Effective Stiffness Tensor Research Articles

Radially transverse isotropic inclusions in isotropic elastic media: Local fields, neutral inclusions, effective elastic properties

Radially transverse isotropic inclusions in homogeneous isotropic elastic host media are considered. Mellin transform method is used for solution of the volume integral equation of the problem for an isolated inclusion subjected to a constant external stress (strain) field. The tensor structure of the solution is revealed with precision to three scalar functions of the radial coordinate, and the system of ordinary differential equations for these functions is derived. For multilayered radially transverse isotropic inclusions with constant elastic coefficients inside layers, explicit solution of these equations is obtained. An efficient numerical algorithm of solution for inclusions with an arbitrary number of the layers is proposed. Neutral inclusions that do not disturb homogeneous external fields applied to the medium are considered. It is shown that an inclusion with an isotropic core and radially transverse isotropic external layer can be weak neutral by appropriate choice of the layer elastic constants. The effective field method is used for determination of the effective elastic stiffness tensor of a homogeneous isotropic medium containing a random set of radially transverse isotropic inclusions.

EFFECTIVE STIFFNESS TENSOR OF COMPOSITE MATERIALS WITH INCLUSIONS OF RANDOM SIZE AND PERIODIC LOCATION OF THEIR CENTERS

In this study, we investigated the possibility of determining the stiffness tensor using theoretical methods. We utilized a periodicity cell, not a representative elementary volume, to solve the problem, in which the periodicity of the inclusion center locations was considered. Three options were employed to estimate the effective deformation properties of composite materials with periodic inclusion center location arrangements and random size values. In this paper, the applicability of the proposed probabilistic approach to determining the effective stiffness tensor on a periodicity cell is shown. This approach allows one to obtain not only the mean values (i.e., the effective stiffness tensor) but also three main central moments-from the second to the fourth order-characterizing the random nature of the effective stiffness tensor. The approach developed and employed in this study to estimate the effective deformation characteristics of composite materials with periodic inclusion center location arrangements, characterized by random radii values, could be practically extended to other physical and mechanical properties.

Finite-volume micromechanics-based multiscale analysis of composite structural model accounting for elastoplastic and ductile damage mechanisms

A hierarchical homogenization framework is proposed for nonlinear elastic-plastic and damage simulation of a unidirectional composite laminate. Proper scale transition rules between the macro- and the micro-scale variables were introduced, wherein the finite-element and finite-volume methods are employed for predicting the structural and material point response, respectively. Specifically, at the macroscopic scale, the ABAQUS/CAE structural analysis provides macroscopic strains for the lower-scale unit cell simulation (top-down) at each load increment. Reversely, at the microscopic scale, the unit cell problem is modelled using extended parametric finite-volume direct averaging micromechanics (FVDAM) through a Meta-UMAT subroutine that links the global finite element solver and the constitutive laws (UMAT subroutines). Homogenization of stresses and effective stiffness tensor obtained from the unit cell predictions is passed back to the ABAQUS/CAE for the higher-scale finite-element structural model computation (bottom-up). The stresses and strains at the global and local scales can be tracked during the incremental loading on the macroscopic structures through continuous homogenization and localization. The predictive capabilities of the proposed technique are verified vis-à-vis the experimental response of the Boron/Aluminum (B/Al) laminate obtained from off-axis tensile and shear loading available in the literature.

Mechanical properties of hierarchical lattice via strain gradient homogenization approach

With the advancement of 3D printing technology, manufacturing metamaterials with extreme mechanical properties is becoming more feasible. Hierarchical lattices appear to be ideal candidates for obtaining desirable lightweight, high specific stiffness, and enhanced specific energy absorption. They can be constructed by architected substructures at multiple length scales. The interpretations of the underlying deformation mechanisms are necessary in order to manipulate with expected mechanical properties. In this paper, experiments were conducted to examine the effective mechanical behaviors of hierarchical lattice metamaterials. A strain gradient homogenization approach has been employed to correlate their morphological characteristics to the resulting material properties. The effective stiffness tensors are identified including the fourth-, fifth-, and sixth-order stiffness tensors. The higher-order inertial parameters are determined by a fitting procedure. Comparisons between direct finite element analyses and the homogenized strain gradient continua have been made for hierarchical lattice materials under static and dynamic loads. It is found that strain gradient homogenization approach can be an accurate, efficient and reliable way of predicting the mechanical behaviors of hierarchical lattice. A systematic analysis of geometrical parameters was conducted to uncover the underlying mechanisms responsible for the enhancement of effective properties. This work offers a novel approach for designing the mechanical properties of hierarchical metamaterials through precise control of secondary structure types and distributions.

Effect of interface roughness on the elastic properties of 3D layered media

In our work, we numerically investigated how the roughness of interfaces between different layers in 3D layered models affects the upscaled and downscaled elastic properties. We considered layered models with random rough interfaces and fixed elastic properties in the background and the middle layer, upscaled them to compute the effective stiffness tensor and then downscaled them to obtain new elastic parameters of the middle layer in the statistically equivalent models with smooth interfaces. Afterward, we investigated the dependence of the obtained results on four input varied parameters (the correlation length, standard deviation of the roughness and two physical elastic parameters of the middle layer). Next, we proposed an algorithm for recovering the covariance matrix of such stiffness tensors for arbitrary values from the four-parameter space, applying bilinear regression to the parameters of the interface and the linear component-by-component interpolation by two nearest points to the material parameters. Verification of this algorithm showed that for limited standard deviation values the errors in restoring the covariance matrix do not exceed 0.07. Thus, smooth interface models with random material parameters in the middle layer can be statistically simulated with arbitrary values of the input parameters with high accuracy.

Effective medium method in solution of the elastic and thermo-elastic homogenization problems for polycrystals with spheroidal transverse isotropic grains

Self-consistent effective medium method is used for determination of the effective elastic and thermo-elastic properties of polycrystalline materials with spheroidal transverse isotropic grains. The grains with arbitrary aspect ratios including very oblate (disks) and very prolate (fibers) spheroids are considered. The function of the Owens-March type is used for description of distribution of grains over orientations. The equations for the effective elastic stiffness tensor and the tensor of effective coefficients of thermo-expansion of polycrystals are derived using integral equations for elastic and thermo-elastic fields in heterogeneous media and the hypotheses of the effective medium method. The numerical algorithm of the method is described in detail. Numerical examples are presented for polycrystalline clay rock materials with strongly anisotropic grains and for polycrystalline zinc. Graphs of the dependences of the effective elastic constants and effective coefficients of thermo-expansion of the polycrystals on the parameters of the orientation distribution function and aspect ratios of grains are presented.

Experimental Investigation and Micromechanical Modeling of Hard Rock in Protective Seam Considering Damage–Friction Coupling Effect

The hard rock in the protective coal seam of the Pingdingshan Mine in China is a typical quasi-brittle material exhibiting complex mechanical characteristics. According to available research on the mechanical property, the inelastic deformation and development of damage are considered related with crack initiation and propagation, which are main causes of the material degradation. In the present study, an original experimental investigation on the rock sample of the Pingdingshan coal mine is firstly carried out to obtain the basic mechanical responses in a conventional triaxial compression test. Based on the homogenization method and thermodynamic theory, a damage–friction coupled model is proposed to simulate the non-linear mechanical behavior. In the framework of micromechanics, the hard rock in a protective coal seam is viewed as a heterogeneous material composed of a homogeneous solid matrix and a large number of randomly distributed microcracks, leading to a Representative Elementary Volume (REV), i.e., the matrix–cracks system. By the use of the Mori–Tanaka homogenization scheme, the effective elastic properties of cracked material are obtained within the framework of micromechanics. The expression of free energy on the characteristic unitary is derived by homogenization methods and the pairwise thermodynamic forces associated with the inelastic strain and damage variables. The local stress tensor is decomposed to hydrostatic and deviatoric parts, and the effective tangent stiffness tensor is derived by considering both the plastic yield law and a specific damage criterion. The associated generalized Coulomb friction criterion and damage criterion are introduced to describe the evolution of inelastic strain and damage, respectively. Prepeak and postpeak triaxial response analysis is carried out by coupled damage–friction analysis to obtain analytical expressions for rock strength and to clarify the basic characteristics of the damage resistance function. Finally, by the use of the returning mapping procedure, the proposed damage–friction constitutive model is applied to simulate the deformation of Pingdingshan hard rock in triaxial compression with respect to different confining pressures. It is observed that the numerical results are in good agreement with the experimental data, which can verify the accuracy and show the obvious advantages of the micromechanic-based model.

Effective elastic properties and wave surfaces of rock materials containing multiple cavities and cracks (effective field approach)

The paper is devoted to simulation of the effective elastic properties of rock materials containing two substantially different types of defects simultaneously: random sets of ellipsoidal pores (cavities) and elliptical cracks. For solution of the homogenization problem and calculation of the effective elastic properties of such materials, the self-consistent effective field method is used. For composites with one type of heterogeneities, the method coincides with the Mori–Tanaka method. The method allows deriving analytical expressions for the effective elastic stiffness tensors of the materials containing pores and cracks of various scales. These tensors have correct symmetry with respect to tensor indices and provide physically reasonable values of the effective elastic constants in wide regions of porosity and crack density. The materials with pores and cracks of substantially different sizes and of close sizes are considered. The method predicts substantially different effective elastic constants of the materials with these microstructures. Comparison of the components of the effective elastic stiffness tensors for the materials with various values of porosity and crack density are presented. Wave surfaces of acoustical waves in the materials containing pores and cracks are constructed. It is shown that for materials with spherical pores and circular cracks, the shapes of these surfaces are close to ellipsoidal. The results of the paper can be used for determination of fracture level in damaged rock materials by acoustical methods.

PP-wave reflection coefficient in stress-induced anisotropic media and amplitude variation with incident angle and azimuth inversion

The PP-wave reflection coefficient (i.e., the ratio of the amplitude of reflected P wave to the amplitude of incident P wave) has crucial practical applications in many field scenarios, such as seismic inversion and in-situ stress monitoring. However, the effect of biaxial stress, frequently encountered in the earth’s interior, on the PP-wave reflection coefficient is seldom studied. To fill this gap, we first formulate the P- and S-wave moduli and density in a biaxial-stress-induced anisotropic medium based on the acoustoelasticity theory. Then, the stress-dependent anisotropy parameters and normal/tangential rock weaknesses are proposed in the context of the stress-induced anisotropy model to obtain the corresponding effective elastic stiffness tensor and its spatial perturbation. Two stress-induced anisotropy factors (SIAF) are established to quantify the magnitude of anisotropy in a stress-induced anisotropic medium. The media with large SIAF tend to demonstrate strong anisotropy when stressed and tend to be more easily fractured from the perspective of engineering application. Furthermore, an approximate formula for the PP-wave reflection coefficient for biaxial-stress-induced anisotropic media is developed with the scattering theory. Numerical results illustrate how the reflection coefficients are affected by horizontal biaxial stress and demonstrate the feasibility and accuracy of our formula. Modeled and field inversion examples illustrate that SIAF can be reasonably inverted with our amplitude variation with incident angle and azimuth inversion method from azimuthal seismic data. Our results help to better understand the effect of biaxial stress on anisotropy and wave propagation in stress-induced anisotropic media.

Prediction of mechanical behavior of rocks with strong strain-softening effects by a deep-learning approach

Rock materials exhibit various mechanical characteristics, and it is difficult to describe the strain–stress relation with strong strain-softening behavior by a single constitutive law. In the present study, a long short term memory (LSTM) deep learning method is proposed to predict the material’s deformation under different loading conditions. Unlike the traditional analysis method focusing on the determination of effective tangent stiffness tensor, the constructed LSTM-based procedure requires only the strain–stress values in certain cases to predict the future mechanical behaviors, even for rock materials with strong strain-softening, indicating that the loading history can be taken into account as time sequence data. In order to validate the accuracy, two applications are provided with the established LSTM model: predicting the deformation of granite and sandstone in conventional triaxial compression tests without introducing any elastoplastic parameters or constitutive laws, where the dataset for training the LSTM model is collected alternatively from an analytical micromechanical damage model and from laboratory experiments by considering a wide range of confining pressure. Comparisons of accuracy and convergence rate with different neural network structures are also carried out to check the best performance of the procedure. Implementation method of the trained LSTM model in Finite Element program as a constitutive relation is also provided and applied to the simulation of sandstone. Comparisons show that the LSTM-FEM method provides a good capacity to predict the mechanical behavior of rocks.

Estimation of in situ stresses from PP-wave azimuthal seismic data in fracture-induced anisotropic media

Horizontally transverse isotropy (HTI) induced by vertical or subvertical aligned fractures is common for unconventional fractured porous shale oil or gas reservoirs. Compared with the unfractured rocks, the seismic response characteristics of PP-wave azimuthal amplitudes are usually disturbed by the fractures and the in situ stresses. Knowledge of fracture properties, as well as in situ stresses, is required to optimize horizontal well planning and hydraulic fracturing during production, and the seismic inversion for in situ stresses from the PP-wave azimuthal amplitude data in fracture-induced anisotropic media is an essential step. Using the linear-slip theory and the effective stress law, we derive the fluid-saturated effective elastic stiffness tensors parameterized by background elastic moduli, the effective stress coefficient of isotropic host rocks, fluid modulus, porosity, and fracture parameters based on the anisotropic Gassmann’s fluid substitution equation. Combining the perturbations in saturated stiffness tensors and scattering theory, we formulate the reflection coefficient equation of PP-wave data as a function of background porosity-related stress parameter and two (i.e., normal and shear) fracture weakness parameters. Following Bayes’ rule, we estimate the porosity-related stress parameter and fracture weaknesses using the inversion method of azimuthal Fourier coefficients. Finally, we compute the effective horizontal and vertical in situ stresses using the estimated elastic moduli, porosity-related stress parameter, and fracture weaknesses. Synthetic and real data sets demonstrate that our proposed inversion approach based on the derived reflection equation provides us another way to obtain reasonable estimates of in situ stresses in a complex-fractured porous shale reservoir.

Effective Elastic Stiffness Tensor and Ultrasonic Velocities for 3D Printed Polycrystals with Pores and Texture

ABSTRACT This paper focuses on the micromechanical modeling of pores and texture in 3D-printed polycrystals. A Gaussian-shape approximation is used to describe the orientation distribution function (ODF) and construct the initial homogenization model of the representative volume element (RVE). Spherical and ellipsoidal pores of varying sizes are added in stages to the RVE for homogenization, utilizing the modified Mori-Tanaka (MT) scheme in conjunction with the stepwise iterative method. Wherein, the mechanical interactions between pores and the spatial distribution of pore locations could be taken into account, when developing a cross-scale model from microstructure to macroelasticity. After obtaining the final elastic stiffness tensor, the Christoffel equation is combined with the Cardano’s formula to derive a closed form solution for ultrasonic velocities depending on microstructure. According to the numerical results, this method can effectively capture the behavior characteristics of pores and texture on the elastic stiffness tensor, average velocity, and velocity distribution.

Converting the Roughness of Model Interfacies into the Statistical Distribution of Elastic Parameters of the Layered Medium

In the paper, the effect of interface roughness on the elastic parameters of a layered medium is studied. Three-dimensional models of a layered medium with different material inside and outside the layer are considered. Initially, the first class of models is statistically generated with rough interfaces between the layers and fixed elastic parameters of the inner layers. Then the effective stiffness tensor is estimated for the equivalent homogeneous model. After that, new elasticity parameters of the inner layer are reconstructed for second class of models with flat interfacies. Thus, the roughness of the model interfacies is mapped into a statistical distribution of the stiffness tensor components for a fixed model geometry. In addition, an algorithm for extending the results of reconstructing the elastic tensors for arbitrary interface roughness parameters using bilinear regression of covariance matrices is proposed. The algorithm verification shows that the error in restoring the covariance matrix does not exceed 7%; i.e., one can use it in statistical modeling the second class of models with flat interfacies for the arbitrary interface roughness values and given physical parameters of layers in the first class of models.

Discretization Approach for the Homogenization of Three-Dimensional Solid-Solid Phononic Crystals in the Quasi-Static Limit: Density and Elastic Moduli

With the application of a homogenization theory, based on the Fourier formalism (which provides efficient and exact formulas by which to determine all the components of the effective stiffness and mass density tensors, valid in the regime of large wavelengths), a new approach to calculate the effective quasi-static response in three-dimensional solid-solid phononic crystals is reported. The formulas derived in this work for calculating the effective elastic parameters show a dependence, in terms of summations over the vectors, of the reciprocal lattice by the discretization of the volume of the inclusion in small parts (e.g., small cubes), to obtain a system of equations from which we define the effective response. In particular, we present the numerical results calculated for several cubic lattices with solid constituents and different shapes of inclusions in the unit cell versus the filling fraction, as well as for fixed values of it. By this means, we analyzed the effect of the type of Bravais lattice of the materials, and the geometry of the inclusions that constitute the three-dimensional phononic array, on the resulting effective anisotropy. Finally, our theory confirms other well-known results with previous homogenization theories as a particular case study. In this regard, the examples and results shown here can be useful for the design of metamaterials with predetermined elastic properties.

Atomistic vs. continuum models of nanoporous elastic solid: Stress fields, size-dependent effective stiffness and surface constants

The method of classical molecular dynamics (MD) is applied to study the stress fields and effective elastic stiffness tensor of nanoporous materials. A continuum model for a nanoporous solid involving a surface layer of sub-nanometer thickness is proposed. Systematic comparisons of the unit cell atomistic MD results are performed with two continuum models, one based on the finite element analysis and another based on the multipole series expansion. This comparison validates the continuum models, shows their applicability limits, and provides reliable identification of the surface elastic constants.

Random Stiffness Tensor of Particulate Composites with Hyper-Elastic Matrix and Imperfect Interface.

The main aim of this study is determination of the basic probabilistic characteristics of the effective stiffness for inelastic particulate composites with spherical reinforcement and an uncertain Gaussian volume fraction of the interphase defects. This is determined using a homogenization method with a cubic single-particle representative volume element (RVE) of such a composite and the finite element method solution. A reinforcing particle is spherical, located centrally in the RVE, surrounded by the thin interphase of constant thickness, and remains in an elastic reversible regime opposite to the matrix, which is hyper-elastic. The interphase defects are represented as semi-spherical voids, which are placed on the outer surface of this particle. The interphase is modeled as hyper-elastic and isotropic, whose effective stiffness is calculated by the spatial averaging of hyper-elastic parameters of the matrix and of the defects. A constitutive relation of the matrix is recovered experimentally by its uniaxial stretch. The 3D homogenization problem solution is based upon a numerical determination of strain energy density in the given RVE under specific uniaxial and biaxial stretches as well as under shear deformations. The analytical relation of the effective composite stiffness to the input uncertain parameter is recovered via the response function method, using a polynomial basis and an optimized order. Probabilistic calculations are completed using three concurrent approaches, namely the iterative stochastic finite element method (SFEM), Monte Carlo simulation and by the semi-analytical method. Previous papers consider the composite fully elastic, which limits the applicability of the resulting effective stiffness tensor computed therein. The current study voids this assumption and defines the composite as fully hyper-elastic, thus extending applicability of this tensor to strains up to 0.25. The most important research finding is that (1) the effective stiffness tensor is sensitive to random interface defects in its hyper-elastic range, (2) its resulting randomness is not close to Gaussian, (3) the semi-analytical method is not perfectly suited to stochastic calculations in this region of strains, as opposed to the linear elastic region, and (4) that the increase in random dispersion of defects volume fraction has a much higher effect on the stochastic characteristics of this stiffness tensor than fluctuation of the strain.

Electrical and mechanical behaviour of metal thin films with deformation-induced cracks predicted by computational homogenisation

Motivated by advances in flexible electronic technologies and by the endeavour to develop non-destructive testing methods, this article analyses the capability of computational multiscale formulations to predict the influence of microscale cracks on effective macroscopic electrical and mechanical material properties. To this end, thin metal films under mechanical load are experimentally analysed by using in-situ confocal laser scanning microscopy (CLSM) and in-situ four point probe resistance measurements. Image processing techniques are then used to generate representative volume elements from the laser intensity images. These discrete representations of the crack pattern at the microscale serve as the basis for the calculation of effective macroscopic electrical conductivity and mechanical stiffness tensors by means of computational homogenisation approaches. A comparison of simulation results with experimental electrical resistance measurements and a detailed study of fundamental numerical properties demonstrates the applicability of the proposed approach. In particular, the (numerical) errors that are induced by the representative volume element size and by the finite element discretisation are studied, and the influence of the filter that is used in the generation process of the representative volume element is analysed.

Reflection coefficients of P‐waves in partially saturated fractured media

ABSTRACTWave‐induced fluid flow, an important mechanism in seismic wave attenuation, is considered in the amplitude variation with the offset technique. In a fractured marine carbonate reservoir, fractures and pores develop simultaneously, such that wave‐induced flow is an important mechanism in attenuation caused by the fluid flow between pores as well as between pores and fractures. However, these two types of attenuation are typically investigated separately or studies assume that the pores and fractures are dry or saturated by the same fluid. Rock pores are usually saturated by a mixture of two or more fluids, such that the fracture and pore filling fluids may be different. During variation with offset reservoir prediction and fluid identification, the influence that the porosity, fracture, and fluid have on the P‐wave reflection coefficient should be considered. The patchy saturation model can describe the seismic wave dissipation caused by the wave‐induced flow between pore fluids, whereas the Norris model can describe seismic wave dissipation caused by the fluid flow between fractures and pores. This study considers both the patchy saturation model and Norris model to calculate the amplitude variation with the offset P‐wave reflection coefficient under different conditions. Combining the saturation and Norris models forms a new model, which is regarded as a transversely isotropic medium with a vertical symmetric axis, and can be extended to any incident angle through the relaxation function and the high‐ and low‐frequency elastic limits of the stiffness tensor, where the complete effective stiffness tensor of the new model can be obtained. Finally, the amplitude variation with offset reservoir technology is used to determine the influence of model parameters on the reflection coefficient of P‐waves under different angles and frequencies, including fracture weakness, fluid saturation and pore fluid saturation.

On mean field homogenization schemes for short fiber reinforced composites: Unified formulation, application and benchmark

This paper revisits the topic of mean field homogenization for short fiber reinforced composite materials. A short glass fiber reinforced thermoplastic polyamide 6.6 with a fiber mass fraction of 35 % is used as an example material for which microstructural analyses and experimental tests were performed. We cover a set of common models: the Self-Consistent (SC), Mori–Tanaka (MT), Ponte Castañeda-Willis (PCW), Interaction Direct Derivative (IDD) and Two-Step (TS) schemes. We first recast them into a unified formulation that permits a thorough theoretical comparison, including the topics of loss of major symmetry as well as connections between the models. We are able to show that the MT, PCW and IDD schemes can be expressed in a surprisingly similar form. This extends to the equations for prediction of effective stiffness and compliance tensors as well as the strain and stress localization tensors. Secondly, we address the resolution of the material microstructure within mean field homogenization schemes, comparing classical and more recent and efficient methods. Last, we benchmark the different mean field schemes and modeling approaches, including comparisons to numerical homogenization by Fast Fourier Transformation (FFT) and experimental data from tensile tests. These show that the MT and TS schemes are both capable of accurately modeling the composite material for complex orientations and homogeneous fiber phases. For possibly inhomogeneous fiber phases only the TS scheme yields both accurate and physically reasonable results. Other models such as the IDD or PCW are of great theoretical importance, but cannot be generally applied for the given material class.

A virtual joint method of analytical form for parallel manipulators

Oriented towards the stiffness optimization of parallel manipulators, a stiffness modeling method based on subspace analysis was proposed. The paper revealed the “shielding effect” of passive joints, i.e. shielding the partial local stiffness of a mechanism on Cartesian space, and accordingly brought forward the concept of Effective Stiffness Tensor (EST), which was then obtained by introducing the orthogonal projector. The method allows a stiffness model of analytical form as well as high computation efficiency, which makes it suitable for applications in the stiffness optimization of parallel manipulators. Proposed method had been applied to the parallel alignment mechanism in China’s large space telescope and verified by dynamic experiments.