A new representation for viscoelastic behavior of materials in two- and three-dimensional problems

The main objective of this study is to develop a set of equations, in which any desirable rheological model can be employed to investigate the behavior of viscoelastic materials (VMs) in two- or three-dimensional settings within the scope of small deformations. Application of the differential-form constitutive equations in 3D problems has been mainly limited to the isotropic materials. On the other hand, the integral-form constitutive equations are mostly unfavorable in terms of computational costs. To the best knowledge of the authors, in general, there is no interrelation between the differential-form and the integral-form viscoelastic constitutive equations in the 3D theory of linear viscoelasticity. This study tends to link the differential and the integral forms of linear viscoelasticity in a general setting. As a consequence, for example, relaxation moduli of a VM can be formulated in terms of model parameters, i.e., spring constants and viscosity coefficients, of any desirable rheological model. Some numerical examples are also provided to support the efficiency of the proposed formulation in 2D and 3D loading scenarios