Enforcing Boundary Conditions in Micro-macro Transition for Second Order Continuum

In recent years the multiscale computational homogenization has been extensively developed. Such macro-modelling does not require any constitutive assumptions at the macro-level. The multi-scale computational homogenization has also been extended for the second order continuum at the macro level. The second-order framework is based on incorporation of the gradient of macroscopic deformation in micro to macro multiscale transition. The introduction of the second-order continuum at macro-scale takes into account the size effect and gives more accurate results in case of insufficient scale separation.

This paper concentrates on some issues of the fully coupled second order homogenization scheme. Attention is focused on micro-macro transitions of the discretized microstructure. In the presented paper a new approach is proposed which can handle any type of boundary conditions (i.e. displacement, periodic and static). The boundary conditions enforce the deformation of representative volume element (RVE) according to given gradient and second gradient of displacements in average sense. After expansion of the displacement vector in geometric centre of RVE and truncation after second order term the continuum boundary conditions can be written in integral form as

1

$$ \begin{array}{*{20}c} {\smallint _\Gamma \delta t \cdot rd\Gamma = 0,} & {\smallint _\Gamma n \otimes rd\Gamma = 0,} & {\smallint _\Gamma n \otimes x \otimes rd\Gamma = 0,} \\ \end{array} $$

where n is the normal vector, r is the microfluctuation of displacement field and dt is statically admissable variation of tractions on boundary. If the first integral satisfies Hill-Mandel theorem, the second and third integral enforce deformation of RVE according to given strain tensor and given gradient of deformation in average sense, correspondingly. After FE discretization of (1), the constraint equation is formulated in matrix from Cq = g, where q is the vector of displacemet-type degrees of freedom. To enforce the constraints the projection matrices are formulated as

2

$$ \left. {Q = I - C^T \left( {CC^T } \right)^{ - 1} } \right)C $$

and applied in the computation of stiffness matrix and load vector. One of advantages of the proposed approach is that the rigid translation and rotation, i.e. 1st order and 2nd order deformations can be applied individually by a sequence of operations. Moreover, enforcing constraints can be performed element by element subassembly procedure.

In the paper analytical and numerical results are presented. Equivalent stiffness operators are derived analytically for the linear and homogeneous material in RVE. The results are compared with theMindlin constitutive model. In the case of heterogeneous non-linear material illustrative examples was numerically computed. The shear layer problem and plate bending in plane strain for a non-linear homogeneous material with an intristic length scale are also presented.