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Über dieses Buch

Considerably simplified models of macroscopic material behavior, such as the idealization for metals of elastic-time independent plastic response with a yield (onset) criterion, have served the engineering profession well for many years. They are still basic to the design and analysis of most structural applications. In the need to use materials more effectively, there are circumstances where those traditional models are not adequate, and constitutive laws that are more physically realistic have to be employed. This is especially relevant to conditions where the inherent time dependence of inelastic deformations, referred to as "viscoplasticity", is pronounced such as at elevated temperatures and for high strain rates. Unified theories of elastic-viscoplastic material behavior, which are primarily applicable for metals and metallic alloys, combine all aspects of inelastic response into a set of time dependent equations with a single inelastic strain rate variable. For such theories, creep under constant stress, stress relaxation under constant strain, and stress-strain relations at constant rates are each special cases of a general formulation. Those equations mayor may not include a yield criterion, but models which do not separate a fully elastic region from the overall response could be considered "unified" in a more general sense. The theories have reached a level of development and maturity where they are being used in a number of sophisticated engineering applications. However, they have not yet become a standard method of material representation for general engineering practice.



1. Formulation of a Unified Constitutive Theory of Elastic-Viscoplastic Behavior

The mechanical behavior of materials is an essential component of technology which has received considerable attention over many years. Of particular interest is the response of materials to mechanical and thermal loadings, the influence of environmental factors, and the conditions and mechanisms of failure. The terms “constitutive equations” and “material modelling” are usually applied to the analytical representation of the material response characteristics prior to total failure. There have been considerable advances on this subject in recent years due to better understanding of the physics of deformation and the advent of efficient computational capability which enables solution of complicated equations.
Sol R. Bodner

2. Specific Applications

Initial interest in applying unified viscoplastic theories was in determining deformations and stresses in structural components at high temperatures subjected to steady and low frequency cyclic loadings. These problems originated in the operation of gas turbine engines and power generation plants. Strain rates were generally less that 1 sec−1 and could be as low as 10−7 sec−1. For these problems, the parameter D0 in the kinetic equation (8) of the B-P model was set to be 104 sec−1 and sets of material constants were generated on that basis. These applications usually involved high temperatures so thermal recovery of hardening was an important component of the evolution equations. A number of more recent applications of the B-P model were concerned with strain rates above 10 sec−1 and were therefore based on the higher value of D0=108 sec−1. These sets of material constants could also be used at lower strain rates making use of modern numerical techniques. Recovery of hardening is usually unimportant in applications at high strain rates so those parameters tend to be omitted in the determination of the high rate material constants.
Sol R. Bodner

3. Commentaries

At this stage, the B-P constitutive theory is well developed and provides a set of equations that adequately represents the main features of rate dependent inelastic behavior of metals and alloys over an extensive range of strain rates and temperatures. Relatively few material parameters appear in the equations and these could generally be related to specific response characteristics which indicates a satisfactory physical basis of the governing equations. As a consequence, the parameters have physical interpretations and in most cases their values can be obtained from a limited band of conventional test data such as stress-strain curves at constant strain rates. Techniques for parameter identification from such test data have been devised. The equations have been incorporated into finite element and finite difference computer programs with applications for a variety of loading and environmental conditions. They appear to be suitable for characterizing components of composite materials for which purpose the lack of a yield criterion and loading/unloading conditions makes them particularly suitable. Those properties also make the equations useful in problems of dynamic fracture mechanics and in formulating failure criteria for ductile materials. The emphasis on the use of load history dependent state variables in the basic equations with suitable evolution equations for those variables permits applications to problems involving complicated thermo-mechanical loading histories and to stability considerations.
Sol R. Bodner


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