A continuum and computational framework for viscoelastodynamics: III. A nonlinear theory

TL;DR

This paper develops a nonlinear viscoelastic theory by extending the Holzapfel-Simo framework to fully nonlinear deformations using Green-Naghdi–type kinematics within Hill's hyperelasticity and generalized-strain formalisms. It presents two complementary energy formalisms, Helmholtz and Gibbs, to derive consistent constitutive relations, including multi-term and multi-relaxation generalizations, and introduces internal state variables to capture non-equilibrium processes. A rigorous numerical strategy is formulated, featuring consistent linearization, nonlinear constitutive integration via mid-point time discretization, and a modular implementation that uses spectral decompositions and projection tensors. The approach is demonstrated through creep, shear, and bearing examples, showing correct relaxation behavior, tunable nonlinear elastic responses, and applicability to layered viscoelastic devices under dynamic loading. The work provides a thermodynamically sound, computation-friendly framework for large-strain viscoelasticity with potential extensions to anisotropy and structure-preserving numerical schemes.

Abstract

We continue our investigation of viscoelasticity by extending the Holzapfel-Simo approach discussed in Part I to the fully nonlinear regime. By scrutinizing the relaxation property for the non-equilibrium stresses, it is revealed that a kinematic assumption akin to the Green-Naghdi type is necessary in the design of the potential. This insight underscores a link between the so-called additive plasticity and the viscoelasticity model under consideration, further inspiring our development of a nonlinear viscoelasticity theory. Our strategy is based on Hill's hyperelasticity framework and leverages the concept of generalized strains. Notably, the adopted kinematic assumption makes the proposed theory fundamentally different from the existing models rooted in the notion of the intermediate configuration. The computation aspects, including the consistent linearization, constitutive integration, and modular implementation, are addressed in detail. A suite of numerical examples is provided to demonstrate the capability of the proposed model in characterizing viscoelastic material behaviors at large strains.

Figures (11)

• Figure 1: Illustration of different scale functions. The Green-Lagrange strain (blue curve) gives a finite value when the stretch approaches zero. The Hencky scale function (red curve) and the Bažant-Itskov scale function (green curve) satisfy the symmetry property \ref{['eq:tension-compression-symmetry']}, as is indicated by the red and green dots, respectively.
• Figure 2: The interpretation of the generalized hyperelastic strain energy function \ref{['eq:psi-def-fused-type-1']} as a device of multiple springs of Hookean type in parallel. Each one of the Hookean spring is described by a single quadratic energy of a specific strain. Their combined mechanical behavior can be viewed as a single spring whose energy is the sum of quadratic strain energies of different strains.
• Figure 3: Fitting of the Ogden model (left) and the model of Hill's class using two Curnier-Rakotomanana strains (right). The dots and lines represent the experimental data and model predictions, respectively. The blue, red, and black colors represent the uniaxial tensile, equi-biaxial tensile, and pure shear responses, respectively. The quality of fit metric $\chi^2$ is adopted from the work of Dal Dal2021. A smaller value of $\chi^2$ indicates a better model fit.
• Figure 4: An illustration of the theory based on the Helmholtz free energy using the spring-dashpot rheological model.
• Figure 5: An illustration of the theory based on the Gibbs free energy using the spring-dashpot rheological model.

Theorems & Definitions (27)

• Remark 1
• Lemma 1
• proof
• Remark 2
• Lemma 2
• Definition 1
• Remark 3
• Remark 4
• Remark 5
• Proposition 1