Mathematical model lateral strain

Before discussing mechanical models or other mathematical representation of viscoelastic behavior, it is very important to note that the preceding section deals only with observed behavior or the experimental response of polymers under laboratory conditions. That is, the viscoelastic properties are defined from observations of real behavior and need not be defined by a particular mathematical model. Mathematical models are developed for the simple purpose of understanding and describing observed behavior. Also, as will be evident later, other loading modes such as constant strain rate and steady state oscillation, etc. can be used to determine viscoelastic properties. 

The above mathematical models (and later derivatives) define constitutive relationships for the plastic strain regime and they all assume a linear elastic behavior terminated by a yield point that is rate dependent. Hence the yield surface of the material is rate dependent. Since the purpose of these models are to develop methods to calculate deformations which are rate dependent beyond the yield point of a material they are often referred to by the term viscoplasticity, (see Perzyna, (1980), Christescu, (1982)). This practice is analogous to referring to methods to calculate deformation beyond the yield point of an ideal rate independent elastic-plastic material as classical plasticity. However, more general theories of viscoplasticity have been developed in some of which no yield stress is necessary. See Bodner, (1975) and Lubliner, (1990) for examples. 

Given a model, the analysis can be performed mathematically. A finite element computer code (Ref 38) for the analysis of finite or infinitesimal strains is now available, modified (Ref 68) to account for shock induced stresses, temp rise (by the assumption of a constant Grueneisen parameter), heat generated by the decompn of the expl and transient heat transfer. Later in this article we report an empirical treatment of propint initiation data (Fig 4)-Analy tic ally obtained data are in fair agreement with exptl results so that further effort along these lines appears justified (Ref 68)... 

The analysis includes three mathematically distinct cases addressing all possible interfacial adhesive stress scenarios (1) fully elastic adhesive throughout the bondline, (2) adhesive plastically strained at only one bondline end, and (3) adhesive exhibiting plastic strains at both ends of the joint. For comparison and validation purposes with the second analytical model and the experimental example provided later, only the first scenario is reviewed herein. Bond configuration and notations adopted are shown in Fig. 10.11. It should be noted that the origin of the x-coordinate is the middle of the joint only for the current mathematical lap-shear stress expressions. However, for other contexts in this chapter, the origin is located at the left end of the lap joint (i.e. near the gap of Fig. 10.10). 

In this section, elementary mechanical models that can describe some aspects of viscoelastic polymeric behavior are presented. Although these simple models cannot represent the behavior of real polymers over their complete history of use, they are very helpful to gain physical understanding of the phenomena of creep, relaxation and other test procedures and to better understand the relationship between stress and strain for a viscoelastic material. Undoubtedly, the first models were developed on the basis of observations and not just as a mathematical exercise. Generalized mechanical models are presented later in Chapter 5. 

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