PSI - Issue 42

Pauline Herr et al. / Procedia Structural Integrity 42 (2022) 498–505 P. Herr et al. / Structural Integrity Procedia 00 (2019) 000–000

499

2

direction of the adhesive layer are usually considered. In addition, the whole adhesive layer is treated as a finite thickness cohesive zone, which typically leads to a thickness dependent mechanical behavior. Nevertheless, they are generally able to model the fracture behavior with good accuracy. The Traction Separation Laws (TSL) for cohesive zone models can be directly measured by experiments, e.g., with the Double Cantilever Beam (DCB) test in peel mode I. This can involve a significant e ff ort if di ff erent types and volume fractions of fillers and thicknesses of the adhesive layer should be tested in order to optimize the mechanical behavior for specific applications. A virtual development process for adhesives, which is based on a multiscale model using computational homogenization, could reduce the number of experiments needed. In such a (fully coupled) multiscale model for structural FE simulations, a Representative Volume Element (RVE) is assigned to each integration point in the FE model at the macroscale and the boundary value problem at the mi croscale, also referred to as homogenization problem, is also solved using another FE model often. This is usually called a FE 2 approach, cf. Matousˇ et al. (2017). The homogenization problem thereby needs to be solved at least once for each load step at the macroscale in each integration point, which requires tremendous computational resources. Spahn et al. (2014) proposed a FE-FFT scheme instead, whereby a Fast Fourier Transform (FFT) based microme chanics solver is used for the homogenization problem. These FFT solvers can achieve a significant reduction of the computational times and memory consumption compared to classical FE simulations; however, the resources required in this approach are still high for complex microstructures. Therefore, surrogate models for the mechanical behavior at the microscale can be used to make multiscale structural FE simulations feasible for a broader range of applications. Data-driven approaches based on Machine Learning (ML) techniques provide high adaptivity regarding various input and output data and are therefore well-suited for surrogate models, as, e.g., shown by Li and Zhuang (2020). In this work, we propose a ML-based surrogate model for cohesive zones, which is able to take into account the e ff ect of di ff erent volume fractions of glass beads as filler on the fracture behavior of an adhesive layer in mode I. The training data for the supervised learning are obtained from FFT simulations with our in-house Fortran code, see Bo¨deker et al. (2022). This involves o ffl ine learning the constitutive behavior with a selection of solutions of the homogenization problem. Furthermore, the ML-based model was implemented in the commercial FE software Abaqus via a UMAT subroutine and its capabilities regarding structural FE simulations were assessed by an exemplary simulation of a DCB test. It should be noted that the material parameters used in this work are chosen purely from experimental experience and literature data, i.e., they are of a realistic order of magnitude, but are not based on experiments. Thus, the results should be rather interpreted qualitatively way and as a proof of concept for the proposed model and date generation strategy. One possibility for an ML-based surrogate cohesive zone model would be a direct link of separations and their history to the resulting traction vector. Nevertheless, this could easily lead to an nonphysical mechanical behavior, de pending on the available training data. Moreover, complex load paths including unloading are required in the training data, which increases the complexity of the data generation. In our implementation of the FFT-based micromechanics solver, we use extrapolations of the solution fields from previous load steps as initial guess, which can significantly reduce the computational time, but cannot be applied if the loading direction changes. Another option is to make some assumptions on internal state variables and their evolution, see also the work by Reimann et al. (2019), which is why we propose a model of the form 2. Procedure

σ = (1 − D M ) K ( u − u

pl ),

(1)

where K is the sti ff ness of the cohesive zone, D M the macroscopic scalar damage variable and u pl the scalar plastic separation. It should be noted that the model in this work is limited to mode I only, and hence is purely scalar. For the evolution of the both state variables D M and u pl , a Feed Forward Neural Network (FFNN) is used each, which has the maximum value of separation in load history u max and the filler volume fraction v f as input data. It is remarked that the evolution equations of state variables in standard cohesive models for adhesive layers are also commonly formulated as functions of a maximum equivalent separation during load history, e.g., Camanho and Da´vila (2002).