This thesis concerns the application of modern methods from the field of mathematical programming for solving certain problems of nonsmooth mechanics. Attention is given to contact problems with friction as well as plasticity problems. First, the discrete, quasistatic, small-displacement, linear elastic, contact problem with Coulomb friction is written as a set of nonsmooth, unconstrained equations. These equations are seen to be B-differentiable which enables Pang's Newton method for B-differentiable equations to be used. In addition, the problem is written as a set of smooth, constrained equations and inequalities. These are then solved by an interior point method. The two algorithms are compared for two-dimensional problems, and it is verified that Pang's Newton method is superior in speed as well as robustness.

This work is then extended in that it is shown that the nonsmooth formulation of frictional contact is in fact semismooth, so that the convergence theory for the Newton method used may be strengthened. The efficiency of the Newton method is enhanced further in that the linear system of equations solved in each iterationis solved more efficiently. Numerical tests for three-dimensional problems show that the unconstrained, nonsmooth Newton method works excellently.

Next, attention is given to the discrete, quasistatic, small-strain, plasticity problem with the von Mises yield function and associative flow rule. This problem is formulated as a set of unconstrained, nonsmooth equations. It is seen that the equations are piecewise smooth (hence semismooth) so that Pang's Newton method can be applied. This method is then compared to the classic radial return method. In addition, it is shown, for linear kinematic and linear isotropic hardening, that the radial return method represents a piecewise smooth mapping, so that Pang'sNewton method with its global convergence theory is applicable. Local quadratic convergence (even to nondifferentiable points) of the standard implementation of the radial return method is established.

Finally, discrete frictional contact problems for elastoplastic bodies are considered. A system of semismooth equations is formulated and solved by Pang's Newton method.