In this work we show how a machine learning algorithm based on a Bayesian optimization framework can be used in conjunction with finite element analysis to autonomously select parameter values for a ductile damage model representative of experimental test data.

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# Themes: Advanced Computational Methods in Fracture

# SINGULAR INTEGRAL EQUATION FOR SOLVING COHESIVE CRACK PROBLEM FOR INITIALLY RIGID TRACTION-SEPARATION RELATION

In case of an initially rigid traction-separation cohesive relation, the total potential energy is not differentiable. This makes the use of variational operator over it questionable. Therefore, the accurate application of FEM is mathematically doubtful. The present work bypasses this issue by modelling the cohesive crack problem as a singular integral equation and solving it using Chebyshev polynomials.

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# REDEFINED J-INTEGRAL AND J-INTEGRAL RANGE DELTA-J AS FINITE STRAIN ELASTIC-PLASTIC CRACK PARAMETERS [Keynote]

The summary of applications of redefined three-dimensional J-integral and J-integral range Delta-J are presented in this paper. The redefined fracture parameters were derived with a rigorous consideration on energy dissipation into a small volume in the vicinity of the crack front. It can be seen as a rigorous extension of two-dimensional T_ε^*-integral to three-dimensional problem. The equation formulations are briefly presented in this paper. Then, their applications will be presented in the conference.

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# THE VIRTUAL ELEMENT METHOD FOR EFFICIENT CRACK TIP LOADING ANALYSIS AND CRACK GROWTH SIMULATION

To precisely model crack growth, accurate calculations of crack front loading and crack deflection angles are essential. These calculations require solutions of the underlying boundary value problems (BVPs), which are typically obtained by applying numerical methods, e.g., the finite element method (FEM). However, since accuracy and computational cost of the analyses are in general competing aspects, compromises often have to be made in order to generate satisfactory results in acceptable times. In contrast, the use of more efficient methods, both for the solution of the BVP as well as for the subsequent crack tip loading analyses, can substantially lower the computational effort while maintaining desired accuracies. The virtual element method (VEM) is a fairly new discretization scheme for the numerical solution of BVPs, and can be interpreted as a generalization of the FEM. Since the VEM can handle arbitrary polytopal meshes in a straightforward manner, it provides a higher degree of flexibility in the discretization process than the FEM, which turns out to be profitable in terms of both computing times and accuracy. This holds in particular for the simulation of crack growth in 2D and 3D, sparing adaptive re-meshing or the construction of discontinuous element shape functions.

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# 3D FRACTURE MECHANICS ANALYSIS OF THERMOMAGNETOELECTROELASTIC ANISOTROPIC SOLIDS ACCOUNTING FOR CRACK FACE CONTACT WITH FRICTION

Thermal expansion of the material usually causes existing cracks to close, resulting in the requirement to consider contact problems. The latter are complicated since one should consider the contact of crack faces accounting for sliding, friction, and unknown contact area. This study tries to solve this task by development of the 3D boundary element approach with iterative solver, which can determine the contact zone, sliding of crack faces and account for friction between them. Moreover, multifield materials and various thermal, mechanical, electric and magnetic boundary and contact conditions can be considered.

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# CYCLIC EFFECTIVE NEAR-FIELD LOADING BASED ON THE DOMAIN INTEGRAL METHOD

This paper presents a modification of the domain integral method for cyclic loading and crack closure to compute the cyclic effective J-Integral as a near-field loading parameter. The path-dependency of the solution is discussed for different reference states of the field quantities in a cycle. It turns out that refering to the crack opening time point the cyclic effective J-Integral is path-independent for a domain outside the active plastic zone. The validity of this procedure is discussed by comparison with a global energy approach and theoretical field solutions for the J-controlled zone.

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# MULTIPHYSICS ANALYSIS OF PHOTOVOLTAIC SOLAR CELLS [Keynote]

The physics of power generation on photovoltaic solar cells is complex, involving multiple fields. In this study, the performance characteristics of Silicon based solar cells are investigated considering mechanical, therma, electrical and chemical fields through a finite element (FE) analysis. A fully coupled thermo-mechanical-electrostatic-chemical relations for a deformable semiconductor are planned to be developed by combining the thermodynamically consistent formulation of the interaction between electric field and polarizable matter with the carrier charge transportation. A functional form of free energy considering the energies due to all the above-mentioned fields will be defined, through which the constitutive relations can be derived. As a result, the influence of the densities of electrons and holes can be coupled to the mechanical equilibrium. Therefore, the FE analysis can be finally extended to estimate the I-V characteristics of a p-n junction. Furthermore, each node of such three dimensional finite element is estimated possesses 14 degrees of freedom, leading to a total of 112 degrees of freedom per element.

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# DEVELOPMENT AND APPLICATION OF THE HYPERCOMPLEX FINITE ELEMENT METHOD FOR LINEAR AND NONLINEAR ENERGY RELEASE RATE CALCULATIONS [Keynote]

The augmentation of existing finite element codes to use complex and hypercomplex variables and algebras provides an accurate and straightforward method to compute the energy release rate for linear and nonlinear solids. The basic concept is to introduce complex nodes defined by real and imaginary nodes. The real nodes define the geometry and the imaginary nodes define the perturbation to the real mesh. The crack is extended using imaginary coordinates surrounding the crack tip. The solution of the complex system of equations then yields a complex displacement with the imaginary displacement equal to the derivative of the displacement with respect to the crack length. Subsequently, the energy release rate (the derivative of the strain energy with respect to the crack length) can be determined using from the complex strains and stresses. The results indicate that the ERR results are as accurate as the J integral but the method has several advantages: there are no contours to interrogate – only one result is provided, the method works for both linear and nonlinear materials with loading and unloading, unlike the J integral, and no integral formulation must be developed and implemented. Numerical examples demonstrate the accuracy of the method.

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# FINITE ELEMENT MODELING FOR PREDICTING OPTIMAL HOLE PROFILE IN A FINITE SQUARE PLATE OF HETEROGENEOUS BRITTLE MATERIAL (WC+CO) UNDER UNIAXIAL COMPRESSION OR UNIAXIAL DISPLACEMENT

The objective of this paper is to develop numerical models to predict and optimize the ratio (D/W) of hole diameter D over plate width W of a square plate with a center hole. The plate is made from tungsten carbide. The geometry of the model was a square plate with a circular hole in the center. FEM simulation was performed for hole diameter to plate width ratio from 0 to 0.71 in terms of fracture strength (Sut or Suc) under uniaxial compression, or uniaxial displacement. SCF values in the simulations showed good fit with analytical values.It is shown that maximum normal tensile stress develops at the upper point along the free edge of vertical hole,and maximum compressive stresses at left and right horizontal points along the free edge of the hole.. The numerical solution of the normal tensile stress distrbution on the “future fracture plane in Mode I” guarantees a certain degree of stability in the crack propagation in heterogeneous brittle materials.This stability, caused the compliance of the plate to remain independent of crack length, and hence

the fracture toughness can be measured by the critical load itself. The results are relevant to the design of inserts cutting tools.

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# MECHANICAL MODEL OF SLIDING FRICTION AND THE STUDY OF THE ONSET OF SLIDING FRICTION

Friction widely exists in our daily life and nature, and the onset of sliding friction plays an important role. However, the underlying physical mechanism of this dynamic process is still unclear. This paper will further explore the physical nature of crack like defects. We reduce the experimental configuration to a slider-substrate model, where the slider can be described using thin long beams and the substrate is considered as an elastic half-space. In this way, the relevant displacement and stress field solutions can be obtained by solving Cauchy singular integral equations. The numerical results can well describe the experimental results. By introducing a critical criterion for static dislocation nucleation, the calculated critical forces are in good agreement with those of the sliding precursor. Based on the model, the dynamics of the sliding precursor is further considered. It is found that the strain field caused by the moving dislocation is in good agreement with the strain field caused by the defect in the experiment, and the transient emission process of the interface edge dislocation is similar to the spatio-temporal dynamic behavior observed in the experiment. These works may contribute to further understanding of the mechanism related to sliding friction processes.

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