Material models elastoplastic materials

Fig. 7. (a) Load-displacement curve of a typical elastoplastic material and (b) the schematic of the indentation model of Oliver and Pharr [40]. S—contact stiffness he— contact depth /imax—indenter displacement at peak load hf—plastic deformation after load removal hs—displacement of the surface at the perimeter of the contact. 

Nonlinear models of rheological behavior can be approximated by step functions, whereby the existence of a finite yield stress G plays a dominant role. Three typical nonlinear models include the Saint-Venant model of ideal plastic behavior, the Prandtl-Reuss model of an elastoplastic material, and the Bingham model of viscoelastic behavior. The first model can be mechanically approximated by a sliding block, the second by a Maxwell element and a sliding block in series, and the third by a dash pot damping element and a sliding block in parallel (Figure 2.14). 

In the Dugdale model the stresses in the cohesion zone are assumed to have a constant value CTq over a length d, so that at equilibrium G = where bj is the crack opening displacement (COD) beyond which interactions disappear. This model is used for elastoplastic materials where CTq is the yield stress and d the plastic zone length, and for polymers where ag is the craze stress and d the craze length it can also be applied to liquid bridges where (Jq is the Laplace pressure (see Section 6). 

Of course, the micromechanical relations (bounds, approximations or fit models) presented in this chapter for the effective elastic properties are by no means restricted to the alumina-zirconia system but can be applied to many types of ceramies and eeramie composites. On the other hand they cannot be expected to be automatically applicable to matrix-inclusion type composites in cases where the matrix consists of nonlinearly elastic materials (polymers), viscoelastic materials (glasses or porcelain at high temperature) or elastoplastic materials (metals). In particular, they cannot be a priori expected to be justified for materials of biological origin, although their application to many of these materials, e g. bone, might be seductive and dictated by practical needs. With respect to the inherent anisotropy and the hierarchical microstructure of these materials [Ontanon et al. 2000], however, any mathematical modeling or description of their composition-structure-property relationships has to be performed with due caution. 

In the linear elastic region, constitutive material models are able to correlate strain to stress without the need to consider the history of stress and strain events a specific object or assembly has been subjected to previously. If an elastoplastic material is subjected to mechanical stress it will also respond with instantaneous elastic deformation but above the yield strength deform plastically at a deformation rate governed by the process of deformation. Again, the behavior is not considered to be time dependent. 

The demonstration of the validity of the continuum-based modelling approach to tablet compaction requires familiarity with fundamental concepts of applied mechanics. Under the theory of such a mechanism, powder compaction can be viewed as a forming event during which large irrecoverable deformation takes place as the state of the material changes from loose packing to near full density. Moreover, it is important to define the three components of the elastoplastic constitutive models which arose from the growing theory of plasticity, that is the deformation of materials such as powder within a die ... 

For the yield stress in compression, deviations from Tabor s relation giving values of 2Yc are found. This is presumably due to the elastic strain of the indented material. A detailed analysis of the H/Yc ratio on the basis of mechanical models of elastoplastic indentation reveals that H/Yc linearly increases with ln[(tan/3Ec)/Yc]. Compression-moulded (chain-folded) PE samples, which present the lowest crystallinity of all the samples investigated, also show the lowest H/Yc ratio as a consequence of the comparatively large elastic strains. 

Cambou B., Jafari k., Elamrani K., 1989 An elastoplastic model for granular material using tree yielding mechanisms. Num. Models in geomech. Numog III. Ed Pretruszizak, Pande-Elsevier, pp. 1-8. 

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. 

Lemos JV (2007) Discrete element modeling of masonry structures. Int J Archit Herit 1(2) 190-213 Lourenfo PB (1996) A matrix formulation for the elastoplastic homogenisation of layered materials. Mech Cohes-Frict Mater 1 273-294 Lourenfo PB (2000) Anisotropic sohening model for masonry plates and shells. J Struct Eng 126(9) 1008-1016... 

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