Capturing size effects in effective models: a variational approach

Abstract:

This work deals with identifying effective models for periodic linear elastic architected materials. These materials naturally possess two scales: the size of the unit-cell and the size of the overall structure. When the scale ratio η is not sufficiently small or when the microscopic constituents have contrasted stiffness properties, these architected materials are known to exhibit size-effects that are missed by classical homogenization.

Our aim in the first part of this work is to propose a homogenization procedure that accurately capture these size-effects. Following previous work on the topic, we propose to handle the size-effects occurring in the bulk by pushing the homogenization procedure to second order. Next we demonstrate how to tackle long-standing issues associated to these second-order models. Although accurate in the bulk region, the second-order corrections produced by homogenization still miss significant boundary effects, which can strongly degrade the quality of the predictions. Besides, in these second-order models, the order of the equilibrium equation is increased and new boundary conditions are required. In continuum mechanics, boundary conditions are usually derived by minimizing the energy of the system. For second-order models such an energy is missing, which makes the identification of appropriate boundary conditions particularly difficult. Deriving such an energy present two main challenges. First, the strain-gradient stiffness appearing in these models can be non-positive. A raw truncation of the bulk energy therefore yields a non-positive strain energy density. This property is highly undesirable as it leads to ill-posed boundary-value problems. Secondly, in order to guarantee the accuracy of the model, one must account for the energy generated by the boundary layers. This produces boundary contributions to the energy, that can be non-positive as well.

We present a new systematic framework that allows to address these two limitations and to identify a proper second-order energy for 1D microstructures. The approach is illustrated on a 1D spring network with next-nearest neighbour interactions. The energy is of the strain-gradient type, and is formulated in terms of a generalized strain. We further show on this example that the model is able to capture size-effects up to order η^2, as expected.

In the second part of this work we consider possible extensions to more complex mechanical behaviors. The existence of a variational formulation of the microscopic problem is key to apply our homogenization procedure. Although such formulations are very common in continuum mechanics, some models resist this variational approach : this is the case of Coulomb’s friction law.

In the second part of this work, we discuss a new variational formulation for this law. Key to our formulation is the introduction of a new internal variable, which is motivated by the real contact state at the microscopic level. This allows us to identify an alternative dissipation potential to describe the frictional behavior. The second key ingredient consists in relaxing the usual normality rule (or flow rule) that dictates the evolution law. By doing so, we obtain a model which coincides exactly with Coulomb friction law in the stick regime. We also demonstrate that the sliding behavior predicted by our model is very similar to that of Coulomb friction. More precisely, the two models differ by terms that scale as the size of the asperities. This new formulation opens the path to homogenization of microscopic problems including friction.

Keywords: second-order homogenization, boundary effects, non-positive strain-gradient moduli, variational approach, friction.

Short bio:

In 2022, Manon Thbaut completed a mechanical engineering degree at École des Ponts and a master’s on multiscale analysis in partnership with Sorbonne. She then pursued a PhD at Ecole Polytechnique (LMS lab), under the supervision of Basile Audoly and Claire Lestringant. During her PhD, she developed a variational framework for higher-order homogenization. She is now a postdoctoral fellow in the Computational Mechanics Group at ETH Zürich (Laura de Lorenzis’s group). She is currently working on a variational formulation of Coulomb friction law.