Data‐Driven simulation of inelastic materials using structured data sets and tangential transition rules

Data‐driven computational mechanics replaces phenomenological constitutive functions by performing numerical simulations based on data sets of representative samples in stress‐strain space. The distance of modeling values, e.g. stresses and strains in Gauss‐points of a finite element calculation, from the data set is minimized with respect to an appropriate metric, subject to equilibrium and compatibility constraints, see [1]. Although this method operates well for non‐linear elastic problems, there are challenges dealing with history‐dependent materials, since one point in stress‐strain space might correspond to different material behaviour. In [2], this issue is treated by including local histories into the data set. However, there is still the necessity to include models for the evolution of internal variables. Thus, a mixed formulation is obtained consisting of a combination of classical and data‐driven modeling.


Data-driven approach
The distance-minimizing data-driven problem, introduced by [1], for a discretized system (e.g. by the finite element method) which undergoes displacements u subjected to applied forces f , reads with an appropriate distance function d(·, ·). In this case m ∈ N is the number of material points, {w e } m e=1 are elements of volume, {B e } m e=1 are strain-displacement operators and D e = {(ε i , σ i ), i = 1, . . . , n ∈ N} are data sets consisting of finite number of local material states. Thus, the aim of the data-driven computation is to find the closest point consistent with the kinematics and equilibrium laws to a material data set.

Extension to inelasticity using tangent spaces
An extension of the data-driven paradigm regards inelastic materials since points in stress-strain space might correspond to different material behaviour. Rather including local histories into the data set as in [2] the data set will be extended by the tangent space i.e.
where ∆ε i and ∆σ i are the tangential strain and stress increment. Furthermore the data sets will be separated in two sets of different material behavior e.g. elastic and inelastic D e = D elastic σ e = σ e + C e ∆ε e , ∀e = 1 . . . , m.
Given a list of modeling {ẑ k e } m e=1 and data points {z k e } m e=1 , the material state at loading step (k + 1) can be calculated by the data-driven solver in four steps: • find values of modeling points {ẑ k+1 e } m e=1 by solving the equations (4), (5) and (6) using the data points {z k e } m e=1 ; • assign local data sets {D e } m e=1 bỹ where σ 0 e is the yield limit and δ y (·) is the yield stress; • if σ e · ∆ε e < 0 for all material points, set σ 0 e := δ y (σ e ).

Numerical results for 3D problem
To visualize the convergence properties of the data-driven extension the problem of a cantilever beam subjected to a load causing plastic deformation is considered. Following [2], a virtual test is used to generate accurate coverage of suitable local material states; in this case 160 data set samples. Thus, a standard data set is considered by randomizing an elasto-plasticity with isotropic hardening data set with isotropic hardening. First the load is applied from 0 to 1 · 10 7 , down to 0 and then raised up again to 1.2 · 10 7 . The material parameters of the reference solid used in the data-driven computing are Young's modulus E = 200 · 10 9 Pa, Poisson's ratio ν = 0.3 and initial yield stress σ 0 = 200 · 10 6 Pa. Figure 1a shows the data-driven based displacement solution at the maximum loading magnitude 1.2 · 10 7 using a data sample containing 10000 points. Figure 1b illustrates the convergence of the data-driven solution towards the reference elasto-plasticity with isotropic hardening model solution by increasing number of material data points.