Elastoplastic materials

V. I. Levitas, A. V. Idesman, E. Stein. Finite element simulation of martensitic phase transitions in elastoplastic materials. Int J Solids Struct 55 855, 1998. 

For most practical purposes, the onset of plastic deformation constitutes failure. In an axially loaded part, the yield point is known from testing (see Tables 2-15 through 2-18), and failure prediction is no problem. However, it is often necessary to use uniaxial tensile data to predict yielding due to a multidimensional state of stress. Many failure theories have been developed for this purpose. For elastoplastic materials (steel, aluminum, brass, etc.), the maximum distortion energy theory or von Mises theory is in general application. With this theory the components of stress are combined into a single effective stress, denoted as uniaxial yielding. Tlie ratio of the measure yield stress to the effective stress is known as the factor of safety. 

Most solids are subjected to permanent deformation or breakup once the applied stresses exceed a certain limit. Hence, most solid particles may be classified into two categories elastoplastic particles and elastic-brittle particles. Typical elastoplastic materials include metals and polymers, while typical elastic-brittle materials include coal, activated carbon, and ceramics. Materials that are elastoplastic at room temperature may become brittle at low temperatures and those that are brittle at room temperature may become plastic at high temperatures. 

Some plastic materials have different tensile and compressive characteristics. For example, polystyrene is tough under compressive load but very brittle in tension. However, for most elastoplastic materials, the stress-strain curves in compression are the same as in tension. Hence, the deformation properties of these materials in tension may also be applied to those in compression, which is of great interest to gas-solid flows. 

For elastoplastic materials, two stresses representing the yield strength and the tensile strength are important to consider. The yield strength is the stress at which appreciable... 

In contrast to the simplicity of elastic deformation, plastic deformation occurs in diverse ways. Figure 1.9 illustrates the stress-strain curves for two typical elastoplastic materials (hardened metal and polymer). Both materials show similar linear relationships between stress and strain for the elastic deformation (i.e., before yield strength) but quite different correlations in the yielding processes before fracture. 

Figure 1.9. Deformation of typical elastoplastic materials (hardened metal and polymer) in the stress-strain diagram (after Guy, 1976).

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)... 

Viscoplastic materials, therefore, share many features with elastoplastic materials but, in addition, exhibit a dependence on the rate of straining. Again, a decomposition of the total strain is convenient, this time into elastic and viscoplastic parts (i.e., e = where e and are the elastic and viscoplastic strains, respectively). In analogy with the elastoplastic case, the boundary of the elastic region may be specified in terms of a yield function. However, whereas the region outside the yield surface was inadmissible in the elastoplastic case, the stress is allowed to lie outside the yield surface in the viscoplastic one. Hence, straining beyond the yield point generally results in the creation of an excess (or extra) stress o that decays toward zero with time, typically as (38)... 

Thermoplastic polyurethanes (TPU) are a versatile family of elastoplastic materials characterized by outstanding toughness and abrasion resistance. These materials are prepared from three principal reactants, a difunctional polyol, a difunctional chain extender and a diisocyanate in accordance with the following reaction ... 

Telechelic polyTHF oligodiols (DuPont, BASF, Quacker Chem.) have found important applications, mostly as soft segments in multiblock elastoplastic materials such as polyurethanes (e.g. Spandex ) and polyesters (Hytrel ). 

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. 

With press formings a preheated sheet of elastoplastic material is placed between the positive and negative halves of a form, and the required shaping occurs when these halves are pressed together (Figure 36-5)... 

On an elastoplastic material the various adherence forces can be computed. If the contact is elastic, the... 

Overburden of the tunnel 100.0 m The rock mass is considered as the idealized elastoplastic material with the Mohr-Coulomb yield criterion, and all the materials of support system are treated as elastic material. The initial stress field of analysis zone is calculated as self-weight... 

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). 

Fig. 2.14 Schematic view of the stress-strain relationship for an elastoplastic material. The symbols Yp and yr denote the yield strain, the plastic strain and the failure strain, respectively. The same notation is used for the stress a.

Besides linear viscoelastic behavior, elastoplastic behavior is also often encountered for food products. In Fig. 2.14, the stress-strain behavior of an elastoplastic material is shown schematically. Because the stress-strain relationship is not linear and the strain does not recover if the yield stress is exceeded, the equations to describe this behavior are much more complicated. 

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. 

The fatigue life of a polymer is generally reduced by an increase in temperature (53), although this is not always true for elastoplastic materials. 

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. 

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