Variational based effective models for inelastic materials

Mechanical systems with inelastic materials and given boundary conditions can be generally defined and solved. However, for large systems, this problem becomes non trivial and computationally costly, especially when microstructures occure. Therefore, multi‐scale modeling is used to capture the behavior of the system.


Variational framework
We will introduce a variational approach for the description of inelastic processes resting on thermodynamical extremum principles. For this purpose let us consider a physical system described by external state variables x and internal state variables z, parametrized as where Ω denotes a suitable parameter space. Let this physical system be defined using only two scalar potentials: a free energy Ψ(∇x, x, z) and a dissipation potential ∆(z,ż), where ∇ denotes the gradient with respect to ξ.
The external state variable x will be given by minimization of system potential energy as in inf x Ω Ψ(∇x, x, z) dξ + f ext (x) x = x 0 on ∂Ω , where f ext (x) is the potential of external driving forces. The evolution of the internal variables is described by the Biotequation resulting from the minimization problem Our goal is to capture the behavior of the system described by Eqs. (2) and (3) using only a finite (small) number of parameters. Assume that our state variables are, as functions of ξ, members of suitable function spaces: x ∈ X, z ∈ Z. Let us moreover introduce linear projection operators to finite-dimensional spaces and the resulting essential parameters.
We would like the kinetics of the system under consideration to be captured by the essential parameters as closely as possible. This occurs when the potentials Ψ and ∆ are invariant under variations within the marginal spaces. Hence,
(3) is approximated by Additionally assume that the potential of external loads in Eq. (2) can be expressed via the essential parameters, f ext (x) = f ess (x ess ), and that the boundary conditions may be expressed in terms of the essential parameters, too, by specifying a projection Bx ess . In this setting, we replace the minimization problem in Eq. (2) with the macroscopic form: In what follows, the two minimization problems in Eqs. (6) and (7) will be shown to capture the kinetics of the macroscopic system, thus approximating the behavior of the original system. Given f ess and x ess0 as functions of time, they allow for the computation of x ess and z ess as functions of time.

Inelastic homogenization in periodic microstructures
Let us consider now a material having a microstructure defined by periodically varying C(r) and σ y (r).
Then Ω may be chosen as a rectangular representative volume element. The average of a quantity f over Ω is then given by The macroscopic material behavior may be determined by subjecting the representative volume element to a macroscopic strain e M = e M (t) = ⟨ε⟩ and calculating the macroscopic stress σ M = σ M (t) = ⟨σ⟩. The imposed macroscopic strain may be realized by decomposing the displacement field as where u per satisfies so-called periodic boundary conditions, i.e. it holds u per (r − ) = u per (r + ) on corresponding boundary points r − , r + on opposite sides of Ω.

The macroscopic model and essential variables
The problem is in principle infinite-dimensional because the plastic strains ε p (r, t) have to be known at all points of Ω for all times. In order to devise a macroscopic model of the system under consideration let us subdivide our representative volume element into N sd distinct sub-domains Ω i such that In every sub-domain we are only interested in mean values of the plastic strains. Which constitute the essential internal parameters and as essential external parameters, we will choose the macroscopic strain.
With the definitions above the macroscopic free energy is given as Introducing Lagrange-parameters µ i for the second constraint we obtain the Lagrangian for the minimization problem above as As stationarity conditions we obtain From the second relation in Eq. (14) we see that the stress σ = µ i is constant in every sub-domain Ω i and that the first condition in Eq. (14) is trivially satisfied. By Substitution we get It can be seen that Ψ macro (e M , e p ) is a quadratic expression in e M and e p . From Eq. (15) we obtain immediately the conjugate driving forces to the essential internal parameters as denotes the constant stress in every sub-domain. The macroscopic dissipation potential is defined as assuming that σ y = σ yi is constant in every sub-domain. The yield conditions in each sub-domain is of the form

Polyhedral sub-domains
Let us consider now microstructures consisting of polyhedral sub-domains bounded by facets F i . Of course, any microstructure may be approximated in this way with arbitrary accuracy by choosing the facets small enough. Suppose the facets posses outward normal vectors n i . Then the average strains in every sub-domain can be calculated according to Note that because of the periodic boundary conditions some of the a j are dependent if F j ⊂ ∂Ω. Let the number of independent amplitude vectors be N a . The macroscopic free energy can now be written as where Ψ rve (e M , e p , a 1 , . . . , a Na ) = The stationarity conditions corresponding to minimization are The macroscopic energy will be quadratic of the form The conjugate driving forces follow as

Non-symmetric RVE with octagon center inclusion
In this example, we examine a non-uniform matrix with 9 different sub-domains. 20 different amplitude vectors are to be computed. Fig. 1 shows the different sub-domains with the corresponding amplitude vectors, in addition to the three different elastoplastic materials, assigned to the sub-domains. The problem is then compared to the results from the finite element method, in which a very fine mesh (40x40) elements/domain was used to capture the micro scale. The implementation was performed considering the averaged strains as an input, and the averaged macroscopic stresses as an output applying the Hill-Mandel homogenization scheme. The stress per time step and stress-strain response averaged over the whole composite in comparison to the FE computations can be inferred from Fig. 2. We can observe a very good compliance with the FE output, where the behavior is captured qualitatively and quantitatively. Nevertheless, a deviation from the exact stiffness is observed in which the model is reflecting Reuss effective stiffness. www.gamm-proceedings.com