Effective Elastic Properties of Fractured Rocks: Dynamic vs. Static Considerations

We present a new technique for static computations of effective elastic properties of multiple fractured rocks. This approach is based on the viscoelastic rotated staggered finite-difference (FD) grid wave propagation technique. Our simulations are used to explain discrepancies of some recent numerical studies. The focus is on scale effects of a so-called representative volume element (RVE). From the point of view of classical micromechanics we review different numerical techniques: Static as well as dynamic numerical experiments. We show that the differential effective medium theory (DEM) is capable of producing satisfactory predictions of effective elastic moduli. For non-dilute crack densities this is not the case for the non-interacting approximation (NIA).     

Evaluation of effective elastic properties of 3D printable interpenetrating phase composites using the meshfree radial point interpolation method

ABSTRACT

Interpenetrating phase composites (IPCs) have recently been fabricated using three-dimensional (3D) printing methods. In a two-phase IPC, the two phases are topologically interconnected and mutually reinforced in the three dimensions. As a result, such IPCs exhibit higher stiffness, strength, and toughness than particle- or fiber-reinforced composites. In the current study, three unit cell models for the IPCs with the simple cubic (SC), face-centered cubic (FCC), and body-centered cubic (BCC) microstructures are developed using the meshfree radial point interpolation method. Radial basis functions with polynomial reproduction are applied to construct shape functions, and the Galerkin method is employed to formulate discretized equations. These unit cell-based meshfree models are used to evaluate effective elastic properties of 3D printable IPCs. The simulation results are compared with those based on the finite element (FE) method and various analytical bounding techniques in micromechanics, including the Voigt–Reuss, Hashin–Shtrikman, and Tuchinskii bounds. It is found that all of the simulation results for the effective Young's modulus and shear modulus fall between the Voigt–Reuss upper and lower bounds for each IPC considered, with the FE models predicting higher values than the meshfree models. In addition, it is seen that the SC microstructure leads to higher effective Young's modulus than the BCC and FCC microstructures. Furthermore, the numerical results reveal that the IPCs with the SC, BCC, and FCC microstructures can be approximated as isotropic materials (with the Zener anisotropic ratio varying between 0.9 and 1.0), with the BCC IPC being the most isotropic one, and the SC IPC being the least isotropic one among the three types of IPCs.     

Computation of the bounds on the elastic moduli of a fiber-reinforced composite by Monte Carlo simulations

Bounds on the effective elastic moduli of a fiber-reinforced composite are studied. The composite is composed of infinitely long, parallel, equal-sized circular cylinders (fibers) randomly embedded in a matrix. Based on the variational principles and by employing the cylindrical inclusion solutions as trial functions, the bounds are determined through Monte Carlo simulations. Comparisons are made with the second-order and third-order perturbation bounds. Because the shape of the inclusion has been explicitly taken into consideration, it is found that the bounds computed by the present approach are the sharpest in terms of the width of the bound pair. Specifically, when the fibers are stiffer than the matrix, the present lower bounds coincide with the second-order lower bounds at low concentrations and are slightly higher than the second-order lower bounds at high concentrations, while the upper bounds are far below the corresponding second-order and third-order perturbation bounds. In many cases, the present bound pair is very close and thus provides accurate prediction on the effective properties of this class of composites.     

Adaptive combined DE/FE algorithm for brittle fracture of plane stress problems

A novel adaptive combined DE/FE algorithm is proposed to simulate the fracture procedure of brittle materials of plane stress problems. The main concept of the approach is that a model is composed of the finite element completely at the initial stage without any discrete element generated until portion of the model grid becoming severely deformed; and then the model is fragmented into two subdomains, the finite element (FE) and the discrete element (DE) subdomains. The interface force between the two subdomains is calculated by using the penalty method. An extrinsic cohesive fracture model is employed to simulate the brittle fracture procedure only in the DE subdomain. The adaptive algorithm may allow for the use of the accurate and efficient FEs in the lower distorted region and the DEs which are automatically generated in the severely deformed FE region . The feasibility of the adaptive algorithm is validated by the impact fracture simulation of a glass beam. The comparison of calculation time consumption shows that the adaptive algorithm has a higher efficiency than the DEM. At last, the impact fracture behavior of a laminated glass beam is simulated, and the cracks propagation is compared with the experimental results showing that the adaptive algorithm can be implemented to capture some fracture characteristics of brittle materials.     

Strict error bounds for linear solid mechanics problems using a subdomain-based flux-free method

We discuss, in this paper, a flux-free method for the computation of strict upper bounds of the energy norm of the error in a Finite Element (FE) computation. The bounds are strict in the sense that they refer to the difference between the displacement computed on the FE mesh and the exact displacement, solution of the continuous equations, rather than to the difference between the displacements computed on two FE meshes, one coarse and one refined. This method is based on the resolution of a series of local problems on patches of elements and does not require the resolution of a previous problem of flux equilibration, as happens with other methods. The paper concentrates more specifically on linear solid mechanics issues, and on the assessment of the energy norm of the error, seen as a necessary tool for the estimation of the error in arbitrary quantities of interest (linear functional outputs). Applications in both 2D and 3D are presented.     

A differential approach to microstructure-dependent bounds for multiphase heterogeneous media

We present a new method of deriving microstructure-dependent bounds on the effective properties of general heterogeneous media. The microstructure is specified by the average Eshelby tensors. In the small contrast limit, we introduce and calculate the expansion coefficient tensors. We then show that the effective tensor satisfies a differential inequality with the initial condition given by the expansion coefficient tensors in the small contrast limit. Using the comparison theorem, we obtain rigorous bounds on the effective tensors of multiphase composites. These new bounds, taking into account the average Eshelby tensors for homogeneous problems, are much tighter than the microstructure-independent Hashin–Shtrikman bounds. Also, these bounds are applicable to non-well-ordered composites and multifunctional composites. We anticipate that this new approach will be useful for the modeling and optimal design of multiphase multifunctional composites.     

Effects of Parallel Crack Distributions on Effective Elastic Properties - a Numerical Study

This paper is concerned with numerical studies of effective elastic properties of cracked solids. We concentrate on two dimensional media containing different patterns of parallel crack distributions. We use the Rotated Staggered Grid (RSG) which allows one to simulate elastic wave propagation very accurately in fractured media. Our aim is to compare the predictions given by several effective medium theories to the effective properties we derive from our numerical experiments. Namely, these are the ``Non-interaction approximation (NIA)'', the ``Differential scheme (DS)'' and an extension of the DS (EDS). According to our results, the DS theory and its extension perform well. Simulations of media containing very few cracks prove that for our setup the effective properties stabilize at low numbers of cracks. Finally, we studied parallel cracks clustered in stacked columns. We found that, as expected, the shielding effects dominate in such patterns.     

A Numerical Comparison of Finite Difference and Finite Element Methods for a Stochastic Differential Equation with Polynomial Chaos

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.     

Closed-form effective elastic constants of frame-like periodic cellular solids by a symbolic object-oriented finite element program

In this study, the effective elastic constants of several 2D and 3D frame-like periodic cellular solids with different unit-cell topologies are analytically derived using the homogenization method based on equivalent strain energy. The analytical expressions of strain energy of a unit cell under different strain modes are determined using a generic symbolic object-oriented finite element (FE) program written in MATLAB. The obtained analytical expressions of the strain energy are then used to symbolically compute the effective elastic constants that include Young’s moduli, Poisson’s ratios, and shear moduli. The obtained analytical effective elastic constants are numerically verified using results from an ordinary numerical FE program. The obtained closed-form effective elastic constants are also compared with some existing solutions from the literature. This study demonstrates that symbolic computation platforms can be properly used to provide efficient methodologies for finding useful analytical solutions of mechanical problems. Without the symbolic object-oriented FE program in this study, elaborate and tedious analytical analysis has to be manually performed for each different unit cell. The symbolic object-oriented FE program provides analytical analysis of unit cells that is accurate and fast. The object-oriented programming technique allows the symbolic FE program in this study to be efficiently implemented.     

Effect of overload on fatigue crack retardation of aerospace Al-alloy laser welds using crack-tip plasticity analysis

Investigations on fatigue crack growth retardation due to single tensile and periodic multiple over load in strength undermatched laser beam welded 3.2 mm thick aerospace grade aluminium alloy 2139-T8 sheets are conducted. The effect of overload on the fatigue crack propagation behaviours of the homogenous base metal and welded panels (200 mm wide, centre cracked) was compared using experimental and FE analysis methods. The effective crack tip plasticity has been determined in homogeneous M(T) specimens using Irwin’s method and in both homogeneous and laser welded specimen by calculating crack tip plastic strain using FE analysis for single tensile overload. The crack retardation due to the overload in welded specimens is described by the Wheeler Model. The crack tip plastic zone size in the welded specimen was determined by FE analysis using maximum plastic zone extension at the mid sheet thickness. The results show that the Wheeler Model can be implemented to the highly heterogeneous undermatched weld to describe the crack retardation in fatigue following single tensile overload. Fatigue crack growth retardation due to single overload is found to be larger than the base metal. However, after periodic multiple overload, shorter crack retardation has occurred for undermatched welds than the base metal.     

Modal effective electromechanical coupling approximate evaluations and simplified analyses: numerical and experimental assessments

This contribution presents numerical and experimental assessments of the modal effective electromechanical coupling coefficient (EMCC) using popular approximate evaluations and simplified analyses of piezoelectric structures. For this purpose, first, a common benchmark, consisting of a cantilever Aluminum (Al) beam with symmetrically surface-bonded two pairs of large piezoceramic (PZT) patches, is retained for the assessment of EMCC different evaluation formulas and plane strain (PStrain) and plane stress (PStress) two-dimensional (2D) analyses using ANSYS \({^\circledR}\) coupled piezoelectric three-dimensional (3D) and 2D finite elements (FE). Then, similarly, an experimental assessment is conducted on two benchmarks consisting of Al long and short cantilevers equipped symmetrically with pairs of small and large PZT patches. It is found that, in order to get EMCC accurate approximate numerical evaluation, it is crucial to consider the patches electrodes equipotential constraints and, in order to get EMCC accurate 2D results with regard to 3D calculations, it is necessary to use PStress kinematics for approximate 2D analysis. Besides, 3D FE and experimental frequencies are shown to be bounded from below by PStress and from above by PStrain 2D FE results. Moreover, EMCC 2D PStress results are found closer to 3D FE and experimental results than PStrain 2D FE ones.     

Fluid Substitution in Porous and Fractured Solids: The Non-Interaction Approximation and Gassmann Theory

We establish an exact equivalence of the non-interaction approximation (NIA) and Gassmann theory in describing the changes in effective elasticity due to variations in the bulk modulus of fluid infill of isolated inclusions that have identical shapes and orientations. If the sizes of inclusions (pores or fractures) are equal, the fluid pressures in all inclusionsare equal too regardless of their hydraulic connectivity. This fact makes Gassmann theory rigorous; therefore, other effective media schemes should comply with it when applied to such microstructures. While the NIA satisfies this requirement, other effective media schemes (e.g., self-consistent and differential) do not.     

Evaluation of a simple finite element method for the calculation of effective electrical conductivity of compression moulded polymer–graphite composites

Finite element (FE) techniques can be used for the calculation of the effective properties of random heterogeneous materials, the required input simply consisting of phase properties and representative three-dimensional models of material microstructure. This approach has been widely exploited in recent years, although limited by the considerable amount of computational power required to obtain statistically accurate results. By using simple microstructural models of compression moulded polymer–graphite composites and a FE code modified for execution on graphical processing units, we show that reliable predictions of electrical properties for these materials can now be obtained in a reasonable computational time and with acceptable accuracy and precision. By using an approach based on design of experiments, we also perform a set of simulations aimed at determining the microstructural details which are most significant for the effective properties of these materials.     

Three-dimensional elasticity solution for static response of simply supported orthotropic cylindrical shells

This paper deals with the analysis of homogeneous and laminated cylindrical panels made of an orthotropic material, such as the fiber reinforced composites, using the three-dimensional elasticity equations. Solution is obtained by utilizing the assumption that the ratio of the panel thickness to its middle surface radius is negligible as compared to unity. However, it is shown that by sub-dividing the panel thickness into sub-layers of smaller thickness and matching the interface displacement and stress continuity conditions, very accurate results can be obtained. The two-dimensional shell theories have been compared for their accuracy in the light of the present three-dimensional elasticity analysis. Numerical results for some orthotropic panels show that the two-dimensional shell theories are very inaccurate when the thickness to length ratio of the panel is more than 1/20. Also, it is observed that the predictions of the two-dimensional shell theories are relatively poor in the case of two-layered panels as compared to three-layered and homogeneous panels.     

FE/FMBE coupling to model fluid–structure interaction

In this paper, a finite element (FE)/fast multipole boundary element (FMBE)‐coupling method is presented for modeling fluid–structure interaction problems numerically. Vibrating structures are assumed to consist of elastic or sound absorbing materials. An FE method (FEM) is used for this part of the solution. This structural sub‐domain is embedded in a homogeneous fluid. The case where the boundary of the structural sub‐domain has a very complex geometry is of special interest. In this case, the BE method (BEM) is a more suitable numerical tool than FEM to account for the sound propagation in the homogeneous fluid. The efficiency of the BEM is increased by using FMBEM. The BE‐surface mesh required is directly generated by the FE‐mesh used to discretize the structural sub‐domain and the absorbing material. This FE/FMBE‐coupling method makes it possible to predict the effects of arbitrarily shaped absorbing materials and vibrating structures on the sound field in the surrounding fluid numerically. The coupling method proposed is used to study the acoustic behavior of the lining of an anechoic chamber and that of an entire anechoic chamber in the low‐frequency range. The numerical results obtained are compared with the experimental data. Copyright © 2008 John Wiley & Sons, Ltd.     

On approximate solutions for the deformation of plane anisotropic beams

Plane deformation of anisotropic beams with narrow rectangular cross sections exhibits coupling of stretching, bending and transverse shearing. For anisotropic cantilever beams with a stiff end-cap under end forces and an end couple, assessments were made for approximate solutions by comparing these with numerically exact finite element (FE) solutions. Specific attention is given to point-wise or approximate satisfaction of the end-fixity conditions. As approximate methodologies, (i) the elementary polynomial form of Airy's stress function for the plane stress problem in a rectangular region, (ii) a Timoshenko-type beam theory, and (iii) the Bernoulli-Euler beam theory were selected. Among these, only the polynomial form of Airy's stress function violates the point-wise end-fixity conditions. Both the polynomial Airy stress function and the Timoshenko-type beam theory successfully model the effects of transverse shear deformation and the coupling of stretching and transverse deflection. Analytical solutions demonstrate that the normal shear coupling effect increases linearly with the thickness-to-span ratios in axial normal stress and axial displacement, while the coupling manifests quadratically in transverse displacement. The comparison of end displacements with the numerically exact FE solutions indicates that the polynomial form of Airy's stress function is no better than the Timoshenko-type beam theory. Similar conclusions were reached for the problem of uniformly loaded cantilever beams. It has been found that the accurate prediction of the deformation of thick anisotropic beams with significant normal-shear coupling requires the use of higher order theories.     

A Comprehensive Comparison of the Analytical and Numerical Prediction of the Thermal History and Solidification Microstructure of Inconel 718 Products Made by Laser Powder-Bed Fusion

The finite-element (FE) model and the Rosenthal equation are used to study the thermal and microstructural phenomena in the laser powder-bed fusion of Inconel 718. A primary aim is to comprehend the advantages and disadvantages of the Rosenthal equation (which provides an analytical alternative to FE analysis), and to investigate the influence of underlying assumptions on estimated results. Various physical characteristics are compared among the FE model, Rosenthal equation, and experiments. The predicted melt pool shapes compared with reported experimental results from the literature show that both the FE model and the analytical (Rosenthal) equation provide a reasonably accurate estimation. At high heat input, under conditions leading to keyholing, the reported melt width is narrower than predicted by the analytical equation. Moreover, a sensitivity analysis based on choices of the absorptivity is performed, which shows that the Rosenthal approach is more sensitive to absorptivity, compared with the FE approach. The primary reason could be the effect of radiative and convective losses, which are assumed to be negligible in the Rosenthal equation. In addition, both methods predict a columnar solidification microstructure, which agrees well with experimental reports, and the primary dendrite arm spacing (PDAS) predicted with the two approaches is comparable with measurements.     

一种快速的间接关联挖掘算法

给出了一个基于候选间接关联反单调性和频繁项目对支持矩阵的不需要生成所有频繁集的直接挖掘项目对之间间接关联的挖掘算法,并在一个Web log的真实数据集上进行了试验,与现有算法的比较表明该算法具有更好的性能。     

On the effective electroelastic properties of microcracked generally anisotropic solids

In this study we first obtain the explicit expressions for the 15 effective reduced elastic compliances of an elastically anisotropic solid containing multiple microcracks with an arbitrary degree of alignment under two-dimensional deformations within the framework of the non-interaction approximation (NIA). Under special situations, our results can reduce to the classical ones derived by Bristow (J Appl Phys 11: 81–85, 1960), and Mauge and Kachanov (J Mech Phys Solids 42(4):561–584, 1994). Some interesting phenomena are also observed. For example, when the undamaged solid is orthotropic, the effective in-plane shear modulus is dependent on the degree of the crack alignment. The NIA method is then extended to obtain the effective electroelastic properties of an anisotropic piezoelectric solid containing two-dimensional insulat- ing, permeable or conducting microcracks with an arbitrary degree of alignment. We also derive a set of fifteen coupled nonlinear equations for the unknown effective reduced elastic compliances of a microcrac- ked, anisotropic, elastic solid by using the generalized self-consistent method (GSCM). The set of coupled nonlinear equations can be solved through iteration.     

Application of the method of integral cross-sections for estimating the effective elastic properties of composite materials

We substantiate estimates of the upper and lower bounds of the effective elasticity modulus of piecewise homogeneous bodies. Using the method of integral cross-sections, we solve the elastic problem concerning the bending of a Kirchhoff inhomogeneous square plate and find the effective cylindrical stiffness of this inhomogeneous plate for various geometric parameters of inclusion, as well as determine the upper and lower bounds of the effective stiffness for the set of problems considered. Odessa State Polytechnic University, Odessa, Ukraine. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 70, No. 5, pp. 807–813, September-October, 1997.