Linear-elastic results

Some linear-elastic results were obtained for direct comparison with experimental values of mechanical properties available in the literature. The materials were rubber-toughened epoxy resin (5) and epoxy resin filled with glass beads (22). The material properties used for these analyses are shown in Table II. 

This phenomenon, typical to viscoelastic response under reverse inputs, yields the representative results shown in Fig. 6.17. Note that the comparative linear elastic results tend toward a horizontal asymptopic value of approximately —0.075 during absorption and recover toward zero following desorption. 

Of course, the above independence takes place provided that / = 0 in the domain with the boundary C. The integral of the form (4.100) is called the Rice-Cherepanov integral. We have to note that the statement obtained is proved for nonlinear boundary conditions (4.91). This statement is similar to the well-known result in the linear elasticity theory with linear boundary conditions prescribed on S (see Bui, Ehrlacher, 1997 Rice, 1968 Rice, Drucker, 1967 Parton, Morozov, 1985 Destuynder, Jaoua, 1981). 

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. 

Note that the lamina failure criterion was not mentioned explicitly in the discussion of Figure 4-36. The entire procedure for strength analysis is independent of the actual lamina failure criterion, but the results of the procedure, the maximum loads and deformations, do depend on the specific lamina failure criterion. Also, the load-deformation behavior is piecewise linear because of the restriction to linear elastic behavior of each lamina. The laminate behavior would be piecewise nonlinear if the laminae behaved in a nonlinear elastic manner. At any rate, the overall behavior of the laminate is nonlinear if one or more laminae fail prior to gross failure of the laminate. In Section 2.9, the Tsai-Hill lamina failure criterion was determined to be the best practical representation of failure... 

Composite 1 with the higher fiber/matrix bonding shows a linear-elastic behavior up to = 160 MPa, which is clearly higher than that of composite 2. The following nonlinear region is small, resulting in an elongation at fracture of 0.32 %. 

The above results are derived from linear elastic fracture mechanics and are strictly valid for ideally brittle materials with the limit of the process zone size going to zero. In order to apply this simple framework of results, Irwin (1957) proposed that the process zone, r be treated as an effective increase in crack length, Sc. With this modification, the fracture toughness becomes... 

In graphic presentation of Kk results, the error bars given for the control are typical of all those data points which do not have their own error bars. In cases where error exceeded 10%, individual error bars are provided and labelled with the corresponding symbol. Such large deviations are thought to result from the violation of the homogeneity criterion of linear elastic fracture mechanics at 15% of certain oligomers. (See, for example, Fig. 7). 

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)... 

These results indicate that in the present linear elastic model, the limiting velocity for the screw dislocation will be the speed of sound as propagated by a shear wave. Even though the linear model will break down as the speed of sound is approached, it is customary to consider c as the limiting velocity and to take the relativistic behavior as a useful indication of the behavior of the dislocation as v — c. It is noted that according to Eq. 11.20, relativistic effects become important only when v approaches c rather closely. 

The purpose of this chapter is to remind the reader of the basis of the theory of elasticity, to outline some of its principal results and to discuss to what extent the classical theory can be applied to polymeric systems. We shall begin by reviewing the definitions of stress and strain and the compliance and stiffness matrices for linear elastic bodies at small strains. We shall then state several important exact solutions of these equations under idealised loading conditions and briefly discuss the changes introduced if realistic loading conditions are considered. We shall then move on to a discussion of viscoelasticity and its application to real materials. 

Although a key characteristic of the mechanical behavior of rubber-like materials is their ability to undergo large elastic deformations, we will present here some important results from the theory of linear elasticity [1], which is valid only for small deformations. These serve our present purposes better than the nonlinear theory, because of their simpler character and physical transparency. 

According to Eqs. (13.145) and (13.148) the fracture stress in plane strain is a factor 1 /(1-v2) 1 /0.84 1.2 higher than in plane stress. Experimentally, however, the difference is much bigger. The reason for this discrepancy is that Griffith s equations were developed in linear fracture mechanics, which is based on the results of linear elasticity theory where the strains are supposed to be infinitesimal and proportional to the stress. 

In linear elasticity or viscoelasticity, the superposition principle states that the resulting effects of the different causes (stress or displacements), acting separately, can be superposed to give the total values due to these combined causes. This principle is a consequence of the linearity of the equations governing the stress, strain, and displacements. 

For linear elastic materials, deflection at the load-point of a four-point bending beam is given by two different contributions, accounting for both bending and shear deformation. In the case of notched specimens, a third term accounting for crack length arises. The resulting analytical expression for the specimen compliance C is as follows ... 

Ki = 0.23 MPaVm according to equation 2 for a = 2.2MPa and notch size = a = 2.5 mm. This results in a yielded strip size of 0.22 mm according to equation 3. This value, of namely 0.14 mm, is much higher than the measured band size. Possible explanations for this discrepancy are the fact that the material is not behaving in a linear elastic manner and/or that only a part of the craze/yielded zone breaks at once. 

Previous work pursued the model analytically, for a linearly elastic [5] (or, later, non-linearly elastic [6]) material with constant thermal properties. The analytical model explained several measured fracture properties of thermoplastics the magnitude of impact fracture toughness and its dependence on impact speed [7] and the absolute magnitude of resistance to rapid crack propagation [8]. Recent results have shown that the impact fracture properties of some amorphous and crosslinked polymers show the same rate dependence [11],... 

The commercial finite element program, Abaqus [17], was used to calculate the stress distribution in an edge delamination sample. A fully three-dimensional model of the combinatorial edge delamination specimen was constructed for the finite element analyses (FEA). For clarity, some of the FEA results and schematics are presented as two-dimensional configurations in this paper (e.g.. Fig. 1). The film and substrate were assumed to be linearly elastic. The ratio of the film stiffness to the substrate stiffness was assumed to be 1/100 to reflect the relative rigidity of the substrate. This ratio also represents a typical organic... 

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