The anisotropic continuum stored energy density (ACSED) functional is applied for accurate constitutive modeling of biological tissues and finite element implementation without the isochoric—volumetric split, the anisotropic—isotropic split, or the anisotropic invariant split. Related stress and elasticity tensors in the reference and current configurations are worked out. A new kinematic model is derived based on the tangent Poisson’s ratio as a cubic polynomial function of stretch. The ACSED model, along with the kinematic model, accurately fits uniaxial extension test data for compressible human skin, bovine articular cartilage, and human aorta samples.
Biological tissues, composed of collagen fibers and extracellular matrices, can be treated as nonlinear elastic fiber-reinforced composites. Significant progress in histological structures, mechanical characterizations, anisotropic constitutive models, and finite element implementations has been made for biological tissues. Finite element commercial packages play an important role in fundamentally solving practical boundary value problems. Essentially, we can better understand relations among histological compositions, anisotropic growth, mechanical properties, and performed functions of biological tissues. Continuum nonlinear solid mechanics was documented by Holzapfel (2000) [
As a mathematical convenience, the three major assumptions of existing anisotropic constitutive models implemented into finite element commercial packages are the isochoric—volumetric split, the anisotropic—isotropic split, and the anisotropic invariant split. First, for the incompressibility condition, the multiplicative decomposition of the deformation gradient into isochoric and volumetric parts is enforced, establishing the isochoric—volumetric split. Second, existing anisotropic stored energy density functionals are generally divided into a deformation part for fiber and a deformation part for matrix, constructing the anisotropic—isotropic split. Third, irreducible invariants are directly applied and widely accepted, popularizing the anisotropic invariant split.
With the isochoric—volumetric split, along with the anisotropic—isotropic split, anisotropic stored energy functionals should be divided into four parts. However, the invalidity of the isochoric—volumetric split for anisotropic nonlinear elastic deformations has been irrefutably demonstrated by Cowin (1990) [
In finite element implementations, the perfect incompressibility causes a numerical singularity. In displacement-based finite element methods, a locking occurs as Poisson’s ratio gets closer to 0.5. As a remedy, the reduced integration scheme is used. Spurious frequencies, however, are introduced due to inexact numerical integration of element stiffness matrices as emphasized by Bathe (1996) [
In practical applications of finite element methods for biological tissues, small changes in the Poisson’s ratio near 0.5 result in significant overestimation or underestimation in stress which is probably due to spurious pressure oscillations. For small stretches, stresses can be predicted by more than or less than an order of magnitude, causing unphysical results as demonstrated by Ní Annaidh et al. (2013) [
Thus, it is desirable to formulate a general constitutive model for biological tissues without the three major splits. The ACSED functional for ith fiber family, Ψ i , based on the energy balance between the stored energy density and stress work done equivalents in continuum mechanics under isothermal processes is covariantly formulated by Zhao (2018) [
P i : F = S i : C = τ i : I = σ i : J I = 2 Ψ i ( I 1 , i , I 2 , i , I 3 , i ) , (1)
where P i , S i , τ i , and σ i are the first Piola-Kirchhoff stress tensor, second Piola-Kirchhoff stress tensor, Kirchhoff stress tensor, and Cauchy stress tensor for ith fiber family, respectively. The corresponding conjugates F , C , I , and J I are the deformation gradient tensor, the right Cauchy—Green tensor, unit tensor, and product of volume ratio and unit tensor, respectively. For i = 1 , 2 , ⋯ , n , the index n is the total number of fiber families. The anisotropic invariants or principal invariant components (PIC) of the right Cauchy—Green tensor C = F T F , I 1, i , I 2, i , and I 3, i , are defined by
I 1 , i = tr ( C A 0 , i ) , I 2 , i = 1 2 [ I 1 I 1 , i − tr ( C 2 A 0 , i ) ] , I 3 , i = det C , (2)
where I 1 = tr C is the first principal invariant for isotropic hyperelastic materials and I 3 , i = I 3 is noticed. Biological tissues are usually modeled as isotropic, transversely isotropic, and orthotropic hyperelastic materials. For the orthotropic symmetry as an example, the structure tensors A 0 , i are defined as
A 0 , i = a 0 , i ⊗ a 0 , i , i = 1 , 2 , 3 , (3)
and a 0 , i is a direction unit vector for fiber i in reference configuration denoted by the subscript 0 and the matrix operators, “tr”, and “det”, denote trace and determinant operations, respectively.
The first-order derivatives of PICs with respect to C are given by
∂ I 1 , i ∂ C = A 0 , i , (4)
∂ I 2 , i ∂ C = 1 2 ( I 1 A 0 , i + I 1 , i I − C A 0 , i − A 0 , i C ) , (5)
∂ I 3 ∂ C = I 3 C − 1 . (6)
The second-order derivatives of PICs with respect to C are further given by
∂ 2 I 1 , i ∂ C 2 = O , (7)
∂ 2 I 2, i ∂ C 2 = 1 2 ( A 0, i ⊗ I + I ⊗ A 0, i − A 0, i ⊙ I − I ⊙ A 0, i ) , (8)
∂ 2 I 3 ∂ C 2 = I 3 ( C − 1 ⊗ C − 1 − C − 1 ⊙ C − 1 ) , (9)
where O is the fourth-order null tensor.
The second Piola—Kirchhoff stress tensor and the Cauchy stress tensor fot ith fiber family desired in finite element implementations can be generally derived based on (1), (4), (5), and (6). The second Piola—Kirchhoff stress tensor in reference configuration turns out to be
S i = 2 ∑ j = 1 3 ∂ Ψ i ∂ I j , i ∂ I j , i ∂ C = 2 ∂ Ψ i ∂ I 1 , i A 0 , i + ∂ Ψ i ∂ I 2 , i ( I 1 A 0 , i + I 1 , i I − C A 0 , i − A 0 , i C ) + 2 I 3 ∂ Ψ i ∂ I 3 C − 1 , (10)
and the Cauchy stress tensor, σ i , in current configuration can be converted by the push-forward operation of the second Piola—Kirchhoff stress tensor (10)2 and readily obtained as
σ i = 1 J F S i F T = 2 ∂ Ψ i J ∂ I 1, i A i + ∂ Ψ i J ∂ I 2, i ( I 1 A i + I 1, i B − B A i − A i B ) + 2 J ∂ Ψ i ∂ I 3 I , (11)
where A i = F A 0, i F T is the structure tensor in current configuration, B = F F T is the left Cauchy—Green tensor, and J = I 3 is the volume ratio or Jacobian.
Substituting (10)2 or (11)2 into (1) yields the partial differential equation for Ψ i
Ψ i = I 1 , i ∂ Ψ i ∂ I 1 , i + 2 I 2 , i ∂ Ψ i ∂ I 2 , i + 3 I 3 ∂ Ψ i ∂ I 3 . (12)
With the Lie group method, the general solution normalized by I 1, i reads
Ψ i / I 1, i = f ( I 2, i 1 2 / I 1, i ) + g ( I 3 1 3 / I 1, i ) , (13)
where functions f and g can be fixed by curvatures of deformations based on differential geometry. For normal, shear, and ellipsoidal stretches, the ACSED functional has been obtained by Zhao (2018) [
Ψ i = c 1 , i I 1 , i + c 2 , i I 2 , i + c 3 , i I 1 , i 3 c 4 , i + 1 I 3 c 4 , i , (14)
where the constitutive parameters for ith fiber family, c 1, i , c 2, i , c 3, i , and c 4, i , are determined by experimental tests.
For biological tissues with simple kinematic relations, constant Poisson’s ratios ν 1 j are used by Blatz and Ko (1962) [
ν 1 j = − ln λ j ln λ 1 or λ j = λ 1 − ν 1 j , j = 2 , 3. (15)
For biological tissues with complex kinematic relations, stretch dependent or tangent Poisson’s ratios are used by Fortes and Nogueira (1989) [
ν 1 j ( λ 1 ) = − d ( ln λ j ) d ( ln λ 1 ) = − λ 1 d λ j λ j d λ 1 , j = 2 , 3 , (16)
where “ln” denotes natural logarithmic function.
The major objectives are to apply the ACSED functional, to develop the kinematic model based on tangent Poisson’s ratio, to derive the stress and elasticity tensors in reference and current configurations, and to fit uniaxial extension and kinematic test data for compressible human skin, bovine articular cartilage, and human aorta samples.
This paper is organized as follows. In Section 2, the stress—stretch relation and the kinematic relation in uniaxial extension mode are derived for constitutive modeling biological tissues. Related stress and elasticity tensors are derived for finite element implementation. In Section 3, the ACSED constitutive model, along with the new kinematic model, is applied to fit uniaxial extension test data for different compressible biological tissues. In Section 4, modeling of biological tissues, the isochoric—volumetric split, and the predictive ACSED model are discussed. In Section 5, conclusions are drawn and the desired future work is provided.
For ith fiber family, P i is related to F by
P i T = ∂ Ψ i ∂ F = ∂ Ψ i ∂ I 1, i ∂ I 1, i ∂ F + ∂ Ψ i ∂ I 2, i ∂ I 2, i ∂ F + ∂ Ψ i ∂ I 3 ∂ I 3 ∂ F . (17)
With all three PICs generally treated as variables, the first-order derivatives of the ACSED functional are worked out as
∂ Ψ i ∂ I 1 , i = c 1 , i + c 3 , i ( 3 c 4 , i + 1 ) I 1 , i 3 c 4 , i I 3 c 4 , i , (18)
∂ Ψ i ∂ I 2 , i = c 2 , i 2 I 2 , i , (19)
∂ Ψ i ∂ I 3 = − c 3 , i c 4 , i I 1 , i 3 c 4 , i + 1 I 3 c 4 , i + 1 , (20)
and the second-order derivatives of the ACSED functional are further obtained as
∂ 2 Ψ i ∂ I 1 , i 2 = 3 c 3 , i c 4 , i ( 3 c 4 , i + 1 ) I 1 , i 3 c 4 , i − 1 I 3 c 4 , i , ∂ 2 Ψ i ∂ I 2 , i 2 = − c 2 , i 4 I 2 , i 3 , (21)
∂ 2 Ψ i ∂ I 3 2 = c 3 , i c 4 , i ( c 4 , i + 1 ) I 1 , i 3 c 4 , i + 1 I 3 c 4 , i + 2 , (22)
∂ 2 Ψ i ∂ I 1 , i ∂ I 2 , i = ∂ 2 Ψ i ∂ I 2 , i ∂ I 3 = 0 , ∂ 2 Ψ i ∂ I 1 , i ∂ I 3 = − c 3 , i c 4 , i ( 3 c 4 , i + 1 ) I 1 , i 3 c 4 , i I 3 c 4 , i + 1 . (23)
Mechanical characterizations of anisotropic hyperelastic materials have been accurately and effectively conducted by uniaxial extension tests. Thus, the ACSED constitutive model in uniaxial extension mode is derived from the Equations (2), (17), (18), (19), and (20).
For intima, media, and advantitia layers of arteries, the structure tensors for two families of preferred fiber directions with angles ± θ are given respectively by
Α 0,1 ( θ ) = ( cos 2 θ cos θ sin θ 0 cos θ sin θ sin 2 θ 0 0 0 0 ) (24)
and
A 0,2 ( − θ ) = ( cos 2 θ − cos θ sin θ 0 − cos θ sin θ sin 2 θ 0 0 0 0 ) (25)
where ± θ measure in-plane symmetric angles between fiber direction and principal stretch direction. In uniaxial extension tests of arteries, the related tensors are C = diag [ λ 1 2 , λ 2 2 , λ 3 2 ] , C 2 = diag [ λ 1 4 , λ 2 4 , λ 3 4 ] . The related PICs are obtained as
I 1 , i = λ 1 2 cos 2 θ + λ 2 2 sin 2 θ , (26)
I 2 , i = 1 2 ( λ 1 2 λ 2 2 + λ 2 2 λ 3 2 sin 2 θ + λ 3 2 λ 1 2 cos 2 θ ) , (27)
I 3 = λ 1 2 λ 2 2 λ 3 2 , (28)
and their derivatives with respect to λ 1 are worked out as
∂ I 1 , i ∂ λ 1 = 2 λ 1 cos 2 θ , ∂ I 2 , i ∂ λ 1 = λ 1 ( λ 2 2 + λ 3 2 cos 2 θ ) , ∂ I 3 ∂ λ 1 = 2 λ 1 λ 2 2 λ 3 2 . (29)
For two fiber families with ± θ , substituting the derivatives (18), (19), and (20), PICs (26), (27), and (28) and derivatives of PICs (29) into (17) and summing up stresses yields the total nominal stress equation
P 11 = ∑ i = 1 2 P 11 , i = 4 c 1 , i λ 1 cos 2 θ + c 2 , i λ 1 ( λ 2 2 + λ 3 2 cos 2 θ ) 1 2 ( λ 1 2 λ 2 2 + λ 2 2 λ 3 2 sin 2 θ + λ 3 2 λ 1 2 cos 2 θ ) + 4 c 3 , i ( λ 1 2 cos 2 θ + λ 2 2 sin 2 θ ) 3 c 4 , i ( λ 1 λ 2 λ 3 ) 2 c 4 , i [ ( 2 c 4 , i + 1 ) λ 1 cos 2 θ − c 4 , i λ 2 2 λ 1 sin 2 θ ] , i = 1 , 2. (30)
The corresponding stress normalization condition reads
4 c 1 , i cos 2 θ + c 2 , i ( 1 + cos 2 θ ) + 4 c 3 , i [ ( 2 c 4 , i + 1 ) cos 2 θ − c 4 , i sin 2 θ ] = 0. (31)
For one fiber family with θ = 0 , the constitutive model (30) combined with the stress normalization condition (31) is simplified as
P 11 , 1 = 2 c 1 , 1 ( λ 1 − 1 ) + c 2 , 1 ( λ 2 2 + λ 3 2 2 − 1 ) + 2 c 3 , 1 ( 2 c 4 , 1 + 1 ) [ λ 1 4 c 4 , 1 + 1 ( λ 2 λ 3 ) 2 c 4 , 1 − 1 ] . (32)
In the constitutive Equations (30) and (32), transverse stretches λ 2 and λ 3 as a function of longitudinal stretch λ 1 are needed. Thus, kinematic relations, λ 2 ( λ 1 ) and λ 3 ( λ 1 ) , need to be modeled. Integrating the equation (16) yields the general kinematic model
λ j = exp [ − ∫ ν 1 j ( λ 1 ) λ 1 d λ 1 ] , j = 2,3, (33)
where “exp” or “e” denotes an exponential function.
In order to relate transverse stretches λ 2 and λ 3 with longitudinal stretch λ 1 , a cubic polynomial is assumed for tangent Poisson’s ratios as
ν 1 j ( λ 1 ) = k 0 j + k 1 j λ 1 + k 2 j λ 1 2 + k 3 j λ 1 3 , j = 2 , 3 , (34)
and substituting (34) into (33), integrating, and applying the normalization condition of λ j | λ 1 = 1 = 1 yields the general kinematic model
λ j = λ 1 − k 0 j e − [ k 1 j ( λ 1 − 1 ) + k 2 j ( λ 1 2 − 1 ) / 2 + k 3 j ( λ 1 3 − 1 ) / 3 ] , j = 2 , 3 , (35)
where parameters k 0 j , k 1 j , k 2 j , and k 3 j need to be determined by kinematic characterization in uniaxial extension tests.
For the ACSED functional with ith fiber family (14), the second Piola-Kirchhoff stress tensor, S i , can be obtained as
S i = 2 c 1 , i A 0 , i + c 2 , i ( I 1 A 0 , i + I 1 , i I − C A 0 , i − A 0 , i C ) 2 I 2 , i + 2 c 3 , i I 1 , i 3 c 4 , i [ ( 3 c 4 , i + 1 ) A 0 , i − c 4 , i I 1 , i C − 1 ] I 3 c 4 , i , (36)
and the corresponding Cauchy stress tensor, σ i , in current configuration can be converted by the push-forward operation (11)1 of the second Piola-Kirchhoff stress tensor (36) as
σ i = 2 c 1 , i A i J + c 2 , i ( I 1 A i + I 1 , i B − B A i − A i B ) 2 I 2 , i J + 2 c 3 , i I 1 , i 3 c 4 , i [ ( 3 c 4 , i + 1 ) A i − c 4 , i I 1 , i I ] I 3 c 4 , i J , (37)
and the total second Piola-Kirchhoff and Cauchy stress tensors for n number of fiber families can respectively be summed up as
S = ∑ i = 1 n S i , and σ = ∑ i = 1 n σ i . (38)
The fourth-order elasticity tensor in reference configuration can be readily derived by
ℂ i = 4 ∑ j = 1 3 [ ∑ k = 1 3 ∂ 2 Ψ i ∂ I j , i ∂ I k , i ( ∂ I j , i ∂ C ⊗ ∂ I k , i ∂ C ) + ∂ Ψ i ∂ I j , i ∂ 2 I j , i ∂ C 2 ] . (39)
Substituting the first-order and second-order derivatives of PICs (4) through (9), simplifying, and rearranging produces
ℂ i = Δ 1, i A 0, i ⊗ A 0, i + Δ 2, i ( A 0, i ⊗ I + I ⊗ A 0, i ) + Δ 3, i ( A 0, i ⊗ H 0, i + H 0, i ⊗ A 0, i ) + Δ 4, i ( A 0, i ⊗ C − 1 + C − 1 ⊗ A 0, i ) + Δ 5, i I ⊗ I + Δ 6, i ( I ⊗ H 0, i + H 0, i ⊗ I ) + Δ 7, i H 0, i ⊗ H 0, i + Δ 8, i ( I ⊗ C − 1 + C − 1 ⊗ I ) + Δ 9, i ( C − 1 ⊗ H 0, i + H 0, i ⊗ C − 1 ) + Δ 10, i C − 1 ⊗ C − 1 + Δ 11, i ( A 0, i ⊙ I + I ⊙ A 0, i ) + Δ 12, i C − 1 ⊙ C − 1 , (40)
where H 0, i = 0.5 ( C A 0, i + A 0, i C ) is an interim second-order tensor for abbreviation and the twelve coefficients Δ 1, i , Δ 2, i , ⋯ , Δ 12, i for the ACSED functional, using (21), (22), and (23), are given by
Δ 1 , i = 12 c 3 , i c 4 , i ( 3 c 4 , i + 1 ) I 1 , i 3 c 4 , i − 1 I 3 c 4 , i − c 2 , i I 1 2 I 2 , i 1.5 , Δ 2 , i = c 2 , i I 2 , i 1.5 − c 2 , i I 1 I 1 , i 4 I 2 , i 1.5 , (41)
Δ 3 , i = c 2 , i I 1 2 I 2 , i 1.5 , Δ 4 , i = − 4 c 3 , i c 4 , i ( 3 c 4 , i + 1 ) I 1 , i 3 c 4 , i I 3 c 4 , i , Δ 5 , i = − c 2 , i I 1 , i 2 4 I 2 , i 1.5 , (42)
Δ 6 , i = c 2 , i I 1 , i 2 I 2 , i 1.5 , Δ 7 , i = − c 2 , i I 2 , i 1.5 , Δ 8 , i = Δ 9 , i = 0 , (43)
Δ 10 , i = 4 c 3 , i c 4 , i 2 I 1 , i 3 c 4 , i + 1 I 3 c 4 , i , Δ 11 , i = − c 2 , i I 2 , i 0.5 , Δ 12 , i = 4 c 3 , i c 4 , i I 1 , i 3 c 4 , i + 1 I 3 c 4 , i . (44)
The elasticity tensor in current configuration, c i | , can be converted from the elasticity tensor in reference configuration, ℂ i , by the following push-forward operation of (40)
c i | = J − 1 ( F ⊙ F ) : ℂ i : ( F T ⊙ F T ) . (45)
With the properties of double contraction operations between fourth-order tensors [
c i | = δ 1, i A i ⊗ A i + δ 2, i ( A i ⊗ B + B ⊗ A i ) + δ 3, i ( A i ⊗ H i + H i ⊗ A i ) + δ 4, i ( A i ⊗ I + I ⊗ A i ) + δ 5, i B ⊗ B + δ 6, i ( B ⊗ H i + H i ⊗ B ) + δ 7, i H i ⊗ H i + δ 8, i ( I ⊗ B + B ⊗ I ) + δ 9, i ( I ⊗ H i + H i ⊗ I ) + δ 10, i I ⊗ I + δ 11, i ( A i ⊙ B + B ⊙ A i ) + δ 12, i I ⊙ I , (46)
where H i = F H 0, i F T = 0.5 ( B A i + A i B ) is for abbreviation and the corresponding coefficients δ 1, i , δ 2, i , ⋯ , δ 12, i for the ACSED functional in current configuration are given by
δ j , i = Δ j , i J − 1 , j = 1 , 2 , ⋯ , 12 , (47)
and the total elasticity tensors for n number of fiber families can be combined as
ℂ = ∑ i = 1 n ℂ i , or c | = ∑ i = 1 n c i | . (48)
The coupled fourth-order elasticity tensors in both reference and current configurations (40) and (46) are derived for the ACSED functional.
Uniaxial extension tests with a full kinematic characterization of compressible human abdominal skin samples have been conducted at the stretch rate of 0.001 s−1 by Wahlsten et al. (2019) [
The kinematic model (35) fits the test data as
λ 2 = e [ − 805.158 ln λ 1 + 2183.826 ( λ 1 − 1 ) − 981.877 ( λ 1 2 − 1 ) + 194.660 ( λ 1 3 − 1 ) ] , (49)
λ 3 = e [ 587.242 ln λ 1 − 1593.726 ( λ 1 − 1 ) + 720.646 ( λ 1 2 − 1 ) − 145.065 ( λ 1 3 − 1 ) ] . (50)
The stretch dependent volume ratio can be readily obtained as
J = λ 1 − 216.916 e [ 590.100 ( λ 1 − 1 ) − 261.231 ( λ 1 2 − 1 ) + 49.595 ( λ 1 3 − 1 ) ] . (51)
The models for tangent Poisson’s ratios are obtained as
ν 12 ( λ 1 ) = 805.158 − 2183.826 λ 1 + 1963.753 λ 1 2 − 583.980 λ 1 3 , (52)
ν 13 ( λ 1 ) = − 587.242 + 1593.726 λ 1 − 1441.291 λ 1 2 + 435.196 λ 1 3 . (53)
The constitutive parameters have been determined by the trial-and-error-on-digit (TED) method and the linear least square (LLSQ) method combined by Zhao (2019) [
Uniaxial extension tests for bovine articular cartilage of humeral joint samples with full kinematic characterizations have been conducted at the stretch rate of 0.01 s−1 by Woo et al. (1979) [
λ 2 = λ 1 − 55.194 e [ 135.311 ( λ 1 − 1 ) − 54.151 ( λ 1 2 − 1 ) + 9.334 ( λ 1 3 − 1 ) ] , (54)
ν 12 ( λ 1 ) = 55.194 − 135.311 λ 1 + 108.302 λ 1 2 − 28.002 λ 1 3 , (55)
and λ 3 = λ 1 − ν 13 with ν 13 = 2.241 for theckness reduction.
The ACSED model (32) fits the nominal stress—stretch test data shown in
Uniaxial extension tests for circumferential and axial strips of anatomically separated intima, media, and advantitia layers from human abdominal aorta, along with the kinematic characterization of circumferential contraction—axial stretch relation, have been studied by Holzapfel (2006) [
In the physiological pressure range from 50 to 200 mmHg, a relative volume gain for most arteries is 2% to 6% as studied by Yosibash et al. (2014) [
The constitutive parameters resolved from fitting uniaxial extension tests of compressible human abdominal skins, bovine articular cartilages, and human abdominal aortas are listed in
| Material | c 1,1 | c 2,1 | c 3,1 | c 4,1 | θ ( ˚ ) | J m |
|---|---|---|---|---|---|---|
| skin | 52.97137 (kPa) | 540.63140 (kPa) | 5.29038 (kPa) | 1.22538 | 0.0 | 0.694 |
| cartilage | 16.67519 (MPa) | 27.13968 (MPa) | 0.00481 (MPa) | 1.19906 | 0.0 | 0.572 |
| aorta | −11.18101 (kPa) | 678.64510 (kPa) | 0.99998 (kPa) | 5.04082 | ±52.2 | 1.040 |
Kinematic relations are indispensable in constitutive modeling compressible biological tissues. Poisson’s ratio plays a crucial role in relating stretches without work done to the stretch with a work done. The historical development on Poisson’s ratio has been reviewed by Greaves (2013) [
By and large, the ACSED functional (14), along with the general kinematic model (35), offers a new approach in constitutive modeling of compressible as well as incompressible biological tissues.
The c 3, i term with I 1, i 3 c 4, i + 1 / I 3 c 4, i in the ACSED functional (14) represents ellipsoidal stretch deformation rather than spherical deformation.
The c 3, i term with I 1, i 3 c 4, i / I 3 c 4, i + 0.5 in the Cauchy stress tensor (37) makes the multiplicative decomposition of the deformation gradient incomplete. Thus, the Cauchy stress tensor cannot be decoupled into a deviatoric stress tensor and a hydrostatic stress tensor.
The assumption of the isochoric—volumetric split is equivalent to assume that hydrostatic stress should only depend on volume change as emphasized by Charrier et al. (1988) [
1 3 tr σ i = 2 3 J ( I 1, i ∂ Ψ i ∂ I 1, i + 2 I 2, i ∂ Ψ i ∂ I 2, i + 3 I 3 ∂ Ψ i ∂ I 3 ) . (56)
Hydrostatic stress in (56) is a function of I 1, i , I 2, i , and I 3 , indicating the assumption of hydrostatic stress only as a function of I 3 is false. Thus, the isochoric—volumetric split is invalid.
The ACSED functional was formulated by three different grouping processes. First, irreducible invariants are grouped into three PICs (2). Second, the PICs are lumped into dimensionless PIC groups (13) based on energy balance through Lie group method by Zhao (2016) [
The ACSED model can accurately fit and predict test data. Fitting capability of ACSED model has been demonstrated here but predictive capability is yet to be conducted. However, both fitting and predicting capabilities for the isotropic continuum stored energy model formulated through the same three grouping processes have been thoroughly examined through Treloar’s test data in uniaxial extension, uniaxial compression, equibiaxial extension, biaxial extension, pure shear, and simple shear modes by Zhao (2021) [
Biological tissues exhibit complex mechanical behaviors. Complexities include highly nonlinear stress—stretch response, anisotropy, inhomogeneity, rate dependency, active and passive responses, pre-stress, residual stress, in vivo and ex vivo characterizations, and others. Both accurate and complete set of experimental tests in multiple deformation modes for anisotropic biological tissues are more difficult to conduct but are still desirable for examining the predictive capability of anisotropic constitutive models.
The ACSED functional was formulated through three grouping processes rather than the three major splits. The one-dimensional normal stretch, two-dimensional shear stretch, and three-dimensional ellipsoidal stretch deformations become the new metric of nonlinear elastic deformation. The stress and elasticity tensors in both reference and current configurations have been derived. With the cubic polynomial for a tangent Poisson’s ratio, the general kinematic model is established. The ACSED model, along with the kinematic model, has been accurately fitted to the uniaxial extension test data of compressible human abdominal skin, bovine articular cartilage, and human abdominal aorta samples. In the future, the predictive capability of the ACSED model should be examined for biological tissues in different deformation modes.
The author is thankful to his family, Jianming and Jiesi Zhao, for their encouragement, support, and help.
The author declares no conflicts of interest regarding the publication of this paper.
Zhao, F.Z. (2022) Anisotropic Constitutive Modeling of Compressible Biological Tissue. Advances in Pure Mathematics, 12, 357-373. https://doi.org/10.4236/apm.2022.125027