Linear elastic assumption

Procedures for performing valid fracture tests on polymers have been published [23] and besides containing details of specific fracture testing methods and procedures, criteria for the validity of the linear elastic assumption also are given. Some of the more common test geometries are shown in Figure 19.4. 

Both quantities K and G are only defined for linear elastic solids and unfortunately many materials are nonlinear and inelastic. Even in materials like metals, which are linear at low strain, pronounced plasticity often occurs around the crack tip on account of the high stresses developed there. In plastics and rubberlike materials, viscoelasticity also occurs, rendering linear elastic assumptions invalid. 

In moie ductile materials the assumptions of linear elastic fracture mechanics (LEFM) are not vahd because the material yields more at the crack tip, so that... 

The use of the single parameter, K, to define the stress field at the crack tip is justified for brittle materials, but its extension to ductile materials is based on the assumption that although some plasticity may occur at the tip the surrounding linear elastic stress field is the controlling parameter. 

The basic assumptions of fracture mechanics are (1) that the material behaves as a linear elastic isotropic continuum and (2) the crack tip inelastic zone size is small with respect to all other dimensions. Here we will consider the limitations of using the term K = YOpos Ttato describe the mechanical driving force for crack extension of small cracks at values of stress that are high with respect to the elastic limit. 

In recent years impact testing of plastics has been rationalised to a certain extent by the use of fracture mechanics. The most successful results have been achieved by assuming that LEFM assumptions (bulk linear elastic behaviour and presence of sharp notch) apply during the Izod and Charpy testing of a plastic. 

Equations (10.23) and (10.24) hold for the /3-phase as well and could be inserted into Eqn. (10.22). The additivity of pt with respect to the elastic and electric potential is based on 1) the assumption of linear elastic theory (which is an approximation) and 2) the low energy density of the electric field (resulting from the low value of the absolute permittivity e0 = 8.8x10 12 C/Vm). In equilibrium, V/i, = 0 and A V, = df-pf = 0. Therefore, in an ionic system with uniform hydrostatic pressure, the explicit equilibrium condition reads Aa/fi=A)... 

Substantial work on the application of fracture mechanics techniques to plastics has occurred since the 1970s (215—222). This is based on eadier work on inorganic glasses, which showed that failure stress is proportional to the square root of the energy required to create the new surfaces as a crack grows and inversely with the square root of the crack size (223). For the use of linear elastic fracture mechanics in plastics, certain assumptions must be met (224) (/) the material is lineady elastic (2) the flaws within the material are sharp and (5) plane strain conditions apply in the crack front region. 

An analytical elastic membrane model was developed by Feng and Yang (1973) to model the compression of an inflated, non-linear elastic, spherical membrane between two parallel surfaces where the internal contents of the cell were taken to be a gas. This model was extended by Lardner and Pujara (1980) to represent the interior of the cell as an incompressible liquid. This latter assumption obviously makes the model more representative of biological cells. Importantly, this model also does not assume that the cell wall tensions are isotropic. The model is based on a choice of cell wall material constitutive relationships (e.g., linear-elastic, Mooney-Rivlin) and governing equations, which link the constitutive equations to the geometry of the cell during compression. 

Petrie and Ito (84) used numerical methods to analyze the dynamic deformation of axisymmetric cylindrical HDPE parisons and estimate final thickness. One of the early and important contributions to parison inflation simulation came from DeLorenzi et al. (85-89), who studied thermoforming and isothermal and nonisothermal parison inflation with both two- and three-dimensional formulation, using FEM with a hyperelastic, solidlike constitutive model. Hyperelastic constitutive models (i.e., models that account for the strains that go beyond the linear elastic into the nonlinear elastic region) were also used, among others, by Charrier (90) and by Marckmann et al. (91), who developed a three-dimensional dynamic FEM procedure using a nonlinear hyperelastic Mooney-Rivlin membrane, and who also used a viscoelastic model (92). However, as was pointed out by Laroche et al. (93), hyperelastic constitutive equations do not allow for time dependence and strain-rate dependence. Thus, their assumption of quasi-static equilibrium during parison inflation, and overpredicts stresses because they cannot account for stress relaxation furthermore, the solutions are prone to numerical instabilities. Hyperelastic models like viscoplastic models do allow for strain hardening, however, which is a very important element of the actual inflation process. 

An important role in the present model is played by the strongly non-linear elastic response of the rubber matrix that transmits the stress between the filler clusters. We refer here to an extended tube model of rubber elasticity, which is based on the following fundamental assumptions. The network chains in a highly entangled polymer network are heavily restricted in their fluctuations due to packing effects. This restriction is described by virtual tubes around the network chains that hinder the fluctuation. When the network elongates, these tubes deform non-affinely with a deformation exponent v=l/2. The tube radius in spatial direction p of the main axis system depends on the deformation ratio as follows ... 

In this context it has to be pointed out that in the original Dugdale model the material behavior is assumed to be linearly elastic and perfectly plastic the latter assumption leads to a uniform stress distribution in the plastic zone. This may be a simplified situation for many materials to model, however, the material behavior in the crack tip region where high inhomogeneous stresses and strains are acting is a rather complex task if nonlinear, rate-dependent effects in the continuum... 

The second approach, due to lrwin is to characterise the stress field surrounding a crack in a stressed body by a stress-field parameter K (the stress intensity factor ). Fracture is then supposed to occur when K achieves a critical value K - Although, like Griffith s equation, this formulation of fracture mechanics is based on the assumptions of linear elasticity, it is found to work quite effectively provided that inelastic deformations are limited to a small zone around the crack tip. Like, however, the critical parameter remains an empirical quantity it cannot be predicted or related explicitly to the hysical properties of the solid. Like,, K. is time and temperature de ndent. 

Linear elastic fracture mechanics (LEFM) describes the behaviour of sharp cracks in linear, perfectly elastic materials. Since polymers are neither linear nor elastic, the utility of the theory may, at first sight, seem doubtful. In fact, the deviations from the theoretical assumptions are such that quite minor modifications to the analysis produce a precise description of crack growth in polymers within the framework of the conventional theory. The considerable resources of the subject may thus be utilised in that testing experience on other materials may be employed, together with the available analytical work. 

Any errors incurred because of history effects will be of the order n, Le. less than about 10% This assumption greatly simplifies the analysis since the linear elastic equations may be employed with the elastic parameters replaced by time dependent forms where the time scale is appropriate to the particular circumstances. In general, Poisson s ratio, v, does not vary greatly and it is sufficient to use E(t) and Oc(if) together with a constant COD criteria, 8. The most convenient method is to write ... 

Recalling our linear elastic constitutive assumption, we may fiuther simplify this result to read... 

This analytical solution review is tractable only for very limited assumptions, such as homogeneity and linearly elastic behavior (not to mention excluding variations that are time- or temperature-dependent). The first deviation that must be examined is the elastic linearity assumption for polishing pads. Polymers, in general, show behavior that lies between that of an elastic solid and a viscous fluid. The term viscoelastic has been applied to this behavior. 

Most of the crack problems that have been solved are based on two-dimensional, linear elasticity (i.e., the infinitesimal or small strain theory for elasticity). Some three-dimensional problems have also been solved however, they are limited principally to axisymmetric cases. Complex variable techniques have served well in the solution of these problems. To gain a better appreciation of the problems of fracture and crack growth, it is important to understand the basic assumptions and ramifications that underlie the stress analysis of cracks. 

For larger strains, while maintaining the assumption that the material is linearly elastic, they showed that [141] ... 

The preceding equations provided a reasonable foundation for predicting DE behavior. Indeed the assumption that DEs behave electronically as variable parallel plate capacitors still holds however, the assumptions of small strains and linear elasticity limit the accuracy of this simple model. More advanced non-linear models have since been developed employing hyperelasticity models such as the Ogden model [144—147], Yeoh model [147, 148], Mooney-Rivlin model [145-146, 149, 150] and others (Fig. 1.11) [147, 151, 152]. Models taking into account the time-dependent viscoelastic nature of the elastomer films [148, 150, 151], the leakage current through the film [151], as well as mechanical hysteresis [153] have also been developed. 

The mean stresses are a fairly good approximation for thin-walled tubes where the variations through the wall are small. However, the range of applicability of the thin-waU assumption depends upon the material properties and geometry. In a linear elastic material, the variation in ag is less than 5% for rlh > 20. When the material is nonlinear or the deformation is large, the variations in stress can be more severe (see Figure 57.10). 

The fundamental principle on which fracture mechanics is based is that cracklike defects exist in all materials and that when critical conditions are attained at the crack tip, the crack will begin to propagate and the material will fracture. In linear elastic fracture mechanics (LEFM) the assumption is made that the material deforms elastically (i.e. is Hookean) at all times, thereby greatly simplifying definition of the elastic energy stored in the material prior to fracture. The most... 

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