Mechanical properties constitutive equation

This chapter is devoted to a short description of low-strain mechanical properties of polymers in the solid state and in the glass transition region, with an emphasis on the effect of crosslinking on these properties. There are three degrees of complexity in the description of this behavior, depending on the number of variables taken into account in the constitutive equations under consideration. 

To avoid the apparent complications with absolute rheologic measurement techniques, a number of investigators (4,5). have used relative measurement systems to make rheologic measurements. The major difference between the relative and absolute measurement techniques is that the fluid mechanics in the relative systems are complex. The constitutive equations needed to find the fundamental rheologic variables cannot be readily solved. Relative measurement systems require the use of Newtonian and non-Newtonian calibrations fluids with known properties to relate torque and rotational speed to the shear rate and shear stress (6). 

Kwon Y, Leonov AI (1995) Stability constraints in the formulation of viscoelastic constitutive equations. J Non-Newton Fluid Mech 58 25—46 Laius LA, Kuvshinskii EV (1963) Structure and mechanical properties of oriented amorphous linear polymers. Fizika Tverdogo Tela 5(11) 3113—3119 (in Russian)... 

To complement the equations obtained from the application of the conservation principles, it is required to use some equations based on physical, chemical, or electrochemical laws, that model the primary mechanisms by which changes within the process are assumed to occur (rates of the processes, calculation of properties, etc.). These equations are called constitutive equations and include four main categories of equations definition of process variables in terms of physical properties, transport rate, chemical and electrochemical kinetics, and thermodynamic equations. 

FTMA has great potential for applications in conjunction with a coupled theory of the effects of both temperature and strain histories on mechanical material properties. This is beyond the scope of present day theoretical capabilities (31.31) and the exact constitutive equation for a given material has to be determined experimentally. 

Polymeric fluids are the most studied of all complex fluids. Their rich rheological behavior is deservedly the topic of numerous books and is much too vast a subject to be covered in detail here. We must therefore limit ourselves to an overview. The interested reader can obtain more thorough presentations in the following references a book by Ferry (1980), which concentrates on the linear viscoelasticity of polymeric fluids, a pair of books by Bird et al. (1987a,b), which cover polymer constitutive equations, molecular models, and elementary fluid mechanics, books by Tanner (1985), by Dealy and Wissbrun (1990), and by Baird and Dimitris (1995), which emphasize kinematics and polymer processing flows, a book by Macosko (1994) focusing on measurement methods and a book by Larson (1988) on polymer constitutive equations. Parts of this present chapter are condensed versions of material from Larson (1988). The static properties of flexible polymer molecules are discussed in Section 2.2.3 their chemistry is described in Flory (1953). 

Both E, in ideal solids, and rj, in ideal liquids, are material functions independent of the size and shape of the material they describe. This holds for isotropic and homogeneous materials, that is, materials for which a property is the same at all directions at any point. Isotropic materials are so characterized because their degree of symmetry is infinite. In contrast, anisotropic materials present a limited number of elements of symmetry, and the lower the number of these elements, the higher the number of material functions necessary to describe the response of the material to a given perturbation. Even isotropic materials need two material functions to describe in a generalized way the relationship between the perturbation and the response. In order to formulate the mechanical behavior of ideal solids and ideal liquids in terms of constitutive equations, it is necessary to establish the concepts of strain and stress. 

It is valid for each continuum independent of the individual material properties and is therefore one of the fundamental equations in fluid mechanics and subsequently also in heat and mass transfer. The movement of a particular substance can only be described by introducing a so-called constitutive equation which links the stress tensor with the movement of a substance. Generally speaking, constitutive equations relate stresses, heat fluxes and diffusion velocities to macroscopic variables such as density, velocity and temperature. These equations also depend on the properties of the substances under consideration. For example, Fourier s law of heat conduction is invoked to relate the heat flux to the temperature gradient using the thermal conductivity. An understanding of the strain tensor is useful for the derivation of the consitutive law for the shear stress. This strain tensor is introduced in the next section. 

What this equation tells us is that a particular state of stress is nothing more than a linear combination (albeit perhaps a tedious one) of the entirety of components of the strain tensor. The tensor Cijn is known as the elastic modulus tensor or stiffness and for a linear elastic material provides nearly a complete description of the material properties related to deformation under mechanical loads. Eqn (2.52) is our first example of a constitutive equation and, as claimed earlier, provides an explicit statement of material response that allows for the emergence of material specificity in the equations of continuum dynamics as embodied in eqn (2.32). In particular, if we substitute the constitutive statement of eqn (2.52) into eqn (2.32) for the equilibrium case in which there are no accelerations, the resulting equilibrium equations for a linear elastic medium are given by... 

A key element in the study of coupled THM processes is the verification of numerical codes and validation of model results against well-conditioned field and laboratory experiments. Here, the challenge lies in providing a set of well-defined conditions for the boundaries, the rates of thermal and mechanical loading, the initial state (of stress, temperature and flow), and constitutive equations for coupling and material properties. Coupled THM experiments in the field require well-considered test designs, robust instrumentation, careful result interpretation, and often have durations of months and years. 

The elastoplastic multiscale analysis requires several computational modules, including (1) a microscale computation module, which consists of a set of numerical solutions for the local constitutive equation of each subphase, (2) a micromechanical computation module, which provides numerical tools to link the mechanical properties of each of the local subphases to the macroscopic responses, and (3) a macroscale computation module, in which the continuum mechanics governing equations are enforced to simulate the overall mechanical response of the material and to identify the local loading conditions over the R VE. Each of these computational modules is discussed in the following. A flowchart of the multiscale analysis is shown in Figure 5.24. 

The physical basis for the design of piezoelectric membranes is based on simple combined electrical and mechanical relations (Gauss law and Hooke s law). The relationship between the electrical and mechanical properties of piezoelectrics is governed by the following constitutive equations ... 

The mechanical property of a homogeneous material is expressed by the constitutive equation which relates the stress tensor a p to the velocity gradient tensor K p, where... 

If there is no external field, the director m in nematics is entirely arbitrary. Thus the equilibrium state of nematics is not unique and can be changed by infinitesimal perturbation. This property, generally called broken symmetry in statistical mechanics, necessitates a special treatment in the mathematical handling of the kinetic equation, and introduces a new type of constitutive equation, unique to the ordered fluid. 

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