . (3.10)Without explicit computation of this distance, the left-GL( n )-invariance and the right-O( n )-invariance ofthe metric g immediately allow us to show the inverse deformation symmetry of the geodesic strain measure:dist geod ( F, SO( n )) = inf Q ∈ SO( n ) dist geod ( F, Q ) = inf Q ∈ SO( n ) dist geod ( F − F, F − Q )= inf Q ∈ SO( n ) dist geod ( , F − Q ) = inf Q ∈ SO( n ) dist geod ( Q T Q, F − Q )= inf Q ∈ SO( n ) dist geod ( Q T , F − ) = dist geod ( F − , SO( n )) . (3.11)This symmetry property demonstrates at once that the Eulerian (spatial) and the
Lagrangian (referential)points of view are equivalent with respect to the geodesic strain measure: in the Eulerian setting, the inverse F − of the deformation gradient appears more naturally , whereas F is used in the Lagrangian frame (cf.Figure 10). Equality (3.11) shows that both points of view can equivalently be taken if the geodesic strainmeasure is used. As we will see later on (Remark 3.5), the equality dist geod ( B, SO( n )) = dist geod ( C, SO( n ))also holds for the right Cauchy-Green deformation tensor C = F T F = U and the Finger tensor B = F F T = V , further indicating the independence of the geodesic strain measure from the chosen frame of reference.This property is, however, not unique to geodesic (or logarithmic) strain measures; for example, the Frobeniusnorm (cid:107) (cid:101) E / ( U ) (cid:107) = (cid:107) U − U − (cid:107) = (cid:107) V − V − (cid:107) of the Baˇzant approximation (cid:101) E / = ( U − U − ), cf. (1.4) and [17], which can be considered a “quasiloga-rithmic” strain measure, fulfils the inverse deformation symmetry as well. However, it is not satisfied for Note that Cauchy originally introduced the tensors C − and B − in his investigations of the nonlinear strain [41, 42, 77,182], where C = F T F = U is the right Cauchy-Green deformation tensor [81, 77] and B = F F T = V is the Finger tensor.Piola also formulated an early nonlinear elastic law in terms of C − , cf. [203, p. 347]. The quantity √ (cid:107) U − U − (cid:107) is suggested as a measure of strain magnitude by Truesdell and Toupin [204, p. 266]. Ω ϕ (Ω) F − Lagrangian frame(material setting) Eulerian frame(spatial setting)
Figure 10: The Lagrangian and the Eulerian point of view are equivalently represented by the geodesic strain measure:dist geod ( F, SO( n )) = dist geod ( F − , SO( n )). the Euclidean distance to SO( n ): in general, (cid:107) U − (cid:107) = dist Euclid ( F, SO( n )) (cid:54) = dist Euclid ( F − , SO( n )) = (cid:107) V − − (cid:107) . (3.12)Now, let F = R U denote the polar decomposition of F with U ∈ Sym + ( n ) and R ∈ SO( n ). In order toestablish a simple upper bound on the geodesic distance dist geod ( F, SO( n )), we construct a particular curve γ R connecting F to its orthogonal factor R ∈ SO( n ) and compute its length L ( γ R ). For γ R ( t ) := R exp((1 − t ) log U ) , where log U ∈ Sym( n ) is the principal matrix logarithm of U , we find γ R (0) = R exp(log U ) = R U = F and γ R (1) = R exp(0) = R ∈ SO( n ) . It is easy to confirm that γ R is in fact a geodesic as given in (3.7) with ξ = log U ∈ Sym( n ). Since γ − R ( t ) ˙ γ R ( t ) = ( R exp((1 − t ) log U )) − R exp((1 − t ) log U ) · ( − log U ) = − log U , the length of γ R is given by L ( γ R ) = (cid:90) (cid:113) g γ R ( t ) ( ˙ γ R ( t ) , ˙ γ R ( t )) d t (3.13)= (cid:90) (cid:113) (cid:104) γ R ( t ) − ˙ γ R ( t ) , γ R ( t ) − ˙ γ R ( t ) (cid:105) µ,µ c ,κ d t = (cid:90) (cid:113) (cid:104)− log U, − log U (cid:105) µ,µ c ,κ d t = (cid:90) (cid:107) log U (cid:107) µ,µ c ,κ d t = (cid:107) log U (cid:107) µ,µ c ,κ . We can thereby establish the upper bound dist ( F, SO( n )) = inf Q ∈ SO( n ) dist ( F, Q ) ≤ dist ( F, R ) (3.14) ≤ L ( γ R ) = (cid:107) log U (cid:107) µ,µ c ,κ = µ (cid:107) dev n log U (cid:107) + κ U )] (3.15)for the geodesic distance of F to SO( n ). 16ur task in the remainder of this section is to show that the right hand side of inequality (3.15) is also alower bound for the (squared) geodesic strain measure, i.e. that, altogether,dist ( F, SO( n )) = µ (cid:107) dev n log U (cid:107) + κ U )] . However, while the orthogonal polar factor R is the element of best approximation in the Euclidean case(for µ = µ c = 1, κ = n ) due to Grioli’s Theorem, it is not clear whether R is indeed the element in SO( n )with the shortest geodesic distance to F (and thus whether equality holds in (3.14)). Furthermore, it is noteven immediately obvious that the geodesic distance between F and R is actually given by the right handside of (3.15), since a shorter connecting geodesic might exist (and hence inequality might hold in (3.15)).Nonetheless, the following fundamental logarithmic minimization property of the orthogonal polar fac-tor, combined with the computations in Section 3.1, allows us to show that (3.15) is indeed also a lower boundfor dist geod ( F, SO( n )). Proposition 3.1.
Let F = R √ F T F be the polar decomposition of F ∈ GL + ( n ) with R ∈ SO( n ) and let (cid:107) . (cid:107) denote the Frobenius norm on R n × n . Then inf Q ∈ SO( n ) (cid:107) sym Log( Q T F ) (cid:107) = (cid:107) sym log( R T F ) (cid:107) = (cid:107) log √ F T F (cid:107) , where inf Q ∈ SO( n ) (cid:107) sym Log( Q T F ) (cid:107) := inf Q ∈ SO( n ) inf {(cid:107) sym X (cid:107) | X ∈ R n × n , exp( X ) = Q T F } is defined as the infimum of (cid:107) sym . (cid:107) over “all real matrix logarithms” of Q T F . Proposition 3.1, which can be seen as the natural logarithmic analogue of Grioli’s Theorem (cf. Section2.2), was first shown for dimensions n = 2 , Corollary 3.2.
Let (cid:107) X (cid:107) µ,µ c ,κ = µ (cid:107) dev n sym X (cid:107) + µ c (cid:107) skew X (cid:107) + κ X )] , µ, µ c , κ > , for all X ∈ R n × n , where (cid:107) . (cid:107) is the Frobenius matrix norm. Then inf Q ∈ SO( n ) (cid:107) sym Log( Q T F ) (cid:107) µ,µ c ,κ = (cid:107) log √ F T F (cid:107) µ,µ c ,κ . Proof.
We first note that the equality det exp( X ) = e tr( X ) holds for all X ∈ R n × n . Since det Q = 1 for all Q ∈ SO( n ), this implies that for all X ∈ R n × n with exp( X ) = Q T F ,tr(sym X ) = tr( X ) = ln(det(exp( X ))) = ln(det( Q T F )) = ln(det F ) . Therefore (cid:107) sym X (cid:107) µ,µ c ,κ = µ (cid:107) dev n sym X (cid:107) + κ X )] = µ (cid:107) sym X (cid:107) + n κ − µ n [tr(sym X )] = µ (cid:107) sym X (cid:107) + n κ − µ n (ln(det F )) Of course, the application of such minimization properties to elasticity theory has a long tradition: Leonhard Euler, inthe appendix “De curvis elasticis” to his 1744 book “Methodus inveniendi lineas curvas maximi minimive proprietate gaudentessive solutio problematis isoperimetrici latissimo sensu accepti” [62, 164], already proclaimed that “[ . . . ] since the fabric of theuniverse is most perfect, and is the work of a most wise creator, nothing whatsoever takes place in the universe in which somerule of maximum and minimum does not appear. ” While (cid:107) Q T XQ (cid:107) µ,µ c ,κ = (cid:107) X (cid:107) µ,µ c ,κ for all X ∈ R n × n and Q ∈ O( n ), the orthogonal invariance requires the equalities (cid:107) QX (cid:107) µ,µ c ,κ = (cid:107) XQ (cid:107) µ,µ c ,κ = (cid:107) X (cid:107) µ,µ c ,κ , which do not hold in general. Observe that µ (cid:107) dev n Y (cid:107) + κ [tr( Y )] = µ (cid:107) Y (cid:107) + n κ − µ n [tr( Y )] for all Y ∈ R n × n . Q ∈ SO( n ) (cid:107) sym Log( Q T F ) (cid:107) µ,µ c ,κ (3.16)= inf Q ∈ SO( n ) inf {(cid:107) sym X (cid:107) µ,µ c ,κ | X ∈ R n × n , exp( X ) = Q T F } = inf Q ∈ SO( n ) inf { µ (cid:107) sym X (cid:107) + n κ − µ n (ln(det F )) | X ∈ R n × n , exp( X ) = Q T F } = µ inf Q ∈ SO( n ) inf {(cid:107) sym X (cid:107) | X ∈ R n × n , exp( X ) = Q T F } + n κ − µ n (ln(det F )) = µ (cid:107) log √ F T F (cid:107) + n κ − µ n (ln(det F )) = µ (cid:107) log √ F T F (cid:107) + n κ − µ n [tr(log √ F T F )] = µ (cid:107) dev n log √ F T F (cid:107) + κ √ F T F )] = (cid:107) log √ F T F (cid:107) µ,µ c ,κ . (cid:4) Note that Corollary 3.2 also implies the slightly weaker statementinf Q ∈ SO( n ) (cid:107) Log( Q T F ) (cid:107) µ,µ c ,κ = (cid:107) log √ F T F (cid:107) µ,µ c ,κ by using the simple estimate (cid:107) X (cid:107) µ,µ c ,κ ≥ (cid:107) sym X (cid:107) µ,µ c ,κ .We are now ready to prove our main result. Theorem 3.3.
Let g be the left- GL( n ) -invariant, right- O( n ) -invariant Riemannian metric on GL( n ) definedby g A ( X, Y ) = (cid:104) A − X, A − Y (cid:105) µ,µ c ,κ , µ, µ c , κ > , for A ∈ GL( n ) and X, Y ∈ R n × n , where (cid:104) X, Y (cid:105) µ,µ c ,κ = µ (cid:104) dev n sym X, dev n sym Y (cid:105) + µ c (cid:104) skew X, skew Y (cid:105) + κ tr( X ) tr( Y ) . (3.17) Then for all F ∈ GL + ( n ) , the geodesic distance of F to the special orthogonal group SO( n ) induced by g isgiven by dist ( F, SO( n )) = µ (cid:107) dev n log U (cid:107) + κ U )] , (3.18) where log is the principal matrix logarithm, tr( X ) = (cid:80) ni =1 X i,i denotes the trace and dev n X = X − n tr( X ) · is the n -dimensional deviatoric part of X ∈ R n × n . The orthogonal factor R ∈ SO( n ) of the polar decomposi-tion F = R U is the unique element of best approximation in
SO( n ) , i.e. dist geod ( F, SO( n )) = dist geod ( F, R ) = dist geod ( R T F, ) = dist geod ( U, ) . In particular, the geodesic distance does not depend on the spin modulus µ c . Remark 3.4 (Uniqueness of the metric) . We remark once more that the Riemannian metric considered inTheorem 3.3 is not chosen arbitrarily: every left-GL( n )-invariant, right-O( n )-invariant Riemannian metricon GL( n ) is of the form given in (3.17) for some choice of parameters µ, µ c , κ > (cid:3) Remark 3.5.
Since the weighted Frobenius norm on the right hand side of equation (3.18) only dependson the eigenvalues of U = √ F T F , the result can also be expressed in terms of the left Biot-stretch tensor V = √ F F T , which has the same eigenvalues as U :dist ( F, SO( n )) = µ (cid:107) dev n log V (cid:107) + κ V )] . (3.19)18pplying the above formula to the case F = P with P ∈ Sym + ( n ), we find √ P T P = √ P P T = P andtherefore dist ( P, SO( n )) = dist ( P, ) = µ (cid:107) dev n log P (cid:107) + κ P )] , (3.20)since is the orthogonal polar factor of P . For the tensors U and V , the right Cauchy-Green deformationtensor C = F T F = U and the Finger tensor B = F F T = V , we thereby obtain the equalitiesdist geod ( B, SO( n )) = dist geod ( B, ) = dist geod ( B − , ) (3.21)= dist geod ( C, ) = dist geod ( C − , ) = dist geod ( C, SO( n ))and dist geod ( V, SO( n )) = dist geod ( V, ) = dist geod ( V − , ) (3.22)= dist geod ( U, ) = dist geod ( U − , ) = dist geod ( U, SO( n )) . Note carefully that, although (3.20) for P ∈ Sym + ( n ) immediately follows from Theorem 3.3, it is not trivialto compute the distance dist geod ( P, ) directly: while the curve given by exp( t log P ) for t ∈ [0 ,
1] is in fact ageodesic [87] connecting to P with squared length µ (cid:107) dev n log P (cid:107) + κ [tr(log P )] , it is not obvious whetheror not a shorter connecting geodesic might exist. Our result ensures that this is in fact not the case. (cid:3) Proof of Theorem 3.3.
Let F ∈ GL + ( n ) and (cid:98) Q ∈ SO( n ). Then according to our previous considerations (cf.Section 3.1) there exists ξ ∈ gl ( n ) withexp(sym ξ − µ c µ skew ξ ) exp((1 + µ c µ ) skew ξ ) = F − (cid:98) Q (3.23)and (cid:107) ξ (cid:107) µ,µ c ,κ = dist geod ( F, (cid:98) Q ) . (3.24)In order to find a lower estimate on (cid:107) ξ (cid:107) µ,µ c ,κ (and thus on dist geod ( F, (cid:98) Q )), we computeexp(sym ξ − µ c µ skew ξ ) exp((1 + µ c µ ) skew ξ ) = F − (cid:98) Q = ⇒ exp((1 + µ c µ ) skew ξ ) − exp(sym ξ − µ c µ skew ξ ) − = (cid:98) Q T F = ⇒ exp( − sym ξ + µ c µ skew ξ ) = exp( (1 + µ c µ ) skew ξ (cid:124) (cid:123)(cid:122) (cid:125) ∈ so ( n ) ) (cid:98) Q T F .
Since exp( W ) ∈ SO( n ) for all skew symmetric W ∈ so ( n ), we find F ξ
SO( n ) GL + ( n ) F b Q dist ( F, SO( n ))Figure 11: The geodesic (intrinsic) distance to SO( n ); neither the element (cid:98) Q of best approximation nor the initial tangent F ξ ∈ T F GL + ( n ) of the connecting geodesic is known beforehand. − sym ξ + µ c µ skew ξ (cid:124) (cid:123)(cid:122) (cid:125) =: Y ) = Q Tξ F (3.25)with Q ξ = (cid:98) Q exp( − (1 + µ c µ ) skew ξ ) ∈ SO( n ); note that sym Y = − sym ξ . According to (3.25), Y = − sym ξ + µ c µ skew ξ is “a logarithm” of Q Tξ F . The weighted Frobenius norm of the symmetric part of Y = − sym ξ + µ c µ skew ξ is therefore bounded below by the infimum of (cid:107) sym X (cid:107) µ,µ c ,κ over “all logarithms” X of Q Tξ F : (cid:107) sym ξ (cid:107) µ,µ c ,κ = (cid:107) sym Y (cid:107) µ,µ c ,κ (3.25) ≥ inf {(cid:107) sym X (cid:107) µ,µ c ,κ | X ∈ R n × n , exp( X ) = Q Tξ F }≥ inf Q ∈ SO( n ) inf {(cid:107) sym X (cid:107) µ,µ c ,κ | X ∈ R n × n , exp( X ) = Q T F } = inf Q ∈ SO( n ) (cid:107) sym Log( Q T F ) (cid:107) µ,µ c ,κ . (3.26)We can now apply Corollary 3.2 to finddist ( F, (cid:98) Q ) = (cid:107) ξ (cid:107) µ,µ c ,κ = µ (cid:107) dev n sym ξ (cid:107) + µ c (cid:107) skew ξ (cid:107) + κ ξ )] ≥ µ (cid:107) dev n sym ξ (cid:107) + κ ξ )] (3.27)= (cid:107) sym ξ (cid:107) µ,µ c ,κ (3.26) ≥ inf Q ∈ SO( n ) (cid:107) sym Log( Q T F ) (cid:107) µ,µ c ,κ Corollary 3.2 = µ (cid:107) log √ F T F (cid:107) µ,µ c ,κ = µ (cid:107) dev n log U (cid:107) + κ U )] for U = √ F T F . Since this inequality is independent of (cid:98) Q and holds for all (cid:98) Q ∈ SO( n ), we obtain the desiredlower bound dist ( F, SO( n )) = inf (cid:98) Q ∈ SO( n ) dist ( F, (cid:98) Q ) ≥ µ (cid:107) dev n log U (cid:107) + κ U )] on the geodesic distance of F to SO( n ). Together with the upper bounddist ( F, SO( n )) ≤ dist ( F, R ) ≤ µ (cid:107) dev n log U (cid:107) + κ U )] already established in (3.15), we finally finddist ( F, SO( n )) = dist ( F, R ) = µ (cid:107) dev n log U (cid:107) + κ U )] . (3.28)By equation (3.28), apart from computing the geodesic distance of F to SO( n ), we have shown thatthe orthogonal polar factor R = polar( F ) is an element of best approximation to F in SO( n ). However,it is not yet clear whether there exists another element of best approximation, i.e. whether there is a (cid:98) Q ∈ SO( n ) with (cid:98) Q (cid:54) = R and dist geod ( F, (cid:98) Q ) = dist geod ( F, R ) = dist geod ( F, SO( n )). For this purpose, we needto compare geodesic distances corresponding to different parameters µ, µ c , κ . We therefore introduce thefollowing notation: for fixed µ, µ c , κ >
0, let dist geod ,µ,µ c ,κ denote the geodesic distance on GL + ( n ) induced bythe left-GL( n )-invariant, right-O( n )-invariant Riemannian metric g (as introduced in (3.5)) with parameters µ, µ c , κ . Furthermore, the length of a curve γ with respect to this metric will be denoted by L µ,µ c ,κ ( γ ). Loosely speaking, we use the term “a logarithm of A ∈ GL + ( n )” to denote any (real) solution X of the matrix equationexp X = A . (cid:98) Q ∈ SO( n ) is an element of best approximation to F with respect to g for some fixedparameters µ, µ c , κ >
0. Then there exists a length minimizing geodesic γ : [0 , → GL + ( n ) connecting (cid:98) Q to F of the form γ ( t ) = (cid:98) Q exp( t (sym ξ − µ c µ skew ξ )) exp( t (1 + µ c µ ) skew ξ )with ξ ∈ R n × n , and the length of γ is given by L µ,µ c ,κ ( γ ) = (cid:107) ξ (cid:107) µ,µ c ,κ = µ (cid:107) dev n sym ξ (cid:107) + µ c (cid:107) skew ξ (cid:107) + κ ξ )] . We first assume that skew ξ (cid:54) = 0. We choose ˜ µ c > µ c < µ c and finddist ,µ, ˜ µ c ,κ ( F, SO( n )) = inf Q ∈ SO( n ) dist ,µ, ˜ µ c ,κ ( F, Q ) (3.29) ≤ dist ,µ, ˜ µ c ,κ ( F, (cid:98) Q ) ≤ L µ, ˜ µ c ,κ ( γ ) , since γ is a curve connecting F to (cid:98) Q ∈ SO( n ); note that although γ is a shortest connecting geodesic withrespect to parameters µ, µ c , κ by assumption, it must not necessarily be a length minimizing curve withrespect to parameters µ, ˜ µ c , κ . Obviously, (cid:107) ξ (cid:107) µ, ˜ µ c ,κ < (cid:107) ξ (cid:107) µ,µ c ,κ if skew ξ (cid:54) = 0, and therefore L µ, ˜ µ c ,κ ( γ ) = (cid:107) ξ (cid:107) µ, ˜ µ c ,κ < (cid:107) ξ (cid:107) µ,µ c ,κ = L µ,µ c ,κ ( γ ) = dist ,µ,µ c ,κ ( F, (cid:98) Q ) . (3.30)By assumption, (cid:98) Q is an element of best approximation to F in SO( n ) for parameters µ, µ c , κ , thusdist ,µ,µ c ,κ ( F, (cid:98) Q ) = dist ,µ,µ c ,κ ( F, SO( n )) (3.31)= µ (cid:107) dev n log U (cid:107) + κ U )] = dist ,µ, ˜ µ c ,κ ( F, SO( n )) , where the last equality utilizes the fact that the distance from F to SO( n ) is independent of the secondparameter ( µ c or ˜ µ c ). Combining (3.29), (3.30) and (3.31), we thereby obtain the contradictiondist ,µ, ˜ µ c ,κ ( F, SO( n )) ≤ L µ, ˜ µ c ,κ ( γ ) < dist ,µ,µ c ,κ ( F, (cid:98) Q ) = dist ,µ, ˜ µ c ,κ ( F, SO( n )) , hence we must have skew ξ = 0. But then γ (1) = (cid:98) Q exp(sym ξ − µ c µ skew ξ ) exp((1 + µ c µ ) skew ξ ) = (cid:98) Q exp(sym ξ ) , and since exp(sym ξ ) ∈ Sym + ( n ), the uniqueness of the polar decomposition F = R U yields exp(sym ξ ) = U and, finally, (cid:98) Q = R . (cid:4) The fact that the orthogonal polar factor R = polar( F ) is the unique element of best approximation to F in SO( n ) with respect to the geodesic distance corresponds directly to the linear case (cf. equality (2.3) inSection 2.1), where the skew symmetric part skew ∇ u of the displacement gradient ∇ u is the element of bestapproximation with respect to the Euclidean distance: for F = + ∇ u we have U = + sym ∇ u + O ( (cid:107)∇ u (cid:107) ) and R = + skew ∇ u + O ( (cid:107)∇ u (cid:107) ) , hence the linear approximation of the orthogonal and the positive definite factor in the polar decompositionis given by skew ∇ u and sym ∇ u , respectively. The geometric connection between the geodesic distance onGL + ( n ) and the Euclidean distance on the tangent space R n × n = gl ( n ) at is illustrated in Figure 12. Remark 3.6.
Using a similar proof, exactly the same result can be shown for the geodesic distance dist geod , right induced by the right- GL( n ) -invariant, left- O( n ) -invariant Riemannian metric [207] g right A ( X, Y ) = (cid:104) XA − , Y A − (cid:105) µ,µ c ,κ n ) R = polar( F )GL + ( n ) T GL + ( n ) = gl ( n ) ∼ = R n × n T SO( n ) = so ( n ) F ∇ u skew ∇ u dist , gl ( ∇ u, so ( n ))= µ || dev n sym ∇ u || + κ [tr ∇ u ] dist ( F, SO( n ))= || U − || = ||√ F T F − || dist ( F, SO( n ))= µ || dev n log U || + κ [tr(log U )] Figure 12: The isotropic Hencky energy of F measures the geodesic distance between F and SO( n ). The linear Euclidean strainmeasure is obtained as the linearization via the tangent space gl ( n ) at . on GL( n ): dist , right ( F, SO( n )) = dist ( F, SO( n )) = µ (cid:107) dev n log U (cid:107) + κ U )] . The right-GL( n )-invariant Riemannian metric can be motivated in a way similar to the left-GL( n )-invariantcase: it corresponds to the requirement that the distance between two deformations F and F should notdepend on the initial shape of Ω, i.e. should not be changed if Ω is homogeneously deformed beforehand (cf.Figure 13). A similar independence from prior deformations (and so-called “pre-stresses”), called “elasticdeterminacy” by L. Prandtl [171], was postulated by H. Hencky in the deduction of his elasticity model; cf.[100, p. 618], [146, p. 19] and Section 4.2. (cid:3) According to Theorem 3.3, the squared geodesic distance between F and SO( n ) with respect to anyleft-GL( n )-invariant, right-O( n )-invariant Riemannian metric on GL( n ) is the isotropic quadratic Henckyenergy W H ( F ) = µ (cid:107) dev n log U (cid:107) + κ U )] , where the parameters µ, κ > was based on a set of axioms including a law of superposition (cf. Section 4.2) for thestress response function [146], an approach previously employed by G. F. Becker [18, 152] in 1893 and laterfollowed in a more general context by H. Richter [176], cf. [177, 175, 178]. A different constitutive model foruniaxial deformations based on logarithmic strain had previously been proposed by Imbert [114] and Hartig[89]. While Ludwik is often credited with the introduction of the uniaxial logarithmic strain, his ubiquitouslycited article [124] (which is even referenced by Hencky himself [102, p. 175]) does not provide a systematicintroduction of such a strain measure.While the energy function W H ( F ) = dist ( F, SO( n )) already defines a measure of strain as describedin Section 1.1, we are also interested in characterizing the two terms (cid:107) dev n log U (cid:107) and | tr(log U ) | as separate partial strain measures . Hencky’s approach is often misrepresented as empirically motivated. Truesdell claims that “
Hencky himself does not givea systematic treatement ” in introducing the logarithmic strain tensor [199, p. 144] and attributes the axiomatic approach toRichter [176] instead [204, p. 270]. Richter’s resulting deviatoric strain tensors dev log U and dev log V are disqualified as“ complicated algebraic functions ” by Truesdell and Toupin [204, p. 270]. F F F F Ω B · Ω F · Ω F · B · Ω F · Ω F · B · Ω dist geod ( F , F ) dist geod ( F · B, F · B ) = Figure 13: The right-GL( n )-invariance of a distance measure on GL( n ): the distance between two homogeneous deformations F , F is not changed by a prior homogeneous deformation B , i.e. dist geod ( F , F ) = dist geod ( F · B, F · B ). Theorem 3.7 (Partial strain measures) . Let ω iso ( F ) := (cid:107) dev n log √ F T F (cid:107) and ω vol ( F ) := | tr(log √ F T F ) | . Then ω iso ( F ) = dist geod , SL( n ) (cid:18) F det F /n , SO( n ) (cid:19) and ω vol ( F ) = √ n · dist geod , R + · (cid:16) (det F ) /n · , (cid:17) , where the geodesic distances dist geod , SL( n ) and dist geod , R + · on the Lie groups SL( n ) = { A ∈ GL( n ) | det A =1 } and R + · are induced by the canonical left-invariant metric ¯ g A ( X, Y ) = (cid:104) A − X, A − Y (cid:105) = tr( X T A − T A − Y ) . Remark 3.8.
Theorem 3.7 states that ω iso and ω vol appear as natural measures of the isochoric and volumetric strain, respectively: if F = F iso F vol is decomposed multiplicatively [73] into an isochoric part F iso = (det F ) − /n · F and a volumetric part F vol = (det F ) /n · , then ω iso ( F ) measures the SL( n )-geodesicdistance of F iso to SO( n ), whereas √ n ω vol ( F ) gives the geodesic distance of F vol to the identity in thegroup R + · of purely volumetric deformations. (cid:3) Proof.
First, observe that the canonical left-invariant metrics on SL( n ) and R + · are obtained by choosing µ = µ c = 1 and κ = n and restricting the corresponding metric g on GL + ( n ) to the submanifolds SL( n ), R + · and their respective tangent spaces. Then for this choice of parameters, every curve in SL( n ) or R + · is a curve of equal length in GL + ( n ) with respect to g . Since the geodesic distance is defined as the infimallength of connecting curves, this immediately impliesdist geod , SL( n ) ( F iso , SO( n )) ≥ dist geod , GL + ( n ) ( F iso , SO( n ))23s well as dist geod , R + · ( F vol , ) ≥ dist geod , GL + ( n ) ( F vol , ) ≥ dist geod , GL + ( n ) ( F vol , SO( n ))for F iso := (det F ) − /n · F and F vol := (det F ) /n · . We can therefore use Theorem 3.3 to obtain the lowerbounds dist , SL( n ) ( F iso , SO( n )) ≥ dist , GL + ( n ) ( F iso , SO( n ))= (cid:107) dev n log (cid:18)(cid:113) F T iso F iso (cid:19) (cid:107) + 1 n (cid:20) tr (cid:18) log (cid:113) F T iso F iso (cid:19)(cid:21) = (cid:107) log (cid:32)(cid:18) det (cid:113) F T iso F iso (cid:19) − /n (cid:113) F T iso F iso (cid:33) (cid:107) + 1 n (cid:20) ln (cid:18) =1 (cid:122) (cid:125)(cid:124) (cid:123) det (cid:113) F T iso F iso (cid:19)(cid:21) = (cid:107) log (cid:18)(cid:113) F T iso F iso (cid:19) (cid:107) = (cid:107) log (cid:16) (det F ) − /n √ F T F (cid:17) (cid:107) = ω ( F ) (3.32)and dist , R + · ( F vol , ) ≥ dist , GL + ( n ) ( F vol , SO( n ))= (cid:107) dev n log (cid:18)(cid:113) F T vol F vol (cid:19) (cid:107) + 1 n [tr(log (cid:18)(cid:113) F T vol F vol (cid:19) )] (3.33)= (cid:107) dev n (cid:16) ln((det F ) /n ) · (cid:17) (cid:107) + 1 n [ln(det (cid:16) (det F ) /n · (cid:17) )] = 1 n [ln(det √ F T F )] = 1 n [tr(log √ F T F )] = 1 n ω ( F ) . To obtain an upper bound on the geodesic distances, we define the two curves γ iso : [0 , → SL( n ) , γ iso ( t ) = R exp( t dev n log U )and γ vol : [0 , → R + · , γ vol ( t ) = e tn tr(log U ) · , where F = R U with R ∈ SO( n ) and U ∈ Sym + ( n ) is the polar decomposition of F . Then γ iso connects(det F ) − /n · F to SO( n ): γ iso (0) = R ∈ SO( n ) ,γ iso (1) = R exp(dev n log U ) = R exp(log U − tr(log U ) n · )= R exp(log U ) exp( − tr(log U ) n · )= R U exp( − ln det Un · ) = (det U ) − /n · F = (det F ) − /n · F , while γ vol connects (det F ) /n · and : γ vol (0) = , γ vol (1)= e n tr(log U ) · = e n ln(det U ) · = (det U ) /n · = (det F ) /n · . For some of the rules of computation employed here involving the matrix logarithm, we refer to Lemma A.1 in the appendix. L ( γ iso ) = (cid:90) (cid:107) γ iso ( t ) − ˙ γ iso ( t ) (cid:107) d t (3.34)= (cid:90) (cid:107) ( R exp( t dev n log U )) − R exp( t dev n log U ) dev n log U (cid:107) d t = (cid:90) (cid:107) dev n log U (cid:107) d t = (cid:107) dev n log √ F T F (cid:107) = ω iso ( F )as well as L ( γ vol ) = (cid:90) (cid:107) γ vol ( t ) − ˙ γ vol ( t ) (cid:107) d t (3.35)= (cid:90) (cid:107) ( e tn tr(log U ) · ) − · tr(log U ) n · e tn tr(log U ) · (cid:107) d t = (cid:90) (cid:107) tr(log U ) n · (cid:107) d t = | tr(log U ) | n · (cid:107) (cid:107) = 1 √ n | tr(log √ F T F ) | = 1 √ n ω vol ( F ) , showing that dist , SL( n ) (cid:16) (det F ) − /n · F, SO( n ) (cid:17) ≤ L ( γ iso ) = ω ( F )and dist , R + · (cid:16) (det F ) /n · , (cid:17) ≤ L ( γ vol ) = 1 n · ω ( F ) , which completes the proof. (cid:4) Remark 3.9.
In addition to the isochoric (distortional) part F iso = (det F ) − /n · F and the volumetric part F vol = (det F ) /n · , we may also consider the cofactor Cof F = (det F ) · F − T of F ∈ GL + ( n ). Theorem 3.3allows us to directly compute (cf. Appendix A.4) the distancedist (Cof F, SO( n )) = µ (cid:107) dev n log U (cid:107) + κ ( n − U )] . (cid:3) + ( n ) Extensive work on the use of Lie group theory and differential geometry in continuum mechanics has alreadybeen done by Roug´ee [181, 180, 182, 183], Moakher [137, 139], Bhatia [26] and, more recently, by Fiala [64,65, 66, 67, 68] (cf. [119, 120, 163, 167, 166]). They all endowed the convex cone Sym + (3) of positive definitesymmetric (3 × (cid:101) g C ( X, Y ) = tr( C − XC − Y ) = (cid:104) XC − , C − Y (cid:105) = (cid:104) C − / X C − / , C − / Y C − / (cid:105) , (4.1)where C ∈ Sym + (3) and X, Y ∈ Sym(3) = T C Sym + (3). Fiala and Roug´ee deduced a motivation of thelogarithmic strain tensor log U via geodesic curves connecting elements of Sym + ( n ). However, their approach Note the subtle difference with our metric g C ( X, Y ) = (cid:104) C − X, C − Y (cid:105) . Pennec [166, p. 368] generalizes (4.1) by using theweighted inner product (cid:104) X, Y (cid:105) ∗ = (cid:104) X, Y (cid:105) + β tr( X ) tr( Y ) with β > − n . + ( n ) already corre-sponds to metric states C = F T F , whereas we consider the full set GL + ( n ) of deformation gradients F (cf.Appendix A.3 and Table 1 in Section 6). This restriction can be viewed as the nonlinear analogue of the apriori restriction to ε = sym ∇ u in the linear case, i.e. the nature of the strain measure is not deduced butpostulated. Note also that the metric (cid:101) g cannot be obtained by restricting our left-GL(3)-invariant, right-O(3)-invariant metric g to Sym + (3). Furthermore, while Fiala and Roug´ee aim to motivate the Henckystrain tensor log U directly, our focus lies on the strain measures ω iso , ω vol and the isotropic Hencky strainenergy W H .The geodesic curves on Sym + ( n ) with respect to (cid:101) g are of the simple form γ ( t ) = C / exp( t · C − / M C − / ) C / (4.2)with C ∈ Sym + ( n ) and M ∈ Sym( n ) = T C Sym + ( n ). These geodesics are defined globally, i.e. Sym + ( n )is geodesically complete. Furthermore, for given C , C ∈ Sym + ( n ), there exists a unique geodesic curveconnecting them; this easily follows from the representation formula (4.2) or from the fact that the curvatureof Sym + ( n ) with (cid:101) g is constant and negative [65, 116, 25]. Note that this implies that, in contrast to GL + ( n )with our metric g , there are no closed geodesics on Sym + ( n ).An explicit formula for the corresponding geodesic distance was given by Moakher: dist geod , Sym + ( n ) ( C , C ) = (cid:107) log( C − / C C − / ) (cid:107) . (4.3)In the special case C = , this distance measure is equal to our geodesic distance on GL + ( n ) induced bythe canonical inner product: Theorem 3.3, applied with parameters µ = µ c = 1 and κ = n to R = and U = C , shows that dist geod , GL + ( n ) ( C , ) = (cid:107) log C (cid:107) = dist geod , Sym + ( n ) ( C , ) . More generally, assume that the two metric states C , C ∈ Sym + ( n ) commute. Then C − C ∈ Sym + ( n ),and the left-GL( n )-invariance of the geodesic distance impliesdist geod , GL + ( n ) ( C , C ) = dist geod , GL + ( n ) ( C − C , ) = (cid:107) log( C − C ) (cid:107) = (cid:107) log( C − / C − / C ) (cid:107) = (cid:107) log( C − / C C − / ) (cid:107) (4.4)= dist geod , Sym + ( n ) ( C , C ) . However, since C − C / ∈ Sym + ( n ) in general, this equality does not hold on all of Sym + ( n ).A different approach towards distance functions on the set Sym + ( n ) was suggested by Arsigny et al. [8, 9,7] who, motivated by applications of geodesic and logarithmic distances in diffusion tensor imaging, directlydefine their Log-Euclidean metric on Sym + ( n ) bydist Log-Euclid ( C , C ) := (cid:107) log C − log C (cid:107) , (4.5) Since Sym + ( n ) is not a Lie group with respect to matrix multiplication, the metric (cid:101) g itself cannot be left- or right-invariantin any suitable sense. While Moakher gives the parametrization stated here, Roug´ee writes the geodesics in the form γ ( t ) = exp( t · Log( C C − )) C with C , C ∈ Sym + ( n ), which can also be written as γ ( t ) = ( C C − ) t C ; a similar formulation is given by Tarantola [197,eq. (2.78)]. For a suitable definition of a matrix logarithm Log on GL + ( n ), these representations are equivalent to (4.2) with M = log( C − / C C − / ) ∈ Sym( n ). Moakher [137, eq. (2.9)] writes this result as (cid:107)
Log( C − C ) (cid:107) = (cid:113)(cid:80) ni =1 ln λ i , where λ i are the eigenvalues of C − C . Theright hand side of this equation is identical to the result stated in (4.3). However, since C − C is not necessarily normal, thereis in general no logarithm Log( C − C ) whose Frobenius norm satisfies this equality. Note that the eigenvalues of the matrix C − C are real and positive due to its similarity to C / ( C − C ) C − / = C − / C C − / ∈ Sym + ( n ). (cid:107) . (cid:107) is the Frobenius matrix norm. If C and C commute, this distance equals the geodesic distanceon GL + ( n ) as well: dist geod , GL + ( n ) ( C , C ) = (cid:107) log( C − C ) (cid:107) = (cid:107) log( C − ) + log( C ) (cid:107) (4.6)= (cid:107) log C − log C (cid:107) = dist Log-Euclid ( C , C ) , where equality in (4.6) holds due to the fact that C and C − commute. Again, this equality does not holdfor arbitrary C and C .Using a similar Riemannian metric, geodesic distance measures can also be applied to the set of positivedefinite symmetric fourth-order elasticity tensors, which can be identified with Sym + (6). Norris and Moakherapplied such a distance function in order to find an isotropic elasticity tensor C : Sym(3) → Sym(3) whichbest approximates a given anisotropic tensor [138, 157].The connection between geodesic distances on the metric states in Sym + ( n ) and logarithmic distancemeasures was also investigated extensively by the late Albert Tarantola [197], a lifelong advocate of logarithmicmeasures in physics. In his view [197, p. 4.3.1], “ . . . the configuration space is the Lie group GL + (3) , and theonly possible measure of strain (as the geodesics of the space) is logarithmic. ” “ At the foundation of all elastic theories lies the definition of strain, and before introducing a newlaw of elasticity we must explain how finite strain is to be measured. ” Heinrich Hencky: The elastic behavior of vulcanized rubber [103].
Apart from the geometric considerations laid out in the previous sections, the Hencky strain tensor E = log U can be characterized via a number of unique properties.For example, the Hencky strain is the only strain tensor (for a suitably narrow definition, cf. [152]) thatsatisfies the law of superposition for coaxial deformations: E ( U · U ) = E ( U ) + E ( U ) (4.7)for all coaxial stretches U and U , i.e. U , U ∈ Sym + ( n ) such that U · U = U · U . This characterizationwas used by Heinrich Hencky [196, 97, 102, 103] in his original introduction of the logarithmic strain tensor[99, 101, 100, 146] and, indeed much earlier, by the geologist George Ferdinand Becker [133], who postulateda similar law of superposition in order to deduce a logarithmic constitutive law of nonlinear elasticity [18,152] (cf. Appendix A.2).In the case n = 1, this superposition principle simply amounts to the fact that the logarithm function f = log satisfies Cauchy’s [40] well-known functional equation f ( λ · λ ) = f ( λ ) + f ( λ ) , (4.8)i.e. that the logarithm is an isomorphism between the multiplicative group ( R + , · ) and the additive group( R , +). This means that for a sequence of incremental one-dimensional deformations, the logarithmic strainse i log can be added in order to obtain the total logarithmic strain e totlog of the composed deformation [72]:e + e + . . . + e n log = log L L + log L L + . . . + log L n L n − = log L n L = e totlog , where L i denotes the length of the (one-dimensional) body after the i -th elongation. This property uniquelycharacterizes the logarithmic strain e log among all differentiable one-dimensional strain mappings e : R + → R with e (cid:48) (1) = 1. 27ince purely volumetric deformations of the form λ · with λ > U ∈ Sym + ( n ),the decomposition property (4.7) allows for a simple additive volumetric-isochoric split of the Hencky straintensor [176]: log U = log (cid:20) U (det U ) /n (cid:124) (cid:123)(cid:122) (cid:125) isochoric · (det U ) /n · (cid:124) (cid:123)(cid:122) (cid:125) volumetric (cid:21) = log (cid:20) U (det U ) /n (cid:21) + log (cid:104) (det U ) /n · (cid:105) = dev n log U (cid:124) (cid:123)(cid:122) (cid:125) isochoric + 1 n tr(log U ) · (cid:124) (cid:123)(cid:122) (cid:125) volumetric . In particular, the incompressibility condition det F = 1 can be easily expressed as tr(log U ) = 0 in terms ofthe logarithmic strain tensor. As indicated in Section 1.1, the quadratic Hencky energy is also of great importance to the concept of hypoelasticity [83, Chapter IX]. It was found that the Truesdell equation [199, 201, 200, 76]dd t (cid:3) [ τ ] = 2 µ D + λ tr( D ) · , D = sym( ˙ F F − ) , (4.9)with constant Lam´e coefficients µ, λ >
0, under the assumption that the stress rate dd t (cid:3) is objective and corotational , is satisfied if and only if dd t (cid:3) is the so-called logarithmic corotational rate dd t log and τ = 2 µ log V + λ tr(log V ) · [211, 209, 158, 172, 173, 212, 213, 214], i.e. if and only if the hypoelastic model is exactly Hencky’shyperelastic constitutive model. Here, τ = det F · σ ( V ) denotes the Kirchhoff stress tensor and D is the uniquerate of stretching tensor (i.e. the symmetric part of the velocity gradient in the spatial setting). A rate dd t (cid:3) is called corotational if it is of the special formdd t (cid:3) [ X ] = ˙ X − Ω X + X Ω with Ω ∈ so (3) , (4.10)which means that the rate is computed with respect to a frame that is rotated. This extra rate of rotationis defined only by the underlying spins of the problem. Upon specialisation, for µ = 1, λ = 0 we obtain [34,eq. 71] dd t log [log V ] = D as the unique solution to (4.9) with a corotational rate. Note that this characterization of the spatiallogarithmic strain tensor log V is by no means exceptional. For example, it is well known that [90, p. 49,Theorem 1.8] (cf. [35]) dd t (cid:77) [ A ] = ˙ A + L T A + AL = D , It is telling to see that equation (4.9) had already been proposed by Hencky himself in [100] for the
Zaremba-Jaumannstress rate (cf. (4.13)). Hencky’s work, however, contains a typographical error [100, eq. (10) and eq. (11e)] changing the orderof indices in his equations (cf. [33]). The strong point of writing (4.9) is that no discussion of any suitable strain tensor isnecessary. A rate dd t (cid:3) is called objective if dd t (cid:3) (cid:2) S ( QB ˙ Q T ) (cid:3) = Q ( dd t (cid:3) [ S ( B )]) Q T for all (not necessarily constant) Q = Q ( t ) ∈ O( n ),where S is any objective stress tensor, and if dd t (cid:3) [ S ] = 0 ⇔ S = 0, i.e. the motion is rigid if and only if dd t (cid:3) [ S ] ≡ Corotational rates are also special cases of Lie derivatives [112, 127]. Cf. Xiao, Bruhns and Meyers [210, p. 90]: “ . . . the logarithmic strain [ does ] possess certain intrinsic far-reaching properties [ which ] establish its favoured position in all possible strain measures ”. A = (cid:98) E − = ( − B − ) is the spatial Almansi strain tensor and dd t (cid:77) is the upper Oldroyd rate (asdefined in (4.14)).The quadratic Hencky model τ = 2 µ log V + λ tr(log V ) · = D log V W H (log V ) (4.11)was generalized in Hill’s generalized linear elasticity laws [108, eq. (2.69)] T r = 2 µ E r + λ tr( E r ) · (4.12)with work-conjugate pairs ( T r , E r ) based on the Lagrangian strain measures given in (1.3); cf. Appendix A.2for examples. The concept of work-conjugacy was introduced by Hill [106] via an invariance requirement; thespatial stress power must be equal to its Lagrangian counterpart:det F · (cid:104) σ, D (cid:105) = (cid:104) T r , ˙ E r (cid:105) , (work-conjugacy)by means of which a material stress tensor is uniquely linked to its (material rate) conjugate strain tensor.Hence it generalizes the virtual work principle and is the foundation of derived methods like the finite elementmethod.For the case of isotropic materials, Hill [106, p. 242] (cf. [109]) shows by spectral decomposition techniquesthat the work-conjugate stress to log U is the back-rotated Cauchy stress σ multiplied by det F , hence (cid:104) σ, D (cid:105) = (cid:104) R T σ R, dd t log U (cid:105) , which is a generalization of Hill’s earlier work [106, 108]. Sansour [185] additionally foundthat the Eshelby-like stress tensor Σ = CS is equally conjugate to log U ; here, S denotes the second Piola-Kirchhoff stress tensor. For anisotropy, however, the conjugate stress exists but follows a much more complexformat than for isotropy [109]. The logarithm of the left stretch log V in contrast exhibits a work conjugatestress tensor only for isotropic materials, namely the Kirchhoff stress tensor τ = det F · σ [162, 109].While hyperelasticity in its potential format avoids rate equations, the use of stress rates (i.e. stressincrements in time) may be useful for the description of inelastic material behavior at finite strains. Sincethe material time derivative of an Eulerian stress tensor is not objective, rates for a tensor X were developed,like the (objective and corotational) Zaremba-Jaumann rate dd t ◦ [ X ] = ˙ X − W X + XW , W = skew
L , L = ˙
F F − , (4.13)or the (objective but not corotational) lower and upper Oldroyd rates dd t (cid:79) [ X ] = ˙ X + L T X + XL and dd t (cid:77) [ X ] = ˙ X − LX − XL T , (4.14)to name but a few (cf. [90, Section 1.7] and [186]). Which one of these or the great number of other objectiverates should be used seems to be rather a matter of taste, hence of arbitrariness or heuristics , but not amatter of theory.The concept of dual variables as introduced by Tsakmakis and Haupt in [91] into continuum mechanicsovercame the arbitrariness of the chosen rate in that it uniquely connects a particular (objective) strain rate Hooke’s law [110] (cf. [141]) famously states that the strain in a deformation depends linearly on the occurring stress (“uttensio, sic vis”). However, for finite deformations, different constitutive laws of elasticity can be obtained from this assumption,depending on the choice of a stress/strain pair. An idealized version of such a linear relation is given by (4.11), i.e. by choosingthe spatial Hencky strain tensor log V and the Kirchhoff stress tensor τ . Since, however, Hooke speaks of extension versus force,the correct interpretation of Hooke’s law is T Biot = 2 µ ( U − ) + λ tr( U − ) · , i.e. the case r = in (4.12). Truesdell and Noll [203, p. 404] declared that “ various such stress rates have been used in the literature. Despite claims andwhole papers to the contrary, any advantage claimed for one such rate over another is pure illusion ”, and that “ the properties ofa material are independent of the choice of flux [ i.e. of the chosen rate ] , which, like the choice of a [ strain tensor ] , is absolutelyimmaterial ” [203, p. 97]. For a shear test in Eulerian elasto-plasticity using the Zaremba-Jaumann rate (4.13), an unphysical artefact of oscillatoryshear stress was observed, first in [122]. A similar oscillatory behavior was observed for hypoelasticity in [52]. Hill [108] used the terms conjugate and dual as synonyms.
29o a stress tensor and, analogously, a stress rate to a strain tensor. The rational rule is that, when stressand strain tensors operate on configurations other than the reference configurations, the physically significantscalar products (cid:104) S , ˙ E (cid:105) , (cid:104) ˙ S , E (cid:105) , (cid:104) S , E (cid:105) and (cid:104) ˙ S , ˙ E (cid:105) (with the second Piola-Kirchhoff stress tensor S andits work-conjugate Green strain tensor E ) must remain invariant, see [91, 90]. For modelling elastic material behavior there is no theoretical reason to prefer one strain tensor over anotherone, and the same is true for stress tensors. As discussed in Section 1.1, stress and strain are immaterial . Primary experimental data (forces, displacements) in material testing are sufficient to calculate any straintensor and any stress tensor and to display any combination thereof in stress-strain curves, while only work-conjugate pairs are physically meaningful.However, for modelling finite-strain elasticity, the quadratic Hencky model W H = µ (cid:107) dev n log V (cid:107) + κ V )] = µ (cid:107) dev n log U (cid:107) + κ U )] ,τ = 2 µ dev n log V + κ tr(log V ) , (4.15)exhibits a number of unique, favorable properties, including its functional simplicity and its dependency ononly two material parameters µ and κ that are determined in the infinitesimal strain regime and remainconstant over the entire strain range. In view of the linear dependency of stress from logarithmic strainin (4.15), it is obvious that any nonlinearity in the stress-strain curves can only be captured in Hencky’smodel by virtue of the nonlinearity in the strain tensor itself. There is a surprisingly large number of differentmaterials, where Hencky’s elasticity relation provides a very good fit to experimental stress-strain data, whichis true for different length scales and strain regimes. In the following we substantiate this claim with someexamples. Nonlinear elasticity on macroscopic scales for a variety of materials.
Anand [3, 4] has shown that theHencky model is in good agreement with experiments on a wide class of materials, e.g. vulcanized naturalrubber, for principal stretches between 0 . .
3. More precisely, this refers to the characteristic that intensile deformation the stiffness becomes increasingly smaller compared with the stiffness at zero strain, whilefor compressive deformation the stiffness becomes increasingly larger.
Nonlinear elasticity in the very small strain regime.
We mention in passing that a qualitatively similardependency of material stiffness on the sign of the strain has been made much earlier in the regime of extremelysmall strains (10 − –10 − ). In Hartig’s law [89] from 1893 this dependency was expressed as d σ d ε = E + b σ ,where E is the elasticity modulus at zero stress and b < cf. the book of Bell[19] and [126] in the context of linear elasticity with initial stress. Hartig also observed that the stress-stretchrelation should have negative curvature in the vicinity of the identity, as shown in Figure 14. Crystalline elasticity on the nanoscale.
Quite in contrast to the strictly stress -based continuum consti-tutive modelling, atomistic theories are based on a concept of interatomic forces . These forces are derivedfrom potentials V according to the potential relation f a = − ∂ x a V , which endows the model with a varia-tional structure. A further discussion of hybrid, atomistic-continuum coupling can be found in [60]. Thereby Cf. Truesdell [199, p. 145]: “
It is important to realize that since each of the several material tensors [. . . ] is an isotropicfunction of any one of the others, an exact description of strain in terms of any one is equivalent to a description in termsof any other ” or Antman [6, p. 423]: “
In place of C , any invertible tensor-valued function of C can be used as a measure ofstrain. ” Rivlin [179] states that strain need never be defined at all, cf. [203, p. 122]. The negative curvature ( b <
0) was already suggested by Jacob Bernoulli in 1705 [21] (cf. [20, p. 276]): “
Homogeneous fibersof the same length and thickness, but loaded with different weights, neither lengthen nor shorten proportional to these weights;but the lengthening or the shortening caused by the small weight is less than the ratio that the first weight has to the second. ” As Bell insists [19, p. 155], a purely linear elastic response to finite strain, corresponding to zero curvature of the stress-straincurve at the identity , is never exhibited by any physical material: “ The experiments of 280 years have demonstrated amplyfor every solid substance examined with sufficient care, that the [ finite engineering ] strain [ U − ] resulting from small appliedstress is not a linear function thereof. ” For molecular dynamics (MD) simulations, a well-established level of sophistication is the modelling by potentials withenvironmental dependence (pair functionals like in the Embedded Atom Method (EAM) account for the energy cost to embedatomic nuclei into the electron gas of variable density) and angular dependence (like for Stillinger-Weber or Tersoff functionals). virial stress theorem [195, Chapter 8] to illustrate the state of thesystem. . . λT Biot
Neo-HookeOgdenHencky
Experimental data for rubber
Figure 14: The Biot stress T Biot corresponding to uniaxialstretches by factor λ of incompressible materials fitted toexperimental measurements by Jones and Treloar [115].The curvature in λ = 1 suggests negative third order con-stants ( b < With their analyses in [53] and [54], D(cid:32)lu˙zewskiand coworkers aim to link the atomistic world to themacroscopic world of continuum mechanics. Theysearch for the “best” strain measure with a view to-wards crystalline elasticity on the nanoscale. Theauthors consider the deformation of a crystal struc-ture and compare the atomistic and continuum ap-proaches. Atomistic calculations are made usingthe Stillinger-Weber potential. The stress-strainbehaviour of the best-known anisotropic hyperelas-tic models are compared with the behaviour of theatomistic one in the uniaxial deformation test. Theresult is that the anisotropic energy based on theHencky strain energy (cid:104) C . log U, log U (cid:105) , where C isthe anisotropic elasticity tensor from linear elastic-ity, gives the best fit to atomistic simulations. Morein detail, this best fit manifests itself in the observa-tion that for considerable compression (up to ≈ ≈ [53].Elastic energy potentials based on logarithmic strain have also recently been motivated via moleculardynamics simulations [93] by Henann and Anand [94]. As indicated in Section 1.1 and shown in Sections 2.1 and 3, strain measures are closely connected to isotropicenergy functions in nonlinear hyperelasticity: similarly to how the linear elastic energy may be obtained asthe square of the Euclidean distance of ∇ u to so ( n ), the nonlinear quadratic Hencky strain energy is thesquared Riemannian distance of ∇ ϕ to SO( n ). For the partial strain measures ω iso ( F ) = (cid:107) dev n log √ F T F (cid:107) and ω vol ( F ) = | tr(log √ F T F ) | defined in Theorem 3.7, the Hencky strain energy W H can be expressed as W H ( F ) = µ ω ( F ) + κ ω ( F ) . (5.1)However, it is not at all obvious why this weighted squared sum should be viewed as the “canonical” energyassociated with the geodesic strain measures: while it is reasonable to view the elastic energy as a quantity de-pending on some strain measure alone, the specific form of this dependence must not be determined by purelygeometric deductions, but must take into account physical constraints as well as empirical observations. Third order elastic constants are corrections to the elasticity tensor in order to improve the response curves beyond theinfinitesimal neighbourhood of the identity. They exist as tabulated values for many materials. Their numerical values dependon the choice of strain measure used which needs to be corrected. D(cid:32)lu˙zewski [53] shows that again the Hencky-strain energy (cid:104) C . log U, log U (cid:105) provides the best overall approximation. G.W. Leibniz, in a letter to Jacob Bernoulli [123, p. 572], stated as early as 1690 that “ the [ constitutive ] relation betweenextension and stretching force should be determined by experiment ”, cf. [19, p. 10].
31 5 25 W H ( λ ) = ln ( λ ) W eH ( λ ) = e ln ( λ ) λ σ H σ eH λ Figure 16: The one-dimensional Hencky energy W H compared to the exponentiated Hencky energy W eH and the correspondingCauchy stresses σ H , σ eH for very large uniaxial stretches λ . Observe the non-convexity of W H and the non-invertibilityof σ H . For a large number of materials, the Hencky energy does indeed provide a very accurate model up tomoderately large elastic deformations [3, 4], i.e. up to stretches of about 40%, with only two constant materialparameters which can be easily determined in the small strain range. For very large strains , however, thesubquadratic growth of the Hencky energy in tension is no longer in agreement with empirical measurements. In a series of articles [154, 155, 153, 80], Neff et al. have therefore introduced the exponentiated Hencky energy W eH ( F ) = µk e k ω ( F ) + κ k e ˆ k ω ( F ) = µk e k (cid:107) dev n log U (cid:107) + κ k e ˆ k [tr(log U )] (5.2)1 2 3 tr( σ eH ) tr( σ H )det F tr( σ ) Figure 15: The equation of state (EOS), i.e. the trace of theCauchy stress corresponding to a purely volumetric de-formation (cf. [168]), for the quadratic and the exponen-tiated Hencky model (with parameter (cid:98) k = 4). with additional dimensionless material parameters k ≥ and (cid:98) k ≥ , which for all values of k, (cid:98) k approx-imates W H for deformation gradients F sufficientlyclose to the identity , but shows a vastly differentbehaviour for (cid:107) F (cid:107) → ∞ , cf. Figure 16.The exponentiated Hencky energy has many ad-vantageous properties over the classical quadraticHencky energy; for example, W eH is coercive on allSobolev spaces W ,p for 1 ≤ p < ∞ , thus cavita-tion is excluded [12, 143]. In the planar case n = 2, W eH is also polyconvex [155, 80] and thus Legendre-Hadamard-elliptic [10], whereas the classical Henckyenergy is not even LH-elliptic (rank-one convex) out-side a moderately large neighbourhood of [36, 144](see also [113], where the loss of ellipticity for ener-gies of the form (cid:107) dev log U (cid:107) β with hardening index0 < β < W eH for n = 2. The elastic range of numerous materials, including vulcanized rubber or skin and other soft tissues, lies well above stretchesof 40%. While the behaviour of elasticity models for extremely large strains might not seem important due to physical restraints andintermingling plasticity effects outside a narrow range of perfect elasticity, it is nevertheless important to formulate an idealized law of elasticity over the whole range of deformations; cf. Hencky [99, p. 215] (as translated in [146, p.2]): “
It is not importantthat such an idealized elastic [ behaviour ] does not actually exist and our ideally elastic material must therefore remain an ideal.Like so many mathematical and geometric concepts, it is a useful ideal, because once its deducible properties are known it canbe used as a comparative rule for assessing the actual elastic behaviour of physical bodies. ” W eH satisfies a number of constitutive inequalities [154] such as the Baker-Ericksen inequality[127], the pressure-compression inequality and the tension-extension inequality as well as Hill’s inequality [107, 160, 161], which is equivalent to the convexity of the elastic energy with respect to the logarithmic straintensor [192].Moreover, for W eH , the Cauchy-stress-stretch relation V (cid:55)→ σ eH ( V ) is invertible (a property hithertounknown for other hyperelastic formulations) and pure Cauchy shear stress corresponds to pure shear strain,as is the case in linear elasticity [154]. The physical meaning of Poisson’s ratio [169, 79] ν = κ − µ κ + µ ) is alsosimilar to the linear case; for example, ν = directly corresponds to incompressibility of the material and ν = 0 implies that no lateral extension or contraction occurs in uniaxial tensions tests. The logarithmic distance measures obtained in Theorems 3.3 and 3.7 show a strong similarity to other geodesicdistance measures on Lie groups. For example, consider the special orthogonal group SO( n ) endowed withthe canonical bi-invariant Riemannian metric ˆ g Q ( X, Y ) = (cid:104) Q T X, Q T Y (cid:105) = (cid:104) X, Y (cid:105) for Q ∈ SO( n ) and X, Y ∈ T Q SO( n ) = Q · so ( n ). Then the geodesic exponential at ∈ SO( n ) is given by thematrix exponential on the Lie algebra so ( n ), i.e. all geodesic curves are one-parameter groups of the form (cid:98) γ ( t ) = Q · exp( t A )with Q ∈ SO( n ) and A ∈ so ( n ) (cf. [136]). It is easy to show that the geodesic distance between Q, R ∈ SO( n )with respect to this metric is given bydist geod , SO( n ) ( Q, R ) = (cid:107) log( Q T R ) (cid:107) , where (cid:107) . (cid:107) is the Frobenius matrix norm and log : SO( n ) → so ( n ) denotes the principal matrix logarithm onSO( n ), which is uniquely defined by the equality exp(log Q ) = Q and the requirement λ i (log Q ) ∈ ( − π, π ] forall Q ∈ SO( n ) and all eigenvalues λ i (log Q ).This result can be extended to the geodesic distance on the conformal special orthogonal group CSO( n )consisting of all angle-preserving linear mappings:CSO( n ) := { c · Q | c > , Q ∈ SO( n ) } , where the bi-invariant metric g CSO( n ) is given by the canonical inner product: g CSO( n ) A ( X, Y ) = (cid:104) A − X, A − Y (cid:105) . (5.3)Then dist , CSO( n ) ( c · Q, d · R ) = (cid:107) log( Q T R ) (cid:107) + 1 n (cid:104) ln (cid:16) cd (cid:17)(cid:105) , where log again denotes the principal matrix logarithm on SO( n ). Note that the punctured complex plane C \ { } can be identified with CSO(2) via the mapping z = a + i b (cid:55)→ Z ∈ CSO(2) = (cid:26)(cid:18) a b − b a (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) a + b (cid:54) = 0 (cid:27) . Hill’s inequality [161] can be stated more generally as (cid:104) dd t ◦ [ τ ] − m [ τ D − D τ ] , D (cid:105) ≥ dd t ◦ is the Zaremba-Jaumann stress rate (4.13) and τ is the Kirchhoff stress tensor. For m = 0, as ˇSilhav´y explains, “ Hill’sinequalities [ . . . ] require the convexity of [ the strain energy W ] in [ terms of the strain tensor log V ] . . . This does not seem tocontradict any theoretical or experimental evidence ” [193, p. 309]. Note that µ c · (cid:98) g is the restriction of our left-GL( n )-invariant, right-O( n )-invariant metric g (as defined in Section 3.1) toSO( n ). .3 Outlook While first applications of the exponentiated Hencky energy, which is based on the partial strain measures ω iso , ω vol introduced here, show promising results, including an accurate modelling of so-called tire-derivedmaterial [140], a more thorough fitting of the new parameter set to experimental data is necessary in orderto assess the range of applicability of W eH towards elastic materials like vulcanized rubber. A differentformulation in terms of the partial strain measures ω iso and ω vol , i.e. an energy function of the form W ( F ) = Ψ( ω iso ( F ) , ω vol ( F )) = Ψ( (cid:107) dev log U (cid:107) , | tr(log U ) | ) (5.4)with Ψ : [0 , ∞ ) → [0 , ∞ ), might even prove to be polyconvex in the three-dimensional case. The main openproblem of finding a polyconvex (or rank-one convex) isochoric energy function F (cid:55)→ (cid:101) Ψ( (cid:107) dev log U (cid:107) ) hasalso been considered by Sendova and Walton [189]. Note that while every isotropic elastic energy W can beexpressed as W ( F ) = h ( K , K , K ) with Criscione’s invariants [46, 45, 51, 208] K = tr(log U ) , K = (cid:107) dev log U (cid:107) and K = det (cid:18) dev log U (cid:107) dev log U (cid:107) (cid:19) , (5.5)not every elastic energy has a representation of the form (5.4); for example, (5.4) implies the tension-compression symmetry W ( F ) = W ( F − ), which is not necessarily satisfied by energy functions in general. In terms of the
Shield transformation [191, 39] W ∗ ( F ) := det F · W ( F − ) , the tension-compression symmetry amounts to the requirement F W ∗ ( F ) = W ( F ) or, for incompressiblematerials, W ∗ ( F ) = W ( F ). Moreover, under the assumption of incompressibility, the symmetry can beimmediately extended to arbitrary deformations ϕ : Ω → ϕ (Ω) and ϕ − : ϕ (Ω) → Ω: if det ∇ ϕ ≡
1, we canapply the substitution rule to find (cid:90) ϕ (Ω) W ( ∇ ( ϕ − )( x )) d x = (cid:90) Ω W ( ∇ ( ϕ − )( ϕ ( x ))) · | det ∇ ϕ ( x ) | d x = (cid:90) Ω W ( ∇ ϕ ( x ) − ) d x = (cid:90) Ω W ( ∇ ϕ ( x )) d x if W ( F − ) = W ( F ) for all F ∈ SL( n ), thus the total energies of the deformations ϕ, ϕ − are equal, cf. Figure17. Since the function F (cid:55)→ e (cid:107) dev log U (cid:107) = e dist , SL(2) (cid:16) F det F / , SO(2) (cid:17) Ideally, the function (cid:101)
Ψ should also satisfy additional requirements, such as monotonicity, convexity and exponential growth. The invariants K and K = tr (cid:0) (dev log U ) (cid:1) as well as (cid:101) K = tr (cid:0) (dev log U ) (cid:1) had already been discussed exhaustivelyby H. Richter in a 1949 ZAMM article [176, § K and K have also been considered by A.I. Lurie [125, p. 189]. Criscionehas shown that the invariants given in (5.5) enjoy a favourable orthogonality condition which is useful when determining materialparameters. The tension-compression symmetry is often expressed as τ ( V − ) = − τ ( V ), where τ ( V ) is the Kirchhoff stress tensorcorresponding to the left Biot stretch V . This condition, which is the natural nonlinear counterpart of the equality σ ( − ε ) = − σ ( ε )in linear elasticity, is equivalent to the condition W ( F − ) = W ( F ) for hyperelastic constitutive models. Truesdell and Noll [203, p. 174] argue that “ . . . there is no foundation for the widespread belief that according to the theoryof elasticity, pressure and tension have equal but opposite effects ”. Examples for isotropic energy functions which do not satisfythis symmetry condition in general but only in the incompressible case can be found in [92]. For an idealized isotropic elasticmaterial, however, the tension-compression symmetry is a plausible requirement (with an obvious additive counterpart in linearelasticity), especially for incompressible bodies. Further properties of the Shield transformation can be found in [193, p.288]; for example, it preserves the polyconvexity,quasiconvexity and rank-one convexity of the original energy. Ω ϕ (Ω) ϕ − Figure 17: The tension-compression symmetry for incompressible materials: if det ∇ ϕ ≡ W ( F − ) = W ( F ) for all F ∈ SL( n ), then (cid:82) Ω W ( ∇ ϕ ( x )) d x = (cid:82) ϕ (Ω) W ( ∇ ( ϕ − )( x )) d x . in planar elasticity is polyconvex [155, 80], it stands to reason that a similar formulation in the three-dimensional case might prove to be polyconvex as well. A first step towards finding such an energy is toidentify where the function W with W ( F ) = e (cid:107) dev log U (cid:107) = e dist , SL(3) (cid:16) F det F / , SO(3) (cid:17) , (5.6)which is not rank-one convex [154], loses its ellipticity properties. For that purpose, it may be useful toconsider the quasiconvex hull of W . There already are a number of promising results for similar energyfunctions; for example, the quasiconvex hull of the mapping F (cid:55)→ dist ( F, SO(2)) = (cid:107) U − (cid:107) can be explicitly computed [194, 56, 57], and the quasiconvex hull of the similar Saint-Venant-Kirchhoff energy W SVK ( F ) = µ (cid:107) C − (cid:107) + λ [tr( C − )] has been given by Le Dret and Raoult [121]. For the mappings F (cid:55)→ dist ( F, SO(3)) or F (cid:55)→ dist ( F, SO( n ))with n ≥
2, however, no explicit representation of the quasiconvex hull is yet known, although it has beenshown that both expressions are not rank-one convex [24].It might also be of interest to calculate the geodesic distance dist geod ( A, B ) for a larger class of matrices
A, B ∈ GL + ( n ): although Theorem 3.3 allows us to explicitly compute the distance dist geod ( , P ) for P ∈ Sym + ( n ) and local results are available for certain special cases [129], it is an open question whetherthere is a general formula for the distance dist geod , GL + ( n ) ( Q, R ) between arbitrary rotations
R, Q ∈ SO( n ) forall parameters µ, µ c , κ >
0. Since restricting our left-GL( n )-invariant, right-O( n )-invariant metric on GL( n )to SO( n ) yields a multiple of the canonical bi-SO( n )-invariant metric on SO( n ), we can computedist , GL + ( n ) ( Q, R ) = µ c · dist , SO( n ) ( Q, R ) = µ c (cid:107) log( Q T R ) (cid:107) if for all Q, R ∈ SO( n ) a shortest geodesic in GL + ( n ) connecting Q and R is already contained within SO( n ),cf. Figure 18. However, whether this is the case depends on the chosen parameters µ, µ c ; a general closed-formsolution for dist geod , GL + ( n ) on SO( n ) is therefore not yet known [128].Moreover, it is not known whether our result can be generalized to anisotropic Riemannian metrics,i.e. if the geodesic distance to SO( n ) can be explicitly computed for a larger class of left-GL( n )-invariant An improved understanding of the geometric structure of mechanical problems could, for example, help to develop newdiscretization methods [184, 85]. O( n ) G L + ( n ) QRµ c · dist SO( n ) ( Q, R ) = µ c k log( Q T R ) k dist GL + ( n ) ( Q, R ) ?dist GL + ( n ) ( Q, R ) = µ c k log( Q T R ) k ? SO( n ) G L + ( n ) QRµ c · dist SO( n ) ( Q, R ) = µ c k log( Q T R ) k dist GL + ( n ) ( Q, R ) ?dist GL + ( n ) ( Q, R ) = µ c k log( Q T R ) k ? Figure 18: If SO( n ) contains a length minimizing geodesic connecting Q, R ∈ SO( n ) with respect to our left-GL( n )-invariant,right-O( n )-invariant metric g on GL( n ), then the GL + ( n )-geodesic distance between Q and R is equal to the well-knownSO( n )-geodesic distance µ c (cid:107) log( Q T R ) (cid:107) . Riemannian metrics which are not necessarily right-O( n )-invariant. A result in this direction would haveimmediate impact on the modelling of finite strain anisotropic elasticity [14, 188, 187]. The difficulties withsuch an extension are twofold: one needs a representation formula for Riemannian metrics which are right-invariant under a given symmetry subgroup of O( n ), as well as an understanding of the corresponding geodesiccurves. We have shown that the squared geodesic distance of the (finite) deformation gradient F ∈ GL + ( n ) to thespecial orthogonal group SO( n ) is the quadratic isotropic Hencky strain energy:dist ( F, SO( n )) = µ (cid:107) dev n log U (cid:107) + κ U )] , if the general linear group is endowed with the left-GL( n )-invariant, right-O( n )-invariant Riemannian metric g A ( X, Y ) = (cid:104) A − X, A − Y (cid:105) µ,µ c ,κ , where (cid:104) X, Y (cid:105) µ,µ c ,κ = µ (cid:104) dev n sym X, dev n sym Y (cid:105) + µ c (cid:104) skew X, skew Y (cid:105) + κ tr( X ) tr( Y )with (cid:104) X, Y (cid:105) = tr( X T Y ). Furthermore, the (partial) logarithmic strain measures ω iso = (cid:107) dev n log U (cid:107) = (cid:107) dev n log √ F T F (cid:107) and ω vol = | tr(log U ) | = | tr(log √ F T F ) | have been characterized as the geodesic distance of F to the special orthogonal group SO( n ) and the identitytensor , respectively: ω iso = (cid:107) dev n log U (cid:107) = dist geod , SL( n ) (cid:18) F det F /n , SO( n ) (cid:19) ,ω vol = | tr(log U ) | = √ n · dist geod , R + · (cid:16) (det F ) /n · , (cid:17) , where the geodesic distances on SL( n ) and R + · are induced by the canonical left invariant metric ¯ g A ( X, Y ) = (cid:104) A − X, A − Y (cid:105) .We thereby show that the two quantities ω iso = (cid:107) dev n log U (cid:107) and ω vol = | tr(log U ) | are purely geometricproperties of the deformation gradient F , similar to the invariants (cid:107) dev n ε (cid:107) and | tr( ε ) | of the infinitesimalstrain tensor ε in the linearized setting.While there have been prior attempts to deductively motivate the use of logarithmic strain in nonlinearelasticity theory, these attempts have usually focussed on the logarithmic Hencky strain tensor E = log U (cid:98) E = log V ) and its status as the “natural” material (or spatial) strain tensor in isotropic elasticity. Wediscussed, for example, a well-known characterization of log V in the hypoelastic context: if the strain rate dd t (cid:3) is objective as well as corotational , and ifdd t (cid:3) [ (cid:98) E ] = D := sym( ˙ F F − )for some strain tensor (cid:98) E , then dd t (cid:3) = dd t log must be the logarithmic rate and (cid:98) E = (cid:98) E = log V must be thespatial Hencky strain tensor.However, as discussed in Section 1.1, all strain tensors are interchangeable: the choice of a specific straintensor in which a constitutive law is to be expressed is not a restriction on the available constitutive relations.Such an approach can therefore not be applied to deduce necessary conditions or a priori properties ofconstitutive laws.Our deductive approach, on the other hand, directly motivates the use of the strain measures ω iso and ω vol from purely differential geometric observations. As we have indicated, the requirement that a constitutivelaw depends only on ω iso and ω vol has direct implications; for example, the tension-compression symmetry W ( F ) = W ( F − ) is satisfied by every hyperelastic potential W which can be expressed in terms of ω iso and ω vol alone.Moreover, as demonstrated in Section 4, similar approaches oftentimes presuppose the role of the positivedefinite factor U = √ F T F as the sole measure of the deformation, whereas this independence from theorthogonal polar factor is obtained deductively in our approach (cf. Table 1). Measure of deformation deduced Measure of deformation postulatedlinear dist ,µ,µ c ,κ ( ∇ u, so ( n ))= µ (cid:107) dev n sym ∇ u (cid:107) + κ ∇ u )] dist , Sym( n ) ,µ,κ ( ε, µ (cid:107) dev n ε (cid:107) + κ ε )] geometricallynonlinear dist ( F, SO( n )) = µ (cid:107)√ F T F − (cid:107) dist , Sym( n ) ( U, ) = µ (cid:107) U − (cid:107) geometricallynonlinear(weighted) not well defined dist , Sym( n ) ,µ,κ ( U, )= µ (cid:107) dev n ( U − ) (cid:107) + κ U − )] geodesic dist ,µ,µ c ,κ ( F, SO( n ))= µ (cid:107) dev n log( √ F T F ) (cid:107) + κ √ F T F )] dist , Sym + ( n ) ,µ,κ ( U, )= µ (cid:107) dev n log U (cid:107) + κ U )] log-Euclidean not well defined dist ,µ,κ ( U, )= dist , Sym( n ) ,µ,κ (log U, µ (cid:107) dev n log U (cid:107) + κ U )] Table 1: Different approaches towards the motivation of different strain tensors and strain measures.
Note also that the specific distance measure dist geod on GL + ( n ) used here is not chosen arbitrarily: therequirements of left-GL( n )-invariance and right-O( n )-invariance, which have been motivated by mechanicalconsiderations, uniquely determine g up to the three parameters µ, µ c , κ >
0. This uniqueness property furtheremphasizes the generality of our results, which yet again strongly suggest that Hencky’s constitutive law Observe that (cid:107) dev n ( U − ) (cid:107) does not measure the isochoric (distortional) part F (det F ) /n of F . Acknowledgements
The second author acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through a Heisen-berg fellowship under grant EI 453/2-1.We are grateful to Prof. Alexander Mielke (Weierstraß-Institut, Berlin) for pertinent discussions ongeodesics in GL( n ); the first parametrization of geodesic curves on SL( n ) known to us is due to him [134]. Wealso thank Prof. Robert Bryant (Duke University) for his helpful remarks regarding geodesics on Lie groupsand invariances of inner products on gl ( n ), as well as a number of friends who helped us with the draft.We also thank Dr. Andreas Fischle (Technische Universit¨at Dresden) who, during long discussions oncontinuum mechanics and differential geometry, inspired many of the ideas laid out in this paper.The first author had the great honour of presenting the main ideas of this paper to Richard Toupin on theoccasion of the Canadian Conference on Nonlinear Solid Mechanics 2013 in the mini-symposium organizedby Francesco dell’Isola and David J. Steigmann, which was dedicated to Toupin. Conflict of Interest
The authors declare that they have no conflict of interest.
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Generalized Continua as Models forMaterials . Springer, 2013, pp. 321–331. Appendix
A.1 Notation • R is the set of real numbers , • R + = (0 , ∞ ) is the set of positive real numbers , • R n is the set of real column vectors of length n , • R n × m is the set of real n × m - matrices , • is the identity tensor ; • X T is the transpose of a matrix X ∈ R n × m , • tr( X ) = (cid:80) ni =1 X i,i is the trace of X ∈ R n × n , • Cof X is the cofactor of X ∈ R n × n , • (cid:104) X, Y (cid:105) = tr( X T Y ) = (cid:80) ni,j =1 X i,j Y i,j is the canonical inner product on R n × n , • (cid:107) X (cid:107) = (cid:112) (cid:104) X, X (cid:105) is the
Frobenius matrix norm on R n × n , • sym X = ( X + X T ) is the symmetric part of X ∈ R n × n , • skew X = ( X − X T ) is the skew-symmetric part of X ∈ R n × n , • dev n X = X − n tr( X ) · is the n -dimensional deviator of X ∈ R n × n , • (cid:104) X, Y (cid:105) µ,µ c ,κ = µ (cid:104) dev n sym X, dev n sym Y (cid:105) + µ c (cid:104) skew X, skew Y (cid:105) + κ tr( X ) tr( Y ) is the weighted inner prod-uct on R n × n , • (cid:107) X (cid:107) µ,µ c ,κ = (cid:112) (cid:104) X, X (cid:105) µ,µ c ,κ is the weighted Frobenius norm on R n × n , • GL( n ) = { A ∈ R n × n | det A (cid:54) = 0 } is the general linear group of all invertible A ∈ R n × n , • GL + ( n ) = { A ∈ R n × n | det A > } is the identity component of GL( n ), • SL( n ) = { A ∈ R n × n | det A = 1 } is the special linear group of all A ∈ GL( n ) with det A = 1, • O( n ) is the orthogonal group of all Q ∈ R n × n with Q T Q = , • SO( n ) is the special orthogonal group of all Q ∈ O( n ) with det Q = 1, • Sym( n ) is the set of symmetric , real n × n -matrices, i.e. S T = S for all S ∈ Sym( n ), • Sym + ( n ) is the set of positive definite , symmetric, real n × n -matrices, i.e. x T P x > P ∈ Sym + ( n ) , (cid:54) = x ∈ R n , • gl ( n ) = R n × n is the Lie algebra of all real n × n -matrices, • so ( n ) = { W ∈ R n × n | W T = − W } is the Lie algebra of skew symmetric , real n × n -matrices, • sl ( n ) = { X ∈ R n × n | tr( X ) = 0 } is the Lie algebra of trace free , real n × n -matrices, i.e. tr( X ) = 0 for all X ∈ sl ( n ), • Ω ⊂ R n is the reference configuration of an elastic body,47 ∇ ϕ = Dϕ is the first derivative of a differentiable function ϕ : Ω ⊂ R n → R n , often called the deformationgradient , • curl v denotes the curl of a vector valued function v : R → R , • Curl p denotes the curl of a matrix valued function p : R → R × , taken row-wise, • ϕ : Ω → R n is a continuously differentiable deformation with ∇ ϕ ( x ) ∈ GL + ( n ) for all x ∈ Ω, • F = ∇ ϕ ∈ GL + ( n ) is the deformation gradient , • U = √ F T F ∈ Sym + ( n ) is the right Biot-stretch tensor , • V = √ F F T ∈ Sym + ( n ) is the left Biot-stretch tensor , • B = F F T = V is the Finger tensor , • C = F T F = U is the right Cauchy-Green deformation tensor , • F = R U = V R is the polar decomposition of F with R = polar( F ) ∈ SO( n ), • E = log U is the material Hencky strain tensor , • (cid:98) E = log V is the spatial Hencky strain tensor , • S = D F W ( F ) is the first Piola-Kirchhoff stress corresponding to an elastic energy W = W ( F ), • S = F − S = 2 D C W ( C ) is the second Piola-Kirchhoff stress corresponding to an elastic energy W = W ( C ) (Doyle-Ericksen formula), • τ = S F T = D log V W (log V ) [125, p. 116] is the Kirchhoff stress tensor , • σ = F τ is the Cauchy stress tensor , • T Biot = U S = D U W ( U ) is the Biot stress tensor corresponding to an elastic energy W = W ( U ), • L = ˙ F F − is the spatial velocity gradient , • D = sym L is the rate of stretching or spatial strain rate tensor , • W = skew L is the spatial continuum spin . 48 .2 Linear stress-strain relations in nonlinear elasticity Many constitutive laws commonly used in applications are expressed in terms of linear relations betweencertain strains and stresses, including Hill’s family of generalized linear elasticity laws (cf. Section 4.2.1) ofthe form T r = 2 µ E r + λ tr( E r ) · (A.1)with work-conjugate pairs ( T r , E r ) based on the Lagrangian strain measures given in (1.3). A widely knownexample of such a constitutive law is the hyperelastic Saint-Venant-Kirchhoff model S = 2 µ E + λ tr( E ) = µ ( C − ) + λ C − ) · for r = 1 and T = S , where S denotes the second Piola-Kirchhoff stress tensor. Similarly, a number ofelasticity laws can be written in the form (cid:98) T r = 2 µ (cid:98) E r + λ tr( (cid:98) E r ) · with a spatial strain tensor (cid:98) E r and a corresponding stress tensor (cid:98) T r . Examples include the Neo-Hooke typemodel σ = 2 µ (cid:98) E + λ tr( (cid:98) E ) = µ ( B − ) + λ B − ) · for r = 1, where T = σ is the Cauchy stress tensor, the Almansi-Signorini model σ = 2 µ (cid:98) E − + λ tr( (cid:98) E − ) = µ ( − B − ) + λ − B − ) · for r = − T − = σ , as well as the hyperelastic Hencky model τ = 2 µ log V + λ tr(log V ) · for r = 0 and (cid:98) T = τ . A thorough comparison of these four constitutive laws can be found in [16].Another example of a postulated linear stress-strain relation is the model T Biot = 2 µ log U + λ tr(log U ) · , where T Biot denotes the
Biot stress tensor , which measures the “ stress per unit initial area before deformation ”[28]. This constitutive relation was first given in an 1893 article by the geologist G. F. Becker [18, 152], whodeduced it from a law of superposition in an approach similar to that of H. Hencky. The same constitutive lawwas considered by Carroll [38] as an example to emphasize the necessity of a hyperelastic formulation in orderto ensure physical plausibility in the description of elastic behaviour. Note that of the constitutive relationslisted in this section, only the Hencky model and the Saint-Venant-Kirchhoff model are indeed hyperelastic(cf. [23, Chapter 7.4]).
A.3 Tensors and tangent spaces
In the more general setting of differential geometry, the linear mappings
F, U, C, V, B and R as well as variousstresses at a single point x in an elastic body Ω are defined as mappings between different tangent spaces :for a point x ∈ Ω and a deformation ϕ , we must then distinguish between the two tangent spaces T x Ω and T ϕ ( x ) ϕ (Ω). The domains and codomains of various linear mappings are listed below and indicated in Figure19. Note that we do not distinguish between tangent and cotangent vector spaces (cf. [63]). F, R : T x Ω → T ϕ ( x ) ϕ (Ω) ,U, C : T x Ω → T x Ω ,V, B : T ϕ ( x ) ϕ (Ω) → T ϕ ( x ) ϕ (Ω) . ( x ( t )) ∇ ϕ ( x ( t )) . ˙ x ( t ) x ( t ) ˙ x ( t )Ω ϕ (Ω) ϕ ˙ x ( t ) ∈ T x ( t ) Ω U, C : T x Ω → T x Ω ∇ ϕ. ˙ x ( t ) ∈ T ϕ ( x ( t )) ϕ (Ω) V, B : T ϕ ( x ) ϕ (Ω) → T ϕ ( x ) ϕ (Ω) F, R : T x Ω → T ϕ ( x ) ϕ (Ω)(two-point tensors) Figure 19: Various linear mappings between the tangent spaces T x Ω and T ϕ ( x ) ϕ (Ω). The right Cauchy-Green tensor C = F T F , in particular, is often interpreted as a Riemannian metric onΩ; Epstein [61, p. 113] explains that “ the right Cauchy-Green tensor is precisely the pull-back of the spatialmetric to the body manifold ”, cf. [127]. If Ω and ϕ (Ω) are embedded in the Euclidean space R n , this connectioncan immediately be seen: while the length of a curve x : [0 , → Ω is given by (cid:82) (cid:112) (cid:104) ˙ x, ˙ x (cid:105) d t , where (cid:104)· , ·(cid:105) isthe canonical inner product on R n , the length of the deformed curve ϕ ◦ x is given by (cf. Figure 19) (cid:90) (cid:113) (cid:104) dd t ( ϕ ◦ x ) , dd t ( ϕ ◦ x ) (cid:105) d t = (cid:90) (cid:112) (cid:104) F ( x ) ˙ x, F ( x ) ˙ x (cid:105) d t = (cid:90) (cid:112) (cid:104) C ( x ) ˙ x, ˙ x (cid:105) d t . The quadratic form g x ( v, v ) = (cid:104) C ( x ) v, v (cid:105) at x ∈ Ω therefore measures the length of the deformed line element
F v at ϕ ( x ) ∈ ϕ (Ω). Thus locally,dist Euclid ,ϕ (Ω) ( ϕ ( x ) , ϕ ( y )) = dist geod , Ω ( x, y ) , where dist Euclid ,ϕ (Ω) ( ϕ ( x ) , ϕ ( y )) = (cid:107) ϕ ( x ) − ϕ ( y ) (cid:107) is the Euclidean distance between ϕ ( x ) , ϕ ( y ) ∈ ϕ (Ω)and dist geod , Ω ( x, y ) denotes the geodesic distance between x, y ∈ Ω with respect to the Riemannian met-ric g x ( v, w ) = (cid:104) C ( x ) v, w (cid:105) .Moreover, this interpretation characterizes the Green-Lagrangian strain tensor E = ( C − ) as a measureof change in length : the difference between the squared length of a line element v ∈ T x Ω in the referenceconfiguration and the squared length of the deformed line element F ( x ) v ∈ T ϕ ( x ) ϕ (Ω) is given by (cid:107) F ( x ) v (cid:107) − (cid:107) v (cid:107) = (cid:104) C ( x ) v, v (cid:105) − (cid:104) v, v (cid:105) = (cid:104) ( C ( x ) − ) v, v (cid:105) = 2 (cid:104) E ( x ) v, v (cid:105) , where (cid:107) . (cid:107) denotes the Euclidean norm on R n . Note that for F ( x ) = + ∇ u ( x ) with the displacementgradient ∇ u ( x ), the expression (cid:107) F ( x ) v (cid:107) can be linearized to (cid:107) F ( x ) v (cid:107) = (cid:107) ( + ∇ u ( x )) v (cid:107) = (cid:104) ( + ∇ u ( x )) v, ( + ∇ u ( x )) v (cid:105) = (cid:104) v, v (cid:105) + 2 (cid:104)∇ u ( x ) v, v (cid:105) + (cid:104)∇ u ( x ) v, ∇ u ( x ) v (cid:105) = (cid:107) v (cid:107) + 2 (cid:104) sym ∇ u ( x ) v, v (cid:105) + (cid:107)∇ u ( x ) v (cid:107) = (cid:107) v (cid:107) + 2 (cid:104) sym ∇ u ( x ) v, v (cid:105) + h . o . t . , . o . t . denotes higher order terms with respect to ∇ u ( x ). Thus (cid:107) F ( x ) v (cid:107) − (cid:107) v (cid:107) = 2 (cid:104) ε ( x ) v, v (cid:105) + h . o . t . , where ε = sym ∇ u is the linear strain tensor. A.4 Additional computations
Let Cof F = (det F ) · F − T denote the cofactor of F ∈ GL + ( n ). Then the geodesic distance of Cof F to SO( n )with respect to the Riemannian metric g introduced in (3.5) can be computed directly by applying Theorem3.3: dist (Cof F, SO( n ))= µ (cid:107) dev n log (cid:113) (Cof F ) T Cof F (cid:107) + κ (cid:104) tr (cid:16) log (cid:113) (Cof F ) T Cof F (cid:17)(cid:105) = µ (cid:107) dev n log (cid:113) (det F ) · F − F − T (cid:107) + κ (cid:104) tr (cid:16) log (cid:113) (det F ) · F − F − T (cid:17)(cid:105) = µ (cid:107) dev n log √ F − F − T (cid:107) + κ (cid:104) tr (cid:16) log (cid:0) (det F ) · (cid:1) + log √ F − F − T (cid:17)(cid:105) = µ (cid:107) dev n log( U − ) (cid:107) + κ (cid:0) (ln det F ) · + log( U − ) (cid:1) ] = µ (cid:107)− dev n log U (cid:107) + κ n · (ln det U ) − tr(log U )] = µ (cid:107) dev n log U (cid:107) + κ ( n − U )] . A.5 The principal matrix logarithm on Sym + ( n ) and the matrix exponential The following lemma states some basic computational rules for the matrix exponential exp : R n × n → GL + ( n )and the principal matrix logarithm log : Sym + ( n ) → Sym( n ) involving the trace operator tr and the deviatoricpart dev n X = X − tr( X ) n · of a matrix X ∈ R n × n . Lemma A.1.
Let X ∈ R n × n , P ∈ Sym + ( n ) and c > . Theni) det(exp( X )) = e tr( X ) , ii) exp(dev n X ) = e − tr( X ) n · exp( X ) , iii) log( c · ) = ln( c ) · log( ) , iv) log((det P ) − /n · P ) = log P − ln(det P ) n · = dev n log P .
Proof.
Equality i) is well known (see e.g. [22]). Equality iii ) follows directly from the fact that exp : Sym( n ) → Sym + ( n ) is bijective and that exp(ln( c ) · ) = e ln( c ) · = c · . Since AB = BA implies exp( AB ) =exp( A ) exp( B ), we findexp(dev n X ) = exp( X − tr( X ) n · ) = exp( X ) · exp( − tr( X ) n · ) = exp( X ) · e − tr( X ) n · , showing ii ). For iv), note thattr(log P ) = ln(det P ) = ⇒ log P − ln(det P ) n · = dev n log P , and exp(dev n log P ) = e − tr(log P ) n · exp(log P )= (cid:16) e ln(det P ) (cid:17) − /n · P = (det P ) − /n · P . according to ii). Then the injectivity of the matrix exponential on Sym( n ) shows iv). (cid:4) .6 A short biography of Heinrich Hencky Hencky at MIT, age 45 [135]
Biographical information on Heinrich Hencky, as laid out in [196, 33, 95]: • November 2, 1885: Hencky is born in Ansbach, Franken, Germany • • • • • • • • • • • • • • • July 6, 1951: Hencky dies in an avalanche at age 65 during mountain climbingHencky received his diploma in civil engineering from TH M¨unchen in 1908 and his Ph.D from TH Darm-stadt in 1913. The title of his thesis was “ ¨Uber den Spannungszustand in rechteckigen, ebenen Platten beigleichm¨aßig verteilter und bei konzentrierter Belastung” (“On the stress state in rectangular flat plates underuniformly distributed and concentrated loading”). In 1915, the main results of his thesis were also publishedin the Zeitschrift f¨ur angewandte Mathematik und Physik [96].After working on plasticity theory and small-deformation elasticity, he began his work on finite elasticdeformations in 1928. In 1929 he introduced the logarithmic strain e log = log (cid:0) final lengthoriginal length (cid:1)(cid:1)