Stiffness Tensor Research Articles

Multi-scale design of multi-material lattice structures through a CAD-compatible topology optimisation algorithm

This work deals with the multi-scale topology optimisation (TO) of multi-material lattice structures. The proposed approach is based on: non-uniform rational basis spline (NURBS) hyper-surfaces to represent the geometric descriptor related to each material phase composing the representative volume element (RVE), an improved multiphase material interpolation (MMI) scheme to penalise the element stiffness tensor of the multi-material RVE, the strain energy-based homogenisation method (SEHM) to carry out the scale transition. In this context, the design requirements are defined at different scales and their gradient is evaluated by exploiting the properties of the NURBS entities and of the SEHM. Moreover, the improved MMI scheme proposed here does not require the introduction of artificial filtering techniques to smooth the topological descriptors of the material phases composing the RVE. The effectiveness of the method is proven on both 2D and 3D problems. Specifically, a sensitivity analysis of the optimised configuration of the RVE to the parameters tuning the shape of the NURBS entity is conducted. Finally, the influence of the starting point and of the macroscopic loads on the optimal solution is investigated.

Identification of higher symmetry in triclinic stiffness tensor: Application to high pressure dependence of elastic anisotropy in deep underground structures

We present an approach of identifying anisotropy axes and the type of symmetry from ultrasonic measurements of 21 elastic constants of the complete stiffness tensor characterizing anisotropic materials. The approach is applied to examine anisotropic elasticity of two crystalline rock samples from two underground research laboratories : (1) the migmatized gneiss from Bukov in the Czech Republic; and (2) the Aare Granite from Grimsel in Switzerland. The full stiffness matrix is measured at selected pressure levels from 0.1 to 100 MPa. We demonstrate that the Bukov migmatized gneiss is orthorhombic , whereas the Grimsel granite is transversely isotropic under atmospheric pressure. The degree of anisotropy of both rocks decreases with applied confining pressure due to closing of preferentially oriented cracks. While the Grimsel granite is very sensitive to pressure and becomes almost isotropic at high pressures, a great portion of anisotropy in the Bukov migmatized gneiss remains even under high pressures due to its texture. The dependence of Young's and shear moduli with pressure emphasizes the importance of taking the pressure into account for underground projects.

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.

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.

Graph-based metamaterials: Deep learning of structure-property relations

The effective properties of metamaterials are tailored through the design of their internal structures. According to their main building block, the family of porous three-dimensional metamaterials is divided into truss-, plate- and shell-lattices. The exploration of their full design-space is hampered in practice by a lack of a systematic method to represent their topologies. Here, we demonstrate for the first time that graph models provide an effective representation of shell-lattices. This new graph representation is then leveraged to obtain deep learning-based structure–property models. Using finite element simulations, the stiffness and heat conductivity tensors are established for more than 40,000 microstructural configurations. We find that a modified crystal graph convolutional neural network model provides an accurate description of the structure–property relations. We anticipate the proposed graph-based modeling framework to be applicable to any man-made periodic microstructure, thereby enabling the design and discovery of new materials exhibiting exceptional mechanical, thermal, electrical or magnetic properties.

Geomechanical effects of natural fractures on fluid flow in a pre-salt field

The discovery of carbonate reservoirs in the Brazilian pre-salt field has raised several engineering challenges. These reservoirs are naturally fractured and much stiffer than conventional reservoirs. Thus, the study of fluid flow through natural fractures has received significant attention from the petroleum industry because the production capacity of these fields is associated with the hydraulic behavior of such fractures. However, pressure changes induced by the oil recovery alter the fracture aperture. In turn, changes in the fracture aperture affect the fluid flow inside the fracture channels, increasing or reducing the production capacity of the reservoir. This work investigates the hydromechanical effect of natural fractures on the reservoir behavior at the production unit Tupi pilot of the Brazilian pre-salt. The enhanced dual-porosity/dual permeability model (EDPDP) is adopted to simulate more realistically the hydromechanical behavior of fractured carbonate rock formation. This approach updates the stiffness and permeability tensors considering the fracture orientation and the stress-induced aperture changes. The shape factor is also improved to represent multi-block domains formed by several multiscale fracture sets with different orientations, apertures, and spacing. The hydromechanical formulation of EDPDP implemented in an in-house framework GeMA (Geo Modeling Analysis) is adopted to study the hydromechanical effect of fractures with multiple lengths on the Tupi pilot. The numerical results demonstrate that the complex fracture network is responsible for fluid migration through a preferential pathway. A parametric analysis of the main parameters that affect reservoir behavior was carried out. The parametric study shows higher pore pressure dissipation for smaller dip angles. Then, horizontal fractures are more sensitive to vertical displacements. In addition, smaller spacing and larger fracture aperture enhance permeability, increasing pore pressure dissipation and mechanical deformation. Finally, numerical results were compared against field measurements showing excellent agreement, demonstrating the applicability of the EDPDP model to simulate naturally fractured reservoirs.

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.

Multiscale stiffness characterisation of both healthy and osteoporotic bone tissue using subject-specific data

Severe bone fractures are often treated by appending internal fixations. In unhealthy or osteoporotic patients, post-implantation bone fractures can occur due to external impact (e.g. from a fall), day-to-day activities in highly-osteoporotic cases and mismatches in the stiffness of bone and the implant’s biomaterial, since this causes stress concentrations. One approach to alleviating this problem is to use biomaterials that closely mimic the effective stiffness of real bone, thereby more seamlessly integrating the fixation. This requires to know the properties target (bone properties) and therefore, it highlights the relevance of the evaluation of the bone’s mechanical properties which is impractical via direct measurement. This work presents a methodology (multistage homogenisation) for predicting the anisotropic stiffness of bone given the porosity and mineral fraction, both of which are more readily obtained than the mechanical properties themselves. Unlike previous work we: (i) account for finger-like morphology of the mineral phase at the nanoscale; (ii) use microscopy data to model the osteon geometry and its curvilinear anisotropy at the microscale, and (iii) use data to define the trabecular (microCT) and cortical (microscopy) bone geometries at the mesoscale. The predicts have been shown to agree favourably with experimental data in the literature as well as previous modelling works. The results are summarised in a database containing anisotropic stiffness tensors applicable to a broad range of degrees of bone health (e.g. mineral fractions and mesoscale porosities); thus, this work is a contribution towards being able to design more robust patient-specific bone implants in practice.

Single‐Crystal Elasticity of Antigorite at High Pressures and Seismic Detection of Serpentinized Slabs

AbstractThe subduction of serpentinized slabs is the dominant process to transport “water” into Earth's mantle, and plays a pivotal role for subduction dynamics. Antigorite, the most abundant serpentine mineral in subduction settings, may imprint a seismic signature on serpentinized slabs, making them seismically distinguishable from the dry, non‐serpentinized ones. However, the complete single‐crystal elasticity of antigorite has not been experimentally constrained at high pressures, hindering the use of seismological approaches to detect serpentinization in subducting slabs. Here, we report the full elastic stiffness tensor of antigorite by single‐crystal Brillouin spectroscopy and X‐ray diffraction up to 7.71(5) GPa. We use our results to model seismic properties of antigorite‐bearing rocks and show that their seismological detectability depends on the geometrical relation between seismic wave paths and foliation of serpentinized rocks. In particular, we demonstrate that seismic shear anisotropy shows low sensitivity to serpentinization for a range of relevant geometries.

Learning the dynamics of metamaterials from diffracted waves with convolutional neural networks

Conventional methods used to identify the dynamical properties of unknown media from scattered mechanical waves rely on analytical or numerical manipulations of the wave equation. These methods show their limitations in scenarios where the analyzed medium is moderately sized and the diffraction from the material edges influences the scattered fields significantly, such as non-destructive diagnostics and metamaterial characterization. Here, we show that convolutional neural networks can interpret the diffracted fields and learn the mapping between the scattered fields and all the effective material parameters including mass density and stiffness tensors from a small set of numerical simulations. Furthermore, networks trained with synthetic data can process physical measurements and are very robust to measurement errors. More importantly, the trained network provides insight into the dynamic behavior of matter including quantitative measures of the scattered field sensitivity to each material property and how the sensitivity changes depending on the material under test.

Comment on “Erratum: ‘Two-dimensional porous graphitic carbon nitride C6N7 monolayer: First-principles calculations’ [Appl. Phys. Lett. 119, 142102 (2021)]” [Appl. Phys. Lett. 120, 189901 (2022)

Recently, Bafekry et al. [Appl. Phys. Lett. 120, 189901 (2022)] reported their density functional theory (DFT) results on the elastic constants of C6N7 monolayer. They predicted non-zero elastic constants along the out-of-plane direction for a single-layered material, which contradicts with basic physics of the stiffness tensor for plane stress condition. Moreover, in their work Young's modulus is erroneously calculated. On the basis of DFT calculations, herein we predicted the C11, C12 and C66 of the C6N7 monolayer to be 286, 73 and 107 GPa, respectively, equivalent with an in-plane Youngs modulus of 267 GPa. Using DFT calculations and a machine learning interatomic potential, we also show that C6N7 monolayer shows isotropic elasticity.

Novel porous structures with non-cubic symmetry: Synthesis, elastic anisotropy, and fatigue life behavior

Natural porous structures are often anisotropic in their elastic properties, i.e., they have directional variations that are related to their topology and geometry. This paper presents the synthesis and simulation framework of novel families of non-cubic porous structures based on implicit modeling of cosine surfaces in the three-dimensional Euclidean space. The synthesis was performed using Field-Driven Design (FDD). An in-depth study of the elastic properties and simulated fatigue compression–compression behavior of a selection of five structures from the orthotropic anisotropy family is presented as case studies exposing the stretching-dominated mechanisms. The apparent properties characterized include Young’s modulus , Poisson’s ratio , shear modulus , bulk modulus , and relative density . A systematic approach to the characterization of average apparent properties was performed according to the schemes of Voigt, Reuss, and Hill. We show the anisotropic variation of the cosine surface–based porous structures using the Universal Elastic Anisotropy Index (AU) and compare it with six well-known triply periodic minimal surface (TPMS) structures by analyzing the stiffness tensor to validate and discuss which individual property (stiffness, rigidity, compressibility) has a higher impact on the final anisotropy value. We also provide a formal curvature analysis, based on the notions of mean and Gaussian curvatures to evidence the ridge-shaped surface in the structures. The proposed porous structures showed advantages when compared to cubic TPMS structures. From the data processing of the five analyzed porous structures, two synthesized structures have AU = 0.394 and 0.478, which are lower values than the structures based on Neovius and Schwarz’s Primitive surfaces with AU = 0.529 and 0.604. In addition, simulation results of cyclic compressive fatigue loading indicate that the fatigue resistance properties of the non-cubic porous structures are higher than the TPMS structures.

A comparative study between fiber orientation closure approximations and a new orthotropic closure

AbstractIn injection and compression molding simulation, orientation tensors provide an efficient way with less computational effort to calculate flow‐induced fiber orientations. In these flow calculations, the solution of any even‐ordered orientation tensor needs the following even higher‐ordered orientation tensor. Therefore, a closure is used in order to approximate the higher‐ordered orientation tensor as a function of the components of the lower‐ordered orientation tensors. There exist many closures of the fourth‐order orientation tensor in terms of the second‐order orientation. This paper gives a review with mathematical details for different closure approximations including simple closures, composite closures, eigenvalue‐ based orthotropic closures, invariant‐based closures, and neural‐network‐ based closures. Moreover, a new closure approximation that assumes an orthotropic stiffness tensor was suggested in this work. The results of the closure approximations were validated through the comparison of Young moduli between the closure approximations and the experimental results for both short‐ and long‐fiber‐reinforced composites.

On the Identification of Orthotropic Elastic Stiffness Using 3D Guided Wavefield Data.

Scanning laser Doppler vibrometry is a widely adopted method to measure the full-field out-of-plane vibrational response of materials in view of detecting defects or estimating stiffness parameters. Recent technological developments have led to performant 3D scanning laser Doppler vibrometers, which give access to both out-of-plane and in-plane vibrational velocity components. In the present study, the effect of using (i) the in-plane component; (ii) the out-of-plane component; and (iii) both the in-plane and out-of-plane components of the recorded vibration velocity on the inverse determination of the stiffness parameters is studied. Input data were gathered from a series of numerical simulations using a finite element model (COMSOL), as well as from broadband experimental measurements by means of a 3D infrared scanning laser Doppler vibrometer. Various materials were studied, including carbon epoxy composite and wood materials. The full-field vibrational velocity response is converted to the frequency-wavenumber domain by means of Fourier transform, from which complex wavenumbers are extracted using the matrix pencil decomposition method. To infer the orthotropic elastic stiffness tensor, an inversion procedure is developed by coupling the semi-analytical finite element (SAFE) as a forward method to the particle swarm optimizer. It is shown that accounting for the in-plane velocity component leads to a more accurate and robust determination of the orthotropic elastic stiffness parameters.

Modelling and simulation of fracture in anisotropic brittle materials by the phase-field method with novel strain decompositions

When brittle and quasi-brittle materials, are concerned, it is essential to account for the fundamental fact that they behave differently according as they undergo tensile or compressive loadings. Due to this fact, in modelling and simulating the fracture of brittle and quasi-brittle materials by the phase-field method (PFM), the strain tensor is usually decomposed into a tensile part and a compressive part, and it is argued that only the strain energy related to the tensile part controls the nucleation and propagation of cracks. The present work improves the phase-field method by incorporating into it a novel strain decomposition strategy giving rise to strain decompositions being orthogonal in the sense of the natural inner product with the elastic stiffness tensor acting as a metric. The improved phase-field method involving only a scalar phase-field variable is applied to model and simulate the fracture of brittle anisotropic materials. The results thus obtained are compared with the relevant analytical and numerical ones reported in the literature. It turns out from these comparisons that the improved phase-field method is accurate and efficient.

Stability of intermediate states in the surface lithiation process of silicene

We have studied using density functional theory the Li-silicene system, mainly in the saturated region (0.11>X(Li)<1.0) which is critical to understand the solid electrolyte interface layer (SEI) formation. The electronic and vibrational properties of the intermediate states of lithiated silicene were calculated. Two models were used for the Li atoms on the silicene surface: Li on top of Si (perpendicular) and Li on top of the hexagon center (centered). The centered configurations have lower Eb values up to X(Li) = 0.33. However, the vibrational properties showed instabilities for lithium in the hexagon center. An intermediate state at X(Li) = 0.38 was found to be dynamically stable. A metallic to semiconductor transition is observed at X(Li) = 0.33, and X(Li) = 0.46. Few ordered vacancies or vacancies in the dilute region exhibit vibrational stability and gap opening. Elastic stiffness tensor for different concentrations of Li in the silicene-Li system were calculated, a critical concentration of X(Li)=0.333 was determined for the change of the mechanical properties behavior. silicel has a slightly higher 2D Young‘s moduli than silicene. However, vacancies in silicel produce a substantial decrement in mechanical properties.

Molecular dynamics modelling of amorphisation induced change in the mechanical properties of β-Li2TiO3

ABSTRACT The objective of this study is to understand the effect of irradiation on the mechanical properties of Lithium metatitanate (β-) at the molecular level. Computational samples with different levels of damage are subjected to tensile and compressive loads to study changes in elastic stiffness tensor and the crushing strength. Three levels of displacement ratios are chosen; stoichiometric, where atom types are chosen at random, secondly only Lithium atoms are displaced, and finally, atoms to be displaced are chosen in the ratio 60% Li 20% O 20% Ti. Further, local structural changes due to the interaction of stress and radiation are also analysed. It is observed that for stoichiometric and 60% Li, 20% Ti and 20% O displacement cases, the material becomes isotropic at around ≃0.5 dpa. However, for explicit Li displacement, the material continued to show an anisotropic response. This observation indicates that the collapse of the Ti/O sublattice was essential for amorphisation and deteriorating stiffness. However, it was found that very small disturbances in the O/Ti sub-lattice were sufficient to cause a drastic reduction in the crushing strength. Therefore, while small damages below 0.5 dpa may not drastically change the stiffness of the material, it could alter the crushing strength significantly.

Construction and validation of the upscaling model of organic-rich shale by considering water-sensitivity effects

Due to the complexity of its microscopic composition, the mechanical changes produced by the hydration process of organic-rich shale are more complicated than those of pure clay minerals. Although predecessors have researched the upscaling model of shale, the construction and evaluation of the water-saturated model remain a long-standing challenge. In this work, the combination of the differential effective medium (DEM) model and the self-consistent (SC) model (DEM-SC) was used to obtain a universal stiffness tensor of the clay matrix based on reverse modeling of ultrasonic pulse velocity (UPV) test results. Subsequently, a new water-sensitivity -based upscaling model was constructed that simultaneously considers: (1) the interlayer swelling, confining, and substitution effects of clay matrix under water-saturated conditions; and (2) the influence of depositional environment on the pore morphology of organic matter and the influence of maturity on its elastic properties. The upscaling model showed that the interlayer swelling effect is the most critical water-sensitivity effect, and the confining and the substitution effect can only compensate for part of the modulus reduction caused by interlayer swelling. In addition, the model behaved better when dealing with problems with a large volume ratio of inclusions (such as marine shale) and achieved good prediction results by controlling the relative errors compared with the microindentation experiments at less than 15%. Finally, the model was further applied to the well scale with better prediction results by considering the water-sensitivity effect, proving that the model has a wide range of application potential in formation evaluation and construction operations in shale gas production.

Isogeometric topology optimization of strain gradient materials

In this paper optimal topologies of isotropic linear elastic strain gradient materials are investigated by means of isogeometric topology optimization. The employment of strain gradient theory allows not only to capture the microstructural effects of materials but also to regularize stress/strain concentration phenomena and to address the so-called wedge forces. Isogeometric analysis is used in order to meet the requirements of higher-order continuity of strain gradient elasticity theory. For the purpose of determining the constitutive parameters of strain gradient materials, a novel experiment is conceived by using the Digital Image Correlation (DIC) technique. The so-called SIMP (Solid Isotropic Material with Penalization) method is applied to interpolate the material stiffness tensors (including the fourth-order and the sixth-order stiffness tensor) by power law with penalty. Benchmark numerical experiments are conducted to illustrate the computational effectiveness and numerical robustness of the proposed method.