Small Strain Material Functions

Sections 3.3.1-3.3.3 describe each of these small strain material functions and show typical data for several theologically dif-... 

It is possible to define other small strain material functions, such as stress growth under constant rate of straining (Example 3.2.1) or recoverable strain after constant strain rate. However, these deformation histories are better suited for large strain studies and are discussed in Section 3.2. The small strain material functions will be seen as limits of the large strain ones. Table 3.3.2 lists some of the interrelations between the various experiments for linear viscoelastic behavior. Note that the limiting low shear rate viscosity rio can be calculated from... 

Preferred geometry for viscous melts for small strain material functions... 

By plotting Hugoniot curves in the pressure-particle velocity plane (P-u diagrams), a number of interactions between surfaces, shocks, and rarefactions were solved graphically. Also, the equation for entropy on the Hugoniot was expanded in terms of specific volume to show that the Hugoniot and isentrope for a material is the same in the limit of small strains. Finally, the Riemann function was derived and used to define the Riemann Invarient. 

The critical gel equation is expected to predict material functions in any small-strain viscoelastic experiment. The definition of small varies from material to material. Venkataraman and Winter [71] explored the strain limit for crosslinking polydimethylsiloxanes and found an upper shear strain of about 2, beyond which the gel started to rupture. For percolating suspensions and physical gels which form a stiff skeleton structure, this strain limit would be orders of magnitude smaller. 

The above equations are generally valid for any isotropic material, including critical gels, as long as the strain amplitude y0 is sufficiently small. The material is completely characterized by the relaxation function G(t) and, in case of a solid, an additional equilibrium modulus Ge. 

When using the generalised Hooke s law strain energy function there are a number of possible strain definitions that can be used depending on the situation. When material deformation is very small the infinitesimal strain approach is a valid approximation with the strain defined as... 

In the example given, the constitutive equation used is a multimode Phan Tien Tanner (PTT). It requires the evaluation of both linear and nonlinear material-response parameters. The linear parameters are a sufficient number of the discrete relaxation spectrum 2, and r]i pairs, which are evaluated from small-strain dynamic experiments. The values of the two nonlinear material-response parameters are evaluated as follows. Three semiarbitrary initial values of the two nonlinear parameters are chosen and the principal normal stress difference field is calculated for each of them using the equation of motion and the multimode PTT. They are compared at each field point (i, j) to the experimentally obtained normal stress difference and used in the following cost function F... 

Equations 1.2 to 1.4 represent material functions under large deformations (e.g., continuous shear of a fluid). One may recall a simple experiment in an introductory physics course where a stress (a) is applied to a rod of length Z, in a tension mode and that results in a small deformation AL. The linear relationship between stress (ct) and strain (j/) (also relative deformation, y = AL/L) is used to define the Young s modulus of elasticity E (Pa) ... 

Elastic materials For elastic materials, the stress and strain are related by a unique functional relationship, which may be linear (linearly elastic material) or nonlinear (hyperelastic materials). The function between stress and strain may generally be assumed to be linear for small strains, and in the one-dimensional case, one obtains Hooke s law (37,38)... 

Elastoplastic materials Elastoplastic materials deform elastically for small strains, but start to deform plastically (permanently) for larger ones. In the small-strain regime, this behavior may be captured by writing the total strain as the sum of elastic and plastic parts (i.e., e = e -I- gP, where e and gP are the elastic and plastic strains, respectively). The stress in the material is generally assumed to depend on the elastic strain only (not on the plastic strain or the strain rate), and hence, no unique functional relationship exists between stress and strain. This fact also implies that energy is dissipated during plastic deformation. The point at which the material starts to deform plastically (the yield locus) is usually specified via a yield condition, which for one-dimensional plasticity may be stated as (38)... 

The crack-driving force G may be estimated from energy considerations. Consider an arbitrarily shaped body containing a crack, with area A, loaded in tension by a force P applied in a direction perpendicular to the crack plane as illustrated in Fig. 2.6. For simplicity, the body is assumed to be pinned at the opposite end. Under load, the stresses in the body will be elastic, except in a small zone near the crack tip i.e., in the crack-tip plastic zone). If the zone of plastic deformation is small relative to the size of the crack and the dimensions of the body, a linear elastic analysis may be justihed as being a good approximation. The stressed body, then, may be characterized by an elastic strain energy function U that depends on the load P and the crack area A i.e., U = U(P, A)), and the elastic constants of the material. 

Boltzmann superposition principle A basis for the description of all linear viscoelastic phenomena. No such theor) is available to serve as a basis for the interpretation of nonlinear phenomena—to describe flows in which neither the strain nor the strain rate is small. As a result, no general valid formula exists for calculating values for one material function on the basis of experimental data from another. However, limited theories have been developed. See kinetic theory viscoelasticity, nonlinear, bomb See plasticator safety. 

Qualitatively, the stress created by imposing oscillatory strains will be a function of the amplitude of strain, its frequency, and the properties of the polymeric solution. By applying sufficiently small strains, we can minimize the strain dependency from the material response. Thus, the amplitude of strain is no longer an important consideration, resulting in a linear viscoelastic response. Small strain measurements can be important for certain application as they do not adversely affect the structure of the fluid. 

Considerable success has also been achieved in fitting the observed elastic behavior of rubbers by strain energy functions that are formulated directly in terms of the extension ratios Xi, X2, X2, instead of in terms of the strain invariants /i, I2 [22]. Although experimental results can be described economically and accurately in this way, the functions employed are empirical and the numerical parameters used as fitting constants do not appear to have any direct physical significance in terms of the molecular structure of the material. On the other hand, the molecular elasticity theory, supplemented by a simple non-Gaussian term whose molecular origin is in principle within reach, seems able to account for the observed behavior at small and moderate strains with comparable success. 

Dynamic analysis is generally used to study the linear viscoelastic properties of polymers. The region of linear viscoelastic behavior is where a material function, such as shear modulus or shear viscosity, is independent of the amplitude of the strain or strain rate. Polymers follow linear viscoelastic behavior when the strain or strain rate is sufficiently small. Thus, if the strain amplitude is sufficiently small, the shear stress can be written as ... 

DMA experiments are performed under conditions of very small strain so that the material response is in the linear viscoelastic range. This means that the magnitude of stress and strain are linearly related and the deformation behavior is completely described by the complex modulus function, which is a function of time only. The theory applies both for the case of a tensile deformation or simple extension and for shear. In the latter case the comparable modulus is with components G ico) and G" co). As a first-order approximation, E = 3G. The theory is developed assuming deformation under isothermal conditions, and temperature does not appear (nor is implicit) as a variable. 

The DDM algorithm for a three-field mixed formulation based on the Hu-Washizu functional (Washizu 1975) has been derived and presented elsewhere (Barbato et al. 2007). This formulation stems from the differentiation of basic principles (equilibrium, compatibility and material constitutive equations), applies to both material and geometric nonlinearities, is valid for both quasi-static and dynamic FE analysis and considers material, geometric and loading sensitivity parameters. This general formulation has also been specialized to frame elements and linear geometry (small displacements and small strains). 

In the course of extensive studies of the creep and recovery behaviour of textile fibres already referred to, Leaderman [13] became one of the first to appreciate that the simple assumptions of linear viscoelasticity might not hold even at small strains. For nylon and cellulosic fibres he discovered that although the creep and recovery curves may be coincident at a given level of stress - a phenomenon associated with linear viscoelasticity (Section 4.2.1) the creep compliance plots indicated a softening of the material as stress increased, except at the shortest times (Figure 10.4). Thus, the creep compliance function is non-linear and of the... 

Schapery [16, 17] has used the theory of the thermodynamics of irreversible processes to produce a model that may be viewed as a further extension of Leaderman s. Schapery continued Leaderman s technique of replacing the stress by a function of stress /(a) in the superposition integral, but also replaced time by a function of time, the reduced time ip. The material is assumed to be linear viscoelastic at small strains, with a creep compliance function of the form [17]... 

The complex modulus is a characteristic property of the material that changes only when the material changes. It is a function of time only, since DMA experiments are performed under conditions of very small strain. Under these... 

The cone and plate viscometer can be used for oscillatory shear measurements as well. In this case, the sample is deformed by an oscillatory driver which may be mechanical or electromagnetic. The amplitude of the sinusoidal deformation is measured by a strain transducer. The force deforming the sample is measured by the small deformation of a relatively rigid spring or tension bar to which is attached a stress transducer. On account of the energy dissipated by the viscoelastic polymer system, a phase difference develops between the stress and the strain. The complex viscosity behavior is determined from the amplitudes of stress and strain and the phase angle between them. The results are usually interpreted in terms of the material functions, p, G, G" and others [33-40]. 

For a viscoelastic liquid in the creep phase, the strain begins at some small value, then builds up rapidly but at a decreasing rate until finally reaching a steady state at which strain simply increases linearly with time. In the recovery phase, a viscoelastic liquid recoils back toward zero and finally reaches equilibrium at some smaller total strain than that at the time of unloading. The response over the entire range of time is linear if the shear stress is chosen small enough. In the linear range, the ratio of shear strain to shear stress in the creep phase is a material function of time alone, the shear creep compliance J t) ... 

Kontou and Spathis [44] carried out an investigation into the relationship between long-term viscoelasticity and viscoplastic responses of two types of ethylene-vinyl acetate metallocene-catalysed linear low-density polyethylene using DSC, DMTA and tensile testing. A relaxation modulus function with respect to time was obtained from values of relaxation spectra and treated as a material property. This relaxation modulus function was used to describe the corresponding tensile data and a constitutive analysis, which accounts for the viscoelastic path at small strains and the viscoplastic path at high strains, was employed to predict the tensile behaviour of the ethylene polymers (see also [45 9]). 

To return linear viscoelasticity, it is required that g(e) approaches unity for small strain. The stress-strain data for Smith s SBR vulcanisate rubber material are plotted in Figure 11.3(a). Log stress against log time plots were obtained for fixed strains and, as shown in Figure 11.3(b), form parallel linear relationships. This suggests via Equation (11.7) that the quantity g(e)/e is independent of time. It was found that for extension ratios up to 2, g(e) = 1 provided that a is understood to denote the true stress. At higher strains, the empirical function... 

Although difficult, it is possible to measure stress vs. strain curves of PSAs. Examples of such work include that of Christenson et al. [.3J and Piau et al. [23J. One can do this at various elongation rates and temperatures and create a material response function. Of course, it is much easier to obtain rheological data at small strains than to obtain tensile stress-strain data. One can assume a shape of the stress vs. strain function (i.e. a constitutive relationship) and then use the small strain data to assign values to the parameters in such a function. In order for a predictive model of peel to be useful, one should be able to use readily obtained rheological parameters like those obtained from linear viscoelastic master curve measurements and predict peel force master curves. 

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