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570 M.S. Bruzzi et al. / International Journal of Plasticity 17 (2001) 565±599 Fig. 1. (a) 2D unit cell, containing a square reinforcing particle, with vertical and horizontal lines of symmetry (b) one quadrant of the 2D unit cell: su cient for modelling. prescribed displacement, U 2 , and the right edge had a uniform solution dependent displacement, U 1 . Similarly in 3D, the left hand face, the bottom face, and the back face of the unit cell octant were constrained to have zero displacements in the x 1 ; x 2 and x 3 directions, respectively, where x 3 is the out of plane axis. A prescribed uniform displacement, U 2 , was applied to the top surface and the right face and the front face had uniform solution dependent displacements, U 1 and U 3 , respectively. The unit cell technique cannot, however, be considered as a pure and correct periodic homogenisation technique, i.e. it cannot model correctly shear loading conditions (at least when using the one quadrant in 2D or one octant in 3D models). This is not relevant in the present context as shear loading conditions are not examined in this work. The `true periodic homogenisation' unit cell technique is discussed in more detail by Bensoussan et al. (1978) and Suquet (1987). Tensile loading in the vertical direction was achieved by applying uniform vertical displacements, U 2 , on the top edge of the 2D quadrant and the top surface of the 3D octant, respectively. The applied cell boundary displacements determined the average tensile engineering strain, " ave 22 , for the material. The boundary reaction forces were averaged over the top boundary to calculate the average tensile engineering stress 22 ave for the material. In the 2D model, the e€ect of modelling using 2D plane strain elements was considered. A simulation was performed, modelling the unreinforced Al 2124 alloy, using the experimental stress±strain data as input properties. Due to the strong plane strain constraint of " 33 ˆ 0, the 2D plane strain simulation predicted the tensile behaviour of the unreinforced alloy to be sti€er with a much higher yield stress. The simulation was repeated using generalised plane strain deformation, which allows uniform strain in the out of plane direction, i.e. " 33 ˆ A (where A is a constant), under conditions of zero average out of plane normal traction. In this case the simple tension stressstrain curve was, of course, recovered. Bearing this in mind, it was considered that generalised plane strain with its relaxation of the out of plane constraint would be more appropriate for the proposed 2D periodic unit cell modelling. Therefore, in the 2D simulations, 4-noded with full integration or 8-noded, with reduced integration, quadrilateral generalised plane strain elements were used to model the matrix. These
M.S. Bruzzi et al. / International Journal of Plasticity 17 (2001) 565±599 571 were described in ABAQUS 1 as element types CGPE6 and CGPE10R, respectively. 3-noded elements or 6-noded triangular elements with reduced integration were used to mesh the reinforcing particles and were labelled in ABAQUS 1 as element types CGPE5 and CGPE8R, respectively. In the 3D simulations, 8-noded linear (C3D8) elements or 20-noded quadratic hexahedron (C3D20R) elements with reduced integration were used to model both the matrix and the reinforcing particles. 2D and 3D periodic unit cell models were used for the cyclic loading simulations. Both isotropic hardening and non-linear isotropic kinematic hardening descriptions were used for the matrix. In the latter case a saturated stress±strain hysteresis loop for the Al 2124 T4 matrix alloy obtained from tests at an overall strain amplitude of 3.1% (Foulquier, 1998) was used to ®t the material constants in Eq. (6) according to the method presented in the ABAQUS 1 , (1997b). This data was chosen since it was the highest strain amplitude data available for the material, potentially re¯exive of the plastic strain levels to be experienced by the matrix in the models. For the cyclic loading, attention was restricted to room temperature behaviour. 2.3. Cell embedding method In order to remove the periodicity constraint, and also to assess the performance of the models in terms of variation in boundary constraints, a simple iterative cell embedding technique was investigated. In this method, the material is simulated by a model consisting of a core, similar to a unit cell, containing separate representation of the matrix and reinforcing particles, surrounded by an outer region with the homogenised material properties of the composite, as shown in Fig. 2. The boundary conditions and loading are then applied to this outer region. The embedding allows a non-uniform deformation of the core boundary. The boundary supports shear traction and the removal of the orthotropic constraint condition allows for rotation at the boundary. This special arrangement allows for a more realistic Fig. 2. Quadrant of 2D embedded cell model.
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Micromechanical modelling of the static and cyclic ... - ResearchGate

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