A 3D PVP co-rotational formulation for large-displacement and small-strain analysis of bi-modulus materials

• An efficient computational method is developed for geometrically nonlinear analysis of 3D bi-modulus materials.
• A unified PVP constitutive equation is proposed to deal with the numerical instability of algorithms.
• Traditional stress iteration is not required and the proposed algorithm shows excellent convergence.
• The proposed algorithm is used to simulate the unilateral contact behavior of staggered bio-composites.

An efficient computational method is developed for large-displacement and small-strain analysis of 3D bi-modulus materials which are often found in civil, composite and biological engineering. Based on the parametric variational principle (PVP), a unified constitutive equation of 3D bi-modulus materials is proposed to deal with the problem of numerical instability. The small-strain bi-modulus problem is transformed into a standard linear complementarity problem that can be solved easily by the classic Lemke׳s algorithm. By using the co-rotational approach, the local PVP formulation is combined with an existing co-rotational formulation and a new tangent stiffness matrix, including a parametric variable, is derived for geometrically nonlinear analysis. Traditional stress iteration is not required for calculation of the nodal internal force when the Newton–Raphson scheme or Arc-length method is employed to solve the material and geometric nonlinear problem. Convergence of the proposed algorithm is improved greatly in contrast to the traditionally iterative solution. Also, the proposed algorithm is used to simulate a unilateral contact behavior of staggered bio-composites and the effective elasticity modulus of composites is determined accurately.