Nonlinear viscoelasticity of strain rate type: an overview

There are some materials in nature that experience deformations that are not elastic. Viscoelastic materials are some of them. We come across many such materials in our daily lives through a number of interesting applications in engineering, material science and medicine. This article concerns itself with modelling of the nonlinear response of a class of viscoelastic solids. In particular, nonlinear viscoelasticity of strain rate type, which can be described by a constitutive relation for the stress function depending not only on the strain but also on the strain rate, is considered. This particular case is not only favourable from a mathematical analysis point of view but also due to experimental observations, knowledge of the strain rate sensitivity of viscoelastic properties is crucial for accurate predictions of the mechanical behaviour of solids in different areas of applications. First, a brief introduction of some basic terminology and preliminaries, including kinematics, material frame-indifference and thermodynamics, is given. Then, considering the governing equations with constitutive relationships between the stress and the strain for the modelling of nonlinear viscoelasticity of strain rate type, the most general model of interest is obtained. Then, the long-term behaviour of solutions is discussed. Finally, some applications of the model are presented.

Noll’s Axioms and Formulation of the Closure Relations for the Subgrid Turbulent Tensor in Large Eddy Simulation

In this paper, the relation between the Noll formulation of the principle of material frame indifference and the principle of turbulent frame indifference in large eddy simulation, is revised. The principle of material frame indifference and the principle of turbulent frame indifference proposed by Hutter and Joenk imposes that both constitutive equations and turbulent closure relations must respect both the requirement of form invariance, and the requirement of frame independence. In this paper, a new rule for the formalization of turbulent closure relations, is proposed. The generalized SGS turbulent stress tensor is related exclusively to the generalized SGS turbulent kinetic energy, which is calculated by means of its balance equation, and the modified Leonard tensor.

Constitutive relations for polar continua based on statistical mechanics and spatial averaging

The purpose of the current work is the formulation of macroscopic constitutive relations, and in particular continuum flux densities, for polar continua from the underlying mass point dynamics. To this end, generic microscopic continuum field and balance relations are derived from phase space transport relations for expectation values of point fields related to additive mass point quantities. Given these, microscopic energy, linear momentum and angular momentum, balance relations are obtained in the context of the split of system forces into non-conservative and conservative parts. In addition, divergence–flux relations are formulated for the conservative part of microscopic supply-rate densities. For the case of angular momentum, two such relations are obtained. One of these is force-based, and the other is torque-based. With the help of physical and material theoretic restrictions (e.g. material frame-indifference), reduced forms of the conservative flux densities are obtained. In the last part of the work, formulation of macroscopic constitutive relations from their microscopic counterparts is investigated in the context of different spatial averaging approaches. In particular, these include (weighted) volume-averaging based on a localization function, surface averaging of normal flux densities based on Cauchy flux theory and volume averaging with respect to centre of mass.

Logarithmic Spin, Logarithmic Rate and Material Frame-Indifferent Generalized Plasticity

In this work, we present a new rate type formulation of large deformation generalized plasticity which is based on the consistent use of the logarithmic rate concept. For this purpose, the basic constitutive equations are initially established in a local rotationally neutralized configuration which is defined by the logarithmic spin. These are then rephrased in their spatial form, by employing some standard concepts from the tensor analysis on manifolds. Such an approach, besides being compatible with the notion of (hyper)elasticity, offers three basic advantages, namely: (i) The principle of material frame-indifference is trivially satisfied. (ii) The structure of the infinitesimal theory remains essentially unaltered. (iii) The formulation does not preclude anisotropic response. A general integration scheme for the computational implementation of generalized plasticity models which are based on the logarithmic rate is also discussed. The performance of the scheme is tested by two representative numerical examples.

Eulerian elastoplasticity: Basic issues and recent results

Traditional formulations of elastoplasticity in the presence of finite strain and large rotation are Eulerian type and widely used; they are based upon, among other things, the additive decomposition of the stretching or the Eulerian strain-rate into elastic and plastic parts. In such formulations, yield functions and objective rate constitutive equations are expressed in terms of objective Eulerian tensor quantities, including the stretching, the Kirchhoff stress, internal state variables, etc. Each of these quantities transforms in a corotational manner under a change of the observing frame. According to the principle of material frame-indifference or objectivity, each constitutive function should be invariant, whenever the observing frame is changed to another one by any given time-dependent rotation. In this work the general form of constitutive equations is discussed. Several frequently used objective rates are analyzed with respect to their serviceability to develop a self-consistent formulation, i.e. to be integrable to deliver an elastic in particular hyperelastic relation for vanishing plastic deformation. This would be of great importance, e.g., for so-called spring back calculations in metal forming.

A POSTERIORI TESTS OF ONE-EQUATION LES MODELING OF ROTATING TURBULENCE

Eight subgrid-scale (SGS) models were evaluated using two flow configurations: homogeneous decaying turbulence, and rotating turbulence forced at large or intermediate scales. Testing was performed for the first configuration through a systematic comparison between direct numerical simulation results and large eddy simulation results of many characteristics, including resolved kinetic energy, SGS energy production, molecular dissipation, and kinetic energy spectrum. The new models, which are based on dynamic structure model and satisfy the consistency of material frame indifference with the SGS stress, showed more accurate results than traditional models. In the forced testing, the new models were better able to capture essential features of rotating turbulence, including cyclonic/anti-cyclonic asymmetry, quasi-2D at large scales, and reverse kinetic energy transfer from small to large scales.