In this chapter, we consider the mathematical modeling of a special type of process, the antiplane shear. One can induce states of antiplane shear in a solid by loading it in a special way. We rarely actually load solids so as to cause them to deform in antiplane shear. However, we will .nd that the governing equations and boundary conditions for antiplane shear problems are beautifully simple, and the solution will have many of the features of the more general case and may help us to solve the more complex problem, too. For this reason, in the recent years considerable attention has been paid to the analysis of antiplane shear deformation within the context of elasticity theory. We start the description of the basic assumptions, then we specialize the equilibrium and constitutive equations in the context of antiplane shear and introduce a Sobolev-type space used in the study of such kind of problems. Further, we consider an elastic antiplane displacement-traction boundary value problem, derive its variational formulation, and prove the existence of the unique weak solution by using the Lax-Milgram theorem. Finally, we present a complete description of the frictional contact conditions used in this book, in the context of antiplane process; to this end, we consider both the case of unstressed and pre-stressed reference con.gurations.
KeywordsContact Problem Slip Rate Frictional Contact Friction Bound Antiplane Shear
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