Laplace Stretch: Eulerian and Lagrangian Formulations
LLaplace Stretch: Eulerian and Lagrangian Formulations
Alan D. Freed, Shahla Zamani, L´aszl´o Szab´o and John D. Clayton
Abstract.
Two triangular factorizations of the deformation gradient tensor are studied. The first,termed the Lagrangian formulation, consists of an upper-triangular stretch premultiplied by arotation tensor. The second, termed the Eulerian formulation, consists of a lower-triangular stretchpostmultiplied by a different rotation tensor. The corresponding stretch tensors are denoted as theLagrangian and Eulerian Laplace stretches, respectively. Kinematics (with physical interpretations)and work conjugate stress measures are analyzed and compared for each formulation. While theLagrangian formulation has been used in prior work for constitutive modeling of anisotropic andhyperelastic materials, the Eulerian formulation, which may be advantageous for modeling isotropicsolids and fluids with no physically identifiable reference configuration, does not seem to have beenused elsewhere in a continuum mechanical setting.
Mathematics Subject Classification (2010).
Primary 74A05; Secondary 15A90.
Keywords. continuum mechanics, kinematics, finite strain, Gram-Schmidt factorization.
1. Introduction
The deformation gradient admits a number of different triangular decompositions, whereby in eachcase the full deformation gradient matrix is decomposed into a product of an orthogonal tensor anda triangular stretch tensor. Restricting analysis to a deformation gradient with positive determinant,each orthogonal tensor is a rotation, and each corresponding stretch, either upper or lower triangu-lar, is unique for its corresponding rotation. The first such decomposition considered herein splitsthe deformation gradient into an upper-triangular stretch followed (i.e., premultiplied) by a rotationtensor. This kinematic construction is referred to here as the Lagrangian formulation of the triangulardecomposition, also known as a Gram-Schmidt factorization. The second such decomposition studiedin this paper splits the deformation gradient tensor into a rotation tensor followed (premultiplied) bya lower-triangular stretch tensor. This construction is referred to as the Eulerian formulation of thetriangular decomposition of deformation.The Lagrangian triangular decomposition was first introduced in the context of continuum me-chanics by McClellan [1, 2]. The corresponding upper-triangular stretch tensor was proven very appeal-ing for modeling anisotropic hyperelastic materials by Srinivasa [3]. Other recent applications of theLagrangian decomposition address shape memory polymers [4], anisotropic composites [5], biologicalmembranes [6], and soft biological tissues [7]. Advantages and drawbacks of using the upper-triangulardecomposition in constitutive models are discussed in these and related works [8]. Notably, the triangu-lar decomposition, unlike the polar decomposition, requires no eigenvector analysis to invoke, and thecomponents of stretch have an obvious physical interpretation that facilitates direct and unambiguousparameterization of constitutive response data.In general, Lagrangian formulations (e.g., constitutive models based on Lagrangian measures ofstrain) are preferred for modeling anisotropic solids, as-well-as certain isotropic solids, that have aclearly defined initial, stress-free, or ‘reference’ state. This is readily apparent for single crystals, for a r X i v : . [ phy s i c s . c l a ss - ph ] M a r Freed et al. , Shahla Zamani, L´aszl´o Szab´o and John D. Claytonexample, whereby a reference state is identified with the regular lattice geometry occupied by atoms intheir minimum energy (ground) state. Hyperelasticity is usually invoked in this context [2, 9], wherebyan energy potential depending on a Lagrangian stretch tensor is prescribed. Eulerian formulations,in contrast, are often preferred for modeling isotropic solids (and fluids) that have no obvious initialor reference state. For example, many biological tissues, in vivo, are perpetually under tension, anda stress-free reference state is never physically realized. Eulerian forms are also used for hypoelasticconstitutive modeling that is often more popular than hyperelasticity for solving initial-boundaryvalue problems numerically. However, prior to the present work, no application of the Eulerian lower-triangular decomposition in the context of continuum mechanics seems to have been reported. Adifferent triangular decomposition of the deformation gradient was invoked by Souchet [10], consistingof a lower-triangular stretch premultiplied (rather than postmultiplied) by a rotation. In that case, thelower-triangular stretch is considered a Lagrangian stretch measure rather than an Eulerian stretchmeasure, as newly studied herein.
2. Deformation
Consider a body B embedded in a three-dimensional, Euclidean, point space oriented against a triadof orthogonal, unit, base vectors ( (cid:126) ı ,(cid:126) , (cid:126) k ). The motion x = χ ( X , t ) of some particle P located in B describes a homeomorphism that takes its original location X = X (cid:126) ı + X (cid:126) + X (cid:126) k belonging to thebody’s reference configuration κ r and places it into another location x = x (cid:126) ı + x (cid:126) + x (cid:126) k where P resides in the body’s current configuration κ t .For convenience, we write these two position vectors as X = X i (cid:126) e i and x = x i (cid:126) e i by selectingan indexing strategy, e.g., ( (cid:126) ı ,(cid:126) , (cid:126) k ) (cid:55)→ ( (cid:126) e ,(cid:126) e ,(cid:126) e ), to ensure that the 1 material direction and the12 material surface embed with the motion, as they are invariant under transformations of Laplacestretch [2]. How to select an appropriate indexing strategy is the topic of Ref. [11]. This selectiontechnique has been applied to our example problems.A deformation gradient F maps the set of all tangent vectors located at particle P in body B from its reference configuration κ r into the current configuration κ t . We assume that a body is simplyconnected and its motion χ is sufficiently differentiable so that F = ∂ χ ( X , t ) /∂ X exists and therefore F ij = ∂χ i ( X , t ) ∂X j = F F F F F F F F F = f r f r f r = (cid:2) f c f c f c (cid:3) (1)where vectors f ri = F ij (cid:126) e j contain the rows of tensor F = F ij (cid:126) e i ⊗ (cid:126) e j , while vectors f ci = F ji (cid:126) e j contain its columns, i = 1 , ,
3, with repeated indices being summed according to Einstein’s summationconvention.It follows straightaway that the right, Cauchy-Green, deformation tensor C .. = F T F = C ij (cid:126) e i ⊗ (cid:126) e j ,which is a Lagrangian description of deformation, has components of C ij = f c · f c f c · f c f c · f c f c · f c f c · f c f c · f c f c · f c f c · f c f c · f c (1.2a)while the left, Cauchy-Green, deformation tensor B .. = FF T = B ij (cid:126) e i ⊗ (cid:126) e j , which is an Euleriandescription of deformation, has components of B ij = f r · f r f r · f r f r · f r f r · f r f r · f r f r · f r f r · f r f r · f r f r · f r (1.2b)both of which are symmetric because, for example, f r · f r = f r · f r where f r · f r = F i F i = F F + F F + F F , etc.aplace Stretch 3
3. Laplace Stretch
Laplace stretch, as it has been used in the literature to date, e.g., [1, 2, 3, 4, 12, 5, 6, 13, 8, 14, 11, 7],derives from a Gram-Schmidt (or QR ) decomposition of the deformation gradient F , where matrix Q is orthogonal, and matrix R is upper triangular.Given a coordinate system with base vectors ( (cid:126) e ,(cid:126) e ,(cid:126) e ), we denote such a decomposition as F = RU , where R = R ij (cid:126) e i ⊗ (cid:126) e j has orthogonal components, and U = U ij (cid:126) e i ⊗ (cid:126) e j has upper-triangular components. We select this calligraphic notation to illustrate its similarities and differenceswith the common polar decomposition F = RU , where R = R ij (cid:126) e i ⊗ (cid:126) e j has orthogonal components,and U = U ij (cid:126) e i ⊗ (cid:126) e j has symmetric components. Lagrangian fields U and U are distinct measures forstretch.A polar decomposition of the deformation gradient, F = RU = VR , produces a Lagrangianmeasure for stretch (the right-stretch tensor U ) and an Eulerian measure for stretch (the left-stretchtensor V ) that share in a common, orthogonal, rotation tensor R . An objective of this document isto develop an Eulerian measure for stretch whose components populate a triangular matrix such that F = R L U = VR E , where U is the Lagrangian Laplace stretch, and where V is the Eulerian Laplacestretch, both with triangular elements. In contrast with the polar rotation R , the Lagrangian R L and Eulerian R E Gram rotations are distinct rotations. The Laplace stretches therefore relate via U = R L T VR E and V = R L U R E T . Here we describe a Gram-Schmidt factorization of the deformation gradient, i.e., F = R L U , wherein U = U ij (cid:126) e i ⊗ (cid:126) e j is called the Lagrangian Laplace stretch, or the right Laplace stretch.Srinivasa [3] applied a Cholesky decomposition to the symmetric, positive-definite, right, Cauchy-Green, deformation tensor C to establish the components of his stretch tensor, denoted here as U = U ij (cid:126) e i ⊗ (cid:126) e j ; in particular, U = (cid:112) C U = C / U U = C / U U = 0 U = (cid:113) C − U U = (cid:0) C − U U (cid:1) / U U = 0 U = 0 U = (cid:113) C − U − U (2)where components of the Lagrangian Laplace stretch U ij are upper triangular. Its inverse U − = U − ij (cid:126) e i ⊗ (cid:126) e j follows straightaway, having components of U − ij = / U −U / U U ( U U − U U ) / U U U / U −U / U U / U (3)thereby requiring that each U ii , no sum on i , to be positive—a condition satisfied because of massconservation. It is easily shown that the Lagrangian Laplace stretch U ij belongs to a group under theoperation of matrix multiplication. This group is comprised of all real, 3 ×
3, upper-triangular matriceswith positive diagonal elements [2]. Having a stretch tensor with this property has proven to be usefulin applications, e.g., [2, 14], as it does here.A Gram factorization of the deformation gradient F = F ij (cid:126) e i ⊗ (cid:126) e j produces a Lagrangian rotationtensor R L = δ ij (cid:126) e Li ⊗ (cid:126) e j = R Lij (cid:126) e i ⊗ (cid:126) e j described by R Lij = (cid:2) (cid:126) e L (cid:126) e L (cid:126) e L (cid:3) (4a) Regarding Lagrangian stretches with triangular elements, McLellan [1, 2] was the first to propose an upper-triangulardecomposition of the deformation gradient. Later, Souchet [10] constructed a stretch tensor with lower-triangular com-ponents. We use Srinivasa’s [3] approach for populating an upper-triangular stretch because, of these three Lagrangianapproaches, his is the simplest framework to apply.
Freed et al. , Shahla Zamani, L´aszl´o Szab´o and John D. Claytonwhose columns constitute unit base vectors that can be constructed via (cid:126) e L = f c (cid:107) f c (cid:107) (4b) (cid:126) e L = f c − ( f c · (cid:126) e L ) (cid:126) e L (cid:107) f c − ( f c · (cid:126) e L ) (cid:126) e L (cid:107) (4c) (cid:126) e L = f c − ( f c · (cid:126) e L ) (cid:126) e L − ( f c · (cid:126) e L ) (cid:126) e L (cid:107) f c − ( f c · (cid:126) e L ) (cid:126) e L − ( f c · (cid:126) e L ) (cid:126) e L (cid:107) (4d)wherein Laplace’s technique for removing successive orthogonal projections [15] is apparent, with norm (cid:107) f c (cid:107) .. = (cid:112) f c · f c , etc. It therefore follows that the Lagrangian Laplace stretch has components whichcan be expressed as U ij = (cid:126) e L · f c (cid:126) e L · f c (cid:126) e L · f c (cid:126) e L · f c (cid:126) e L · f c (cid:126) e L · f c (5)that provide a means of geometric interpretation for this measure of stretch. Components U ij ofthe Lagrangian Laplace stretch U = U ij (cid:126) e i ⊗ (cid:126) e j evaluated in a reference frame ( (cid:126) e ,(cid:126) e ,(cid:126) e ) are alsoprojections of column vectors f ci extracted from a deformation gradient F = F ij (cid:126) e i ⊗ (cid:126) e j that areprojected onto its Lagrangian coordinate axes ( (cid:126) e L ,(cid:126) e L ,(cid:126) e L ). Now we describe a Gram-Schmidt like factorization of the deformation gradient, viz., F = VR E ,wherein V = V ij (cid:126) e i ⊗ (cid:126) e j is called the Eulerian Laplace stretch, or the left Laplace stretch.Applying a Cholesky factorization to the symmetric, positive-definite, left, Cauchy-Green, defor-mation tensor B .. = FF T = VV T with components B = B ij (cid:126) e i ⊗ (cid:126) e j one can construct a stretch tensor V = V ij (cid:126) e i ⊗ (cid:126) e j whereby V = (cid:112) B V = 0 V = 0 V = B / V V = (cid:113) B − V V = 0 V = B / V V = (cid:0) B − V V (cid:1) / V V = (cid:113) B − V − V (6)where we now select the lower-triangular matrix from the Cholesky decomposition to quantify thecomponents of our new stretch tensor. Its inverse V − = V − ij (cid:126) e i ⊗ (cid:126) e j follows straightaway, it havingcomponents of V − ij = / V −V / V V / V V V − V V ) / V V V −V / V V / V (7)thereby requiring each V ii , no sum on i , to be positive—a condition satisfied because of mass conserva-tion. It is easily shown that the Eulerian Laplace stretch V ij belongs to a group under the operation ofmatrix multiplication. This group is comprised of all real, 3 ×
3, lower-triangular matrices with positivediagonal elements. The Eulerian and Lagrangian Laplace stretches belong to different mathematicalgroups.A Gram-like factorization of the deformation gradient F = F ij (cid:126) e i ⊗ (cid:126) e j can also describe anEulerian rotation tensor R E = δ ij (cid:126) e i ⊗ (cid:126) e Ej = R Eij (cid:126) e i ⊗ (cid:126) e j constructed as R Eij = (cid:126) e E (cid:126) e E (cid:126) e E = (cid:2) (cid:126) e E (cid:126) e E (cid:126) e E (cid:3) T (8a) The Gram factorization of a square matrix results in an orthogonal matrix and an upper-triangular matrix. Herewe apply the same strategy, but we secure a different orthogonal matrix and a lower-triangular matrix; hence, theterminology ‘Gram like’. aplace Stretch 5whose rows constitute unit base vectors that can be constructed via (cid:126) e E = f r (cid:107) f r (cid:107) (8b) (cid:126) e E = f r − ( f r · (cid:126) e E ) (cid:126) e E (cid:107) f r − ( f r · (cid:126) e E ) (cid:126) e E (cid:107) (8c) (cid:126) e E = f r − ( f r · (cid:126) e E ) (cid:126) e E − ( f r · (cid:126) e E ) (cid:126) e E (cid:107) f r − ( f r · (cid:126) e E ) (cid:126) e E − ( f r · (cid:126) e E ) (cid:126) e E (cid:107) (8d)where, again, Laplace’s solution strategy for removing successive orthogonal projections [15] is appar-ent. It follows that the Eulerian Laplace stretch has components which can be expressed as V ij = f r · (cid:126) e E f r · (cid:126) e E f r · (cid:126) e E f r · (cid:126) e E f r · (cid:126) e E f r · (cid:126) e E (9)that provide a means of geometric interpretation for this measure of stretch. Components V ij of theEulerian Laplace stretch V = V ij (cid:126) e i ⊗ (cid:126) e j evaluated in a reference frame ( (cid:126) e ,(cid:126) e ,(cid:126) e ) are also projectionsof row vectors f ri extracted from a deformation gradient F = F ij (cid:126) e i ⊗ (cid:126) e j that are projected onto itsEulerian coordinate axes ( (cid:126) e E ,(cid:126) e E ,(cid:126) e E ).Obviously, rotations R L and R E are distinct, as are stretches U and V , given that the deforma-tion gradient F decomposes as F = R L U = VR E whose stretch tensors have triangular components U ij and V ij .
4. Physical Interpretation of Laplace Stretch Components
Each Laplace stretch has six, independent, physical attributes. There are three, orthogonal, elongationratios a , b and c , and there are three, orthogonal, simple shears α , β and γ . Their Lagrangian interpre-tations are quantified in a coordinate system with base vectors ( (cid:126) e L ,(cid:126) e L ,(cid:126) e L ), and are distinguished withan underline, viz., a , b , c , α , β and γ . Their Eulerian interpretations are quantified in a coordinatesystem with base vectors ( (cid:126) e E ,(cid:126) e E ,(cid:126) e E ), and are distinguished with an overline, viz., a , b , c , α , β and γ .In general, Lagrangian stretch attributes are distinct from their Eulerian counterparts. However, theirgeometric interpretations are the same. They differ only in their coordinate systems through whichthey are evaluated. The Lagrangian Laplace stretch has geometric interpretations that arise from Eqn. (5) whereby onecan assign [12] U ij = a aγ aβ b bα c = a b
00 0 c β α γ
00 1 00 0 1 (10a)with an inverse of U − ij = /a − γ/b − ( β − αγ ) /c /b − α/c /c (10b)whose constituents are measured in a coordinate frame with base vectors [13] (cid:126) e L = f c (cid:14) a (11a) (cid:126) e L = (cid:0) f c − γ f c (cid:1) (cid:14) b (11b) (cid:126) e L = (cid:0) f c − α f c − ( β − αγ ) f c (cid:1) (cid:14) c (11c) Freed et al. , Shahla Zamani, L´aszl´o Szab´o and John D. Clayton e ββ a γ γ α bc . e L L e L γα a b c α βγ Figure 1.
A geometric interpretation for Lagrangian Laplace stretch.all of which are described in terms of physical attributes defined as a .. = U , b .. = U , c .. = U , α .. = U U , β .. = U U , γ .. = U U (12)where a , b and c are elongations, while α , β and γ are magnitudes of shear, i.e., they are the extentsof shear at unit elongation. From conservation of mass, the elongations must be positive ( a ∈ R + , b ∈ R + , c ∈ R + ), while the shears may be of either sign ( α ∈ R , β ∈ R , γ ∈ R ).According to Eqn. (10), the Lagrangian Laplace stretch arises from the following sequence ofdeformations: it starts with an in-plane shear γ , followed by two out-of-plane shears α and β , andthen finishes with three elongations a , b and c , as illustrated in Fig. 4.1. Two vectors remain invariantunder mappings of the Lagrangian Laplace stretch; they are: vector (cid:126) e L establishes the direction ofin-plane shear, while vector (cid:126) e L × (cid:126) e L points normal to the plane of in-plane shear [2]. The Eulerian Laplace stretch has geometric interpretations that arise from Eqn. (9) whereby one canassign V ij = a aγ b aβ bα c = γ β α a b
00 0 c (13a)with an inverse of V − ij = /a − γ/b /b − ( β − αγ ) /c − α/c /c (13b)whose constituents are measured in a coordinate frame with base vectors (cid:126) e E = f r (cid:14) a (14a) (cid:126) e E = (cid:0) f r − γ f r (cid:1) (cid:14) b (14b) (cid:126) e E = (cid:0) f r − α f r − ( β − αγ ) f r (cid:1) (cid:14) c (14c)aplace Stretch 7 e . e e E EE a c b c αβ ca b γβ c α cab ca α β γ bc c b Figure 2.
A geometric interpretation for Eulerian Laplace stretch.all of which are described in terms of physical attributes defined as a .. = V , b .. = V , c .. = V , α .. = V V , β .. = V V , γ .. = V V (15)where a , b and c are elongations, while α , β and γ are magnitudes of shear, i.e., they are the extentsof shear at unit elongation. From conservation of mass, the elongations must be positive ( a ∈ R + , b ∈ R + , c ∈ R + ), while the shears may be of either sign ( α ∈ R , β ∈ R , γ ∈ R ).According to Eqn. (13), the Eulerian Laplace stretch arises from the following sequence of de-formations: it starts with three elongations a , b and c , followed by two out-of-plane shears α and β ,and then finishes with an in-plane shear γ , as illustrated in Fig. 4.2. This sequence of deformationsis the reverse of that occurring with the Lagrangian Laplace stretch. Two vectors remain invariantunder mappings of the Eulerian Laplace stretch, too; they are: vector (cid:126) e E establishes the direction ofin-plane shear, and vector (cid:126) e E × (cid:126) e E points normal to the plane of in-plane shear.
5. Examples
Any motion χ ( X , t ) described by the following deformation gradient quantified in an orthonormalcoordinate system with base vectors ( (cid:126) e ,(cid:126) e ,(cid:126) e ) is said to be shear free; specifically, F ij = λ λ
00 0 λ ∴ B ij = C ij = λ λ
00 0 λ (16)where λ , λ and λ are the three principal stretches that, in this case, obey a = a = λ , b = b = λ and c = c = λ . The Laplace stretch tensors and their Gram rotations have components of U ij = V ij = λ λ
00 0 λ with R Lij = R Eij = . (17)Consequently, there is no distinction between the triangular Laplace stretches U and V and the sym-metric polar stretches U and V for this class of motions. The elongations a , b and c of Laplace stretchequate with the eigenvalues λ , λ and λ of polar stretch. This relationship between elongations andprincipal stretches disappears in the presence of shear [16]. Freed et al. , Shahla Zamani, L´aszl´o Szab´o and John D. Clayton Any motion χ ( X , t ) described by the following deformation gradient quantified in an orthonormalcoordinate system with base vectors ( (cid:126) e ,(cid:126) e ,(cid:126) e ) is said to be a pure shear [12], specifically F ij = 1 √ √ λ λ − λ − λ − (18)where λ is the stretch of pure shear. This motion is described by Cauchy-Green deformation tensorswith components of B ij = λ
00 0 λ − and C ij = 12 λ + λ − λ − λ − λ − λ − λ + λ − (19)that produce a Lagrangian Laplace stretch and its Gram rotation of U ij = 1 (cid:113) ( λ + λ − ) (cid:113) ( λ + λ − ) 0 00 ( λ + λ − ) ( λ − λ − )0 0 1 (20a)and R Lij = 1 √ λ + λ − √ λ + λ − λ λ − − λ − λ (20b)along with an Eulerian Laplace stretch and its Gram rotation of V ij = λ
00 0 λ − and R Eij = 1 √ √ − (21)where R E rotates the Eulerian coordinate frame ( (cid:126) e E ,(cid:126) e E ,(cid:126) e E ) about the background frame ( (cid:126) e ,(cid:126) e ,(cid:126) e )by a fixed 45 ◦ in the 23 plane; whereas, R L rotates the Lagrangian coordinate frame ( (cid:126) e L ,(cid:126) e L ,(cid:126) e L ) fromthe Eulerian frame ( (cid:126) e E ,(cid:126) e E ,(cid:126) e E ) at λ = 1 towards the background frame ( (cid:126) e ,(cid:126) e ,(cid:126) e ) as λ → ∞ .The above components for Eulerian Laplace stretch V ij support Lodge’s statement that pureshear is not a shearing deformation; it is a shear-free deformation in disguise [17, 18]. Lodge justifiesthis position by pointing out that the eigenvectors for stretch do not rotate in a body during pureshears like they do during simple shears.Here the elongations relate as a = a = 1 while b = (cid:112) ( λ + λ − ) / b = λ with c =1 / (cid:112) ( λ + λ − ) / c = λ − , whereas the shears relate as α = ( λ − λ − ) / ( λ + λ − ) and α = 0with β = β = γ = γ = 0. Any motion χ ( X , t ) described by the following deformation gradient quantified in an orthonormalcoordinate system with base vectors ( (cid:126) e ,(cid:126) e ,(cid:126) e ) constitutes a shearing motion; specifically, F ij = β (22)whose Cauchy-Green deformation tensors have components of B ij = β β β and C ij = β β β (23)aplace Stretch 9with its Lagrangian Laplace stretch and rotation having components of U ij = β and R Lij = (24)along with its Eulerian Laplace stretch and rotation having components of V ij = (cid:112) β β/ (cid:112) β / (cid:112) β (25a)and R Eij = / (cid:112) β β/ (cid:112) β − β/ (cid:112) β / (cid:112) β (25b)with the Eulerian Laplace stretch V ij having diagonal elements akin to those of pure shear (cf. Eqn. 21),plus an off-diagonal simple shearing that is attenuated by the extent of pure shearing present.From a rheometric viewpoint, making stress a function of the Eulerian Laplace stretch wouldenable first- and second-normal stress differences to occur, with the first exceeding the second inmagnitude, and they being of opposite sign. A Weisenberg effect would occur, because of a compressivestretch that would set up in the hoop direction. Furthermore, the shear stress would thin, becauseof an effect that (cid:112) γ would have on the shear strain γ/ (cid:112) γ . All of these ‘effects’ occur inpolymeric liquids [19].Here the elongations relate as a = 1 and a = (cid:112) β while b = b = 1 with c = 1 and c = 1 / (cid:112) β , whereas the shears relate as β = β and β = β/ (1 + β ) with α = α = γ = γ = 0.
6. Frameworks for Constitutive Development
A time rate-of-change in the work being done at a particle by tractions applied to its body resultsin a source for internal power caused by stresses, often evaluated per unit mass. Here we constructsets of thermodynamic conjugate pairs for both the Lagrangian and Eulerian frameworks when usingLaplace stretch as one’s kinematic variable. The constituents of these pairs relate to one anothervia constitutive equations (a topic for future papers). To facilitate such endeavors, bijective mapsare derived that convert stress and velocity-gradient tensor components into their associated thermo-dynamic stresses and strain rates, the latter of which are scalar fields.
In terms of Lagrangian fields, stress power ˙ W can be written as ρ tr( S ˙ E ) wherein S is the secondPiola-Kirchhoff stress, E .. = ( C − I ) is the Green strain, and ρ is the initial mass density at a particleof interest in a body.It is easily verified that˙ W = ρ tr( S ˙ E ) = ρ tr( SL L ) where S .. = U S U T , L L .. = ˙ U U − (26)given that F = R L U . The Lagrangian stress S is symmetric because the second Piola-Kirchhoff stress S is symmetric, and the Lagrangian velocity gradient L L is upper-triangular—a consequence of thegroup that stretch U belongs to. The above expression for stress power reduces to a sum of six scalarcontributions; specifically ρ ˙ W = S L L + S L L + S L L + S L L + S L L + S L L (27)0 Freed et al. , Shahla Zamani, L´aszl´o Szab´o and John D. Claytonwherein L Lij = ˙ U ik U − kj = ˙ a/a a ˙ γ/b a ( ˙ β − α ˙ γ ) /c b/b b ˙ α/c c/c (28)and we observe that the diagonal rates are logarithmic, while the off-diagonal rates are not logarithmic.(A very different triangular velocity gradient, viz., Eqn. (36), arises in the Eulerian constructionthat follows.) How to construct proper finite differences to approximate derivatives for the physicalattributes of Laplace stretch is discussed in Ref. [13].Expressing Eqn. (27) in terms of thermodynamic conjugate pairs is not a unique process, cf.Ref. [20]. Here we shall consider a pairing described by ρ ˙ W = π ˙ δ + (cid:88) i =1 (cid:0) σ i ˙ ε i + τ i ˙ γ i (cid:1) (29)whose seven, conjugate, stress-strain pairs are defined as follows: a uniform bulk response is governedby a Lagrangian pressure π and a Lagrangian dilatation δ defined by π .. = S + S + S δ .. = ln (cid:115) aa bb cc ˙ δ = 13 (cid:32) ˙ aa + ˙ bb + ˙ cc (cid:33) (30a)while the squeeze (pure shear) responses are governed by Lagrangian normal-stress differences σ i andLagrangian squeezes ε i defined by σ = S − S ε = ln (cid:115) aa b b ˙ ε = 13 (cid:32) ˙ aa − ˙ bb (cid:33) (30b) σ = S − S ε = ln (cid:115) bb c c ˙ ε = 13 (cid:32) ˙ bb − ˙ cc (cid:33) (30c) σ = S − S ε = ln (cid:114) cc a a ˙ ε = 13 (cid:18) ˙ cc − ˙ aa (cid:19) (30d)of which two are independent because σ = − ( σ + σ ) and ε = − ( ε + ε ), while the (simple) shearresponses are governed by Lagrangian shear stresses τ i and Lagrangian shear strains γ i defined by τ = bc S γ = α − α ˙ γ = ˙ α (30e) τ = ac S γ = β − β ˙ γ = ˙ β (30f) τ = ab S − aαc S γ = γ − γ ˙ γ = ˙ γ (30g)wherein a , b and c are their initial elongation ratios, and where α , β and γ are their initialshears.Bijective maps exist to transform tensor components into thermodynamic stress–strain-rate at-tributes that, for isotropic materials, are described by πσ σ τ τ τ = − − b/c a/c
00 0 0 0 − aα/c a/b S S S S S S (31a) See Ref. [20] for one way to extend this approach to anisotropic materials. aplace Stretch 11with σ = − σ − σ , and ˙ δ ˙ ε ˙ ε ˙ γ ˙ γ ˙ γ = / / / / − / / − / c/b c/a bα/a b/a L L L L L L L L L L L L (31b)with ˙ ε = − ˙ ε − ˙ ε .These strain-rate attributes can be integrated to get the Lagrangian thermodynamic strains δ , ε , ε , ε , γ , γ and γ by choosing initial conditions of δ | = ε | = ε | = ε | = γ | = γ | = γ | = 0provided that the initial elongation ratios have been specified as a , b and c and that the initialmagnitudes of shear have been specified as α , β and γ .At this juncture, constitutive equations between stress-strain attributes of the thermodynamicconjugate pairs ( π, δ ), ( σ , ε ), ( σ , ε ), ( τ , γ ), ( τ , γ ) and ( τ , γ ) are to be introduced (a topic forfuture works) to solve for the Lagrangian thermodynamic stresses π , σ , σ , σ , τ , τ and τ . Theseupdated stress attributes map into our Lagrangian stress components S ij as S S S S = S S = S S = S = / / / / − / / / − / − / c/b c/a
00 0 0 0 bα/a b/a πσ σ τ τ τ (32)from which the second Piola-Kirchhoff stress S = S ij (cid:126) e i ⊗ (cid:126) e j is retrieved via S = U − SU − T , i.e., S ij = U − ik S k(cid:96) U − j(cid:96) , and from here any commonly used stress tensor can be gotten.Although σ and ˙ ε are not needed from a constitutive perspective, they are required to correctlycalculate stress power. In terms of Eulerian fields, stress power ˙ W can be written as ρ tr( τ D ) wherein τ = FSF T is theKirchhoff stress, which relates to Cauchy stress T via τ .. = det( F ) T = ρ ρ T , and where D .. = ( L + L T ) = F − T ˙ EF − is the symmetric part of velocity gradient L .. = ˙ FF − , with ρ being the current massdensity.It can be shown that ˙ W = ρ tr( τ D ) = ρ tr (cid:0) τ L E (cid:1) (33a)given that F = VR E , where this Eulerian velocity gradient L E is defined by L E .. = ◦ VV − wherein ◦ V .. = ˙ V + V Ω E − Ω E V (33b)with ◦ V being an objective co-rotational derivative for this measure of stretch, and Ω E .. = ˙ R E R E T being a spin of an Eulerian coordinate axes ( (cid:126) e E ,(cid:126) e E ,(cid:126) e E ) about the reference axes ( (cid:126) e ,(cid:126) e ,(cid:126) e ).Consequently, stress power ρ ˙ W = tr (cid:0) τ L E (cid:1) arises from two sources in this Eulerian construction,viz. ˙ W = ˙ W + ˙ W . The first is energetic, i.e.,˙ W = ρ tr (cid:0) τ ˙ VV − (cid:1) (34a)while the second satifies objectivity, viz.,˙ W = ρ tr (cid:0) τ V Ω E V − (cid:1) (34b)2 Freed et al. , Shahla Zamani, L´aszl´o Szab´o and John D. Claytonnoting that tr( τ Ω E ) = 0. Thermodynamic stress-strain conjugate pairs can be established in terms ofthe energetic expression (34a). The objective correction (34b) is required to quantify the work beingdone, but it plays no role when creating our Eulerian stress-strain attributes, as every term in thissum has a component of spin in it; therefore, ˙ W = 0 whenever Ω E = .Because ˙ VV − = ˙ V ik V − kj (cid:126) e i ⊗ (cid:126) e j has components that are lower triangular, a consequence ofthe group that tensor V belongs to, the first contribution to stress power put forward in Eqn. (34a)reduces to a sum of six scalar contributions; specifically, ρ ˙ W = τ ˙ V i V − i + τ ˙ V i V − i + τ ˙ V i V − i + τ ˙ V i V − i + τ ˙ V i V − i + τ ˙ V i V − i (35)wherein ˙ V ik V − kj = ˙ aa γ + γ (cid:16) ˙ aa − ˙ bb (cid:17) ˙ bb β − γ ˙ α + β (cid:16) ˙ aa − ˙ cc (cid:17) − αγ (cid:16) ˙ bb − ˙ cc (cid:17) ˙ α + α (cid:16) ˙ bb − ˙ cc (cid:17) ˙ cc (36)which is strikingly different from that of its Lagrangian counterpart ˙ U U − found in Eqn. (28). Presenthere are the squeeze rates ˙ ε = (cid:0) ˙ a/a − ˙ b/b (cid:1) , etc., which appear in the off-diagonal terms, along withtheir corresponding shear rates, e.g., ˙ γ , thereby substantiating our assumed construction of conjugatepairs.In Eqn. (36), a clear delineation exists between pure and simple shearing deformations. Sucha delineation does not arise whenever one uses symmetric measures for stretch, where an isotropic-deviatoric decomposition is the extent to which such fields can be deconstructed.Expressing Eqn. (35) in terms of Eulerian, thermodynamic, conjugate pairs, analogous to thoseconsidered for the Lagrangian frame, one can write ρ ˙ W = π ˙ δ + (cid:88) i =1 (cid:0) σ i ˙ ε i + τ i ˙ γ i (cid:1) (37)whose seven, conjugate, stress-strain pairs are defined as follows: a uniform bulk response is governedby an Eulerian pressure π and an Eulerian dilatation δ defined by π .. = τ + τ + τ δ .. = ln (cid:115) aa bb cc ˙ δ = 13 (cid:32) ˙ aa + ˙ bb + ˙ cc (cid:33) (38a)while the squeeze (pure shear) responses are governed by Eulerian normal-stress differences σ i andEulerian squeezes ε i defined by σ = τ − τ + 3 γτ ε = ln (cid:115) aa b b ˙ ε = 13 (cid:32) ˙ aa − ˙ bb (cid:33) (38b) σ = (cid:26) τ − τ + 3 α ( τ − γτ ) ε = ln (cid:115) bb c c ˙ ε = 13 (cid:32) ˙ bb − ˙ cc (cid:33) (38c) σ = − τ + τ − βτ ε = ln (cid:114) cc a a ˙ ε = 13 (cid:18) ˙ cc − ˙ aa (cid:19) (38d) Curiously, U − ˙ U has components akin to Eqn. (36), except its components are upper triangular instead of lowertriangular, and are expressed in terms of the Lagrangian stretch attributes instead of their Eulerian counterparts. aplace Stretch 13of which only two are independent, while the (simple) shear responses are governed by Eulerian shearstresses τ i and strains γ i defined by τ = τ − γτ γ = α − α ˙ γ = ˙ α (38e) τ = τ γ = β − β ˙ γ = ˙ β (38f) τ = τ γ = γ − γ ˙ γ = ˙ γ (38g)wherein a , b and c are their initial elongation ratios, and where α , β and γ are their initial shearoffsets.The sets of thermodynamic conjugate pairs for the Lagrangian and Eulerian frameworks aretaken to be the same. Each set is composed of three modes: one pair to describe uniform dilatation,three pairs to describe pure shears, and three pairs to describe simple shears. In both cases, only twoof the three pure-shear pairs are independent, thereby resulting in sets of six, independent, conjugatepairs that have direct connections with the six independent components of stress and stretch rate.Bijective maps exist to transform tensor components into thermodynamic stress–strain-rate at-tributes that, for isotropic materials, are described by πσ σ τ τ τ = − γ − α − αγ
00 0 0 1 − γ
00 0 0 0 1 00 0 0 0 0 1 τ τ τ τ τ τ (39a)with σ = − σ − σ + 3 (cid:0) ατ − βτ + γτ (cid:1) (39b)which arises from the constraint equation σ − γτ σ − ατ σ + 3 βτ = − − − τ τ τ and where ˙ δ ˙ ε ˙ ε ˙ γ ˙ γ ˙ γ = / / / / − / / − / − α α − β β γ − γ γ ˙ V i V − i ˙ V i V − i ˙ V i V − i ˙ V i V − i ˙ V i V − i ˙ V i V − i (39c)with ˙ ε = − ˙ ε − ˙ ε . (39d)These strain rates can be integrated to get the Eulerian thermodynamic strains δ , ε , ε , ε , γ , γ and γ by using initial conditions of δ | = ε | = ε | = ε | = γ | = γ | = γ | = 0 provided thatthe initial elongation ratios have been specified as a , b and c and that the initial magnitudes ofshear have been specified as α , β and γ .At this juncture, constitutive equations between the Eulerian thermodynamic conjugate pairs( π, δ ), ( σ , ε ), ( σ , ε ), ( τ , γ ), ( τ , γ ) and ( τ , γ ) are to be introduced (again, a topic for future4 Freed et al. , Shahla Zamani, L´aszl´o Szab´o and John D. Claytonwork) to solve for the Eulerian thermodynamic stresses π , σ , σ , τ , τ and τ . After the thermo-dynamic stresses have been updated they can be mapped back into the components of Kirchhoff stress τ ij via τ τ τ τ = τ τ = τ τ = τ = / / / − α − γ / − / / − α γ / − / − / α γ γ
00 0 0 0 1 00 0 0 0 0 1 πσ σ τ τ τ (40)from which any commonly used stress tensor can be easily gotten.Although σ and ˙ ε are not needed from a constitutive perspective, they are required to calculatestress power. Also, to correctly compute stress power, Eqns. (34a or 37 & 34b) must both contribute,the former because of straining and the latter because of coordinate spin. A numerical strategy basedupon quaternion theory to acquire spin tensors from rotation tensors by using finite difference schemescan be found in Ref. [14].
7. Conclusions
Lagrangian and Eulerian triangular decompositions of deformation have been analyzed and compared.Physically observable stretch/strain components comprising the triangular Laplace stretch of each de-composition have been derived and then highlighted in several example problems involving homoge-neous deformations. Consideration of stress power, i.e., rate of working done by each stretch rate, hasenabled derivation of work conjugate stress-stretch tensors as-well-as thermodynamically conjugatescalar pairs of stress-strain attributes with physical meaning. Significantly, the Eulerian formulationcontaining an Eulerian, lower-triangular, stretch tensor has not been developed elsewhere in the me-chanics literature, to the authors’ knowledge. The current results provide a theoretical foundation forconstruction of constitutive models to be undertaken in future work.
Acknowledgement
This research was inspired by a conversation that ADF had with Prof. Michael Sacks at the2019 Annual Meeting of the Society for Engineering Science held at Washington University in St.Louis, where he encouraged the author to develop of an Eulerian constitutive framework suitable forbiomechanics.SZ was funded by the U.S. Army Research Laboratory, Aberdeen, MD.
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