A design sensitivity analysis technique for composites undergoing finite elastoplastic deformation

Abstract

Aditi Chattopadhyay* and Ruijiang Guo** Department of Mechanical and Aerospace Engineering Arizona State University Tempe, AZ 85287-6106 The paper develops the theory of nonlinear structural design sensitivity analysis for composites undergoing finite elastoplastic deformation. The rate (time-independent) constitutive model, which is objective, is employed to account for the plastic material behavior undergoing finite deformations. The reference volume concept is used to unify the shape and nonshape design problems. A higher order approximation procedure of the integration of the rate constitutive equations is used for the response analysis. The direct differentiation approach is adopted to obtain the design sensitivity equations. A method of partial differentiation of the rate constitutive equations, which yields a set of linear differential equations in the partial derivatives of stresses and internal variables with respect to the design variable, is used. The presentation of the general theory is followed by a numerical example. A composite laminated beam is used as an example and results obtained using the theory developed are compared with those from finite difference. Nomenclature 'b = body force vector per unit mass at load level t C = 4th order linear elastic modulus tensor d = design variable vector YJ, : J ~ = Jacobian and area metric of the transformation Ox+'x , respectively ;.I, AJ,= Jacobian and area metric of the transformation 'x+Ox, respectively Ti = ply thickness of laminated beam of the ith ply ' u = displacement field vector at load level t x = coordinate vector in the fixed reference voulme t~ = coordinate vector at load level t 'E = strain tensor at load level t ei = ply angle of the laminated beam of the ith ply ' P = mass per unit volume at load level t '(3 = Cauchy stress tensor at load level t (P = curvature of the beam at load level t * Assistant Professor, Senior Member AIAA, Member ASME, AHS, SPIE **Graduate Research Assistant, Student Member AIAA '5 = internal variable vector at load level t.