Elastic behavior constitutive relations

The formulation of proper constitutive relations is a complex problem and is the basis of the science of rheology, which cannot be covered here. This section presents only four relatively simple constitutive relations that have proved to be practically useful to chemical engineers. Elastic fluid behavior is expressly excluded from consideration. The following equations are a listing of these constitutive relations many others are possible ... 

The dependence of mechanical behavior on constitution in Zr02-Ni system results from the variation of microstructure and its distribution. In the regions rich in Ni or PSZ, the mechanical performance is controlled by continuous matrix component and displays elasto-plastic or linear elastic characteristics, respectively. The non-linear elastic behavior at 60 vol% PSZ is related to the connectivity transition of matrix component. 

Elastic Modulao. The mechanical behavior is in general terms concerned with the deformation that occurs under loading. Generalized equations that relate stress to strain are called constitutive relations. The simplest form of such a relation is Hooke s Law which relates the stress s to the strain e for rmiaxial deformation of the ideal elastic isotropic solid ... 

The Maxwell Model. In the above development, discussion moves from elastic behavior to viscoelastic descriptions of material behavior. In a simple sense, viscoelasticity is the behavior exhibited by a material that has both viscous and elastic elements in its response to a deformation or load. In early days, this was often represented by elastic or viscous mechanical elements combined in different ways (9-12). The simplest models are two element models that contain a viscous element (dashpot) and an elastic element (spring). The dashpot is assumed to follow a Newtonian fluid constitutive law in which the stress is related directly to the strain rate by the following expression ... 

Constitutive Description of Polymer Melt Behavior K-BKZ and DE Descriptions. Although there are many nonlinear constitutive models that have been proposed, the focus here is on the K-BKZ model because it is relatively simple in structure, can be related conceptually to finite elasticity descriptions of elastic behavior, and because, in the mind of the current author and others (82), the model captures the major features of nonlinear viscoelastic behavior of polymeric fluids. In addition, the reptation model as proposed by Doi and Edwards provides a molecular basis for understanding the K-BKZ model. The following sections first describe the K-BKZ model, followed by a description of the DE model. 

If the deformations are not kept small, but are carried to the point where the elastic behavior is nonlinear, equations 38 and 39 do not hold. For soft polymeric solids, deviations from linearity appear sooner i.e., at smaller strains) in extension than in shear, because of the geometrical effects of Hnite deformations. At substantial deformations, the relations between creep and recovery are much more complicated than those given above, and require formulation by nonlinear constitutive relationships. 

Polymeric materials, such as rubber, exhibit a mechanical response which cannot be properly described neither by means of elastic nor viscous effects only. In particular, elastic effects account for materials which are able to store mechanical energy with no dissipation. On the other hand, a viscous fluid in a hydrostatic stress state dissipates energy, but is unable to store it. As the experimental results reported in Part 1 have shown, filled rubber present both the characteristics of a viscous fluid and of an elastic solid. Viscoelastic constitutive relations have been introduced with the intent of describing the behavior of such materials able to both store and dissipate mechanical energy. 

The purpose of this chapter has been to give a description of some of the most useful contact mechanics expressions as they relate to studies of adhesion. The primary assumption regarding the properties of the materials themselves is that a linear constitutive model is obeyed throughout the strained region, with the possible exception of a relatively small cohesive zone at the contact edge. Many of the results obtained for simple linear elastic behavior are analytic. Linear viscoelasticity can be handled as well, although in this case numerical approaches... 

It is clear that viscoelastic fluids require a constitutive equation that is capable of describing time-dependent rheological properties, normal stresses, elastic recovery, and an extensional viscosity which is independent of the shear viscosity. It is not clear at this point exactly as to how a constitutive equation for a viscoelastic fluid, when coupled with the equations of motion, leads to the prediction of behavior (i.e., velocity and stress fields) which is any different from that calculated for a Newtonian fluid. As the constitutive relations for polymeric fluids lead to nonlinear differential equations that cannot easily be solved, it is difficult to show how their use affects calculations. Furthermore, it is not clear how using a constitutive equation, which predicts normal stress differences, leads to predictions of velocity and stress fields which are significantly different from those predicted by using a Newtonian fluid model. Finally, there are numerous possibilities of constitutive relations from which to choose. The question is then When and how does one use a viscoelastic constitutive relation in design calculations especially when sophisticated numerical methods such as finite element methods are not available to the student at this point For the... 

The methods utilized to measure the viscoelastic functions are often close to the stress patterns occurring in certain conditions of use of polymeric materials. Consequently, information of technological importance can be obtained from knowledge of these functions. Even the so-called ultimate properties imply molecular mechanisms that are closely related to those involved in viscoelastic behavior. Chapters 16 and 17 deal with the stress-strain multiaxial problems in viscoelasticity. Application of the boundary problems for engineering apphcations is made on the basis of the integral and differential constitutive stress-strain relationships. Several problems of the classical theory of elasticity are revisited as viscoelastic problems. Two special cases that are of special interest from the experimental point of view are studied viscoelastic beams in flexion and viscoelastic rods in torsion. 

The role of constitutive equations is to instruct us in the relation between the forces within our continuum and the deformations that attend them. More prosaically, if we examine the governing equations derived from the balance of linear momentum, it is found that we have more unknowns than we do equations to determine them. Spanning this information gap is the role played by constitutive models. From the standpoint of building effective theories of material behavior, the construction of realistic and tractable constitutive models is one of our greatest challenges. In the sections that follow we will use the example of linear elasticity as a paradigm for the description of constitutive response. Having made our initial foray into this theory, we will examine in turn some of the ideas that attend the treatment of permanent deformation where the development of microscopically motivated constitutive models is much less mature. 

Stress is related to strain through constitutive equations. Metals and ceramics typically possess a direct relationship between stress and strain the elastic modulus (2) Polymers, however, may exhibit complex viscoelastic behavior, possessing characteristics of both liquids and solids (4.). Their stress-strain behavior depends on temperature, degree of cure, and thermal history the behavior is made even more complicated in curing systems since material properties change from a low molecular weight liquid to a highly crosslinked solid polymer (2). ... 

The viscoelasticity properties are also important, because they can supply information directly related to the form of the macromolecules. The models of the linear viscoelasticity are developed from two elements a spring and a dashpot. Two of those elements in line constitute the Maxwell model and in parallel the Kelvin model (or Vogt).20 Normally, those models don t represent the behavior of complex materials satisfactorily. Other models such as the Burgers model, where the Maxwell and Kelvin models are connected in line, are used to determine the modulus of elasticity (Yj and Y2) and the coefficients of viscosity ( and t]2).21... 

The most modern picture of membrane deformation recognizes that the membrane is a composite of two layers with distinct mechanical behavior. The membrane bilayer, composed of phospholipids and integral membrane proteins, exhibits a large elastic resistance to area dilation but is fluid in surface shear. The membrane skeleton, composed of a network of structural proteins at the cytoplasmic surface of the bilayer, is locally compressible and exhibits an elastic resistance to surface shear. The assumption that the membrane skeleton is locally incompressible is no longer applied. This assumption had been challenged over the years on the basis of theoretical considerations, but only very recently has experimental evidence emerged that shows definitively that the membrane skeleton is compressible. This has led to a new constitutive model for membrane behavior [Mohandas and Evans, 1994]. The principal stress resultants in the membrane skeleton are related to the membrane deformation by ... 

If this deformation field does not fulfill the geometrical compatibility, a strain tensor related to stress is generated. The constitutive equation, which represents the mechanical behavior of the material, relates this strain tensor and the stress tensor. Due to the memory effect of wood, this tensor has to be divided into two parts (1) an elastic strain, connected to the actual stress tensor and (2) a memory strain, which includes all the strain due to the history of that point (e can deal with plasticity, creep, mechanosorption, etc.). 

The stress field in the specimen being tested is related to the applied force. When a material is stressed, the deformations are controlled by what is known as the constitutive behavior of the material. For example, some materials respond to stress linear elastically, and others behave elasto-plastically. The linear elastic stress-strain relationship is called Hooke s Law. AE however, are more strongly dependent on the irreversible (nonelastic) deformations in a material. Therefore, this method is only capable of detecting the formation of new cracks and the progression of existing... 

The first stage in the calculation is to choose a constitutive equation that relates the applied stresses to the resulting strains. For an elastic material, the behavior is described by two independent elastic constants, such as the shear modulus G and the bulk modulus K. The constitutive equation for an isotropic linear elastic solid has the form (28)... 

In the isotropic solid, the unstressed state is chosen as the fixed reference configuration. One way to allow for fluid-like behavior is then to let the unstressed configuration evolve in time. VanArsdale [31] has developed constitutive equations for suspensions using this concept of elastic fluids. Working with the deformation tensor B =F F , VanArsdale uses a new deformation tensor b that is similar to B but which, instead, characterizes the deformation from the evolving unstressed state. The evolution is expressed by the relation for the lower convected... 

As a factor related to chain motion, one may also consider polymer elastic modulus. As indicated in Properties of Polymers (39), at one time it was hoped that mechanical studies of polymers could be entirely replaced by electrical measurements. There are indeed close similarities between the general shapes and temperature-dependences of the mechanical and dielectric loss curves, but the quantitative connection between these phenomena is not as simple as was origindly believed. Electrical measurements constitute a useful addition to, but not a substitute for, mechanical studies. However, the elastic modulus and the dielectric constant are related to similar physic behavior. These parameters are response functions obtained by stimulating a material and measuring the subsequent relaxation phenomena. Therefore, similar to the dielectric constant, the elastic modulus depends not only on chain motion but also on free space in the polymer matrix. In fact, relaxation phenomenon of... 

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