A Brief Review of Elasticity and Viscoelasticity for Solids

Abstract

Abstract. There are a number of interesting applications where modeling elastic and/or viscoelastic materials is fundamental, including uses in civil engineering, the food industry, land mine detection and ultrasonic imaging. Here we provide an overview of the subject for both elastic and viscoelastic materials in order to understand the behavior of these materials. We begin with a brief introduction of some basic terminology and relationships in continuum mechanics, and a review of equations of motion in a continuum in both Lagrangian and Eulerian forms. To complete the set of equations, we then proceed to present and discuss a number of specific forms for the constitutive relationships between stress and strain proposed in the literature for both elastic and viscoelastic materials. In addition, we discuss some applications for these constitutive equations. Finally, we give a computational example describing the motion of soil experiencing dynamic loading by incorporating a specific form of constitutive equation into the equation of motion.

Mechanochemical pattern formation in simple models of active viscoelastic fluids and solids

“…(1) is computed from these elements, see e.g. [42]. It is important to note that those springs and dashpots do not have a direct physical interpretation.…”

Contact interaction of two rectangular plates made from different materials with an account of physical nonlinearity

“…, N , is given. Then, the system (21) defines a system of functions ψ (1) j (y), which are substituted into the system (22), and a new choice with regard to the variable x is defined as ϕ (2) j (x). Then, using the chosen functions, the system (21) yields a new function ψ (3) j (y) with respect toy, and so on.…”

“…Owing to MVI, a solution to Eq. (16) can be searched in the form of (17), where the functions ϕ i (x) and ψ i (y) are defined through the following equations (22) in the following way. A certain system of N functions with regard to one of the variables, let us say ϕ 0 j (x) j = 1, 2, .…”

“…A deeper state of the art of mathematical approaches can be found in references [22][23][24]. More recently, quasi-static and dynamical frictional non-standard contact states belonging to a new class, including also effects of memory, have been studied in piezoelectric materials [25,26].…”

Analysis of a contact problem with normal damped response and unilateral constraint

“…Various results, examples and mechanical interpretations in the study of viscoelastic materials of the form (1), can be found in [4] and the references therein. Such kind of constitutive laws were used in the literature in order to model the behavior of real materials like rubbers, rocks, metals, pastes and polymers.…”