The problem of pulling a fiber imbedded in a matrix material can be idealized (in the...

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The problem of pulling a fiber imbedded in a matrix material can be idealized (in the interest of gaining qualitative understanding of the stress distributions at the fiber-matrix interface) as one of studying the following problem [8]: consider a hollow circular cylinder with outer radius a, inner radius b, and length L. The outer surface of the hollow cylinder is assumed to be fixed and its inner surface ideally bonded to a rigid circular cylindrical core of radius b and length L, as shown in Fig. P1.18. Suppose that an axial force F = Pz is applied to the rigid core along its centroidal axis. (a) Find the axial displacement 8 of the rigid core by assuming the following displacement field in the hollow cylinder: Urug=0, Uz = Uz (r) (b) Find the relationship between the applied load P and displacement 8 of the rigid core. (c) Determine the work done by the load P. Here the hollow cylinder represents the matrix around the fiber while the fiber is idealized as the rigid core. x2 0 X1 b The problem of pulling a fiber imbedded in a matrix material can be idealized (in the interest of gaining qualitative understanding of the stress distributions at the fiber-matrix interface) as one of studying the following problem [8]: consider a hollow circular cylinder with outer radius a, inner radius b, and length L. The outer surface of the hollow cylinder is assumed to be fixed and its inner surface ideally bonded to a rigid circular cylindrical core of radius b and length L, as shown in Fig. P1.18. Suppose that an axial force F = Pz is applied to the rigid core along its centroidal axis. (a) Find the axial displacement 8 of the rigid core by assuming the following displacement field in the hollow cylinder: Urug=0, Uz = Uz (r) (b) Find the relationship between the applied load P and displacement 8 of the rigid core. (c) Determine the work done by the load P. Here the hollow cylinder represents the matrix around the fiber while the fiber is idealized as the rigid core. x2 0 X1 b