Nonlinear viscoelastic, thermodynamically consistent, models for biological soft tissue

Abstract

The mechanical behavior of most biological soft tissue is nonlinear viscoelastic rather than elastic. Many of the models previously proposed for soft tissue involve ad hoc systems of springs and dashpots or require measurement of time-dependent constitutive coefficient functions. The model proposed here is a system of evolution differential equations, which are determined by the long-term behavior of the material as represented by an energy function of the type used for elasticity. The necessary empirical data is time independent and therefore easier to obtain. These evolution equations, which represent non-equilibrium, transient responses such as creep, stress relaxation, or variable loading, are derived from a maximum energy dissipation principle, which supplements the second law of thermodynamics. The evolution model can represent both creep and stress relaxation, depending on the choice of control variables, because of the assumption that a unique long-term manifold exists for both processes. It succeeds, with one set of material constants, in reproducing the loading–unloading hysteresis for soft tissue. The models are thermodynamically consistent so that, given data, they may be extended to the temperature-dependent behavior of biological tissue, such as the change in temperature during uniaxial loading. The Holzapfel et al. three-dimensional two-layer elastic model for healthy artery tissue is shown to generate evolution equations by this construction for biaxial loading of a flat specimen. A simplified version of the Shah–Humphrey model for the elastodynamical behavior of a saccular aneurysm is extended to viscoelastic behavior.

Appendix

1.1 Evolution equation when the strain energy is a function of a tensor

Let E be a second order tensor of state variables, and let H be the second order tensor of conjugate control variables. Assume that the energy function has the form

$$ \psi ({\mathbf{E}};{\mathbf{H}}) = \phi ({\mathbf{E}}) + {\mathbf{E}}:{\mathbf{H}}. $$

The affinities are the second order tensor, X=∂ψ/∂E. The affinities are a function of E, so that X=h(E). Assume that h is invertible so that ψ(E; H) can be written in terms of the affinities for fixed control variables, $\bar \psi ({\mathbf{X}};{\mathbf{H}}).$

The gradient dissipation condition (Eq. 5) is that the evolution of the affinities obeys ${\mathbf{\dot X}} = - k\partial \bar \psi /\partial {\mathbf{X}}$ for fixed controls, where ${\mathbf{\dot X}}$ is a second order tensor. By the chain rule,

$$ {\mathbf{\dot X}} = \frac{{\partial {\mathbf{X}}}} {{\partial {\mathbf{E}}}}{\mathbf{\dot E}}. $$

(43)

Here ∂X/∂E is a fourth order tensor. Again by the chain rule,

$$ \frac{{\partial \bar \psi }} {{\partial {\mathbf{X}}}} = \frac{{\partial \psi }} {{\partial {\mathbf{E}}}}\frac{{\partial {\mathbf{E}}}} {{\partial {\mathbf{X}}}}. $$

(44)

Substitution of Eqs. 43 and 44 into ${\mathbf{\dot X}} = - k\partial \bar \psi /\partial {\mathbf{X}}$ yields the evolution equation in terms of the state variables,

$$ {\mathbf{\dot E}} = - k\left( {\frac{{\partial {\mathbf{X}}}} {{\partial {\mathbf{E}}}}} \right)^{ - 2} \frac{{\partial \psi }} {{\partial {\mathbf{E}}}} = - k\left( {\frac{{\partial {\mathbf{X}}}} {{\partial {\mathbf{E}}}}} \right)^{ - 2} \left( {\frac{{\partial \phi }} {{\partial {\mathbf{E}}}} + {\mathbf{H}}} \right). $$

(45)

This form reduces to that of Eq. 6 in the case that E has only diagonal non-zero entries. Then H also has only diagonal non-zero entries. The affinity is a second order tensor defined by

$$ ({\mathbf{X}})_{ij} = \left( {\frac{{\partial \psi }} {{\partial {\mathbf{E}}}}} \right)_{ij} = \frac{{\partial \psi }} {{\partial {\mathbf{E}}}}_{ij} . $$

(46)

If ψ only depends on E₁₁, E₂₂, and E₃₃, then the only non-zero terms of X are the three terms X _ii .

The fourth order tensor ∂X/∂E is defined by

$$ \left( {\frac{{\partial {\mathbf{X}}}} {{\partial {\mathbf{E}}}}} \right)_{ijkm} = \frac{{\partial {\mathbf{X}}_{ij} }} {{\partial {\mathbf{E}}_{km} }}. $$

(47)

The only non-zero terms of this tensor, if only the diagonal entries of E are non-zero, have the form

$$ \left( {\frac{{\partial {\mathbf{X}}}} {{\partial {\mathbf{E}}}}} \right)_{iikk} = \frac{{\partial {\mathbf{X}}_{ii} }} {{\partial {\mathbf{E}}_{kk} }} = \frac{{\partial ^2 \psi }} {{\partial {\mathbf{E}}_{ii} \partial {\mathbf{E}}_{kk} }}. $$

(48)

The evolution equation therefore has the form of Eq. 6.