Assessment of temporal integration schemes for the sensitivity analysis of frictional contact/impact response of axisymmetric composite structures

An assessment is made of three temporal integration schemes for the sensitivity analysis of the frictional contact/impact response of axisymmetric composite structures. The structures considered consist of an arbitrary number of perfectly-bonded homogeneous anisotropic layers. The material of each layer is assumed to be hyperelastic, and the effect of geometric non-linearity is included. Sensitivity coefficients measure the sensitivity of the response to variations in the different material, lamination and geometric parameters of the structure. A displacement finite element model is used for the spatial discretization. Normal contact conditions are incorporated into the formulation by using a perturbed Lagrangian approach with fundamental unknowns consisting of both the nodal displacements and the Lagrange multipliers associated with the contact conditions. The Lagrange multipliers are allowed to be discontinuous at interelement boundaries. Tangential contact conditions are incorporated by using either a penalty method or a Lagrange multiplier technique, in conjunction with the classical Coulomb's friction model. The three temporal integration schemes considered are: the implicit Newmark and Houbolt schemes, and the explicit central difference method. In the case of the implicit methods, the Newton-Raphson iterative technique is used for the solution of the resulting non-linear algebraic equations, and for the determination of the contact region, contact conditions (sliding or sticking), and the contact pressures. Sensitivity coefficients are evaluated by using a direct differentiation approach in conjunction with the incremental equations. Numerical results are presented for the frictional contact of a composite spherical cap impacting a rigid plate, showing the effects of each of the following factors on the accuracy of the predicted response and sensitivity coefficients: (a) incorporating the normal contact conditions, (b) the magnitude of the penalty parameter in the normal direction (for the perturbed Lagrangian method), and (c) the time step size for the response and the sensitivity analyses.