Inelastic response

Finally, several chapters are provided which summarize the applications of shock-compression techniques to the study of material properties, and which illustrate the multidisciplinary nature of shock-wave applications. These applications include the inelastic response of materials, usually resulting from the extreme impact loads produced by colliding bodies, but also resulting from intense radiation loading. 

In metals, inelastic deformation occurs at the crack tip, yielding a plastic zone. Smith [34] has argued that the elastic stress intensity factor is adequate to describe the crack tip field condition if the inelastic zone is limited in size compared with the near crack tip field, which is then assumed to dominate the crack tip inelastic response. He suggested that the inelastic zone be 1/5 of the size of the near crack tip elastic field (a/10). This restriction is in accordance with the generally accepted limitation on the maximum size of the plastic zone allowed in a valid fracture toughness test [35,36]. For the case of crack propagation, the minimum crack size for which continuum considerations hold should be at least 50 x (r ,J. 

In general, the predicted displacement using both LBNL s elastic and CEA s elasto-brittle (weakly inelastic) models are within the ranges of field measurements, except for very close to the drift wall. However, in a few individual anchors, displacement values are more than 50% larger than predicted by the elastic material behaviour. The increased displacement in these anchors may be explained by inelastic responses leading to a better agreement with the ubiquitous joint model (e.g. Anchor 4 in Figure 5a). 

Richeton, J., Ahzi, S., Vecchio, K. S., Jiang, F. C., and Makradi, A. (2007) Modeling and validation of the large deformation inelastic response of amorphous polymers over a wide range of temperatures and strain rates, Int. J. Solids Structures, 44, 7938-7954. 

The pioneering analytical solution by Eshelby [59], for an ellipsoidal inclusion embedded in an infinite elastic medium, has been extended to nonlinear cases in the literature. For example, the secant approach by Berveiller and Zaoui [63] and the self-consistent tangent method by HiU [64] and Hutchinson [65] are generalizations of this method for elastoplastic problems. The limitation of these analytical methods persists in their inability to simulate complex material stractures, which result in inelastic responses that are too stiff [62,66]. Also, accurate stress redistribution in an inelastic analysis cannot be captured by these models [67]. Several models have been developed to resolve these issues in the literature, such as the above-mentioned tangent [64,66,68,69], secant [63,70], and affine [67,71] methods. 

The well-established elastic-predictor/plastic-corrector return mapping algorithm can be utilized to obtain the inelastic responses of the microscale amorphous and crystalline phases. Here, we only outline the steps to be used. A detailed description of this solution algorithm can be foimd in References [103] to [105]. The return mapping technique is capable of handling both associative and nonassociative flow rules with variant tangent stiffnesses and results in a consistent solution approach [105]. It is noted that this algorithm is applicable to the material, intermediate, or spatial formulations. 

Fiber-based frame analysis is one of the most advanced methodologies to model the nonlinear behavior of beams and columns under combined axial and bending loads. The Mid-America Earthquake Center analysis environment ZEUS-NL (Elnashai et al., 2002), is a compntational tool for the analysis of two and three dimensional frames. In ZEUS-NL, elements capable of modeling material and geometric nonlinearity are available. The forces and moments at a section are obtained by integrating the inelastic responses of individnal fibers. The Eularian approach towards geometric nonlinearity is employed at the element level. Therefore, full account is taken of the spread of... 

Ordonez, D., Foti, D., Bozzo, L. (2002). Comparative study of the inelastic response of base isolated buildings. Earthquake Engineering Structural Dynamics, 32(1), 151-164. doi 10.1002/eqe.224... 

The range of material behavior considered next is broadened significantly by appeal to the notion of a plastic rate equation as a model for any possible physical mechanism of deformation that may be operative. The ideas will be developed for general states of stress, but will be applied primarily for the case of thin films in equi-biaxial tension. Constitutive relationships that serve as models for inelastic response of materials for a wide variety of physical mechanisms of deformation have been compiled by Frost and Ashby (1982). These constitutive equations are represented as scalar equations expressing the inelastic equivalent strain rate /3e in terms of the effective stress (Tm/ /3 and temperature T. These strain rate and stress measures are denoted by 7 and as by Frost and Ashby (1982), and the rate equations representing models of material behavior all take the form... 

Fig. 41. Response of YbPdjSij as measured with 5 = 12.5meV. This illustrates how the quasielastic response at higher temperatures changes to an inelastic response (top panel) at T=5 K. The hatched areas are due to both nuclear incoherent scattering and to phonons. The area that is not hatched between the fiill and dashed lines represents the scattering from the impurity phase. (From Weber et al. 1989b.)...

Certainly the clearest conclusion from the examples of this chapter is the total absence of sharp features in the inelastic response function of anomalous lanthanide and metallic actinide materials. This contrasts strongly with the sharp dispersionless crystal-field excitations observed in most lanthanide compounds, in which the exchange interactions are weak (fig, 2), and with the sharp spin-wave excitations found in systems with strong exchange interactions. In many of the early studies with neutron inelastic scattering, for example of the heavy lanthanides or transition metals and their compounds, the width of the excitations was never an issue. It was almost always limited by the instrumental resolution, although it should be stressed that this resolution is relatively poor compared to that obtained by optical techniques. However, the situation is completely different in the materials discussed in this chapter. Now the dominant factor is often the width indeed in some materials the width of the over-damped response function is almost the only remaining parameter with which to characterize the response. 

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