Plasticity of Pressure-Sensitive Materials

The modeling of the behavior of pressure-sensitive materials is embedded in the general continuum mechanics. The basic equations of continuum mechanics can be split into the material-independent and the material-dependent equations. The starting point is the introduction of the kinematics based on pure mathematical considerations. In addition, the velocities and the accelerations of the relevant kinematical variables are presented. The next section is devoted to the introduction of the action on the continuum and the inner reaction. Starting with such properties like forces and stresses finally the static equilibrium is stated. The last part of the material-independent equations is the introduction of the balances. Limiting our discussions by thermo-mechanical actions only, the balance of mass, momentum, moment of momentum, energy and entropy are deduced. The specific properties and features of the pressure-sensitive materials are presented in the next sections. Within this chapter the general ideas of material modeling (deductive approach) are given. Finally, some examples of special constitutive equations for incompressible and compressible materials are presented. These examples are mostly related to rubber-like materials.

Models for isotropic materials based on the equivalent stress concept are discussed. At first, so-called classical models which are useful in the case of absolutely brittle or ideal ductile materials are presented. Tests for basic stress states are suggested. At second, standard models describing the intermediate range between the absolutely brittle and ideal-ductile behavior are introduced. Any criterion is expressed by various mathematical equations formulated, for example, in terms of invariants. At the same time the criteria can be visualized which simplifies the application. At third, in the main part pressure-insensitive, pressure-sensitive and combined models are separated. Fitting methods based on mathematical, physical and geometrical criteria are necessary. Finally, three examples (gray cast iron, poly(oxymethylene) (POM) and poly(vinyl chloride) (PVC) hard foam) demonstrates the application of different approaches in modeling certain limit behavior. Two appendices are necessary for a better understanding of this chapter: in Chap. 2 applied invariants are briefly introduced and a table of discussed in this chapter criteria with references is given.

Cellular metals, e.g., made by solidification of molten metal foam, have interesting mechanical properties, among them high specific strength and stiffness coupled with inflammability and good damping properties. This makes them interesting for engineering applications which require the prediction of the onset of yielding under multi-axial stress states and the development of plastic strains over a strain range that may extend into the regime of full compaction of the foam micro-structure, as it is the case in applications for crash protection. This chapter investigates the micro-mechanical deformation mechanisms which govern the elasto-plastic behavior of cellular metals on the macro-mechanical level, where the cellular structure can be treated as a homogeneous material if the difference between the cell size and the component size is large enough. If this is the case suitable constitutive models can be applied for predicting the onset of macroscopic yielding, the evolution of plastic strains and the hardening behavior. Thus, a review of the most important material models proposed for simulating the effective elasto-plastic behavior of isotropic cellular metals is presented. This behavior is characterized by a distinct pressure sensitivity, which sets apart the behavior of cellular metals from the one of solid metals as described by classical (e.g., von Mises) theory of plasticity.

A thin soft elasto-plastic interphase between two different media is under consideration. The intermediate layer is assumed to be of infinitesimal thickness and is modeled by nonlinear transmission conditions which incorporate the elasto-plastic material behavior of the layer. The case of pressure-independent (von Mises) as well as pressure-dependent yield condition is theoretically treated. Finite element analysis of a bimaterial structure with such an imperfect elasto-plastic interface (von Mises) shows the efficiency of the approach and illustrates some restrictions of its application.

In the case of several rigid plastic models, the equivalent strain rate (quadratic invariant of the strain rate tensor) approaches infinity in the vicinity of maximum friction surfaces. The strain rate intensity factor is the coefficient of the leading singular term in a series expansion of the equivalent strain rate in the vicinity of such surfaces. This coefficient controls the magnitude of the equivalent strain rate in a narrow material layer near maximum friction surfaces. On the other hand, the equivalent strain rate is involved in many conventional equations describing the evolution of parameters characterizing material properties. Experimental data show that a narrow layer in which material properties are quite different from those in the bulk often appears in the vicinity of surfaces with high friction in metal forming processes. This experimental fact is in qualitative agreement with the aforementioned evolution equations involving the equivalent strain rate. However, when the maximum friction law is adopted, direct use of such equations is impossible since the equivalent strain rate in singular. A possible way to overcome this difficulty is to develop a new type of evolution equations involving the strain rate intensity factor instead of the equivalent strain rate. This approach is somewhat similar to the conventional approach in the mechanics of cracks when fracture criteria from the strength of materials are replaced with criteria based on the stress intensity factor in the vicinity of crack tips. The development of the new approach requires a special experimental program to establish relations between the magnitude of the strain rate intensity factor and the evolution of material properties in a narrow material layer near surfaces with high friction as well as a theoretical method to deal with singular solutions for rigid plastic solids. Since no numerical method has been yet developed to determine the strain rate intensity factor, the present chapter focuses on analytical and semi-analytical solutions from which the dependence of the strain rate intensity factor on process and material parameters are found. In particular, the effect of pressure-dependency of the yield criterion on the strain rate intensity factor is emphasized using the double shearing model.

In this chapter the formulation for damage known as Gurson model is presented. The original formulation, set in a micro-mechanical context, and different adjustments of phenomenological nature are described. The range of the parameters of the model and their influence on the representation are described. The main computational details for the implementation of the model by means of the finite element method are presented and examples of application are given.