Strain-hardening response of thin films

No single choice of constitutive relation for a material can be expected to provide an accurate description of behavior in all circumstances. On the other hand, some structure of constitutive representation of the material must be adopted as a basis for comparison and correlation of data and, ultimately, to make predictions of material behavior for prescribed circumstances. A few of the unifying concepts in constitutive theory for rate-independent deformation of solids, primarily metals, that exhibit permanent deformation are summarized. Once a constitutive structure is established, the concepts are specialized to the states of stress and deformation typical of thin metal films. 

The first of the concepts of broad applicability is the existence of a yield surface in stress space this same quantity is known by a variety of names, including yield locus, the elastic limit surface and yield function. The function defining the surface is denoted by and its role is to identify the boundary of the elastic range in a space spanned by the stress components (7jj. In the most general circumstances to be considered here, this function takes the form 

A concept that has been central to the development of relationships between plastic strain rate and current state of stress, which are the flow rules of plasticity, underlies the postulate of a maximum plastic resistance. This postulate can be stated in the following way. Consider an elastic-plastic material under circumstances in which the state of stress aij satisfies the yield condition = 0. In geometrical terms, is a point on the 

Immediate consequences of the postulate are that the yield function defines a convex surface in stress spacef and that the plastic strain rate e . is normal to the yield surface at any point at which the surface is smooth. Generalization of the feature of normality to surfaces with an edge or an apex is straightforward. The feature of normality implies that 

To render the response rate-independent, the quantity A must be linear in ij. In approximate terms, any stress rate that is directed from the yield surface into the elastic range of response, or that is tangent to the yield surface with aijd(f /daij = 0, induces no plastic deformation. This implies that only the component of ij acting in the direction of the outward yield surface normal contributes to plastic strain. It follows that (7.54) must have the form 

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Suppose that a thin film of elastic-plastic material bonded to an elastic substrate is subjected to temperature cycling, as in the example discussed in Section 7.5.2. Furthermore, suppose that the film material has precisely the same uniaxial stress-strain response under monotonic loading as does the material considered there. However, in the present context, the film is assumed to strain harden according to the kinematic hardening idealization. Under these circumstances of temperature cycling, with the conditions... 

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