Development of 3D Trefftz Voronoi Cells with Ellipsoidal Voids &/or Elastic/Rigid Inclusions for Micromechanical Modeling of Heterogeneous Materials

In this paper, as an extension to the authors's work in [Dong and Atluri (2011a,b, 2012a,b,c)], three-dimensional Trefftz Voronoi Cells (TVCs) with ellipsoidal voids/inclusions are developed for micromechanical modeling of heterogeneous materials. Several types of TVCs are developed, depending on the types of heterogeneity in each Voronoi Cell(VC). Each TVC can include alternatively an ellipsoidal void, an ellipsoidal elastic inclusion, or an ellipsoidal rigid inclusion. In all of these cases, an inter-VC compatible displacement field is assumed at each surface of the polyhedral VC, with Barycentric coordinates as nodal shape functions. The Trefftz trial displacement fields in each VC are expressed in terms of the Papkovich-Neuber solution. Ellipsoidal harmonics are used as the Papkovich-Neuber potentials to derive the Trefftz trial displacement fields. Characteristic lengths are used for each VC to scale the Trefftz trial functions, in order to avoid solving systems of ill-conditioned equations. Two approaches for developing VC stiffness matrices are used. The differences between these two approaches are that, the compatibility between the independently assumed fields in the interior of the VC with those at the outer- as well as the inner-boundary, are enforced alternatively, by Lagrange multipliers in multi-field boundary variational principles, or by collocation at a finite number of preselected points. These VCs are named as TVC-BVP and TVC-C respectively. Several three-dimensional computational micromechanics problems are solved using these TVCs. Computational results demonstrate that both TVC-BVP and TVC-C can efficiently predict the overall properties of composite/porous materials. They can also accurately capture the stress concentration around ellipsoidal voids/inclusions, which can be used in future to study the damage of materials, in combination of tools of modeling micro-crack initiation and propagation. Therefore, we consider that the 3D TVCs developed in this study are very suitable for ground-breaking micromechanical study of heterogeneous materials

On Improving the Celebrated Paris’ Power Law for Fatigue, by Using Moving Least Squares

In this study, we propose to approximate the a－n relation as well as the da/dn－∆K relation, in fatigue crack propagation, by using the Moving Least Squares (MLS) method. This simple approach can avoid the internal inconsistencies caused by the celebrated Paris’ power law approximation of the da/dn－∆K relation, as well as the error caused by a simple numerical differentiation of the noisy data for a－n measurements in standard fatigue tests. Efficient, accurate and automatic simulations of fatigue crack propagation can, in general, be realized by using the currently developed MLS law as the “fatigue engine” [da/dn versus ∆K], and using a high-performance “fracture engine” [computing the K-factors] such as the Finite Element Alternating Method. In the present paper, the “fatigue engine” based on the present MLS law, and the “fracture engine” based on the SafeFlaw computer program developed earlier by the authors, in conjunction with the COTS software ANSYS, were used for predicting the total life of arbitrarily cracked structures. By comparing the numerical simulations with experimental tests, it is demonstrated that the current approach can give excellent predictions of the total fatigue life of a cracked structure, while the celebrated Paris’ Power Law may miscalculate the total fatigue life by a very large amount

A Fuzzy-set-based Joint Distribution Adaptation Method for Regression and its Application to Online Damage Quantification for Structural Digital Twin

Online damage quantification suffers from insufficient labeled data. In this context, adopting the domain adaptation on historical labeled data from similar structures/damages to assist the current diagnosis task would be beneficial. However, most domain adaptation methods are designed for classification and cannot efficiently address damage quantification, a regression problem with continuous real-valued labels. This study first proposes a novel domain adaptation method, the Online Fuzzy-set-based Joint Distribution Adaptation for Regression, to address this challenge. By converting the continuous real-valued labels to fuzzy class labels via fuzzy sets, the conditional distribution discrepancy is measured, and domain adaptation can simultaneously consider the marginal and conditional distribution for the regression task. Furthermore, a framework of online damage quantification integrated with the proposed domain adaptation method is presented. The method has been verified with an example of a damaged helicopter panel, in which domain adaptations are conducted across different damage locations and from simulation to experiment, proving the accuracy of damage quantification can be improved significantly even in a noisy environment. It is expected that the proposed approach to be applied to the fleet-level digital twin considering the individual differences.Comment: 29 pages, 10 figure

Stochastic Macro Material Properties, Through Direct Stochastic Modeling of Heterogeneous Microstructures with Randomness of Constituent Properties and Topologies, by Using Trefftz Computational Grains (TCG)

In this paper, a simple and reliable procedure of stochastic computation is combined with the highly accurate and efficient Trefftz Computational Grains (TCG), for a direct numerical simulation (DNS) of heterogeneous materials with microscopic randomness. Material properties of each material phase, and geometrical properties such as particles sizes and distribution, are considered to be stochastic with either a uniform or normal probabilistic distributions. The objective here is to determine how this microscopic randomness propagates to the macroscopic scale, and affects the stochastic characteristics of macroscopic material properties. Four steps are included in this procedure: (1) using the Latin hypercube sampling, to generate discrete experimental points considering each contributing factor (material parameters and volume fraction of each phase, etc.); (2) randomly generating Representative Volume Elements (RVEs) of the microstructure for each discrete experimental point, and compute the effective macro-scale material properties at these points, using the computationally most efficient Trefftz Computational Grains; (3) relating the macro-scale material properties to the microscale random variables using the Kriging method; (4) taking advantage of the approximate macro-micro relation, and using the Monte Carlo simulation, to establish the probabilistic distribution of the macro-scale material properties. By considering the Al/SiC composite as an example, we give step-by step demonstration of the procedure, and give some comparisons with experimental results. The obtained probabilistic distributions of the effective macro-scale material properties have fundamental engineering merits, which can be used for reliability-based material optimization, and integrated-design of micro- as well as macro-structures. The studies in this paper are germane to the concepts of the Materials Genome Initiative (MGI), and Integrated Materials Science, Mathematics, Modeling, and Engineering (IMSMME)

Spherical nano-inhomogeneity with the Steigmann-Ogden interface model under general uniform far-field stress loading

An explicit solution, considering the interface bending resistance as described by the Steigmann-Ogden interface model, is derived for the problem of a spherical nano-inhomogeneity (nanoscale void/inclusion) embedded in an infinite linear-elastic matrix under a general uniform far-field-stress (including tensile and shear stresses). The Papkovich-Neuber (P-N) general solutions, which are expressed in terms of spherical harmonics, are used to derive the analytical solution. A superposition technique is used to overcome the mathematical complexity brought on by the assumed interfacial residual stress in the Steigmann-Ogden interface model. Numerical examples show that the stress field, considering the interface bending resistance as with the Steigmann-Ogden interface model, differs significantly from that considering only the interface stretching resistance as with the Gurtin-Murdoch interface model. In addition to the size-dependency, another interesting phenomenon is observed: some stress components are invariant to interface bending stiffness parameters along a certain circle in the inclusion/matrix. Moreover, a characteristic line for the interface bending stiffness parameters is presented, near which the stress concentration becomes quite severe. Finally, the derived analytical solution with the Steigmann-Ogden interface model is provided in the supplemental MATLAB code, which can be easily executed, and used as a benchmark for semi-analytical solutions and numerical solutions in future studies.Comment: arXiv admin note: text overlap with arXiv:1907.0059

Application of the MLPG Mixed Collocation Method for Solving Inverse Problems of Linear Isotropic/Anisotropic Elasticity with Simply/Multiply-Connected Domains

In this paper, a novel Meshless Local Petrov-Galerkin (MLPG) Mixed Collocation Method is developed for solving the inverse Cauchy problem of linear elasticity, wherein both the tractions as well as displacements are prescribed/measured at a small portion of the boundary of an elastic body. The elastic body may be isotropic/anisotropic and simply connected or multiply-connected. In the MLPG mixed collocation method, the same meshless basis function is used to interpolate both the displacement as well as the stress fields. The nodal stresses are expressed in terms of nodal displacements by enforcing the constitutive relation between stress and the displacement gradient tensor at each nodal point. The equations of linear momentum balance are satisfied at each node using collocation method. The displacement as well as traction boundary conditions are also enforced at each measurement location along the boundary where the conditions are over specified on displacement as well as tractions. The current method is very simple because the inverse problem is directly solved in a fashion similar to a direct problem, without resorting to any iterative optimization. The current method is also very general because it can be applied to arbitrary simply/multiply connected bodies composed of arbitrary isotropic/anisotropic material, and it can also be adapted to solve inverse problems of other physics such as heat transfer, electro-magnetics, etc. Several numerical examples demonstrate the effectiveness and robustness of the current method, even when the prescribed displacement/tractions are corrupted with measurement noises. The extension of the current method to solve nonlinear inverse problems will be straightforward within the framework of incremental loading, which will be explored in future studies

Are “Higher-Order” and “Layer-wise Zig-Zag” Plate & Shell Theories Necessary for Functionally Graded Materials and Structures?

Similar to the very vast prior literature on analyzing laminated composite structures, "higher-order" and "layer-wise higher-order" plate and shell theories for functionally-graded (FG) materials and structures are also widely popularized in the literature of the past two decades. However, such higher-order theories involve (1) postulating very complex assumptions for plate/shell kinematics in the thickness direction, (2) defining generalized variables of displacements, strains, and stresses, and (3) developing very complex governing equilibrium, compatibility, and constitutive equations in terms of newly-defined generalized kinematic and generalized kinetic variables. Their industrial applications are thus hindered by their inherent complexity, and the fact that it is difficult for end-users (front-line structural engineers) to completely understand all the newly-defined generalized DOFs for FEM in the higher-order and layer-wise theories. In an entirely different way, very simple 20-node and 27-node 3-D continuum solid-shell elements are developed in this paper, based on the simple theory of 3D solid mechanics, for static and dynamic analyses of functionally-graded plates and shells. A simple Over-Integration (a 4-point Gauss integration in the thickness direction) is used to evaluate the stiffness matrices of each element, while only a single element is used in the thickness direction without increasing the number of degrees of freedom. A stress-recovery approach is used to compute the distribution of transverse stresses by considering the equations of 3D elasticity in Cartesian as well as cylindrical polar coordinates. Comprehensive numerical results are presented for static and dynamic analyses of FG plates and shells, which agree well, either with the existing solutions in the published literature, or with the computationally very expensive solutions obtained by using simple 3D isoparametric elements (with standard Gauss Quadrature) available in NASTRAN (wherein many 3D elements are used in the thickness direction to capture the varying material properties). The effects of the material gradient index, the span-to-thickness ratio, the aspect ratio and the boundary conditions are also studied in the solutions of FG structures. Because the proposed methodology merely involves: (2) standard displacement DOFs at each node, (2) involves a simple 4-point Gaussian over-integration in the thickness direction, (3) relies only on the simple theory of solid mechanics, and (4) is capable of accurately and efficiently predicting the static and dynamical behavior of FG structures in a very simple and cost-effective manner, it is thus believed by the authors that the painstaking and cumbersome development of "higher-order" or "layer-wise higher-order" theories is not entirely necessary for the analyses of FG plates and shells

Are Higher-Order Theories and Layer-wise Zig-Zag Theories Necessary for N-Layer Composite Laminates?

Although “higher-order” and layer-wise “higher-order” plate and shell theories for composite laminates are widely popularized in the current literature, they involve (1) postulating very complex assumptions of plate/shell kinematics in the thickness direction, (2) defining generalized variables of displacements, strains, and stresses, and (3) developing very complex governing equilibrium, compatibility, and constitutive equations in terms of newly-defined generalized kinemaic and generalized kinetic variables. Their industrial applications are thus hindered by their inherent complexity, and the fact that it is difficult for end-users (front-line structural engineers) to completely understand all the newly-defined FEM DOFs in higher-order and layer-wise theories. In an entirely different way, the authors developed very simple lowest-order (8-node hexahedral), and higher-order (32-node hexahedral) 3-D continuum solid-shell elements, based on the theory of 3D solid mechanics, for static and dynamic analyses of composite laminates. The shear-locking of the lower-order 8-node hexahedral element is alleviated by independently assuming locking-free strain fields for each element. Over-integration is used to evaluate the element stiffness matrices of laminated structures with an arbitrary number of laminae, while only one element is used in the thickness direction without increasing the number of degrees of freedom. A stress-recovery approach is used to compute the distribution of transverse stresses by considering the equations of 3D elasticity. Comprehensive numerical results are presented for static, free vibration, and transient analyses of different laminated plates and shells, which agree well with existing solutions in the published literature, or solutions of very-expensive 3D models (where 3D elements are used to model each layer) by using commercial FEM codes. Because the proposed methodology merely involves simple displacement DOFs at each node, relies only on the simple theory of solid mechanics, and is capable of accurately and efficiently predicting the static and dynamical behavior of composite laminates in a very simple and cost-effective manner, it is thus believed by the authors that the development of “higher-order” or “layer-wise higher-order” theories are not entirely necessary for analyses of laminated plates and shells