172 Computational Mechanics of Composite Materials elasticity homogenisation method,the approximation of the effective yield limit stresses of a composite is proposed as a quite simple closed form function (3.70) or,in terms of the effective yield surface,in the following form: (o) aow-bzy (3.71) where m(y=)=3 uV(u)and Vis any estimate of the viscosity compliance tensor defined using the viscosities u and u2.A review of the most recent theories in this field can be found in [381],for instance. The main aim of computational experiment presented is to determine the global nonlinear homogenised constitutive law for two component composites with elastoplastic components;the FEM based program ABAQUS [1]is used in all computational procedures.However the method presented can be implemented in any nonlinear FEM plane strain/stress code such as [60],for instance.The numerical experiments are carried out in the microstructural(RVE)level,and that is why the global response of the composite is predicted starting from the behaviour of the periodicity cell.The numerical micromechanical model consists of a three-component periodicity cell with horizontal and vertical symmetry axes and dimensions 3.0 cm (horizontal)X 2.13 cm (vertical)(cf.Figure 3.1 and 3.2). The composite is made of epoxy matrix and metal reinforcement with material properties of the components collected in Table 1.The void embedded into the steel casting simulates a lack of any matrix in the periodicity cell.Some nonzero material data are introduced to avoid numerical singularities during the homogenisation problem solution. The 10-node biquadratic,quadrilateral hybrid linear pressure reduced integration plane strain finite elements with 4 integration Gaussian points are used to discretise the cell.Periodic boundary conditions are imposed to ensure periodic character of the entire structure behaviour.A suitable formulation of displacement boundary conditions has the following form: 4,=e(y(P2)-y(B) (3.72) where=represents the displacement function components,E is the global total strain imposed on the periodicity cell,while y(P)and y(P)denote coordinates of the points lying on the opposite sides of the RVE.172 Computational Mechanics of Composite Materials elasticity homogenisation method, the approximation of the effective yield limit stresses of a composite is proposed as a quite simple closed form function 1 2 σ = σ σ (eff) (3.70) or, in terms of the effective yield surface, in the following form: ( ) [ ] ( ) ⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎣ ⎡ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ Φ = + − ≥ 2 (2) 1 (2) (1) 2 1 0 max ( ) y y y y c c y σ m y σ σ σ σ σ (3.71) where ( ) 3 2 1 = = µ µ m y ( ) 2 1 2 µ V µ ,µ and V is any estimate of the viscosity compliance tensor defined using the viscosities µ1 and µ2. A review of the most recent theories in this field can be found in [381], for instance. The main aim of computational experiment presented is to determine the global nonlinear homogenised constitutive law for two component composites with elastoplastic components; the FEM based program ABAQUS [1] is used in all computational procedures. However the method presented can be implemented in any nonlinear FEM plane strain/stress code such as [60], for instance. The numerical experiments are carried out in the microstructural (RVE) level, and that is why the global response of the composite is predicted starting from the behaviour of the periodicity cell. The numerical micromechanical model consists of a three-component periodicity cell with horizontal and vertical symmetry axes and dimensions 3.0 cm (horizontal) × 2.13 cm (vertical) (cf. Figure 3.1 and 3.2). The composite is made of epoxy matrix and metal reinforcement with material properties of the components collected in Table 1. The void embedded into the steel casting simulates a lack of any matrix in the periodicity cell. Some nonzero material data are introduced to avoid numerical singularities during the homogenisation problem solution. The 10-node biquadratic, quadrilateral hybrid linear pressure reduced integration plane strain finite elements with 4 integration Gaussian points are used to discretise the cell. Periodic boundary conditions are imposed to ensure periodic character of the entire structure behaviour. A suitable formulation of displacement boundary conditions has the following form: ( ( ) ( )) 2 P1 u y P y i = ε ij − (3.72) where { } 1 2 ui = u ,u represents the displacement function components, ij ε is the global total strain imposed on the periodicity cell, while ) (P1 y and ) (P2 y denote coordinates of the points lying on the opposite sides of the RVE