Constitutive equations material objectivity

Several generalizations of the inelastic theory to large deformations are developed in Section 5.4. In one the stretching (velocity strain) tensor is substituted for the strain rate. In order to make the resulting constitutive equations objective, i.e., invariant to relative rotation between the material and the coordinate frame, the stress rate must be replaced by one of a class of indifferent (objective) stress rates, and the moduli and elastic limit functions must be isotropic. In the elastic case, the constitutive equations reduce to the equation of hypoelastidty. The corresponding inelastic equations are therefore termed hypoinelastic. 

It is expected that constitutive equations should be invariant to relative rigid rotation and translation between the material and the coordinate frame with respect to which the motion is measured, a property termed objectivity. In order to investigate this invariance, the coordinate transformation... 

One physical restriction, translated into a mathematical requirement, must be satisfied that is that the simple fluid relation must be objective, which means that its predictions should not depend on whether the fluid rotates as a rigid body or deforms. This can be achieved by casting the constitutive equation (expressing its terms) in special frames. One is the co-rotational frame, which follows (translates with) each particle and rotates with it. The other is the co-deformational frame, which translates, rotates, and deforms with the flowing particles. In either frame, the observer is oblivious to rigid-body rotation. Thus, a constitutive equation cast in either frame is objective or, as it is commonly expressed, obeys the principle of material objectivity . Both can be transformed into fixed (laboratory) frame in which the balance equations appear and where experimental results are obtained. The transformations are similar to, but more complex than, those from the substantial frame to the fixed (see Chapter 2). Finally, a co-rotational constitutive equation can be transformed to a co-deformational one. 

In the various formulations of the mathematical theory of linear viscoelasticity, one should differentiate clearly the measurable and non-measurable fimctions, especially when it comes to modelling apart from the material functions quoted above, one may also define non measurable viscoelastic functions which Eu-e pure mathematical objects, such as the distribution of relaxation times, the distribution of retardation times, and tiie memory function. These mathematical tools may prove to be useful in some situations for example, a discrete distribution of relaxation times is easy to handle numerically when working with constitutive equations of the difierential type, but one has to keep in mind that the relaxation times derived numerically by optimization methods have no direct physical meaning. Furthermore, the use of the distribution of relaxation times is useless and costs precision when one wishes simply to go back and forth from the time domain to the frequency domain. This warning is important, given the large use (and sometimes overuse) of these distribution functions. 

The principle of material frame indifference (objectivity) states that variables, for which constitutive equations are needed, cannot depend on the coordinate frame (i.e.. Euclidean space, plus time) in which the variables are expressed. 

We call the fields (3.114)-(3.116) fulfilling the balances of mass (3.63), (3.65), momentum (3.76), moment of momentum (3.93), and energy (3.107) a thermodynamic process, because only these are of practical interest. Then we denote the fields (3.114) as the thermokinetic process and the fields (3.115) as the responses (we limit to the models with symmetric T (3.93) in more general models we must introduce also the torque M into responses (3.115), cf. Rems. 17, 32). The fields (3.116) are controlled from the outside (at least in principle). Just constitutive equations, which express the difference among materials, represent the missing equations and are relations between (3.114) and (3.115) [6, 7, 9, 10, 23, 34, 38, 40, 41, 44, 45], Referring to Sect. 2.1 we briefiy recall that constitutive equations are definitions of ideal materials which approximate real materials in the circumstances studied (i.e at chosen time and space scales). Constitutive equations may be proposed in rational thermodynamics using the constitutive principles of determinism, local action, memory, equipresence, objectivity, symmetry, and admissibility. 

Further reduction of constitutive equations (3.119) may be achieved by the constitutive principle of frame indifference or the principle of objectivity, the material properties and therefore also constitutive equations must be independent of the choice of frame. This principle is a generalization of common experience with mate-... 

Summary. A procedure really specific for the rational thermodynamics is introduced in this section in the form of several principles put forward to derive the thermodynamically consistent constitutive equations. In their most general form, the constitutive equations were proposed as functions (3.118) on the basis of the principles of determinism, local action, differential memory, and equipresence. They were further reduced to the form (3.121) considering the same material throughout the body and applying the principle of objectivity. Because of our interest in fluids only, the constitutive equations were further modified to this material type by means of the principle of material symmetry giving the final form (3.127). Two special types of fluid were defined by (3.129) and (3.130). 

Without going into details we note also that sufficient conditions of minima of a may be discussed and further simpliflcations of these results using objectivity (3.124) and material symmetry may be obtained using Cauchy-Green tensors, Piola-Kirchhoff stress tensors, the known linearized constitutive equations of solids follow, e.g. Hook and Fourier laws with tensor (transport) coefficients which are reduced to scalars in isotropic solids (e.g. Cauchy law of deformation with Lame coefficients) [6, 7, 9, 13, 14]. 

Now we restrict such constitutive equations—responses (4.120) as functions of (4.126)—by the principle of objectivity (or (material) frame indifference), cf. Sect. 3.5 constitutive equations cannot depend explicitly on (non-objective) x and t... 

The general principles of the development of nonlinear models of mass transfer in elastically deformed materials were developed in studies. The general formulation of constitutive equations and the use of non-traditional thermodynamic parameters such as partial stress tensors and diffusion forces lead to significant difficulties in attempts to apply the theory to the description of specific objects. Probably, because of this, the theory is little used for the solution of applied problems. 

It is a common observation that different materials undergo different deformatiorrs when subjected to identical stress. Our next objective then is to formulate material related equations, which cormect the stresses occurring in a solid body with its motion or, more precisely, its strain. These eqrtatiotts reflect the specific material properties of the bodies rmder consideratiore They are actually the equations of state, well known in the context of gases, but extended to a more general class of materials, in our case crystals. They are usually referred to as the constitutive equations. 

For each given isochoric flow problem, p must then be determined by the balance laws and the particular boundary conditions of that problem, while the extra stress tensor S must be supplied by a constitutive equation for the material. The Cauchy and extra stress tensors are both objective. 

Applying the principle of material objectivity plus the dissipation inequality, we are led to the constitutive equation for an incompressible elastic solid... 

A fundamental principle of classical physics is that material properties must be independent of the frame of reference or observer. This axiom is commonly called the principle of material frame-indifference or objectivity [270, pp.41-44]. This principle states that constitutive equations (discussed in greater detail in Section 4.2.3) must be invariant under changes of frame of reference. Under the motion defined by... 

An optimum seeking method is a systematic way of manipulating a set of variables to find the values of the variables to maximize or minimize some criterial. Their most popular uses have been economic ones such as profitability or costs or technical criterial such as conversion of raw materials or product recovery. For the applications described in this paper, the optimization criterial are the minimization of the squared deviations from zero of the equations chosen to constitute the objective function. The equations chosen to formulate the objective function can be written as... 

The present approach conflicts with Copenhagen view tenets quoted in Section 3.2. The concept of object is replaced by the elementary constitutive materials, viz. electron and nuclei sustaining quantum states. The parameters defining charge spin and mass enter those differential equations used to calculate model quantum states (time-independent eigenvalue Schrodinger equation or relativistic equations [5]). 

Analysis of these effects is difficult and time consuming. Much recent work has utilized two-dimensional, finite-difference computer codes which require as input extensive material properties, e.g., yield and failure criteria, and constitutive laws. These codes solve the equations of motion for boundary conditions corresponding to given impact geometry and velocities. They have been widely and successfully used to predict the response of metals to high rate impact (2), but extension of this technique to polymeric materials has not been totally successful, partly because of the necessity to incorporate rate effects into the material properties. In this work we examined the strain rate and temperature sensitivity of the yield and fracture behavior of a series of rubber-modified acrylic materials. These materials have commercial and military importance for impact protection since as much as a twofold improvement in high rate impact resistance can be achieved with the proper rubber content. The objective of the study was to develop rate-sensitive yield and failure criteria in a form which could be incorporated into the computer codes. Other material properties (such as the influence of a hydrostatic pressure component on yield and failure and the relaxation spectra necessary to define viscoelastic wave propagation) are necssary before the material description is complete, but these areas will be left for later papers. 

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