Elastic strain/stress mathematics

Elastic Strain and Stress in a Crystallite - Mathematical Background... 

Leonov model This constitutive equation is based on irreversible thermodynamic arguments resulting from the theory of rubber elasticity [5]. Mathematically it relates the stress to the elastic strain stored in the polymer melt as ... 

Table 10-56 gives values for the modulus of elasticity for nonmetals however, no specific stress-limiting criteria or methods of stress analysis are presented. Stress-strain behavior of most nonmetals differs considerably from that of metals and is less well-defined for mathematic analysis. The piping system should be designed and laid out so that flexural stresses resulting from displacement due to expansion, contraction, and other movement are minimized. This concept requires special attention to supports, terminals, and other restraints. 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Actually, our assumption about the way in which the plate material relaxes is obviously rather crude, and a rigorous mathematical solution of the elastic stresses and strains around the crack indicates that our estimate of 81i is too low by exactly a factor of 2. Thus, correctly, we have... 

Poison s ratio It is the proportion of lateral strain to longitudinal strain under conditions of uniform longitudinal stress within the proportional or elastic limit. When the material s deformation is within the elastic range it results in a lateral to longitudinal strain that will always be constant. In mathematical terms, Poisson s ratio is the diameter of the test specimen before and after elongation divided by the length of the specimen before and after elongation. Poisson s ratio will have more than one value if the material is not isotropic... 

A8. The Helmholtz elastic free energy relation of the composite network contains a separate term for each of the two networks as in eq. 5. However, the precise mathematical form of the strain dependence is not critical at small deformations. Although all the assumptions seem to be reasonably fulfilled, a simpler method, which would require fewer assumptions, would obviously be desirable. A simpler method can be used if we just want to compare the equilibrium contribution from chain engangling in the cross-linked polymer to the stress-relaxation modulus of the uncross-linked polymer. The new method is described in Part 3. 

In this introduction, the viscoelastic properties of polymers are represented as the summation of mechanical analog responses to applied stress. This discussion is thus only intended to be very introductory. Any in-depth discussion of polymer viscoelasticity involves the use of tensors, and this high-level mathematics topic is beyond the scope of what will be presented in this book. Earlier in the chapter the concept of elastic and viscous properties of polymers was briefly introduced. A purely viscous response can be represented by a mechanical dash pot, as shown in Fig. 3.10(a). This purely viscous response is normally the response of interest in routine extruder calculations. For those familiar with the suspension of an automobile, this would represent the shock absorber in the front suspension. If a stress is applied to this element it will continue to elongate as long as the stress is applied. When the stress is removed there will be no recovery in the strain that has occurred. The next mechanical element is the spring (Fig. 3.10[b]), and it represents a purely elastic response of the polymer. If a stress is applied to this element, the element will elongate until the strain and the force are in equilibrium with the stress, and then the element will remain at that strain until the stress is removed. The strain is inversely proportional to the spring modulus. The initial strain and the total strain recovery upon removal of the stress are considered to be instantaneous. 

Atomic force microscopy and attenuated total reflection infrared spectroscopy were used to study the changes occurring in the micromorphology of a single strut of flexible polyurethane foam. A mathematical model of the deformation and orientation in the rubbery phase, but which takes account of the harder domains, is presented which may be successfully used to predict the shapes of the stress-strain curves for solid polyurethane elastomers with different hard phase contents. It may also be used for low density polyethylene at different temperatures. Yield and rubber crosslink density are given as explanations of departure from ideal elastic behaviour. 17 refs. 

The TA Instruments CSL2 rheometer can perform low frequency oscillatory measurements as well as steady-state viscosity determinations, even though it has a simple mechanical system. The sinusoidal wave form is generated mathematically in the computer rather than with an electromechanical drive system. The stress is controlled, and the resulting strain is determined and stored in memory. The computer analyzes the wave form and calculates the viscosity and elasticity of the specimen at the frequency of the test. As of this writing (1996), the oscillation software covers a frequency range of 10-4 -40 Hz. This range could be increased as faster software and computers become available. 

This consists of experimental measurements of stress-strain relations and analysis of the data in terms of the mathematical theory of elastic continua. Rivlin7-10 was the first to pply the finite (or large) deformation theory to the phenomenologic analysis of rubber elasticity. He correctly pointed out the above-mentioned restrictions on W, and proposed an empirical form... 

The mathematical relationship between the stress and the strain depends on material properties, temperature, and the rate of deformation. Many materials such as metals, ceramics, crystalline polymers, and wood behave elastically at small stresses. For tensile elastic deformation, the linear relation between the stress, a, and strain, e, is described by Hooke s law as... 

Therefore under a constant stress, the modeled material will instantaneously deform to some strain, which is the elastic portion of the strain, and after that it will continue to deform and asynptotically approach a steady-state strain. This last portion is the viscous part of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions and is rather difficult to calculate. 

A mathematical model was developed by Smith et al. (1998) to extract data on the elastic Young s modulus and some criteria of failure of stationary phase Baker s yeast from compression testing data. A mean Young s modulus of 150+ 15 MPa with a corresponding mean von Mises strain at failure of 0.75 + 0.08 and a mean von Mises stress at failure of 70+ 4 MPa (Smith et al., 2000b) were obtained. 

Transport phenomena modeling. This type of modeling is applicable when the process is well understood and quantification is possible using physical laws such as the heat, momentum, or diffusion transport equations or others. These cases can be analyzed with principles of transport phenomena and the laws governing the physicochemical changes of matter. Transport phenomena models apply to many cases of heat conduction or mass diffusion or to the flow of fluids under laminar flow conditions. Equivalent principles can be used for other problems, such as the mathematical theory of elasticity for the analysis of mechanical, thermal, or pressure stress and strain in beams, plates, or solids. 

If we now perform a creep experiment, applying a constant stress, a0 at time t = 0 and removing it after a time f, then the strain/ time plot shown at the top of Figure 13-89 is obtained. First, the elastic component of the model (spring) deforms instantaneously a certain amount, then the viscous component (dashpot) deforms linearly with time. When the stress is removed only the elastic part of the deformation is regained. Mathematically, we can take Maxwell s equation (Equation 13-85) and impose the creep experiment condition of constant stress da/dt = 0, which gives us Equation 13-84. In other words, the Maxwell model predicts that creep should be constant with time, which it isn t Creep is characterized by a retarded elastic response. 

Stress-Strain Relations for Viscoelastic Materials. The viscoelastic behaviour of an elastomer varies with temperature, pressure, and rate of strain. This elastic behaviour varies when stresses are repeatedly reversed. Hence any single mathematical model can only be expected to approximate the elastic behaviour of actual substances under limited conditions 2J. ... 

In order to better understand the shear modulus (and the time dependence of the shear modulus) we shall describe it mathematically. Elastic behavior is the instantaneous response to a stress as shown in Figure 4.11. When a stress is imposed, the material deforms instantaneously. When the stress is removed the material relaxes completely. The amount of deformation (strain) is related to the stress (pressure) by the shear modulus according to ... 

Finally, the reader should appreciate a significant difference between the way in which a and r were introduced. In Section 1.2.1 the strain tensor was defined on purely mathematical grounds, whereas the conjugate stress tensor was introduced by purely physical reasoning (i.e., force balance). However, both elastic body arc explicitly excluded. As far as rr is concerned, this is effected by introducing the displacement of mass elements (r) as a vector field and by defining displacement tensor V (r) (see Ekjs. (1.1) and (1-6)] for the stress tensor r, the symmetry property stated in Eki. (1-14) serves to eliminate rotations. 

In mathematical terms, the implementation of the eigenstrain concept is carried out by representing the geometrical state of interest by some distribution of stress-free strains which can be mapped onto an equivalent set of body forces, the solution for which can be obtained using the relevant Green function. To illustrate the problem with a concrete example, we follow Eshelby in thinking of an elastic inclusion. The key point is that this inclusion is subject to some internal strain e, such as arises from a structural transformation, and for which there is no associated stress. 

Linear elastic fracture mechanics (LEFM) has been used successfully for characterization of the toughness of brittle materials. The driving force of the crack advance is described by the parameters such as the stress intensity factor (K) and the strain energy release rate (G). Unstable crack propagates when the energy stored in the sample is larger than the work required for creation of two fracture surfaces. Thus, fracmre occurs when the strain energy release rate exceeds the critical value. Mathematically, it can be written as... 

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