A commented translation of Hans Richter's early work "The isotropic law of elasticity"
Kai Graban, Eva Schweickert, Robert J. Martin, Patrizio Neff
aa r X i v : . [ m a t h . HO ] A p r A commented translation of Hans Richter’s early work“The isotropic law of elasticity”
Kai Graban and Eva Schweickert and Robert J. Martin and Patrizio Neff April 11, 2019
Abstract
We provide a faithful translation of Hans Richter’s important 1948 paper “Das isotrope Elastizit¨atsge-setz” from its original German version into English. Our introduction summarizes Richter’s achievements.
Key words: nonlinear elasticity, isotropic tensor functions, hyperelasticity, logarithmic stretch, volumetric-isochoric split, Hooke’s law, finite deformations, isotropy
AMS 2010 subject classification: 74B20, 01A75
Introduction
Shortly after the second world war, in a series of papers [23, 24, 25, 26] from 1948 − − F = V R into a stretch V ∈ Sym + (3) and a rotation R ∈ SO(3). For Richter, the “physical stress tensor” is the Cauchy stress tensor σ ∈ Sym(3). From thecoaxiality between σ and V for an isotropic response, he deduces the representation formula for isotropictensor functions (the Richter representation , see (2.6)) σ = g ( I , I , I ) · + g ( I , I , I ) · V + g ( I , I , I ) · V (0.1)(predating the Rivlin-Ericksen representation theorem [27] by 7 years) where g i , i = 1 , , I ν , ν = 1 , ,
3, with I = tr( V ) , I = 12 tr( V ) , I = det V. Kai Graban, Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen Eva Schweickert, Lehrstuhl f¨ur Nichtlineare Analysis und Modellierung, Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany; email: [email protected] Robert J. Martin, Lehrstuhl f¨ur Nichtlineare Analysis und Modellierung, Fakult¨at f¨ur Mathematik, Universit¨at Duisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany; email: [email protected] Patrizio Neff, Head of Lehrstuhl f¨ur Nichtlineare Analysis und Modellierung, Fakult¨at f¨ur Mathematik, Universit¨atDuisburg-Essen, Thea-Leymann Str. 9, 45127 Essen, Germany, email: patrizio.neff@uni-due.de L = log V without citing the previous work ofHencky [7, 9, 8, 10, 16, 18, 19, 17, 21]. He then turns to the question of what happens if the relation (0.1) isderived from a stored energy W ( I , I , I ), i.e. when (0.1) is consistent with hyperelasticity. He obtains thecorrect representation (see (3.11) in his text) σ = ∂W∂I · + 1 I · ∂W∂I · V + 1 I · ∂W∂I · V , W = W ( I , I , I ) . (0.2)In the next section, Richter introduces the multiplicative split of the elastic stretch V into volume preserving(isochoric) parts and volume change (see (4.1)) V = V (det V ) · (det V ) · (0.3)and he observes that the logarithmic stretch tensor additively separates both contributions by using theclassical deviator operation (see (4.2)) such thatlog V = dev log V + 13 tr(log V ) · = log (cid:18) V (det V ) (cid:19) + 13 log det V · , dev X = X −
13 tr( X ) · . (0.4)He also observes that the invariants based on the logarithmic stretch tensor satisfy certain algebraic relations,cf. [3]. In Richter’s fifth section, he introduces the volumetric-isochoric split W ( F ) = W iso (dev log V ) + W vol (tr(log V ))= W iso (cid:18) log (cid:18) V (det V ) (cid:19)(cid:19) + W vol (log det V ) = f W iso (cid:18) V (det V ) (cid:19) + f W vol (det V )of the stored energy (often erroneously attributed to [6]) and he immediately obtains the important result: An isotropic energy is additively split into volumetric and isochoric parts if and only if the mean Cauchystress tr σ is only a function of the relative volume change det V . In that case,
13 tr σ = 1det V · W ′ vol (log det V ) = f W ′ vol (det V ) . (0.5)This result has been rediscovered and re-derived multiple times, e.g. in [2, 13, 28, 4, 5, 14]. In addition,Richter shows that this property of the volumetric-isochoric split is invariant under a change of the referencetemperature. Finally, he poses the question whether a linear relation between σ and V in the form (theHooke’s law as he perceives it) σ = 2 µ ( V − ) + λ tr( V − ) · , (0.6)where µ > λ is the second Lam´e parameter, can be consistent with hyperelas-ticity. A short calculus reveals that (0.6) is hyperelastic if and only if 2 µ = λ , i.e. for Poisson ratio ν = (which is approximately satisfied for many metals, e.g. aluminium). For all other values of ν , Hooke’s lawis incompatible with the hyperelastic approach and Richter proposes to use instead (the quadratic Henckyenergy [7, 15]) W ( F ) = µ k dev log V k + 2 µ + 3 λ (log V )with the induced stress-strain law σ · det F = τ = 2 µ log V + λ tr(log V ) · , (0.7)2here τ is the Kirchhoff stress tensor.We will briefly discuss the constitutive relation (0.6). In order to check hyperlasticity of the Cauchystress-stretch relation in this case, we use the representation, consistent with (0.2), σ ( V ) = 2 µJ D V W ( V ) · V, J = det V, (0.8)and consider the energy W ( F ) = 2 µ det V [tr( V ) − σ ( V ) = 2 µ ( V − ) + 2 µ tr( V − ) · .Since tr[ σ ( V )] = 8 µ tr( V − ) and tr[ σ ( α )] = 24 µ ( α − µ = λ is injective (but not bijective, since for tr( σ ) = − K + < − µ there does not exist a stretch V ∈ Sym + (3)such that tr( σ ( V )) = − K + ). Furthermore, note that[tr( V )] = ( λ + λ + λ ) = λ + λ + λ + 2( λ λ + λ λ + λ λ ) = tr( B ) + 2 tr(Cof B ) , (0.9)where λ i are the singular values of the deformation gradient F . Then2 µ det V [tr( V ) −
4] = 2 µ p det( B ) { p tr( B ) + 2 tr(Cof B ) − } = 2 µ p I { p I + 2 I − } = W ( I , I , I ) , I = tr( B ) , I = tr(Cof B ) , I = det B. (0.10)For this energy, the weak empirical inequalities [30] ∂W∂I > ∂W∂I > σ i = 2 µ · ( λ i − λ + λ + λ − < ∂σ i ∂λ i = 2 µ · (1 + 1) = 4 µ and 0 < ( σ i − σ j )( λ i − λ j ) = 2 µ ( λ i − λ j ) respectively, are satisfied as well. We also note that W ( V ) = 2 µ · det V · [tr( V ) −
4] is the Shield-transformation[29] of W ∗ ( F ) = 2 µ · [tr( V − ) − W ∗ ( F ) = 2 µ (cid:18) λ + 1 λ + 1 λ − (cid:19) = g ( λ , λ , λ ) (0.11)has the Valanis-Landel form [31] and g is convex in ( λ , λ , λ ); the TE-inequalities are satisfied as well.Richter’s paper is not only written in German, but his notation strongly relies on German fraktur letters,which makes reading his original work rather challenging. In our faithful translation of his paper, we havetherefore updated the notation to more current conventions; a complete list of notational changes is providedin Appendix A. Richter’s original equation numbering has been maintained throughout. References [1] M. Baker and J. L. Ericksen. “Inequalities restricting the form of the stress-deformation relation for isotropic elastic solidsand Reiner-Rivlin fluids”.
J. Washington Acad. Sci.
44 (1954), pp. 33–35.[2] P Charrier, B. Dacorogna, B Hanouzet, and P Laborde. “An existence theorem for slightly compressible materials innonlinear elasticity”.
SIAM Journal on Mathematical Analysis
Journal of the Mechanics and Physics of Solids
Journalof Computational Physics
270 (2014), pp. 300–324. This calculus shows that the Valanis-Landel form is not invariant under the Shield transformation. In addition, the mapping U T Biot = D U W ( U ) of the stretch U = √ F T F to the Biot stress tensor T Biot is strictly monotone.
5] S. Federico. “Volumetric-distortional decomposition of deformation and elasticity tensor”.
Mathematics and Mechanicsof Solids
Transactions of the Faraday Society
57 (1961), pp. 829–838.[7] H. Hencky. “ ¨Uber die Form des Elastizit¨atsgesetzes bei ideal elastischen Stoffen”.
Zeitschrift f¨ur technische Physik , pp. 215–220.[8] H. Hencky. “Das Superpositionsgesetz eines endlich deformierten relaxationsf¨ahigen elastischen Kontinuums und seineBedeutung f¨ur eine exakte Ableitung der Gleichungen f¨ur die z¨ahe Fl¨ussigkeit in der Eulerschen Form”.
Annalen der Physik ,pp. 617–630.[9] H. Hencky. “Welche Umst¨ande bedingen die Verfestigung bei der bildsamen Verformung von festen isotropen K¨orpern?”
Zeitschrift f¨ur Physik
55 (1929). available at ,pp. 145–155.[10] H. Hencky. “The law of elasticity for isotropic and quasi-isotropic substances by finite deformations”.
Journal of Rheology ,pp. 169–176.[11] J. Lankeit, P. Neff, and Y. Nakatsukasa. “The minimization of matrix logarithms: On a fundamental property of theunitary polar factor”.
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449 (2014), pp. 28–42. doi : .[12] R. J. Martin and P. Neff. “Some remarks on the monotonicity of primary matrix functions on the set of symmetricmatrices”. Archive of Applied Mechanics doi : .[13] J. Murphy and G. Rogerson. “Modelling slight compressibility for hyperelastic anisotropic materials”. Journal of Elasticity
Mathematics and Mechanics of Solids arXiv:1402.4027 (2014).[16] P. Neff, B. Eidel, and R. J. Martin. “Geometry of logarithmic strain measures in solid mechanics”.
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Continuum Mechanics and Thermodynamics doi : .[18] P. Neff, I.-D. Ghiba, and J. Lankeit. “The exponentiated Hencky-logarithmic strain energy. Part I: Constitutive issuesand rank-one convexity”. Journal of Elasticity doi : .[19] P. Neff, J. Lankeit, I.-D. Ghiba, R. J. Martin, and D. J. Steigmann. “The exponentiated Hencky-logarithmic strain energy.Part II: coercivity, planar polyconvexity and existence of minimizers”. Zeitschrift f¨ur angewandte Mathematik und Physik doi : .[20] P. Neff, J. Lankeit, and A. Madeo. “On Grioli’s minimum property and its relation to Cauchy’s polar decomposition”. International Journal of Engineering Science
80 (2014), pp. 209–217. doi : .[21] P. Neff, I. M¨unch, and R. J. Martin. “Rediscovering G. F. Becker’s early axiomatic deduction of a multiaxial nonlinearstress–strain relation based on logarithmic strain”. Mathematics and Mechanics of Solids doi : .[22] P. Neff, Y. Nakatsukasa, and A. Fischle. “A logarithmic minimization property of the unitary polar factor in thespectral and Frobenius norms”. SIAM Journal on Matrix Analysis and Applications doi : .[23] H. Richter. “Das isotrope Elastizit¨atsgesetz”. Zeitschrift f¨ur Angewandte Mathematik und Mechanik
Zeitschriftf¨ur Angewandte Mathematik und Mechanik issn : 1521-4001.[25] H. Richter. “Zum Logarithmus einer Matrix”.
Archiv der Mathematik issn : 0003-889X. doi : .[26] H. Richter. “Zur Elastizit¨atstheorie endlicher Verformungen”. Mathematische Nachrichten
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29] R. T. Shield. “Inverse deformation results in finite elasticity”.
Zeitschrift f¨ur angewandte Mathematik und Physik to appear in International Journal of Non-Linear Mechanics (2019). available at arXiv:1812.03053. doi : .[31] K. C. Valanis and R. F. Landel. “The strain-energy Function of a hyperelastic material in terms of the extension ratios”. Journal of Applied Physics he isotropic law of elasticity By Hans Richter in Haltingen (L¨orrach)Zeitschrift f¨ur Angewandte Mathematik und Mechanik, Vol. 28, 1948, page 205 − Abstract
From the demand of the isotropy and of the existence of the thermodynamic potentials a generalform of the three-dimensional law of elasticity is stated. In doing so, the logarithmic matrix of relativeelongations is used, which permits the separation of the variation of the volume and that of the shapeby simply forming the deviator. The resilience energy is exactly the sum of the energy of the variationof the volume and that of the shape, if the average tension depends only on the variation of the volume.For finite deformations, the law of
Hooke is permissible only in the case ν = .Aus der Forderung der Isotropie und der Existenz der thermodynamischen Potentiale wird f¨ur das r¨aum-liche Elastizit¨atsgesetz eine allgemeine Form angegeben, wobei die logarithmische Dehnungsmatrix ver-wendet wird, bei der die Trennung in Volum- und Gestalt¨anderung durch gew¨ohnliche Deviatorbildungm¨oglich ist. Die elastische Energie ist genau dann die Summe aus Volum- und Gestalt¨anderungsenergie,wenn die mittlere Spannung nur von der Volum¨anderung abh¨angt. Das Hookesche
Gesetz ist f¨ur endlicheVerzerrungen nur bei ν = zul¨assig.En supposant l’isotropie et l’existence des potentiels thermodynamiques, on donne une forme g´en´eralede la loi de l’´elasticit´e en se servant d’une matrix logarithmique d’allongement. Ce proc´ed´e permet unes´eparation des changements de volume et de forme par une simple formation de d´eviateur. Si la tensionmoyenne ne d´epend que du changement de volume, l’´energie d’´elasticit´e est la somme des ´energies dechangement du volume et de la forme. La loi de Hooke n’est admissible que pour ν = . In generalization of Hooke’s law, a material is called purely elastic if the Cauchy stresses depend in a uniquelyreversible way on the stretches. Strictly speaking, however, it is necessary to discuss the heat transfer whichoccurs in the tensile test; in particular, it is necessary to distinguish between an adiabatic and isothermal lawof elasticity. This choice also clarifies what is meant by strains, since strains on the adiabat resp. isothermcan be referred e.g. to the initial state, for which the stresses disappear completely. The strains can also bereferred to a stress-free initial state at an arbitrarily chosen initial temperature Θ instead. Then the stress-free state at another temperature Θ corresponds, in the case of an isotropic material, to uniform stretchesin all directions, i.e. the thermal expansion. In this manner the law of thermal expansion is included inthe elastic law. Of course, the affected material must be assumed not to change permanently by changes intemperature within the considered temperature range.Thus, we assume a stress-free state at a temperature Θ . Let the deformation of the material into anotherstate be characterized by the matrix F and the related stresses by the stress tensor σ . We call the materialideally elastic if σ depends uniquely on F and Θ. The material is said to be isotropic if this dependence isinvariant under Euclidean rotations.When solving the problem of finding the most general form of this dependence, one appropriately operateswith matrices, where the following abbreviations are used: X T is the matrix obtained by reflecting X over its main diagonal. ( X ) ik is the entry in the i -th row andthe k -th column of X . det X is the determinant of X . tr X is the sum of the elements on the main diagonal F is the Jacobian matrix: d b x = F d x . σ is the physical stress tensor at the point b x . X : called the trace of X . is the identity tensor. If f ( x ) = P a n · x n , then, assuming convergence, f ( X ) = P a n X n .Recall the following simple statements: tr( X · Y ) = tr( Y · X ) . (1.1)tr( X · d log Y ) = tr( X · Y − · d Y ) (1.2)if X commutes with Y , but not necessarily with d Y .log(det X ) = tr(log X ) , (1.3)if log X is well defined.For a pure rotation R : RR T = . (1.4)For a pure stretch V : V = V T . Every X can be represented in the form: X = V · R , (1.5)where the multiplication is to be read in its functional notation from right to left.
According to (1.5), F can be interpreted as a rotation R followed by a stretch V , where the principal stretchdirections of the latter are rotated against those of the coordinate axes. For the case of isotropic materials,the application of R must not have any influence on σ . Therefore, σ is a function of V and Θ. For given F ,we can find V by using (1.4) and (1.5) by F F T = V RR T V T = V . (2.1)The most general coaxial relation between σ and V which fulfills the invariance under rotations is now,obviously, σ = f ( V ; I , I , I , Θ) , (2.2)where the I ν are the invariants of V .Instead of V , one can also use a uniquely invertible function of V . As we will see later on, it is appropriateto use the “logarithmic stretch” L = log V , (2.3)which is always defined because of the positive eigenvalues of V . We denote the invariants of L by j = tr( L ) , k = tr( L ) and l = tr( L ) . (2.4)Further, from (1.3) and (2.1) we obtain: j = 12 tr(log( F F T )) = 12 log(det( F F T )) = log(det F ) .Instead of (2.2), we can now write σ = f ( L ; j, k, l, Θ) . (2.5) It is easy to see that here, one of the invariants I ν can be omitted, in contrast to the subsequent formula (2.7). σ ), tr( σ L ) and tr( σ L ) are functions of j , k , l and Θ due to (2.5). If we now define the invariants f , f and f as the solutions to the system of equationstr( σ ) = f tr( ) + f tr( L ) + f tr( L )tr( σ L ) = f tr( L ) + f tr( L ) + f tr( L )tr( σ L ) = f tr( L ) + f tr( L ) + f tr( L )with, in general, non-vanishing determinant, then we have for X = f · + f · L + f · L : tr( σ L ν ) = tr( X L ν ) with ν = 0 , , . Since σ is coaxial to L , it is completely determined by tr( σ ), tr( σ L ) and tr( σ L ). Therefore, σ ≡ X holds;i.e. σ = f ( j, k, l, Θ) · + f ( j, k, l, Θ) · L + f ( j, k, l, Θ) · L . (2.6)Hence, we have found the most general isotropic relation. Using V instead of L , we would correspondinglyobtain: σ = g ( I ν , Θ) · + g ( I ν , Θ) · V + g ( I ν , Θ) · V . (2.7) The internal energy of the material per unit volume in the initial state is denoted by E = E ( j, k, l, Θ) ; (3.1)the entropy is denoted by S = S ( j, k, l, Θ) . (3.2)Then the free energy W takes the form W = E − Θ · S = W ( j, k, l, Θ) . (3.3)If d A is now the differential of the work done by the element of volume, thend A = − d E + Θ · d S = − d W − S · dΘ . (3.4)Thus for isothermal elastic changes, we haved A = − (d W ) Θ=const . ; (3.5)whereas for adiabatic changes d A = − (d E ) S =const . , (3.6)where Θ has to be eliminated in (3.1) and (3.2), so that E appears as a function of j , k , l and S .In order to calculate d A , we transition from a deformation F to the neighboring deformation F + d F . Sincea pure rotation has no influence on d A , we can assume that F is a pure stretch. Let e , e and e be the unitvectors in the principal stretch directions of V , which can be interpreted as coordinate vectors. Let σ , σ and σ be the components of σ in these directions. We can use the rectangular parallelepiped spanned by V e , V e and V e as the volume element, which is generated by the stretch V applied to the unit cube. Let us8ow consider the side which starts from V e and which is spanned by V e and V e . Besides an infinitesimaltilting and change of the surface, this side undergoes a displacement in the e -direction with the magnitude e · (( V + d F ) e − V e ) = e d F e = (d F ) in the transition from V to V + d F . The work done on theconsidered side is therefore − σ · (d F ) · ( V ) · ( V ) = − det( V ) · (d F ) · σ ( V ) . Thus the entire work done by the volume element isd A = − det( V ) · X v =1 (d F ) vv · σ v ( V ) vv = − det( V ) · tr( σ V − d F ) . (3.7)The deformation V + d F now corresponds to a stretch V + d V , where due to (2.1),( V + d V ) = ( V + d F )( V +d F T )or V · d V + d V · V = V · d F T +d F · V .
Multiplying the left side of the equation by σ V − , taking the trace and using (1.1), we find2 tr( σ V − d V ) = tr( σ V − d F T ) + tr( σ V − d F ) = 2 tr( σ V − d F ) , since σ is symmetric and coaxial to V . From (3.7) we therefore obtaind A = − det( V ) · tr( σ V − d V ) . (3.8)Hence, due to (1.2), (1.3) and (2.4): d A = − e j · tr( σ d L ) . (3.8*)If we substitute this expression into the isothermal relation (3.5) and use (2.6), then it follows: e j · [ f tr(d L ) + f tr( L d L ) + f tr( L d L )] = ∂W∂j d j + ∂W∂k d k + ∂W∂l d l . Since, by (2.4), d j = tr(d L ) , d k = 2 tr( L d L ) and d l = 3 tr( L d L ) , we finally conclude that e j f = ∂W∂j , e j f = 2 ∂W∂k , e j f = 3 ∂W∂l and therefore, with (2.6), σ e j = ∂W∂j · + 2 ∂W∂k · L + 3 ∂W∂l · L , W = W ( j, k, l, Θ) . (3.9)Accordingly, from (3.6) we obtain for the adiabatic law: σ e j = ∂E∂j · + 2 ∂E∂k · L + 3 ∂E∂l · L , E = E ( j, k, l, S ) . (3.10)9f we want to omit the introduction of L and use V directly when formulating the law of elasticity, then weappropriately use the following as the invariants of V : I = tr( V ) , I = 12 tr( V ) , I = det( V ) . Furthermore, according to (2.7), (3.5) and (3.8), an analogous computation leads to the law of elasticity inthe form σ = ∂W∂I · + 1 I · ∂W∂I · V + 1 I · ∂W∂I · V , W = W ( I ν , Θ) (3.11)and a corresponding formulation with E ( I , I , I , S ) instead of W . The introduction of the logarithmic stretch L now proves to be not only appropriate to formulate the lawof elasticity as simple as possible, but using L also allows for the decomposition of a deformation intoa shape change and volume change by simply taking the deviatoric part, i.e. the same approach as forinfinitesimal strains, whereas a corresponding decomposition in terms of V is highly inconvenient. To seethis, we decompose the general stretch V into a shape-changing stretch V g and a volume-changing stretch V v ,i.e. we demand: V = V g · V v = V v · V g with det V g = 1 and V v = β · with β > . (4.1)Obviously, (4.1) uniquely determines such a decomposition for each V with det V >
0; namely, for given V , β = √ det V and V g = β − · V .
Since V g commutes with V v , we can take the logarithm of (4.1): L = L g + L v with L g = log V g and L v = log V v . (4.2)Then, by (1.3), we obtain:tr( L g ) = log(det V g ) = 0 , L v = log β · , tr( L v ) = 3 log β . If, in general, we denote by dev D the deviator corresponding to the symmetric matrix D , i.e.dev D = D −
13 tr D · , (4.3)we can finally write: L g = dev L and L v = 13 j · . (4.4)Thus the change of shape is indeed characterized by the deviator of L . For infinitesimal strains we have L ≈ V − , so that dev L turns into the usual deformation deviator.If we now introduce the invariants of dev L : y = tr((dev L ) ) and z = tr((dev L ) ) , (4.5)then y = k − j and z = l − j k + 29 j .
10e can use j , y and z instead of j , k and l as variables. Then j characterizes the change of volume, whereas y and z characterize the change of shape. As one can easily calculate, (3.9) leads to the formula13 e j tr σ = ∂W∂je j · dev σ = − y ∂W∂z · + 2 ∂W∂y · dev L + 3 ∂W∂z (dev L ) (4.6)where, in contrast to (3.9), W = W ( j, y, z, Θ) now holds.A corresponding formula results from (3.10).Without proof, let us remark that y and z cannot take on all possible values independently of each other,but are restricted by the condition 0 ≤ z y ≤ . In the elasticity theory of infinitesimal strains the elastic energy can be interpreted as the sum of the energyof the volume and shape change. Since the change of volume is represented by j and the change of shape isrepresented by y and z , this decomposition is possible for the case of finite strains if and only if W = W vol ( j, Θ) + W iso ( y, z, Θ) , resp. E = E vol ( j, S ) + E iso ( y, z, S ) ) (5.1)holds. Then with (4.6): 13 e j · tr σ = ∂W vol ∂j ( j, Θ) . Thus the average stress depends only on j , i.e. on the change of volume. If, vice versa, tr σ depends only on j , then by (4.6) we obtain ∂ W∂j ∂y = ∂ W∂j ∂z = 0 , which also leads to the form of W in (5.1). Consequently, we can state: The elastic energy can be decomposedinto the energy of change of volume and of change of shape if and only if the mean stress depends only onthe change of volume.
We referred the deformations to the stress-free state at a certain temperature Θ . Now we assume anothertemperature Θ to be used as initial temperature instead of Θ . For σ = 0, the temperature Θ correspondsto a certain deformation V with log V = L . V is a scalar multiple of the identity tensor; thus dev L = 0, y = z = 0. Then with (4.6): ∂W∂j ( j , , , Θ ) = 0 , j = ϕ (Θ ) . (6.1)Since b F = F V − is the matrix corresponding to the deformation F with respect to the new initial state, wethus have b V = V V − , b L = L − L and hence b j = j − j , b y = y , b z = z . (6.2)In formula (4.6), we can now replace j by b j if we simultaneously substitute W with c W (cid:16)b j, y, z, Θ (cid:17) = e − j · W (cid:16)b j + j , y, z, Θ (cid:17) = e − ϕ (cid:0) Θ (cid:1) · W (cid:16)b j + ϕ (cid:0) Θ (cid:1) , y, z, Θ (cid:17) . (6.3)In particular, it follows that the decomposition of the elastic energy, which was discussed in Section 5, isindependent of the choice of the reference temperature. Due to the formulae found previously, one can impose a wide variety of requirements on the law of elasticity,in particular with respect to the dependence on temperature, and verify if these requirements can be satisfied.Let us now consider the question whether the common law by
Hooke remains valid for finite strains.Using the Lam´e constants,
Hooke ’s law takes the form σ = λ · tr( V − ) · + 2 µ · ( V − ) (7.1)or σ = ( λ · I − λ − µ ) · + 2 µ · V . (7.2)It is obvious that (7.1) is actually derived from the general formula (3.9) for small L .In order for the isothermal law of elasticity (7.2) to remain valid for finite strains, the following equationsmust be fulfilled according to (3.11): λI − λ − µ = ∂W∂I , µI = ∂W∂I and 0 = ∂W∂I . This is only possible if λ = 2 µ , which corresponds to the Poisson ratio ν = . For all other values of ν ,Hooke’s law cannot be used for finite strains. Instead, one can use the corresponding logarithmic law σ e j = λj · + 2 µL , (7.3)which, in the isothermal case, corresponds to the decomposable energy W = λ j + µk = (cid:18) λ µ (cid:19) · j + µ · y. Received 2. February 1948 List of Symbols
Our notation Richter’s notation meaning X , Y A , B arbitrary 3 × X T A transpose of X ( X ) ik ( A ) ik entry in the i -th row and the k -th column of X det X | A | determinant of X tr X A trace of X E identity tensor X − A − inverse of XF A Jacobian matrix (state of strain) R R pure Euclidean rotation V S pure stretch σ P stress tensor (state of stress)Θ Θ temperature I , I , I I , I , I invariants of VL L logarithmic stretch: L = log Vj , k , l j , k , l invariants of L : j = tr( L ), k = tr( L ), l = tr( L ) f , f , f f , f , f coefficient functions g , g , g g , g , g coefficient functions X X X = f · + f · L + f · L E u internal energy
S s entropy