Exercícios - Resmat 2 2021-1

5.1 Consider a transversely constrained, elastic medium such that v≡0 and w≡0. The planes x=0 and x=a are subjected to a uniformly distributed pressure of magnitude p₀. Find the stress, strain and displacement fields. (See the figure.) 5.2 Consider a transversely constrained, elastic medium such that v≡0 and w≡0. The plane x=0 is fixed, and the plane which is originally at x=a is displaced a distance δ in the x direction. Find the resulting fields of stress, strain, and displacement. (See the figure.) 5.3 An elastic solid of unbounded extent has a spherical cavity with radius a. The cavity contains a gas under pressure p₀. Find the displacement, stress, and strain fields in the solid. 5.4 A solid, elastic sphere is subjected to a uniform radial pressure p₀ on its surface. Find the displacement, stress, and strain fields in the sphere. 5.5 Find the stress and displacement fields in a very large solid with a spherical void centered at r=0. The solid is subjected to the radial stress τᵣ= T far from the origin. Assume point symmetry with respect to r=0. What is the stress concentration factor associated with τᵣ₀? 5.6 An unbounded elastic medium contains a spherical cavity. The cavity is lined by an elastic, spherical shell which is bonded to the surrounding medium. The cavity contains a gas under pressure of magnitude p₀. (See the figure.) Investigate the stress and displacement fields in the surrounding medium as a function of the material properties of the liner. In general, the material properties of the liner are different from those of the surrounding medium. 5.7 With reference to the analysis in Section 5.1, consider a thin spherical shell such that b-a=r<a (see Fig. 5.1). The shell is filled with a gas under pressure p₀. Show that in this case the stresses τøθ and τøø are (approximately) constant through the thickness t and are given by τøθ=τøø=pa/2t. 5.8 An elastic solid of unbounded extend has a circular cylindrical cavity with radius a. The cavity contains a gas under pressure p₀. Find the displacement, stress, and strain fields in the solid. Assume a condition of plane strain. In other words, assume that the displacement component parallel to the cavity axis vanishes. 5.9 A solid, elastic cylinder is subjected to a uniform radial pressure p₀ on its cylindrical surface. Find the displacement, stress, and strain fields in the cylinder. Assume a condition of plane strain, in other words, assume that the displacement component in the direction of the cylinder axis vanishes.