Stiffness Tensor Research Articles

On the mechanical and mathematical properties of the stiffness and compliance coupling tensors of composite anisotropic laminates

The problem of determining the general and relevant mechanical and mathematical properties of the stiffness, [Formula: see text], and compliance, [Formula: see text], tensors describing the bending-extension coupling of anisotropic laminates is addressed in this paper. A set of theoretical results is proposed, showing some peculiarities of these tensors, from the mathematical and mechanical point of view. The theoretical analysis and the mechanical results are obtained through an extensive use of the so-called polar formalism, introduced as early as 1979 by Prof. G. Verchery.

Estimation of directional single crystal elastic properties from nano-indentation by correlation with EBSD and first-principle calculations

In this study, a two-step optimisation process for the estimation of the single crystal stiffness tensor from the indentation modulus is presented. This was accomplished by using a 2-dimensional data correlation method for nano-indentation, EBSD and ab initio data, as well as EDS for phase separation in a multi-phase material. Here, a single-phase Fe-24Ni-0.4C austenitic steel and a two-phase Fe-9.5Ni-0.5Co Seymchan meteorite were used as example materials.In a first step, the combination of high-speed nano-indentation mapping data with elemental and orientation distribution maps allowed the estimation of indentation moduli along specific crystal surface planes normal directions, (001), (011) and (111) using a least squares optimisation based on starting values from DFT or experimentally determined stiffness tensors. A second global optimisation step to estimate the single crystal stiffness tensor using pre-solved solution parameters of the Vlassak-Nix equations yielded reasonable correspondence between the experimentally determined stiffness tensors from correlative nano-indentation data and other methods. The presented method demonstrates generally the possibility to derive directionally sensitive elastic properties from high-speed nano-indentation for cubic materials statistically.

NbSbO4 Kristalinin Yapısal, Elastik ve Piezoelektrik Özelliklerinin İncelenmesi

The structural, elastic, and piezoelectric properties of the NbSbO4 crystal were calculated based on the density functional theory. These properties were calculated using the ABINIT package program under both the generalized gradient approximation and the local density approximation. The elastic stiffness tensor and the elastic compliance tensor for the NbSbO4 crystal were calculated in the ground state. Voigt Bulk Modulus, Reuss Bulk Modulus, Hill Bulk Modulus, Voigt Shear Modulus, Reuss Shear Modulus, Hill Shear Modulus, Young Modulus, Poisson Ratio, Flexibility Coefficient, Debye temperature, Longitudinal sound wave velocity for NbSbO4 crystal using elastic stiffness and elastic compliance tensor, Transverse sound wave velocity and Average speed of sound were calculated. Then, the ground state piezoelectric tensor of the NbSbO4 crystal was calculated. Accordingly, 2D longitudinal surfaces and 3D representation surfaces of the piezoelectric tensor were obtained using MTEX software. The properties obtained with both the generalized gradient approximation and the local density approximation are compared. As a result of the calculations, it was understood that the material was a flexible and formable material in both approximations.

Calculation of the macroscopic mechanical properties of polyamide 6.6 underwater influence using molecular dynamics simulations, hyperelastic material modeling of the amorphous fraction, and homogenization schemes

AbstractThe macroscopic mechanical properties of water‐influenced polyamide 6.6 is analyzed by combining molecular dynamics simulation and experimental measurements of the media concentration and the degree of crystallization. The mechanical properties of α‐crystalline and amorphous polyamide 6.6 are analyzed separately by using molecular dynamics simulations. The anisotropic linear‐elastic stiffness tensor of the crystalline material is determined by unidirectional tensile and shear stress simulations. The deformations of amorphous simulation cells with different water concentrations are fitted in a compressible hyperelastic Yeoh material model. The media concentration of tensile specimens conditioned in air with 50% relative humidity at 23°C and in water is determined after different conditioning times. The degree of crystallization of the material is determined using Differential Scanning Calorimetry. Both parameters are used to estimate the macroscopic mechanical properties of polyamide 6.6. The homogenization schemes by Reuss and Ahzi are compared to calculate the resulting macroscopic properties. Tensile tests are used to validate the calculated properties: The weakening of the material due to water molecules in the amorphous phase can be shown, but it is underestimated in the methods used.

Geometrically nonlinear static deflection of stiffened composite plates: A fifth-order equivalent model

Nonlinear reduced-order models (NLROM) have been developed to predict the geometrically nonlinear behaviour of structures with low computational cost. NLROMs built using non-intrusive methods are compatible with commercial FE software and have the potential for wide application in engineering. The challenge of non-intrusive NLROMs is to accurately reproduce the internal force of structures. In literature, the internal force has been assumed to be a cubic polynomial function of generalized coordinates. However, this assumption is effective only for structures with small rigid rotations but not for structures with significant rigid rotations such as cantilever structures. In this paper, we propose a fifth-order polynomial approximation of the internal force for such cases. The nonlinear stiffness tensors of the reduced-order model are built using explicit formulas, ensuring applicability to all structures. Our proposed reduced-order model accurately predicts the nonlinear static deflection of composite plates with complex geometries.

Linear viscoelastic transversely isotropic model based on the spectral decomposition of elasticity tensors

The linear viscoelasticity is still a useful model in the engineering for studying the behavior of materials loaded with different loading rates (frequencies). Certain types of materials reveal also an anisotropic behavior: fiber reinforced composites, asphalt concrete mixtures, or wood, to name a few. In general, researchers try to identify experimentally the dependence of engineering constants like: directional Young’s moduli and Poisson’s ratios on loading velocity by means of creep or harmonic oscillatory tests. This approach is appealing from the experimental point of view. However, from the modeling perspective, this is not the case. The engineering constants emerge in nonlinear manner in the relationship between the strain and stress via fourth order stiffness tensor components. This is especially true in higher order anisotropies, yet even in isotropy Poisson’s ratio appears nonlinearly in the stiffness tensor. Several models for the linear viscoelasticity of anisotropic materials already exist in the literature that try to tackle this issue. In this paper, we propose a linear viscoelasticity model for anisotropic materials based on the spectral decomposition of the stiffness tensor. The proposed model offers several advantages: the natural choice of the stiffness tensor eigenvalues as time-dependent variables, state variables with clear interpretation as creep strains, and reduced burden of storage utilizing the orthogonality of the eigenspaces. We implemented the model in the finite-element method system AceGen/AceFEM, and calibrated parameters of the model with the experimental data available in the literature, thus proving the adequacy of the proposed model to describe anisotropic viscoelastic materials.

Estimation of the stiffness tensor from Lamb wave velocity profiles measured on steel with different texture

It is important that steel meets the designated material parameters after production. Ideally, relevant parameters should be measured inline to have the ability to control the process during production. This requires a method that determines material parameters, such as the texture and various stiffness matrix components. These material properties can be derived from ultrasonic Lamb wave velocity measurements for different angles relative to the rolling direction. Objectives A method for estimating relevant stiffness matrix components needs to be developed, tested on simulation results and applied on measurements. Work performed A set of 10 cold-rolled Interstitial Free (IF) steel samples is provided by Tata Steel. These samples have different microstructures and thus different elastic properties. The ultrasonic Lamb wave velocity is measured for each of those samples over 360°. For each sample, a digital representative volume element (RVE) is generated using periodic Voronoi diagrams with a very large number of grains (10,000 – 100,000), based on through-thickness microstructural characterization by Electron Backscatter Diffraction (EBSD). The set of 10 RVEs is used in 3D Finite Difference simulations where grains are included individually and modelled as single crystals with specific orientations. The wave propagation of the S0 and SH0 modes is simulated. An inversion scheme has been developed to estimate a large number of components of the stiffness matrix from a velocity profile. Results Velocity profiles obtained from simulations are compared with those from measurements. For each sample, the results are very similar. The inversion scheme has been tested on simulations and applied to measurements from the IF samples. Results indicate that an acceptable accuracy can be achieved, although not all components are resolved equally well in the measurements. Conclusions The proposed inversion scheme enables the estimation of all relevant stiffness tensor components from a Lamb wave velocity profile with sufficient accuracy.

On thermal coefficients of thermo-magneto-electro-elastic N-laminates

In linear context, it is known that for n-phase 1D laminated structures, the effective magneto-electro-elastic (MEE) stiffness and compliance moduli tensors are exactly obtained from solving the generalized stress-strain relations from the “consistency phase boundary conditions” (CBC) of uniform gstresses and uniform gstrains normally and transversally to the layers respectively. This exact solution for laminates, although not referring to any homogenization framework, identifies with all (Hashin-Shtrikman, Mori-Tanaka, Self-Consistent, …) estimates which coincide at this order. Recently, the author has shown that it furthermore obeys a “generic” explicit simple form resulting from an essential property of the strain and stress dual Green operators (GOs) for infinite layers directionally stacked to form an infinite medium. This solution form is here examined in the extended affine context when thermal (T) gstresses and gstrains are accounted for as eigen-fields additional to the MEE ones. Explicating effective TMEE-coupled properties from this route corresponds with a specific form of the TMEE generalized Levin's (1967) formula for laminates, based on solely knowing the dual strain and stress laminate GOs. The obtained stress and strain thermal eigen-vector terms from this affine approach are examined in terms of their elastic, electric and magnetic components for the well documented case of two-phase BaTiO3–CoFe2O4 TI laminates.

Effective Mechanical Properties of Auxetic Materials: Numerical Predictions Using Variational Asymptotic Method Based Homogenization

Abstract In this work, the variational asymptotic method (VAM) based homogenization framework is used for the first time to determine the equivalent elastic stiffness tensor of auxetic materials. The proposed method allows the structural elements of the auxetic unit cell to naturally incorporate rotational degrees-of-freedom, without any ad-hoc assumptions. The overall macroscale homogenized response of the unit cells is considered to be fully anisotropic; specific possible responses, representative of orthotropy or transverse isotropy naturally emerge from the VAM-based homogenization, due to the arrangements of the structural elements making up the unit cell. For all the auxetic unit cell geometries considered in this study, the predictions obtained from the in-house python-based implementation of the VAM-based homogenization framework are validated using commercial finite element software (abaqus) and open literature. The results demonstrate the versatility and the computational efficiency of the VAM-based homogenization framework to describe auxetic metamaterials.

Micro–macro approach of anisotropic damage: A semi-analytical constitutive model of porous cracked rock

A semi-analytical constitutive model of a porous cracked rock is proposed based on the micro–macro approach within a thermodynamic framework. The representative volume element (RVE) of a porous cracked rock is divided into two subproblems, P1 and P2. The porosity evolution equation of a spherical pore is proposed for P1, whereas P2 is an inelastic problem containing numerous anisotropic penny-shaped cracks. First, the relationship between microscopic and macroscopic strains is established through homogenization. This relationship is introduced into the macroscopic free energy equation. A crack propagation criterion based on the Griffith release rate is proposed, and the evolution equations of the shear and tensile crack densities are proposed based on the surface energy function. Subsequently, the Gaussian numerical integration method is used to obtain a semi-analytical explicit expression of the macroscopic stiffness tensor. Finally, based on the proposed crack propagation criterion, a strength-prediction method is developed. A series of uniaxial and triaxial experimental data for different rocks from other references is used to validate the proposed constitutive model and strength prediction method. The results obtained using the proposed model agree well with the experimental curves obtained from uniaxial tension and triaxial compression tests performed on different rocks. The macroscopic mechanical behavior and evolution of microscopic parameters for different rocks with an increase in confining pressure are effectively demonstrated by the proposed model. Strength prediction method can accurately describe the failure strength of different rocks. Furthermore, a sensibility analysis of model parameters revealed that initial porosity e0 and dilation coefficient ζ have a deteriorating effect on the failure strength, and material constant η has an enhancing effect on the failure strength.

Pair Density Waves from Local Band Geometry.

A band-projection formalism is developed for calculating the superfluid weight in two-dimensional multiorbital superconductors with an orbital-dependent pairing. It is discovered that, in this case, the band geometric superfluid stiffness tensor can be locally nonpositive definite in some regions of the Brillouin zone. When these regions are large enough or include nodal singularities, the total superfluid weight becomes nonpositive definite due to pairing fluctuations, resulting in the transition of a BCS state to a pair density wave (PDW). This geometric BCS-PDW transition is studied in the context of two-orbital superconductors, and proof of the existence of a geometric BCS-PDW transition in a generic topological flat band is established.

Visualizing Higher-Order 3D Tensors by Multipole Lines.

Applications like physics, medicine, earth sciences, mechanical engineering, geo-engineering, bio-engineering use tensorial data. For example, tensors are used in formulating the balance equations of charge, mass, momentum, or energy as well as the constitutive relations that complement them. Some of these tensors (i.e. stiffness tensor, strain gradient, photo-elastic tensor) are of order higher than two. Currently, there are nearly no visualization techniques for such data beyond glyphs. An important reason for this is the limit of currently used tensor decomposition techniques. In this article, we propose to use the deviatoric decomposition to draw lines describing tensors of arbitrary order in three dimensions. The deviatoric decomposition splits a three-dimensional tensor of any order with any type of index symmetry into totally symmetric, traceless tensors. These tensors, called deviators, can be described by a unique set of directions (called multipoles by J. C. Maxwell) and scalars. These multipoles allow the definition of multipole lines which can be computed in a similar fashion to tensor lines and allow a line-based visualization of three-dimensional tensors of any order. We give examples for the visualization of symmetric, second-order tensor fields as well as fourth-order tensor fields. To allow an interpretation of the multipole lines, we analyze the connection between the multipoles and the eigenvectors/eigenvalues in the second-order case. For the fourth-order stiffness tensor, we prove relations between multipoles and the eigenvectors of the second-order right Cauchy-Green tensor and present different interpretations of the multipole lines.

Mechanical Perspective of CsSe Half‐Metallic Ferromagnet from DFT

AbstractCsSe is reported as a half‐metallic ferromagnet compound which is optimized using the first principle calculation, but it is yet to prepared experimentally. CsCl‐type and NaCl‐type structures are preferred in literature to form from their electronic structures. Whether these two structures physically withstand or not is still unclear. Their mechanical properties can give a clear picture. The present study works in this direction and uncovers the mechanical perspective using DFT (Density Functional Theory) framework. The elastic stiffness tensor is obtained first, evaluated the mechanical parameters and then revealed how the mechanical formation is possible. Overall study elucidates to the evidence of why the CsSe compound occurs in CsCl‐type and how it has withstood at a room temperature using the elastic and mechanical properties through DFT. The stability of CsCl‐type has been confirmed from the phonon dispersion by DFPT (Density Functional Perturbation Theory) in this paper. The study also conducted the examination of the CsCl‐type's spintronic nature.

Nonlinear finite volume discretization of geomechanical problem

AbstractElliptic differential operators describe a wide range of processes in mechanics relevant to geo‐energy applications. Extensively used in reservoir modeling, the Finite Volume Method with TPFA can be consistently applied to discretize only a specific type of application under severe assumptions. In this paper, we introduce a positivity preserving Nonlinear Two Point Stress Approximation (NTPSA) based on the recently developed collocated Finite Volume scheme for linear elastic mechanics. The gradient reconstruction is different from the one used in Nonlinear TPFA, but a similar form of weighting scheme is employed to reconstruct the traction vector at each interface. The convergence of the scheme is tested with a homogeneous anisotropic stiffness tensor. The motivation behind the implementation of a new discretization framework in mechanics is to develop a uniform discretization technique preserving monotonicity for generic poromechanics applications.

Active metamaterials for realizing odd mass density

Solids built out of active components have exhibited odd elastic stiffness tensors whose active moduli appear in the antisymmetric part and which give rise to non-Hermitian static and dynamic phenomena. Here, we present a class of active metamaterial featured with an odd mass density tensor whose asymmetric part arises from active and nonconservative forces. The odd mass density is realized using metamaterials with inner resonators connected by asymmetric and programmable feed-forward control on acceleration and active forces along the two perpendicular directions. The active forces produce unbalanced off-diagonal mass density coupling terms, leading to non-Hermiticity. The odd mass is then experimentally validated through a one-dimensional nonsymmetric wave coupling where propagating transverse waves are coupled with longitudinal ones whereas the reverse is forbidden. We reveal that the two-dimensional active metamaterials with the odd mass can perform in either energy-unbroken or energy-broken phases separated by exceptional points along principal directions of the mass density. The odd mass density contributes to the wave anisotropy in the energy-unbroken phase and directional wave energy gain in the energy-broken phase. We also numerically illustrate and experimentally demonstrate the two-dimensional wave propagation phenomena that arise from the odd mass in active solids. Finally, the existence of non-Hermitian skin effect is discussed in which boundaries host an extensive number of localized modes. It is our hope that the emergent concept of the odd mass can open up a new research platform for mechanical non-Hermitian system and pave the ways for developing next-generation wave steering devices.

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.

Strain orthogonal decomposition implemented within the phase field method with interfacial damage to model fracture in multi-phase heterogeneous materials

In recent times, the phase field method is a robust simulation tool that can predict crack formation and propagation in structures. In brittle and quasi-brittle materials, the strain tensor is decomposed into negative and positive parts corresponding to the compression and tension behaviors when the structure is loaded. In the previous studies related to the phase field method, the strain tensor decompositions do not satisfy the orthogonal condition of these two parts, even if the elastic stiffness tensor is isotropic. This problem leads to inaccuracies in the mechanical behavior of the materials. In this paper, the strain tensor orthogonal condition is applied to the bulk strain tensor parts representing the internal strain of the component phases. This orthogonal condition is implemented into the phase field method with interfacial damage to simulate crack propagation in a sample containing multi-phase heterogeneous materials. Furthermore, the shape and area fraction of these phases are recognized by a supplemental sub-function which was studied in our recent paper from image formats. Through several examples, the obtained results are compared with the relevant numerical ones to demonstrate the correctness and effectiveness of the proposed method.

PS reflection moveout in a homogeneous anisotropic layer of arbitrary symmetry and tilt

Use of unconverted reflected PP waves leads to moveout formulae, which relate the moveout to only a limited number of parameters specifying anisotropy. Use of converted reflected PS waves extends the number of parameters to, generally, a complete set of twenty-one parameters. It leads, on the other hand, to more complicated formulae and phenomena unknown from studies of unconverted PP waves. In anisotropic media one has to deal with the existence of two S waves and their singularities, acoustic approximation cannot be used. We present and test approximate formulae for the reflection moveout of PS waves converted at a horizontal reflector underlying a homogeneous layer of arbitrary anisotropy symmetry and orientation. For their derivation, we use the combination of weak-anisotropy approximation and A-parameters specifying anisotropy. The complete set of twenty-one A-parameters represents an alternative to twenty-one independent elements of the stiffness tensor. In addition to the moveout formulae for separate PS1 and PS2 waves, we also present moveout formula for a P to common S wave whose traveltimes represent an average of traveltimes of PS1 and PS2 waves. Use of the P to common S wave reflection leads to several simplifications. Presented tests indicate that for most offsets and models with P-wave anisotropy up-to to 26% and S-wave anisotropy over 25%, the relative traveltime errors do not exceed 2.5%.

Controlling Displacement Fields in Polar Willis Solids via Gauge Transformations.

Rigid-body displacement and deformation constitute the total displacement field of a solid. Harnessing the former calls for well-organized kinematic elements, and controlling the latter allows for creation of shape-morphing materials. A solid capable of simultaneously controlling both rigid-body displacement and deformation remains unknown. Here, we exploit gauge transformations to show how the total displacement field in elastostatic polar Willis solids can be harnessed at will and how those solids can be realized in the form of lattice metamaterials. The transformation method we develop leverages a displacement gauge in linear transformation elasticity, giving rise to polarity and Willis coupling such that the resulting solids not only break minor symmetries of the stiffness tensor, but display cross coupling between stress and displacement. We realize those solids using a combination of tailored geometries, grounded springs, and a set of coupled gears and numerically demonstrate a range of satisfactory, and peculiar, displacement control functions. Our work provides an analytical framework for the inverse design of grounded polar Willis metamaterials to achieve arbitrary displacement control functions by design.

Selective laser melted Ti6Al4V split-P TPMS lattices for bone tissue engineering

This study investigates Ti6Al4V Split-P triply periodic minimal surface (TPMS) structures produced by selective laser melting for the first time. The designs include two different cell morphologies (CM) with five different relative densities (RD). Scanning electron microscopy is utilized to assess the manufacturability and accuracy of the 3D printed Split-P lattice structures. The quasi-static mechanical responses are then studied to identify the failure mechanisms of each lattice type. Afterward, the stress-strain behavior and plateau stress responses are explored to evaluate the load-bearing capacity and energy absorption of five Split-P lattices. Furthermore, finite element analysis is performed to gain insight into the elasto-plastic behavior of the Split-P structures, and unit cell homogenization is employed to determine the equivalent stiffness tensor. The results demonstrate that the elastic modulus, yield strength, and ultimate strength of the 3D printed Split-P lattices range from 1.50 to 3.50 GPa, 57.95 to 152.74 MPa, and 93 to 170 MPa, respectively. Depending on RD and CM, load-bearing capacity range from 0.04 to 0.17. The energy absorption capacity and plateau stress of lattices vary from 22 to 61 MJ/m3 and 30 to 63 MPa, respectively. Gibson-Ashby power law represents the elastic modulus and yield stress of the Split-P lattices as a function of RD and reveals a stretch-dominated deformation behavior. Unit cell homogenization on Split-P structures with zero isovalue shows isotropy in mechanical properties and anisotropy for lattices with non-zero isovalue. Ti6Al4V Split-P lattices with the highest surface area and surface area-to-volume ratio, among other TPMS, can achieve mechanical properties close to those of trabecular and cortical bones, making them suitable for bone implants in load-bearing applications.