A Comprehensive Theory of Yielding and Failure for Isotropic Materials

A theory of yielding and failure for homogeneous and isotropic materials is given. The theory is calibrated by two independent, measurable properties and from those it predicts possible failure for any given state of stress. It also differentiates between ductile yielding and brittle failure. The explicit ductile-brittle criterion depends not only upon the material specification through the two properties, but also and equally importantly depends upon the type of imposed stress state. The Mises criterion is a special (limiting) case of the present theory. A close examination of this case shows that the Mises material idealization does not necessarily imply ductile behavior under all conditions, only under most conditions. When the first invariant of the yield/failure stress state is sufficiently large relative to the distortional part, brittle failure will be expected to occur. For general material types, it is shown that it is possible to have a state of spreading plastic flow, but as the elastic-plastic boundary advances, the conditions for yielding on it can change over to conditions for brittle failure because of the evolving stress state. The general theory is of a three dimensional form and it applies to full density materials for which the yield/failure strength in uniaxial tension is less than or at most equal to the magnitude of that in uniaxial compression. Introduction and Objective The failure of materials generates research at all length scales from the electronic state to the atomic scale to nano to micro and on to macro (macroscopic) scales. The resulting information is considerably enhanced when the effects at the various scales can be interrelated. Usually the intention is to produce a particular result at the macro scale with this behavior being controlled by the mechanisms operative at the smaller scales. However, it can be difficult to confidently and securely approach the intended macro scale when its characterization is so shrouded in doubt and uncertainty. It would be highly advantageous to have a complete and comprehensive account of failure at the macro scale, one which transcends the various materials classes. The alternative is to have descriptors that are unique to each class or sub-class of materials but with great uncertainty as to the range and limits of validity of each characterization. This latter condition represents the current status. The objective here is to present and then probe a reasonably complete macroscopic theory of yielding and failure for homogeneous and isotropic materials. Much of the formalism will be synthesized here from various publications that have recently appeared but of necessity have been given in somewhat fragmented and unrelated forms. Using this new formulation, new results will be found for the yielding, plastic flow and failure in several important problems or classes of problems. This communication completes the cycle of recent papers on the failure of materials mentioned above. In whatever direction future work in the field goes, the present work may help to stimulate further interest and related activity. It is an evident irony of the history of mechanics that the many books written on the strength of materials basically had almost nothing to do with that subject. Such books were virtually confined to the linear range of elastic behavior with minimal or no attention to failure. This occurred because the understanding of failure as an organized discipline was non-existent. This state continued until the advent of fracture mechanics, about which more will be said later. The sparse historical scene of successful research upon materials failure did have one major prominence and this account should begin by acknowledging the subtle but profound contribution of Coulomb[1]. Mohr [2] put Coulomb’s failure result into a form allowing easy utility. The fact that the CoulombMohr failure form does not successfully account for many of the physical effects does not detract from the efforts of either scientist. In the time frames of their separate works, their grasp of the problem was completely beyond compare. Later, the Mises criterion was given, Mises [3], but only as an adjunct to a special case of the Coulomb criterion, namely the Tresca form. Both the Mises and Tresca criteria apply only to the yielding of very ductile metals. The Coulomb-Mohr form was intended to apply across the spectrum of materials types, as is the interest here. A history of strength and failure treatments has been given by Paul [4], and a brief historical summary by Christensen [5]. At the most elementary level it is sometimes said that ceramics are brittle, many but not all metals are ductile, and some types of polymers are ductile and others are brittle. While there is a degree of truth in this assertion, it can be extremely misleading, or even worse. An example will be given later wherein a material commonly considered as being completely ductile when placed in a particularly important special state of stress fails in a brittle manner. Relative to failure, all materials can behave either in a ductile or a brittle manner depending upon the state of stress that they are under and other environmental influences. The two terms, yielding and failure, have imprecise definitions that usually allow a wide latitude of interpretation. This imprecision underlies an uncertain basis of operation. The obvious exception to this situation was the development of fracture mechanics. Fracture mechanics was one of the technical achievements of modern mechanics. A typical application of fracture mechanics involves determining the stability, under imposed stress, of imperfections and stress risers such as cracks and edge notches, holes, attachments etc. In contrast, the means of applying fracture mechanics to the problem of the failure of homogeneous materials under uniform stress states has been far less clear. There are many opinions on this subject, but little substance beyond general statements. Thus, for the failure of homogeneous materials, the integration of fracture mechanics into a more general formalism has not been successfully accomplished in the past. Failure criteria have usually been formulated in terms of stresses, but over the historical time span, failure criteria have occasionally been postulated in terms of strains. The view here is that trying to specify failure in terms of strains is inappropriate and internally inconsistent. Stress must be used in order to have compatibility with fracture mechanics in the brittle range and with dislocation mechanics in the ductile range. To not have union with these two anchor points of physical reality would be extremely serious. Furthermore, force (stress) is the greatly preferred form for molecular dynamics simulations. Stress, not strain, will be used here for these well grounded reasons. In the modern era there have been many attempts to find criteria more general than just that for the perfectly ductile response or alternatively the fracture controlled response. The references cited above give many previous references to such works. A sampling of these efforts should include the following. Drucker and Prager [6] gave a two-parameter yield criterion of conical form in principle stress space. Paul [4] proposed a three-parameter pyramidal type yield surface. Wronski and Pick [7] applied Paul’s criteria to polymers. Raghava, Caddell and Yeh [8] proposed a criterion similar to parts of the present forms, and applied it to polymers. Stassi [9] also discussed similar forms, but without application. Pae [10] applied a three-parameter criterion to polymers. Wilson [11] applied the Drucker-Prager criterion to metals. Jaeger and Cook [12] discussed many three or more parameter models for application to geological materials. None of these approaches possess the combined attributes of involving only a few adjustable parameters, preferably only two, along with the power and flexibility to recreate many different physical effects for many different types of materials. Also, none of these works deal with the essential problem of providing indicators (for any stress state of interest) that differentiate a materials capability for undergoing ductile flow as opposed to the undesirable outcome of brittle failure. The present work will give specific meanings to the terms yielding and failure, ones that will offer a useful distinction between them. There are two technical keys to the following developments. These are: (i) the explicit integration of a fracture mechanism into the yielding versus failure formalism, and (ii) the derivation of an explicit criterion that determines whether a failure mode in any particular state of stress is expected to be of ductile or brittle nature. Not surprisingly, these developments (i) and (ii) are found to be interrelated, but still take separate forms in the final set of equations and conditions. Next the governing relations of this new theory will be given. Conditions for Yielding and Failure The references from which various aspects at this new theory of yielding and failure are collected are Christensen [5, 13-17]. The initial work in 1997 identified a nondimensional properties grouping that spanned the range from completely ductile to very brittle behaviors. The year 2000 work was a view of ductile versus brittle behavior, approached in a very mathematical formalism. A broader treatment in 2004 brought in an explicit fracture mechanism and involved comparison with experimental results for a wide variety of materials types. In 2005 an explicit criterion was derived for distinguishing expected brittle failure behaviors from those of plastic yielding, expressed in terms of the imposed stress state and a particular material characteristic. The first 2006 work was an examination of plastic flow potentials, needed when the condition of ductile behavior exists. The second 2006 work involved a detailed comparison with the Coulomb-Mohr and Drucker-Prager [6] theories. The collection of these individual works and particularly the present amalgam of all of them serve to form this comprehensive account of yielding and brittle failure. The mathematical conditions to be given in this section for yielding and failure are primarily taken from the above references, but for the background details, the particular references should be consulted. A uniform notation and terminology will be adopted. In this section and the following sections some previously open issues will be clarified and closed, some important special problems will be examined and some critical interpretations given. The governing yield/failure function for isotropic materials was found by taking a polynomial expansion, through terms of 2 nd degree, of the invariants of the stress tensor. This procedure gives (1) where sij is the deviatoric stress tensor and ! " # ij is non-dimensional stress with !" # ii+ 3 2 (1+! )s # ij s # ij $ 1