A non-linear programming approach to kinematic shakedown analysis of composite materials

• Published in: International journal for numerical methods in engineering Vol. 66; no. 1; pp. 117 - 146
• Main Authors: Li, H. X., Yu, H. S.
• Format: Journal Article
• Language: English
• Published: Chichester, UK John Wiley & Sons, Ltd 02.04.2006; Wiley
• Subjects: Composite materials; Computational techniques; Dissipation; Exact sciences and technology; finite element method; Fundamental areas of phenomenology (including applications); homogenization theory; Homogenizing; Inelasticity (thermoplasticity, viscoplasticity...); Kinematics; Mathematical analysis; Mathematical methods in physics; Mathematical programming; non-linear programming; Nonlinearity; Numerical analysis; Physics; Shakedown analysis; Solid mechanics; Structural and continuum mechanics; shakedown analysis; Associated plasticity; non-linear programming; Iterative method; Non linear programming; Modeling; Composite material; Periodic structure; Finite element method; Inelasticity; Kinematic theory; Energy dissipation; Cyclic load; Kinematics; composite materials; Representative volume element; Plastic flow; Yield criterion; Homogenization; Large displacement; Engineering design; Upper bound; Shakedown; Periodic medium; homogenization theory; Non linear effect; Microstructure; Ellipsoid
• ISSN: 0029-5981; 1097-0207
• DOI: 10.1002/nme.1547

Using a Representative volume element (RVE) to represent the microstructure of periodic composite materials, this paper develops a non‐linear numerical technique to calculate the macroscopic shakedown domains of composites subjected to cyclic loads. The shakedown analysis is performed using homogenization theory and the displacement‐based finite element method. With the aid of homogenization theory, the classical kinematic shakedown theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. By means of non‐linear mathematical programming techniques, a finite element formulation of kinematic shakedown analysis is then developed leading to a non‐linear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a composite is then obtained. An effective, direct iterative algorithm is proposed to solve the non‐linear programming problem. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples. This can serve as a useful numerical tool for developing engineering design methods involving composite materials. Copyright © 2005 John Wiley & Sons, Ltd.