2.5. Numerical Homogenization and Boundary Conditions

In the method of two-scale numerical homogenization, the response of a material at the macro-scale is determined by an analysis of the behavior of the material’s internal structure at the micro-scale. In the case of nonlinear material behavior, achieving the equilibrium at each scale together with the compatibility of information between the two-scales requires multiple exchanges of needed information between the scales after solving a corresponding nonlinear boundary value problems formulated at each of the scales. On the micro-scale level, the distributions of micro-stresses and micro-strains were calculated, which via homogenization provide the needed information of averaged macroscopic quantities to the macro-scale. In this contribution, our analysis was restricted to the solution of the local nonlinear BVP defined on the two-dimensional representative volume element (RVE). When the characteristic microscopic length was one order smaller than the characteristic macroscopic length, we could take into consideration only effects of the first order.

The analysis was carried out within the realm of linear kinematics. The notation that a bar above a symbol denotes a macroscopic variable or quantity was used. Let x¯ denote the vector of coordinates of a material point at the macroscopic level, and let x stand for coordinates of points within the RVE of volume V defined in the material point. Further, let us assume that the microscopic displacement field u can be additively split into a linear part ε¯x and a fluctuating part r:

where ε¯(x¯,t) is the macro-strain tensor and t is a time-like parameter. At down-scaling (macro-micro transition) and solving the local BVP at RVE, the strain tensor ε¯ is treated as a known quantity (data). At up-scaling (micro-macro transition), the elements of the macro-strain tensor ε¯ can be defined as mean values of the corresponding micro-strains ε averaged over the RVE:

where n is the unit outward normal of the boundary of RVE, Γ=∂V, and ⊗ is a tensor product of vectors. The Formula (6) shows that the macroscopic strain tensor can be expressed by micro-displacements u at the boundary of the RVE. However, Equation (6) is valid provided that the zero gradient condition of the microscopic displacement fluctuation field r is satisfied:

Fulfilling the above condition ensures that deformation of the RVE boundary in the medium sense is in accordance with the pre-set macro-strain ε¯.

Macro-stresses can be defined, similar to macro-strains, as mean values of the micro-stresses σ:

This relationship can be derived from Hill’s theorem [25], which says that the work done by macro-stresses on the corresponding macro-strains is equal to the mean value of the work performed by micro-stresses on the corresponding micro-strains:

where the symbol 〈•〉 stands for averaging over the RVE volume.

The finite element method is applied for numerical calculations, with the finite element designated as the CPS4R Abaqus system library [56]. The RVE area was discretized with 2500 finite elements, each with dimensions of 0.2 × 0.2 mm; see Figure 2.

Numerical tests were carried out enforcing three types of boundary conditions on the boundary of RVE: linear displacement boundary conditions (DBC), uniform traction boundary conditions (TBC), and periodic boundary conditions (PBC). Before starting the numerical simulations of the inhomogeneous nonlinear BVP for high-performance concrete, the work and proper interaction between the Homtools and the Abaqus packages were verified with the example of RVE for a homogeneous material with a hole (Section 3.1).

This method consists of applying on the boundary of RVE the displacement field that would occur if the strain were uniform inside the RVE. For the considered linear kinematics, the boundary conditions can be defined by the formulas in Equations (5) and (7):

in which ε(u) is the micro-strain field and x is the position vector of point x on boundary Γ. There is no restriction concerning the use of this method, except that no rigid part must intersect the boundary, and holes are permitted.

This method consists of applying on the boundary of RVE the stress vector field that would occur if the stress were uniform inside the RVE. For the considered linear kinematics, the boundary conditions can be defined as follows:

where σ is the Cauchy stress tensor and n denotes the outward unit normal. There is no restriction concerning the use of this method, except that no holes must intersect the boundary.

Enforcing the PBC is theoretically relevant for periodic media, which can be defined by a periodicity cell and the associated periodicity vector of translation. The periodic homogenization process consists of assuming that the strains and stresses are periodic at the level of the periodicity cell (which is defined as the RVE). The periodicity of stresses and strains leads to specific periodic boundary conditions for the localization problem on the RVE. In order to introduce periodic boundary conditions, the boundary of RVE is decomposed into two opposing parts, Γ+ and Γ−, such that Γ=Γ+∪Γ−. Each point x+ on Γ+ is associated with a unique point x− on Γ−, and the unit normal vectors at these boundaries satisfy n−=−n+. Then, the PBCs are defined as follows:

where t±=σn± is the stress vector. For this type of boundary condition, u is forced to be periodic and t is forced to be antiperiodic. Note that LDBCs satisfy the periodicity of u only whereas TBCs satisfy the antiperiodicity of t only. There is no restriction concerning the use of this method; periodic holes and rigid parts intersecting the boundary are permitted.