A boundary element model for nonlinear viscoelasticity

The boundary element methodology is applied to the analysis of non-linear viscoelastic solids. The adopted non-linear model uses the same relaxation moduli as the respective linear relations but with a time shift depending on the volumetric strain. Nonlinearity introduces an irreducible domain integral into the original integral equation derived for linear viscoelastic solids. This necessitates the evaluation of domain strains, which relies on proper differentiation of an integral with a strong kernel singularity. A time domain formulation is implemented through a numerical integration algorithm. The effectiveness of the developed numerical tool is demonstrated through the analysis of a plate with a central crack. The results are compared with respective predictions by the finite element method. Introduction Many polymers exhibit highly nonlinear viscoelastic behaviour in areas of stress or strain concentrations such as those arising from the presence of cracks. Material non-linearity manifests itself as considerable strain softening near the crack tip. The development of numerical techniques for the implementation of relevant constitutive models describing such behaviour has been an important research objective. Non-linear viscoelastic solutions, based on the finite element method (FEM), have been formulated and tested for efficiency and stability [1]. The boundary element method (BEM) has been extensively and very effectively used in modelling linear viscoelastic behaviour [2]. It has, in particular, been found a reliable tool for the analysis of viscoelastic fracture mechanics problems [3]. It seems, however, that there has not been any previous attempt to extend such formulations to modelling the nonlinear behaviour of polymers. Various constitutive models have been proposed for representing nonlinear viscoelasticity in polymers [4]. Schapery [5] proposed a quite general and frequently applied model, which includes the principle of time-stress superposition. The latter is accounted for through the definition of ‘reduced time’, a concept originally introduced to account for temperature variation [6]. Based on experimental studies, Knauss and Emri [7, 8] linked the timestress superposition model to the concept of free volume. This constitutive model has been applied to various problems [9, 10] and found to be a very effective analysis tool for assessing the effect of nonlinearity on the behaviour of polymer materials. The non-linear visco-elastic model employed in the present BEM formulation is based on the reduced time concept, which is, in turn, considered as a function of mechanical free-volume changes. The relaxation moduli of linear visco-elasticity are thus employed in the Boltzmann constitutive equations with a time shift depending on the volumetric strain. The difference between the actual stress tensor and its linear counterpart generates an irreducible domain integral into the original integral equation derived for linear viscoelastic solids. Domain strains are obtained by differentiation of a domain integral with a strong kernel singularity resulting in a singular integral and a regular free term. A time domain formulation is implemented through a numerical integration algorithm. The effectiveness of the developed numerical tool is demonstrated through the analysis of a plate with a central crack subjected to remote tension. The results are compared with respective predictions by the finite element method. Background theory The linear viscoelastic model adopted in earlier BEM formulations [11] is, in accordance with Boltzmann's principle, of hereditary integral type σij = Gijkl(t)εkl(0) + 0 ( ) ( ) d t kl ijkl G t ε τ τ τ τ ∂ − ∫ ∂ (1) where σij, εij are the stress and small strain tensors, respectively, and Gijkl(t) the relaxation moduli in the general case of an anisotropic medium. The problem is described relative to a Cartesian frame of reference xi, i =1,2,3, adopting the summation convention for repeated indices. Introducing the notation for the Stieltjes convolution of two functions [12], eq (1) can be more concisely written as σij = Gijkl ∗ dεkl (2) The nonlinear constitutive equations adopted here are [9] σij = ( ) [ ( ) ( )] d t kl ijkl G t ε τ ζ ζ τ τ τ −∞ ∂ − ∂ ∫ = Gijkl[ζ (t)]εkl(0) + 0 ( ) [ ( ) ( )] d t kl ijkl G t ε τ ζ ζ τ τ τ ∂ − ∂ ∫ (3) where ζ(t) is the reduced or intrinsic time, which may account for the effect of temperature [6], moisture and pressure variations on the relaxation moduli. A general definition of ζ(t) is