A polyconvex hyperelastic model for fiber-reinforced materials in application to soft tissues

Abstract

In this paper a generalized anisotropic hyperelastic constitutive model for fiber-reinforced materials is proposed. Collagen fiber alignment in biological tissues is taken into account by means of structural tensors, where orthotropic and transversely isotropic material symmetries appear as special cases. The model is capable to describe the anisotropic stress response of soft tissues at large strains and is applied for example to different types of arteries. The proposed strain energy function is polyconvex and coercive. This guarantees the existence of a global minimizer of the total elastic energy, which is important in the context of a boundary value problem.

Appendix

Relations between the generalized structural tensors (15) and (18)

In the following, the relations between the weight factors w ^(r) _i , i = 1,...,n, associated with principal material directions (18) and the factors v ^(r) _i , i = 0,1,...,n, related to matrix and fibers (15) are given for some basic fiber constellations.

Unidirectional alignment of one family of fibers leads to transverse isotropy with respect to the fiber direction. Setting n = 1 in (15)₁ and insertion of (14)₁, (15)₂ leads in view of (17)₂ to

$$ \tilde{\mathbf L}_r=v_0^{(r)}{\mathbf L}_0+v_1^{(r)}{\mathbf L}_1 =\frac{1}{3}\left(1-v_1^{(r)}\right){\mathbf I}+v_1^{(r)}{\mathbf L}_1 =\frac{1}{3}\left(1+2v_1^{(r)}\right)\widehat{\mathbf L}_1 +\frac{2}{3}\left(1-v_1^{(r)}\right)\widehat{\mathbf L}_2. $$

In a fiber reinforced material, orthotropy may be the result of different fiber configurations. We first consider the case where fibers are aligned in three mutually orthogonal fiber directions coinciding with the principal material directions, so that $\user2{l}_i=\user2{m}_i, i=1,2,3$. Then, in view of (15,16,18) one obtains

$$ \tilde{\mathbf L}_r =\frac{1} {3}\sum\limits_{i=1}^{3}\left[1-v_1^{(r)}-v_2^{(r)}-v_3^{(r)}+3v_i^{(r)} \right]\widehat{\mathbf L}_i. $$

A material with two orthogonal fiber families is likewise orthotropic, where the principal directions are given by the two fiber directions, so that $\user2{l}_1=\user2{m}_1$ and $\user2{l}_2=\user2{m}_2$, and the direction normal to the plane in which the fibers lie. The generalized structural tensors (18) are thus given by

$$ \tilde{\mathbf L}_r =\frac{1} {3}\left[1+2v_1^{(r)}-v_2^{(r)}\right] \widehat{\mathbf L}_1 + \frac{1} {3}\left[1-v_1^{(r)}+2v_2^{(r)}\right]\widehat{\mathbf L}_2 + \frac{1}{3}\left[1-v_1^{(r)}-v_2^{(r)}\right]\widehat{\mathbf L}_3. $$

Two equivalent families of fibers (v ^(r)₂  = v ^(r)₁ ) aligned in two arbitrary directions result in orthotropy. The principal material directions are given by the bisectors of the two fiber directions and the normal to the plane in which the fibers lie. The fiber directions can be expressed in terms of the principal material directions by

$$ \user2{m}_1=\cos \alpha\,\user2{l}_1+\sin \alpha \,\user2{l}_2,\,\,\user2{m}_2=\cos \alpha\,\user2{l}_1-\sin \alpha \,\user2{l}_2, $$

where the angle between the fibers is 2α. Hence, the generalized structural tensors read as

$$ \tilde{\mathbf L}_r= \frac{1} {3}\left[1+(6\cos^2\alpha-2)v_1^{(r)}\right]\widehat{\mathbf L}_1 +\frac{1}{3}\left[1+(6\sin^2\alpha-2)v_1^{(r)}\right]\widehat{\mathbf L}_2 +\frac{1}{3}\left[1-2v_1^{(r)}\right]\widehat{\mathbf L}_3. $$

Finally, if there is a third fiber family i = 3 aligned normal to the plane spanned by the mechanically equivalent fiber families, we have

$$ \tilde{\mathbf L}_r= \frac{1} {3}\left[1+(6\cos^2\alpha-2)v_1^{(r)}-v_3^{(r)}\right] \widehat{\mathbf L}_1 +\frac{1}{3}\left[1+(6\sin^2\alpha-2)v_1^{(r)}-v_3^{(r)}\right]\widehat{\mathbf L}_2 +\frac{1}{3}\left[1-2v_1^{(r)}+2v_3^{(r)}\right]\widehat{\mathbf L}_3. $$

Additionally, superposition of several of the given cases may preserve the orthotropic material symmetry.