Effective Elastic Stiffness Research Articles

Designing spongy-bone-like cellular materials: Matched topology and anisotropy

Bone is a natural material with properties such as high specific stiffness and strength. These exceptional mechanical properties are attributed to the meso-scale structure and elastic anisotropy of spongy bone. Replicating the topological traits and mechanical properties of spongy bone presents a novel opportunity to develop high-performance cellular materials. To achieve this, we propose an innovative framework for designing biomimetic cellular materials that match the trabecular structure and elastic anisotropy of spongy bone. This framework introduces a forward-flow design process that utilizes gradient-based feature tuning on a low-dimensional feature vector, transforming the complex inverse design problem into an efficient iterative process. A key innovation in our approach is the use of a pre-trained generative model, SliceGAN, to reconstruct 3D unit cells from 2D micro-CT images. This significantly reduces the cost and time associated with traditional layer-by-layer CT scans typically required for 3D training data. Numerical homogenization is then used to determine the effective elastic stiffness matrix, and a Fourier neural operator is trained to predict these matrices efficiently, greatly enhancing the computational efficiency of the design process. Using this framework, we successfully designed unit cells with topological traits and elastic anisotropy that closely approximate those of natural spongy bone. This opens new avenues for developing spongy-bone-mimetic cellular materials with exceptional mechanical properties. Moreover, the framework's versatility allows it to be extended to the design of other bio-inspired cellular materials.

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.

A Pragmatic Design of Elastic Metamaterial with Extreme Anisotropic Stiffness

Traditional elastic metamaterials encounter difficulties when trying to independently adjust the operating frequency in two perpendicular directions. In this research, we present a practical active elastic metamaterial with the capability to achieve extreme stiffness anisotropy. Our approach involves incorporating piezoelectric patches, coupled with adjustable conductance, into the microstructure unit cell to control the effective elastic stiffness in both primary directions. We conduct simulations to manipulate wave propagation within such metamaterials and conduct experimental investigations to validate the design. The obtained results closely align with mathematical analysis and numerical predictions, demonstrating the effectiveness of our proposed metamaterial. This novel approach offers a versatile and innovative design methodology for elastic metamaterials.

Effective Stiffness and Seismic Response Modification Models Recommended for Cantilever Circular Columns of RC Bridges, First Part: Serviceability Limit State

This paper’s main aim was to explain the process of characterising the structural over-strength factor (R), seismic behaviour factor (Q), and the effective elastic stiffness, Keff, of cantilever-reinforced concrete (RC) urban bridge columns with solid circular cross-sections for use in seismic design under the Serviceability Limit State (SLS). Similarly, mathematical models have been proposed to determine the average values of effective stiffness and seismic response modification factors suggested for cantilever-reinforced concrete bridge columns at SLS. This is because multiple design codes stipulate that cantilever RC bridge columns must meet the SLS requirements. Therefore, to comply, the lateral displacement ductility demand must not exceed unity after a moderate or small earthquake. While the behaviour of the materials remains in the elastic range, this performance criterion can be conservative. If the materials undergo small deformations, the slight damage can be quickly repaired to meet the SLS.

Effective Stiffness and Seismic Response Modification Models Recommended for Cantilever Circular Columns of RC Bridges, Second Part: Collapse Prevention Limit State

This paper aims to assess the structural overstrength (R), seismic behaviour factor (Q), and effective inertia factor (keff), of which the effective elastic stiffness (Keff) is a function, i.e., Keff (keff), of reinforced concrete (RC) cantilever columns with solid circular sections in urban bridges. The R and Q factors proposed in the design codes have been based mainly on an empirical basis. However, some studies suggest that, in the case of RC bridge columns, both factors are determined by a range of geometric and structural parameters. These results indicate that equations incorporating the main geometric and structural parameters of the columns, rather than the use of fixed values, are the appropriate method for estimating the values of R, Q and keff for the columns. Therefore, the results confirm the relation between R, Q and keff with the mechanical properties of the concrete, the aspect ratio length–depth, the axial load–strength ratio of a column, as well as its longitudinal reinforcement steel ratio. Finally, the research proposes models for estimating R, Q and keff that are recommended for the collapse prevention limit state of cantilevered circular columns of RC bridges.

Bayesian optimisation of hexagonal honeycomb metamaterial

Periodic mechanical metamaterials, such as hexagonal honeycombs, have traditionally been designed with uniform cell walls to simplify manufacturing and modelling. However, recent research has suggested that varying strut thickness within the lattice could improve its mechanical properties. To fully explore this design space, we developed a computational framework that leverages Bayesian optimisation to identify configurations with increased uniaxial effective elastic stiffness and plastic or buckling strength. The best topologies found, representative of relative densities with distinct failure modes, were additively manufactured and tested, resulting in a 54% increase in stiffness without compromising the buckling strength for slender architectures, and a 63% increase in elastic modulus and a 88% increase in plastic strength for higher volume fractions. Our results demonstrate the potential of Bayesian optimisation and solid material redistribution to enhance the performance of mechanical metamaterials.

Graph neural network for predicting the effective properties of polycrystalline materials: A comprehensive analysis

We develop a polycrystal graph neural network (PGNN) model for predicting the effective properties of polycrystalline materials, using the Li7La3Zr2O12 ceramic as an example. A large-scale dataset with >5000 different three-dimensional polycrystalline microstructures of finite-width grain boundary is generated by Voronoi tessellation and processing of the electron backscatter diffraction images. The effective ion conductivities and elastic stiffness coefficients of these microstructures are calculated by high-throughput physics-based simulations. The optimized PGNN model achieves a low error of <1.4% in predicting all three diagonal components of the effective Li-ion conductivity matrix, outperforming a linear regression model and two baseline convolutional neural network models. Sequential forward selection method is used to quantify the relative importance of selecting individual grain (boundary) features to improving the property prediction accuracy, through which both the critical and unwanted node (edge) feature can be determined. The extrapolation performance of the trained PGNN model is also investigated. The transfer learning performance is evaluated by using the PGNN model pretrained for predicting conductivities to predict the elastic properties of the same set of microstructures.:

Novel metamaterial structures with negative thermal expansion and tunable mechanical properties

The thermal expansion and stiffness characteristics are significant in determining the responses of lightweight mechanical metamaterials in a thermal environment. This paper proposed a novel trapezoid unit with negative thermal expansion (NTE) by three conventional materials with positive thermal expansion, and constructed a series of lightweight metamaterials by sharing edges and sharing vertexes. The effective linear coefficients of thermal expansion (CTE) of these metamaterial structures are theoretically explored by considering axial and bending deformations. Based on Euler beam theory, the effective linear elastic stiffnesses of these metamaterials are given accurately by the stiffness matrix method. The accuracy of the theoretical effective CTE and stiffness is well verified by numerical modeling. The effective linear CTE, elastic stiffness, and Poisson's ratio of these metamaterials can be programmable by controlling geometrical parameters and material properties. The NTE effect can be achieved in ET, EQ, EH, and VT metamaterials due to the presence of re-entrant structures, in which the ET metamaterial can fulfill the maximum magnitude NTE. The NTE and negative Poisson's ratio can simultaneously be realized in EQ metamaterial. Significantly, a broader range of CTE from negative to positive values can be fulfilled in these metamaterials compared with the conventional triangle unit. Furthermore, the high specific stiffness with extremely low thermal stress can be obtained in EQ, VT, and VH metamaterials, which benefits the stability controlling and enhancing natural frequencies of structures in a thermal environment.

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.

Enhanced energy absorption performance of 3D printed 2D auxetic lattices

Auxetic lattices have attracted increasing attention due to their unusual mechanical behavior and potential for an array of applications. However, a narrow window of stiffness realizable for a given cell topology limits their applications. In this study, a pair of novel 2D re-entrant auxetic lattices capable of exhibiting enhanced stiffness and energy absorption is proposed by introducing vertical ligaments into conventional re-entrant structures. These modified re-entrant auxetic lattices were realized via fused deposition additive manufacturing. The deformation patterns and the energy absorption characteristics of 3D printed auxetic lattices under quasi-static compression were investigated both via Finite Element (FE) simulations and experiments. The effective elastic stiffness of the proposed lattices was theoretically estimated. The FE results corroborated by experiments, elucidate the role of different sub-cells on the effective mechanical properties of the proposed auxetic lattices. The results indicate that the proposed structures — Type A and B variants, exhibit enhanced stiffness (+355%) and superior energy absorption (+165%) in comparison to conventional 2D re-entrant lattices of the same mass. Furthermore, the findings of the study suggest that the strength, stiffness, energy absorption capacity and Poisson’s ratio of 2D auxetic lattices can be tailored by tuning the sub-cell properties and cell wall thickness.

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.

The effect of porosity on the elastic properties of cortical bone and ultrasound propagation

This paper focuses on the modeling of the effect of different porosity on the overall elastic properties of saturated cortical bone and ultrasound wave propagation in such bone. We first utilize micromechanical model of Zhou, Cui and Sevostianov (2020) to model anisotropic effective elastic stiffness of saturated cortical bone and evaluate the effect of partial porosities. It is shown that the moduli C33satand C44satof saturated bone are more sensitive to canalicular porosity, while the moduli C11satand C66satof saturated bone are more sensitive to Haversian porosity. This model has been validated by experimental data. We compare the extents of anisotropy of the saturated bones for different levels of partial porosity. Last but not the least, we apply this micromechanical model to the elastic field of saturated cortical bone to evaluate the effect of partial porosity on the velocity of different bulk waves and guided waves. Results show that the velocity of the fast longitudinal wave and the slower SH wave along the horizontal direction are more sensitive to Haversian porosity ϕHav, but the velocity of the fast longitudinal wave and SH wave along the vertical direction and the velocity of the SV wave along the horizontal direction are more sensitive to the canalicular porosity ϕCan. Further, it is found that, the plateau region of the axisymmetric longitudinal guided waves mode L(0, 1) and L(0, 2) phase velocity are more sensitive to canalicular porosity ϕCan, and the phase velocity of the lowest nonaxisymmetric flexural mode F(1, 1) in low frequencie is more sensitive to Haversian porosity ϕHav.

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.

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.

A practical approach to develop a proper anisotropic rock physics model for media comprising of multiple fracture sets in the absence of sophisticated laboratory/wireline data

For economical production from a fractured reservoir, a characteristic analysis of the fracture parameters like its density and orientation within the reservoir is essential to improve the fluid flow during extraction. This study deals with the development of a proper anisotropic rock physics model for a media with multiple fracture sets to study the spatial distribution of important fracture parameters i.e., fracture density and orientation in the absence of sophisticated laboratory/wireline and pre-stack seismic data. The crest of hydrocarbon producing fault-bounded Balkassar Anticline in Northern Potwar, Upper Indus Basin, Pakistan is selected as a case study representing a potential zone for development of fractures at reservoir level (Sakesar Limestone). The methodology consists of the interpretation of 3D post-stack seismic and conventional wireline log data to demarcate the reservoir containing fractures. The Ant-tracking discrete fracture network (DFN) attribute is applied on 3D post-stack seismic data to obtain an initial estimate about the presence of fracture corridors and their orientations. Based on this initial estimate, a proper rock physics model has been developed utilizing inverse Gassmann relations, T-matrix approximation, and Brown and Korringa relations. The output from the developed rock physics model has been displayed in the form of 13 effective independent elastic stiffness constants (monoclinic symmetry–representing media comprising of multiple fracture sets) as a function of fracture densities and azimuthal fracture orientations. A clear decreasing trend in effective elastic stiffness constants with increasing fracture densities can be observed. Similarly, a periodic trend of effective elastic stiffness constants with fracture orientations can be observed. These trends are more or less expected, but they would have been difficult to quantify without a proper rock physics model. The use of independent effective elastic constants for the generation of synthetic seismic amplitude versus angle and azimuth (AVAZ) data and its correlation with observed seismic AVAZ data in a geostatistical sense has been discussed.

Cyclic Behavior of Reinforced Concrete Columns with Unstressed Steel Strands as Longitudinal Reinforcement

The use of unstressed Grade 1860 MPa seven-wire steel strands as longitudinal reinforcement has the potential to reduce steel congestion in reinforced concrete members. However, no studies which use unstressed steel strands as the longitudinal reinforcement in concrete columns have been reported in the literature. Thus, in this study, five large-scale column specimens: four of which used unstressed steel strands and one which used conventional Grade 420 MPa deformed bars as longitudinal reinforcement were subjected to cyclic loading under constant axial load to investigate the effectiveness in using strands. Columns with unstressed steel strands as longitudinal reinforcement exhibited drift capacities ranging between 4.96% and 5.70%, which is well over the conventional expectation of 3.50% drift requirement for columns in special moment frames. Also, the specimens exhibited lower energy dissipation but enhanced recentering capability owing to the fact that the strands have significantly higher tensile yield capacity compared to deformed bars. A reduction in effective elastic stiffness was also observed as the postcracking stiffness of specimens with strands dropped due to the low longitudinal reinforcement ratio of 0.78% provided. The test results indicated the relative ineffectiveness of strands in compression than in tension, where the strand bulged and unwound once the surrounding confinement deteriorated. Despite this, strands did straighten up and continued to sustain tensile stresses under load reversal. Based on the experimental observations, two constitutive models, one which considers strain hardening and the other which assumes an elasto-perfectly plastic response in tension was proposed for strands. The constitutive model with strain hardening was able to produce reasonably accurate estimates of flexural strength when used in a detailed moment-curvature analysis. Both constitutive models when used in a simplified analysis, which used equivalent stress block for concrete, was able to consistently produce conservative estimates of flexural strength, which is suitable for design purpose.

Effect of agglomeration and dispersion on the elastic properties of polymer nanocomposites: A Monte Carlo finite element analysis

Abstract Owing to their superior mechanical and physical properties, carbon nanotubes (CNT) seem to hold a great promise as an ideal reinforcing material for composites of high-strength and low-density. In most of the experimental results up to date, however, only modest improvements in the strength and stiffness have been achieved by incorporating carbon nanotubes in polymers. In the present paper, the influence of single wall carbon nanotube agglomeration on the effective stiffness is analyzed by using an Eshelby’s inclusion model. Analytical expressions are derived for the effective elastic stiffness of single wall carbon nanotubereinforced composites with the effects of agglomeration. The present study not only provides the important relationship between the effective properties and the morphology of CNT-reinforced composites, but also may be useful for improving and tailoring their mechanical properties. In addition, a multiscale Monte Carlo finite element method (MCFEM) was used for determining mechanical properties of polymer nanocomposites (PNC) that consist of polymers reinforced with single-walled carbon nanotubes (SWCNT). Specifically, the method uses a multiscale homogenization approach to link the structural variability at the nano/micro scales with the local constitutive behavior. Subsequently, the method incorporates a FE scheme to determine the Young’s modulus and Poisson ratio of PNC. The use of the computed properties in macroscale modeling is validated by comparison with experimental tensile test data.

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.

Seismic Characterization of Decoupled Orthorhombic Fractures Based on Observed Surface Azimuthal Amplitude Data

Commonly-used seismic fracture characterization method assumes one single fracture set, and this article aims to relax the limitation and demonstrate the feasibility of seismic characterization for two sets of orthorhombic fractures. Elastic properties of orthorhombic fractures can be studied using a model of horizontal and vertical fractures as linear-slip interfaces embedded in an isotropic background. High fracture density in hydrocarbon reservoirs is the “sweet spot” of effective permeability for fluid flow, and seismic characterization of orthorhombic fractures using the variations in reflection amplitude versus offset and azimuth (AVOAz) helps to optimize the production of naturally fractured reservoirs. Based on an effective orthorhombic linear-slip model, we first construct the effective elastic stiffness matrix in terms of elastic moduli and decoupled horizontal and vertical fracture densities under the assumption of weak anisotropy (WA) or small fracture densities. Using the scattering function and first-order perturbation in effective elastic stiffness components, we derive a WA and linearized PP-wave reflection coefficient containing decoupled orthorhombic fracture densities to describe the characteristics of horizontal and vertical fracture sets. We then propose an efficient AVOAz inversion method to perform the seismic characterization of decoupled orthorhombic fracture densities based on observed surface azimuthal amplitude data in a Bayesian framework. We finally apply the inversion approach to synthetic and field datasets to demonstrate its feasibility and reasonability in orthorhombic fracture characterization.