A Hypercomplex Multi-physics Approach for Fracture Mechanics and Inverse Material Parameter Determination




Ramirez Tamayo, Daniel

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The hypercomplex-variable finite element method, ZFEM, has the unique capability of computing arbitrary-order derivatives of a structure's response variable using a single finite element run through the use of hypercomplex algebras. In this research, first-order ZFEM analyses have been used to conduct structural integrity assessment of structures under different loading conditions such as mechanical, thermal, mixed-mode, interface cracks, bonded interfaces, and functionally graded materials. If the physical phenomenon is embodied within the finite element formulation, then the energy release rate can be computed straightforwardly using the proposed method as ZFEM obviates the use of corrective terms when extended into new physics or materials. The energy release rate, an important fracture parameter, is computed as the first-order derivative of the mechanical strain energy with respect to the crack area. When extended to hypercomplex algebras, ZFEM can be used to compute multiple order derivatives of the strain energy. If higher-order derivatives are considered during a crack propagation analysis, fewer crack increments are required. However, current available methods to solve a hypercomplex-valued system of equations are computationally expensive. Thus, the proposed method aims to develop efficient solution techniques within a commercial finite element software. In particular, through the systematic "hypercomplexification" of user-defined elements, UELs, in Abaqus. Once an existing finite element code is "hypercomplexified", highly accurate arbitrary-order derivatives of the nodal displacements with respect to any input parameter can be obtained. For the particular case of a finite element implementation dealing with cohesive elements, this information is useful for inversely determining the cohesive fracture parameters that govern the interfacial behavior of a joint from experimental or synthetic data using a finite element-based approach. First, the use of ZFEM to accurately compute first- and second-order derivatives of the strain energy in fracture mechanics analyses with different loading conditions or physics is verified. The results indicate that ZFEM's results are of the same accuracy as the J-integral formulation (or its modified versions) but ZFEM obviates the use of corrective terms. Then, the advantages of having accurate first-order derivatives during an optimization algorithm to inversely determine material parameters are demonstrated for several examples with experimental data. Finally, the process to implement ZFEM into a commercial finite element software is discussed. In addition to the particulars of the implementation, a new scheme for the solution of hypercomplex-valued system of equations in Abaqus is presented. The results indicate that the new method has the same accuracy as previous versions of ZFEM but exhibits superior computational efficiency.


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finite element methods, fracture mechanics, hypercomplex variables, inverse determination of parameters, optimization



Mechanical Engineering