The Influence of the Fluctuation Term on the Accuracy of Multiscale Analysis Results

Resumo

This work investigates the influence of the fluctuation term in the formulation of the homogenized constitutive tensor within a multiscale approach applied to the analysis of heterogeneous media. A phase-field constitutive model is used, coupled with the multilevel finite element method, which allows for the simultaneous consideration of macro and micro scales. The phase-field model enables the diffuse representation of fractures, capturing the evolution of the displacement field and crack propagation without the need for an explicit description of the fracture geometry. This makes the model particularly useful for materials like concrete, which may exhibit complex cracking patterns.At the macroscopic scale, the material's constitutive behavior is not directly defined; it must be obtained from solving the microscopic scale, where the local material properties, such as variations in stress and strain, are better represented. The connection between the scales requires defining homogenized parameters that adequately represents the material’s behavior. Literature suggests that this transition cannot be done through a simple average of the local constitutive tensors, as this approach disregards important fluctuations in the displacement field.To ensure theoretical consistency, the inclusion of a fluctuation term is required, which depends on the inverse of the stiffness matrix of the microscale problem. However, calculating this term results in a significant computational cost. With the multiscale implementation in place, comparative tests were conducted between simulations with and without the fluctuation tensor, evaluating both the accuracy of the results and the impact on processing time. All implementations were carried out using INSANE, an open-source platform developed by the Department of Structural Engineering (DEES) at the Federal University of Minas Gerais (UFMG).