A generalized strain approach to anisotropic elasticity

This work proposes a generalized Lagrangian strain function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\alpha$$\end{document}fα (that depends on modified stretches) and a volumetric strain function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\alpha$$\end{document}gα (that depends on the determinant of the deformation tensor) to characterize isotropic/anisotropic strain energy functions. With the aid of a spectral approach, the single-variable strain functions enable the development of strain energy functions that are consistent with their infinitesimal counterparts, including the development of a strain energy function for the general anisotropic material that contains the general 4th order classical stiffness tensor. The generality of the single-variable strain functions sets a platform for future development of adequate specific forms of the isotropic/anisotropic strain energy function; future modellers only require to construct specific forms of the functions \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_\alpha$$\end{document}fα and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g_\alpha$$\end{document}gα to model their strain energy functions. The spectral invariants used in the constitutive equation have a clear physical interpretation, which is attractive, in aiding experiment design and the construction of specific forms of the strain energy. Some previous strain energy functions that appeared in the literature can be considered as special cases of the proposed generalized strain energy function. The resulting constitutive equations can be easily converted, to allow the mechanical influence of compressed fibres to be excluded or partial excluded and to model fibre dispersion in collagenous soft tissues. Implementation of the constitutive equations in Finite Element software is discussed. The suggested crude specific strain function forms are able to fit the theory well with experimental data and managed to predict several sets of experimental data.

Remark. Valanis and Landel 36 strain energy function where i s are principal stretches, play an important role in modelling incompressible isotropic solids. The function r is arbitrary and this set a platform for modelling specific types of incompressibe isotropic elastic solids. Numerous specific forms of r , that are able to successfully model the mechanical behaviour of incompressible isotropic solids, have been proposed in the literature, see for example Ogden 2 and Shariff 37 . The single variable function r depends on an invariant with a clear physical meaning and this makes the Valanis and Landel form experimentally attractive 17 . The Valanis and Landel form impels us to develop anisotropic constitutive equations, which depends on single variable aribtrary functions that will set a platform for future modelling of specific types of anisotropic elastic solids. Our constitutive equations are developed via generalized strain single-variable functions. We overtly emphasize that, in this paper, we are not particularly concerned in obtaining specific forms of the proposed generalized strain functions. A rigourous construction of specific forms such those found in references 2,14,37 requires a lot of work and it is beyond the scope of this paper. As mentioned above, the generalized constitutive equations described here will set a platform (analogous to the "generalized" Valanis and Landel form for isotropic elastic solids) for future modelling of specific types of anisotropic elastic solids.

Preliminaries
In this paper, summation convention is not used and all subscripts i,j and k take the values 1, 2, 3, unless stated otherwise. Vectors and tensors are written in lowercase and uppercase bold fonts, respectively. Only quasistatic deformations and time-independent fields are considered. The mechanical body forces are assumed to be negligible. The deformation gradient is denoted by F and C = F T F = U 2 , respectively, where U is the right stretch tensor.

General strain energy function
A general strain energy function for an elastic solid can be expressed as The facilitate the construction of an incompressible material, regarded as a material recovered from the corresponding compressible material by taking the incompressible limit 38 , we use the modified stretch tensor where J = det F > 0 and det is the determinant of a tensor. Hence, we express The spectral representation where * i = J − 1 3 i , ⊗ denotes the dyadic product and, i and u i are an eigenvalue and an orthonormal eigenvector of U , respectively. In view of (6),

Stress. The Cauchy stress for a compressible solid is and for an incompressible solid
where p is the Lagrange multiplier associated with the constraint det F = 1 and I is the identity tensor. Following the work of 10,17 , the Cauchy stress T with respect to the Eulerian orthonormal basis {v 1 , v 2 , v 3 } , where v i = Ru i and R = FU −1 takes the form where for compressible elastic solids and in the case for an incompressible solid ( J = 1 ), we have, The nominal stress It is clear from the above that hydrostatic stress for a compressible material In the incompressibility limit, the value of lim J→1 ∂W ∂J and the appropriate properties of W are discussed in Shariff and Parker 38 . We note that all the proposed strain energy functions in this paper are consistent with the strain energy functions proposed by Shariff and Parker 38 . The deformation dependent bulk modulus is defined as 39 The ground-state bulk modulus is defined as

Generalized strain
Consider a set of general class of Lagrangean strain tensor F (α) , similar to that defined by Hill 1 , where N = {1, 2, 3, . . .} is the set of natural numbers excluding 0 and f α : (0, ∞) → R is a monotonic increasing function, i.e, f ′ α ( * i ) > 0 , such that . where ᾱ is a real number. We strongly emphasize that we are not concerned with proposing prototypes of the strain function f α , such as those expressed in (21) and (22). An objective of this paper is to construct constitutive equations that depend explicitly on the arbitrary functions f α and g α (defined below), and are consistent with infinitesimal elasticity. We define a volumetric strain sistent with infinitesimal elasticity is facilitated via infinitesimal strain elasticity. Hence, in sections "Isotropic" to "General anisotropy", we start the construction of a finite strain constitutive equation with the development of its infinitesimal strain energy function counterpart. For the sake of generality, the general constitutive equations given below contain numerous functions of f α and g α , which may seem unappealing. However, in many occasions only a few f α and g α functions are required to model anisotropic solids (see "Example of specific forms of fα and gα used in experimental fitting" section).

Isotropic
Let W (I) represents the strain energy for an isotropic elastic solid. We then have where Q is an arbitrary rotational tensor and (35) implies that the strain energy W (I) can be symmetrically express in terms the principal stretches (spectral invariants) i .

Infinitesimal strain. The strain energy function for infinitesimal strain deformations is
where µ and χ are, respectively, the ground state shear and bulk moduli and is the Lame's constant. For the purpose of this paper, we express where h = tr E . In the case of an incompressible solid, (37) is reduced to Finite strain. A finite strain energy function that is consistent with its infinitesimal counterpart (37) is proposed, i.e., where the "higher order" term φ (I) (which depends on i ) satisfies the P-property and the conditions at F = I . We note that, in view of (40), the function φ (I) does not contribute to the infinitesimal strain energy function. In the case of an incompressible material, we propose   for appropriate values of the material constants µ r and α r . It is worth noting that Remark: In sections "Transversely Isotropic with a unit preferred direction a" and "Two preferred direction elastic solid" below, we discuss elastic solids with one and two preferred directions. In some of these solids, the mechanical influence of compressed fibres is negligible or is different from stretched fibres and in some soft tissue solids, the influence of fibre dispersion could be relevant in modelling constitutive equations: In Appendices A and B (Supplementary information), we illustrate how the strain energy functions developed in sections "Transversely Isotropic with a unit preferred direction a" and "Two preferred direction elastic solid" can be easily amended to take account of these influences. .

Transversely Isotropic with a unit preferred direction a
Let W (T) represents the strain energy for a transversely isotropic elastic solid. We then have Following the work of Shariff 14 , we can express the strain energy function in terms of the spectral invariants Since, a is a unit vector, we have, and hence only 5 of the 6 invariants in (56) are independent 31,32 .

Infinitesimal strain. The infinitesimal strain energy function is 41
where A = a ⊗ a . The material constants ā i ( i = 1, 2, . . . , 5 ) can be be described in terms of physical parameters as shown below: where Here ν p is the Poisson ratio in a particular direction on the plane of symmetry, when the material is extended in a direction on the plane of symmetry perpendicular to the particular direction, ν a is the Poisson ratio in the preferred direction when the material is extended in the plane of symmetry, E p is the Young's Modulus in the plane of symmetry normal to the preferred direction a , µ a is the shear modulus in the preferred direction and E a is the Young's modulus in the preferred direction. Take note that we have also the relation where ν zp is the Poisson ratio in any direction on the plane of symmetry, when the material is extended in the preferred direction. We can express (58) in the form where The infinitesimal hydrostatic stress The ground-state bulk modulus then takes the form It can be easily shown that, in the incompressible limit, as ν zp → 0.5 and 1 − ν a − ν p → 0 42 the ground-state bulk modulus χ → ∞ . It is clear from (65) that, since, exists, we have (2) .  (63) and (69), we easily construct a finite strain energy that it is consistent with its infinitesimal counterpart, i.e.
or, alternatively, where the higher order function φ (I) (for convenient, we use the same expression for all anisotropic material discussed in this paper, although they are, generally, different functions.) has the properties given in "Finite strain" section and φ (I) depends on the spectral invariants i and a i . The weighted Cauchy stress takes the form

Two preferred direction elastic solid
Consider an elastic material with preferred unit directions a and b , where the unit vectors a and b are independent. The strain energy Hence, we can express W (P) in terms of the spectral invariants 17,30 We note that in view of (57) and the relations only 7 of the 9 invariants in (74) are independent and they formed the minimal/irreducible integrity basis 30,32 . Infinitesimal strain. Modifying the work of Shariff and Bustamante 30 , we have the strain energy The mean hydrostatic stress is The bulk modulus Finite strain. Following the work of sections "Isotropic" and "Transversely Isotropic with a unit preferred direction a" sections, we propose the strain energy function where η i = (a · b)a i ι i , φ (I) has the properties given in "Finite strain" section and is a function of i , a i and ι i . The weighted Cauchy stress takes the form An orthotropic strain energy function can be easily obtained from (82) by letting η i = 0.

General anisotropy
Consider the strain energy W (G) that depends of the 4th order classical stiffness tensor C , i.e., Note that W g must be invariant with respect to the rotation Q , i.e. Infinitesimal strain. The strain energy for a general anisotropic elastic solid is (85)  where φ (I) depends on the spectral invariants i and c ijkl . For example, when specialized to a transversely isotropic material, we have from (71) In this case the spectral components take the form As mentioned earlier, it is important that W (G) satisfies the P-property. It is clear that, in view of (92), that the symmetrical part of the P-property is satisfied. We now show that W (G) has a unique value when two or more of the principal stretches have a same value. Consider the case when * 1 = * 2 = . In this case the principal directions u 1 and u 2 are not unique but u 3 (99) χ = tr CI 9 . (100) (105) c ijkl =ā 1 (δ ik δ jl + δ jk δ il ) + 2ā 2 δ ij δ kl +ā 3 2 (a i a k δ jl + a i a l δ jk + a j a k δ il + a j a l δ ik ) + 2ā 4 a i a j a k a l +ā 5 (a k a l δ ij + a i a j δ kl ) .
(106)  112) and (113) that W (G) has a unique value when * 1 = * 2 = , since it is independent of the eigenvectors u 1 and u 2 . Following the above method, it is straightfoward to show that W (G) has a unique value when any two of the principal stretches have a same value. In the case when * 1 = * 2 = * 3 = , the value of W (G) is unique since, in this case, The weighted Cauchy stress is where

Example of specific forms of f α and g α used in experimental fitting
In this section, we suggest specific forms of the strain functions f α and g α to fit experiment data. We strongly emphasize that we are not interested in constructing the (or an) optimal form of f α and g α for a particular material; we are only interested in giving examples of specific forms of the proposed strain functions that can be used to fit experimental data. Constructing an optimal form of the strain functions for a particular material similar to the previous work of Shariff 12,14,37 will be done in the near future. We also emphasize that the specific forms are mainly constructed via visual curve fitting. Since, we are dealing with many types of isotropic and anisotropic materials, curve fitting exercises (such as those found in references [45][46][47] for isotropic solids only), for all the isotropic/anisotropic solids mentioned below require a considerable amount of work and analysis, and it is outside the scope of this paper.
Only strain energy functions with φ (I) = 0 are discussed in this section.
Isotropic. In the case of a compressible material, we use the simple strain functions based on the Hencky strain energy function 40 to fit the simple tension data of an isotropic polyurethane foam material used in Blatz and Ko 48 experiment, where four sets of data are used. The nominal axial stress in the 3-direction is The values of the lateral stretch 1 = 2 is obtained in terms of 3 via the zero lateral stress condition, i.e., (108) (117) to fit the simple tension data in Fig. 1; these are the same values that are obtained in Blatz and Ko 48 experiment. Fig. 1 shows that our theory reasonably fit the nominal stress vs. axial stretch curve and the behaviour of the lateral stretch 1 = 2 in terms of the axial stretch 3 is predicted quite well in Fig. 2. For incompressible materials ( J = 1 ), we give, below, specific forms for the strain functions to fit the experimental data of Treloar 49  (120) µ = 32 psi , ν = 0.25  where m = 0.5, 1, 2 corresponds, respectively, to uniaxial, pure shear and equibiaxial deformatons. It is clear in Fig. 3, that the theoretical curves for the three different deformations fit the experiment data very well when the shear modulus has the value µ = 11.009 3 kg/cm 2 . To fit Jones and Treloar 50 biaxial experimental data, we use µ = 0.4(MPa) and the strain functions The biaxial deformation experiment require the stress difference where σ 1 and σ 2 are principal Cauchy stresses. Fig. 4 indicates that the strain functions (124) fits the experiment extremely well.
Transversely isotropic. We compare our theory with the axial compression experiment of Jin et al. 51 on compressible rectangular slabs of transversely isotropic Marcellus shale. Although the measured experimental strains are infinitesimal, the stress-strain behaviour is very mildly nonlinear 42 . Since, the strains are infinitesimal, the Cauchy and nominal stress are indistinguishable. The nominal stress-strain relation is required for the curve fitting and the rock is compressed in the 3-direction. In general, 1 = 2 and their values are obtained via the relations To compare with the experimental data given in this section for incompressible materials, we consider a strain energy function of the form 14 where the constants µ T and µ L , represent the elastic shear moduli in the ground state. The other ground state elastic constant β 41 can be related to an elastic constant which has more direct physical interpretations, such as the extension moduli. Since the ground-state constant values when the fibre tension is different from when fibre compression (see Appendix A in Supplementary information), we have, In this section we compare our theory with the uniaxial tension and compression experiment of Chui et al. 52 and Takaza et al. 53 multiple angle uniaxial experiment on soft tissue. We note that in Chui et al. 52 the uniaxial stretch in the fibre direction is the stiffest, where else Takaza et al. 53 experiment indicates that the transverse stress is (131) E a =16.12 ± 1.29 GPa , E p = 37.72 ± 7.04 GPa , ν zp =0.35 ± 0.15 , ν p = 0.25 ± 0.01 , µ a = 6.87 ± 1.19 GPa .
(132)  www.nature.com/scientificreports/ the stiffest. In soft tissues, the initial large extension is generally achieved at relatively low levels of stress with subsequent stiffening at higher levels of extension. This behaviour is due to the recruitment of collagen fibres as they become uncrimped and reach their natural lengths. The inverse error function erf −1 (x) seems a good candidate to describe the above mentioned soft tissue stress-strain behaviour since it has low initial gradients followed by high gradients at higher values of x. In view of this, for simplicity, to fit the experiments, we use the strain functions where α 1−3 � = 0 are dimensionless material parameters. The tensor and vector components used below are with respect to the Cartesian basis {g 1 , g 2 , g 3 } . The stressstrain relations are based on that given in (13). We first consider Chui et al. 52  Orthotropic. Here, we only consider fitting our theory with the incompressible simple shear experimental data of Dokos et al. 54 on passive myocardium, where the material can be considered to be orthotropic 55 . We consider the strain energy 12 where the strain function (134) σ 12 = 2 l 1 (γ s 2 + cs) + l 2 (γ c 2 − cs) + l 3 γ cs ,    www.nature.com/scientificreports/ and Let a and b represent the fibre and sheet directions 55 , respectively, of the passive myocardium. In Figure 10, there are six sets of data, however, the experimental data corresponding to the fibre/sheet directions of the passive myocardium with Cartesian components [1, 0, 0] T /[0, 0, 1] T and [0, 0, 1] T /[1, 0, 0] T are indistinguishable. We note that no experiment is perfect. This indistinguishable behaviour could be caused by minute errors or approximations in the experiment or it could be the actual behaviour of the myocardium specimen or other unknown factors.
We first fit the five sets of data that correspond to fibre/sheet directions with Cartesian components (from top to bottom in Figure 10 Figure 10 that very good agreement is indicated between the model and the experimental data.
Using the fitted values, we then predict the set of data that corresponds to the fibre/sheet directions with components [0, 0, 1] T /[1, 0, 0] T . The predicted curve shown in Fig. 11 is also in good agreement with the experimental data.

Finite element implementation
In order to obtain numerical solutions for nonlinear isotropic and anisotropic elastic problems, a finite element software, such as Abaqus 56 , requires the end users to supply an explicit expression for the consistent tangent modulus tensor for an invariant-based potential function. In many cases, the consistent (algorithmic) tangent modulus tensor is used in the finite element code, where τ (J) is the Jaumann rate of the Kirchhoff stress and D is the deformation rate tensor. The consistent tangent modulus tensor requires, among others, the explicit expression for the 4th-order tensor (see, for example, 57 )  containing "classical" tensor invariants that can be written explicitly in terms of C were obtained in the literature; this due to the fact that the second derivative of the classical invariants with respect to C (a 4th order tensor) is easily obtained because they can be expressed explicitly in terms of C . However, the consistent tangent modulus tensor for a potential function W (e) , containing spectral invariants that cannot be written explicitly in terms of C but can be written explicitly in terms of the eigenvalues and eigenvectors of C , has only recently developed by Shariff 11 and, in view of this, the spectral formulation of the consistent tangent modulus tensor, developed by Shariff 11 , is not so well-known; hence, it may be mistakenly assumed that the spectral consistent tangent modulus tensor cannot be explicitly evaluated and that the proposed model cannot be used in a Finite Element commercial software. So the objective of this section is to report that the spectral consistent tangent modulus tensor can be evaluated using the results given in Shariff 11 . For isotropic, transversely isotropic and two-preferred direction materials, described above, the strain energy function W (e) contains, invariants of the form where G is a second order tensor. In Shariff 11 , the tangent modulus tensor (142) for a strain energy function that contains invariants of the form (144) is explicitly formulated. In the case of the strain energy function W (G) , given in (103), for a general anisotropic material, the corresponding tangent modulus tensor (142) can be derived using the results given in Shariff 10 ; however, due to its complex derivation, we will not derive it here.

Conclusion
In this contribution, we define a generalized strain function that is similar to the Hill's 1 strain function and a volumetric function, where they are used to characterize strain energy functions in isotropic or anisotropic elasticity. These strain functions are single variable functions that depend on an invariant with a clear physical meaning, which facilitates the construction of specific forms of the strain energy function, in the sense that a function of a single variable with a clear physical interpretation is easily manageable and this is indicated in "Example of specific forms of fα and gα used in experimental fitting" section; they also facilitate the construction of strain energy functions that are consistent with infinitesimal elasticity as described in sections "Isotropic" to "General anisotropy". The proposed generalized strain functions enable the development of a strain energy function for a general anisotropic material that contains the general 4th order classical stiffness tensor. Having a clear physical interpretation, the spectral invariants are attractive in aiding experiment design. Some previous strain energy functions that appeared in the literature can be considered as special cases of the proposed generalized strain energy functions. The proposed constitutive equations can be easily converted to allow the mechanical influence (u i · Gu i )g( * i ) and J , www.nature.com/scientificreports/ of compressed fibres to be excluded or partial excluded and to model fibre dispersion in collagenous soft tissues, and they can be easily implemented in a Finite Element software. In "Example of specific forms of fα and gα used in experimental fitting" section, we show that the suggested crude specific strain function forms fitted well with experimental data and managed to predict several sets of experimental data. The single-variable-function constitutive equations are expected to set a platform for future modelling of various types of anisotropic elastic solids; future modellers only require to construct specific forms of the functions f α and g α to model their strain energy functions. The extent of the proposed model applicability to different anisotropic needs to be assessed by comparing it with relevant experimental data of a much wider class of anisotropic materials.