Examples Of Constitutive Equations Research Articles

Constitutive equations for thixotropic fluids

To distinguish it clearly from nonlinear viscoelasticity, we define “ideal thixotropy” as “a time-dependent viscous response to the history of the strain rate, with fading memory of that history,” endowing such fluids with memory but no elasticity. An “ideal thixotropic fluid” has instantaneous stress relaxation upon cessation of flow and no elastic recoil on removal of stress. We describe “nonideal thixotropic” fluids as those whose viscoelastic time scales governing stress relaxation are much shorter than those governing the thixotropic response. This ensures that a clear distinction can be maintained between “thixotropy” and “nonlinear viscoelasticity.” The stress tensor for an ideal thixotropic fluid can in general be expressed as a contraction product of a fourth rank viscosity tensor with the velocity gradient tensor, in which the viscosity tensor depends on the history of the flow. We show examples of constitutive equations that meet the definitions of ideal thixotropy or nonideal thixotropy. We also show examples of constitutive equations that have been designated as “thixotropic” by virtue of containing an equation for evolution of a “structure parameter,” but whose behavior is indistinguishable from that of ordinary nonlinear viscoelasticity, and so should not be considered thixotropic.

Compressible rigid viscoplastic fluids

In this paper, a general methodology for constructing fluid-type constitutive equations for description of the behavior of porous media when subjected to large deformations and high strain rates is proposed. The elastic deformations are neglected and the representation of the dynamic state of stress in the material is obtained from classic yield conditions for plastic solids using the viscoplastic regularization and/or a stress superposition method. Examples of constitutive equations obtained using both procedures are presented. We show that irrespective of the method used, it is not possible to obtain consistent viscoplastic fluid formulations starting from certain yield functions. Comparisons between the response of the proposed models for uniaxial compression conditions is provided. Extensions of these models that include coupling between damage, plastic and viscous effects are proposed. Finally, some results of numerical modeling of penetration into concrete using a compressible rigid-viscoplastic model are presented.

On Mathematical Aspects of Dual Variables in Continuum Mechanics. Part 2: Applications in Nonlinear Solid Mechanics

AbstractContinuing the approach of Part 1 of this paper we apply our results to dual variables appearing in continuum mechanics. We investigate the kinematics and dynamics of continuous bodies. As a consequence, this leads to a mixed variant formulation of kinematics and Cauchy's law for the stress tensor. The mixed variant formation of kinematics is advantageous, for instance, in finite deformation plasticity. Finally, we give some examples of constitutive equations which demonstrate in an exemplaric manner the additional mathematical structure gained through our approach.

How to use damage mechanics

The background of continuum damage mechanics is first presented in the framework of thermodynamics with some examples of constitutive equations for ductile damage, creep damage and fatigue damage. After the general scheme of structural calculations for macro-crack initiation, through non-coupled or coupled strain damage equations, some examples of “simple applications” are given: fracture limits of metal forming, surface initial damage in fatigue, creep fatigue interaction, and bifurcation of cracks.

On the numerical implementation of inelastic time dependent and time independent, finite strain constitutive equations in structural mechanics

A number of complex issues are resolved in a way that allows the incorporation of finite strain, inelastic material behavior into the piecewise numerical construction of solutions in solid mechanics. Without recourse to extensive continuum mechanics preliminaries, an elementary time independent plasticity model, an elementary time dependent creep model, and a viscoelastic model are introduced as examples of constitutive equations which are routinely used in engineering calculations. The constitutive equations are all suitable for problems involving large deformations and finite strains. The plasticity and creep models are in rate form and use the symmetric part of the velocity gradient or the stretching to compute the co-rotational time derivative of the Cauchy stress. The viscoelastic model computes the current value of the Cauchy stress from a hereditary integral of a ‘materially invariant’ form of the stretching history. The current configuration is selected for evaluation of equilibrium as opposed to either the reference configuration or the last established equilibrium configuration. The process of strain incrementation is examined in some depth. The stretching, evaluated at the mid-interval and multiplied by the time step, is identified as the appropriate finite strain increment to use with the selected forms of the constitutive equations. Discussed is the conversion of rotation rates based on the spin into incremental orthogonal rotations. These rotations are used to update stresses and state variables due to rigid body rotation during the load increment. Comments and references to the literature are directed at numerical integration of the constitutive equations with an emphasis on doing this accurately, if not exactly, for any time step and stretching. This material taken collectively provides an approach to numerical implementation which is marked by its simplicity.