Third-order Elasticity Research Articles

Measurement of the Static Nonlinear Third‐Order Elastic Moduli of Rocks: Problems and Applicability

AbstractThe third‐order elastic (TOE) model has been used to describe the widely observed nonlinear mechanical behaviors of earth materials. In addition to linear elastic constants (λ, μ), three nonlinear elastic moduli (A, B, C) are required for isotropic rocks. Contrary to previous research on dynamic TOE moduli, this study followed the protocol to measure strain and stress under uniaxial and hydrostatic compressive tests statically, which were later used to invert for the full set of TOE moduli for four standard rock types with differing pore structures of a porous oolitic limestone, quartz‐rich sandstones, and a dense crystalline basalt. The applicability of the TOE model to characterize nonlinearity depends on the fulfillment of path‐independence and small‐strain assumptions. Using the measured static TOE moduli, the finite element model demonstrates that the stress in the vicinity of the wellbore is more amplified than the stress in the linear elastic case, which leads to a wider zone of rock failure around the wellbore. Due to the long‐standing discrepancy between static and dynamic moduli, the rarely reported full set of static TOE moduli is necessary and will benefit future research in understanding the effect of rock nonlinearity on geophysical and geomechanical applications, such as long‐term safe storage of CO2 and generating process of geohazards.

Cross-modulation in guided wave propagation: how does it relate to the Luxemburg-Gorky effect?

The well-known Luxemburg-Gorky effect in radio waves has also been observed in elastic waves recently, which points to new possibilities for incipient damage detection. However, how the cross-modulation phenomenon of guided waves in a weakly nonlinear medium is related to the Luxemburg-Gorky effect remains an open question. This issue is investigated in this paper. Considering the third-order nonlinear elasticity of a plate waveguide, a theoretical framework is proposed to analyze the influence of the mode combination and mixing direction of a pair of single-frequency and modulated waves on the cross-modulated component generation. In particular, a codirectional shear-horizontal wave mixing scheme is highlighted which enables the generation of internally-resonant cross-modulated components at all frequencies. After verification by finite element simulation, mechanisms underpinning the cross-modulated components in the codirectional shear-horizontal wave mixing scheme are revealed through tactical tuning of the higher-order material elastic constants. Experiments are conducted to further confirm the phenomena and substantiate their relevance to the Luxemburg-Gorky effect. It is established that the cross-modulated components of guided waves can be generated and practically measured in a weakly nonlinear plate via both pure and mixed mechanisms as a result of the cubic nonlinearity instead of the quadratic nonlinearity. Compared with the conventional two-wave mixing methods based on quadratic nonlinearity, the cross-modulated components exhibit higher sensitivity to material microstructural changes, which is conducive to incipient damage detection. Although the observed nonlinear cross-modulation in guided waves shows similarities with the Luxemburg-Gorky effect, they stem from different mechanisms: the former from nonlinear elasticity and the latter nonlinear dissipation.

DFT-continuum characterization of third-order elasticity of sI methane hydrates under pressure

Methane gas hydrates (GHs) are polyhedral crystalline guest-host materials found under high pressure and low-temperature conditions, which can serve as an energy source. Previous work on methane GH material physics was limited to simple linear models, which only involves second-order elasticity. However, this is not fully suited to high-stress load conditions in technological applications and fundamental material physics. For other material systems, it has been demonstrated that third-order elasticity and pressure derivatives of second-order elasticity have a strong and hence significant correlation. To narrow a critical theory-simulation gap in gas hydrates materials research, in this work we expand prior work from second-order elastic constants (SOECs) to third-order elastic constants (TOECs). By using the open-source Python tool Elastic3rd and the DFT calculation software Vienna Ab initio Simulation Package (VASP), we found that the non-linear fitting involving TOECs gave a better overall prediction and a smaller root-mean-square deviation on pressure-strain evaluation when compared with linear fitting. In addition, the non-linear fitting provides robust results on the piezo-effect on the shear constant C44 and the ductile-to-brittle transition (P = −0.5 GPa). These results are not achievable from previous work based on a linear model and these findings prove that non-linear models, including TOECs, are needed under high pressures. In addition, this research includes a detailed analysis of the calculation of TOECs and mechanical properties to study pressure stability limits and ductile-brittle transitions. Together the results, findings, and analyses from this work are a novel and significant contribution to the material physics knowledge of gas hydrates and hydrogen-bonded crystalline materials.

Ab initio calculations of third-order elastic coefficients

Third-order elasticity (TOE) theory predicts strain-induced changes in second-order elastic coefficients (SOECs) and can model elastic wave propagation in stressed media. Although third-order elastic tensors have been determined based on first principles in previous studies, their current definition is based on an expansion of thermodynamic energy in terms of the Lagrangian strain near the natural, or zero pressure, reference state. This definition is inconvenient for predictions of SOECs under significant initial stresses. Therefore, when TOE theory is necessary to study the strain dependence of elasticity, the seismological community has resorted to an empirical version of the theory. This study reviews the thermodynamic definition of the third-order elastic tensor and proposes using an ``effective'' third-order elastic tensor. An explicit expression for the effective third-order elastic tensor is given and verified. We extend the ab initio approach to calculate third-order elastic tensors under finite pressure and apply it to two cubic systems, namely, NaCl and MgO. As applications and validations, we evaluate (a) strain-induced changes in SOECs and (b) pressure derivatives of SOECs based on ab initio calculations. Good agreement between third-order elasticity-based predictions and numerically calculated values confirms the validity of our theory.

On Strong Ellipticity and Infinitesimal Stability in Third-Order Nonlinear Strain Gradient Elasticity

On Strong Ellipticity and Infinitesimal Stability in Third-Order Nonlinear Strain Gradient Elasticity

Nonlocal continuum mechanics structures: The virtual powers method vs the extra fluxes topic

In this paper, once and for all, we want to strongly affirm that the thermodynamic framework for complex materials based on extra fluxes can lead to more severe restrictions with respect to the virtual powers scheme. Even if formally converging to the same final constitutive results, it seems only the consequence of an inappropriate approach to the problem itself. To achieve this goal, we proceed with a quick critical comparison between the different steps of the two methodologies: by way of example, we address novel constitutive settings for the third-order infinitesimal elasticity and viscoelasticity with an extension to thermo-elasticity, based on nonlocal revisited Green-Naghdi like thermal properties. Finally, suitable thermodynamic restrictions are derived from the generalized Clausius-Duhem inequality.

Third-order elastic constants and biaxial relaxation coefficient in wurtzite group-III nitrides by hybrid-density functional theory calculations

We determine the second-order elastic constants (SOECs) and the third-order elastic constants (TOECs) for wurtzite AlN, GaN, and InN using the hybrid-density functional theory calculations with the plane wave basis sets. We apply the analytical formulas for the deformation gradient tensors as functions of the Lagrangian strain in order to eliminate the truncation errors in the Taylor expansion series of the deformation gradients and to facilitate the calculation of the Lagrangian stress. We show that the convergence criteria for the calculation of the TOECs with respect to the k-points density and the plane wave cutoff energy are similar for the strain–energy method and the strain–stress approach. The strain–energy method turns out to be more stable against the numerical errors than the strain–stress approach, which requires smaller tolerance for the precision of the self-consistent calculations. The SOECs, extracted by the method of least squares, are consistent with the experimental data and the previous ab initio calculations. Then, we investigate the biaxial relaxation coefficient for AlN, GaN, and InN, subjected to biaxial stress in the plane perpendicular to the c axis of the wurtzite structure. This coefficient determines the relationship between the in-plane and out-of-plane strain components in thin films and quantum wells grown on c-plane substrates. We demonstrate that for InN and AlN, the biaxial relaxation coefficient increases significantly with the in-plane strain, whereas it shows the opposite behavior in GaN. These results are well described by the third-order elasticity theory and they cannot be modeled by the linear theory of elasticity, which predicts no dependence of the biaxial relaxation coefficient on the in-plane strain. Therefore, the obtained TOECs should prove very useful for the modelling of strain-related phenomena in heterostructures, nanostructures and devices made of the group-III nitride semiconductors.

Stress-dependent elasticity and wave propagation — New insights and connections

To establish a consistent framework for seismic wave propagation that accommodates the effects of stress changes, it is critical to take into account the different definitions of stress and their corresponding effects on seismic quantities (e.g., wave speeds) as dictated by continuum mechanics. Revisiting this fundamental theoretical foundation, we first emphasize the role of stress within various forms of the wave equation resulting from different choices of stress definitions. Subsequently, using this basis, we investigate connections among existing theories that describe the variation of elastic moduli as a function of changes in stress. We find that there is a direct connection between predicting stress-induced elastic changes with the well-known third-order elasticity tensor and the recently proposed adiabatic pressure derivatives of elastic moduli. However, each of these approaches has different merits and drawbacks in terms of experimental validation as well as in their use. In addition, we investigate the connection with another general approach that relies on micromechanical structures (e.g., cracks and pores). Although this can be done algebraically, it remains unclear as to which definition of stress and which corresponding constitutive relationship should be considered in practical scenarios. We support our analysis with validations using previously published benchmark experimental data.

Classical nonlinear simulation of low-order modes of Lamb waves in plate

For the classical nonlinear research of low-order modes of Lamb waves, this paper firstly introduces the classical nonlinearity derived from the intrinsic nonlinear induced low-order Lamb waves (S0 and A0 modes). Theoretical and numerical calculations are studied in two aspects. The influence of nonlinear effects on the nonlinear effects of classical nonlinear low-order Lamb waves and the cumulative growth effect are analyzed by finite element simulation. The results show that the nonlinear effect produced by the superelastic material model is greater than the geometric nonlinearity, and the linear elastic material model does not produce nonlinear effects. In addition, as the third-order elasticity increases in the material, the amplitude of the second harmonic gradually increases. The second harmonic generated by the A0 mode with phase velocity mismatch is the S0 mode. It can be seen that the group velocity matching is not a necessary condition for generating the second harmonic. Since the phase velocity matching is not satisfied, there is no cumulative growth effect; The higher the S0 mode phase velocity matching degree, the more obvious the cumulative growth effect, and the second harmonic is the S0 mode.

Poroacoustoelasticity for rocks with a dual-pore structure

Acoustoelasticity describes the interaction of acoustic waves with nonlinear elastic deformations, particularly the change of wave velocity due to initial stresses or strains in a predeformed body. The theory extends the strain energy to cubic terms (third-order elasticity) and allows for finite strains to model deformations at high confining pressures. However, the theory considers equant (stiff) pores but neglects the effects of soft (compliant) pores, such as microfractures, cracks, and grain contacts. Our main contribution is to include these effects. Application of the novel poroacoustoelasticity theory to ultrasonic measurements on carbonate samples at varying confining pressures provides a better fit for the measured data of pressure dependence of wave velocity. We have quantified the contribution of the compliant pores to the nonlinear behavior of the wave velocity and determined the relation between the threshold pressure (beyond which the theories with and without compliant pores yield the same velocity) and porosity and permeability. The extension of poroacoustoelasticity theory by incorporating a dual-pore structure provides better description for stress dependence of wave velocity in fluid-saturated heterogeneous rocks, which can be applicable in further field studies regarding reservoir characterization and in situ stress estimation.

Third-order nonlocal elasticity in buckling and vibration of functionally graded nanoplates on Winkler-Pasternak media

The focus of the present work is to present an analytical approach for buckling and free vibrations analysis of thick functionally graded nanoplates embedded in a Winkler-Pasternak medium. The equations of motion are derived according to both the third-order shear deformation theory, proposed by Reddy, and the nonlocal elasticity Eringen's model. For the first time, the equations are solved analytically for plates with two simply supported opposite edges, the solutions also turning helpful as shape functions in the analysis of structures with more complex geometries and boundary conditions. Sensitivity analyses are finally performed to highlight the role of nonlocal parameters, aspect and side-to-thickness ratios, boundary conditions, and functionally graded material properties in the overall response of plates and cylindrical shells. It is felt that the proposed strategy could be usefully adopted as benchmark solutions in numerical routines as well as for predicting some unexpected behaviors, for instance, in terms of buckling load, in thick nanoplates on elastic foundations.

Higher-Order Peridynamic Material Correspondence Models for Elasticity

Higher-order peridynamic material correspondence model can be developed based on the formulation of higher-order deformation gradient and constitutive correspondence with generalized continuum theories. In this paper, we present formulations of higher-order peridynamic material correspondence models adopting the material constitutive relations from the strain gradient theories. Similar to the formulation of the first-order deformation gradient, the weighted least squares technique is employed to construct the second-order and the third-order deformation gradients. Force density states are then derived as the Frechet derivatives of the free energy density with respect to the deformation states. Connections to the second-order and the third-order strain gradient elasticity theories are established by realizing the relationships between the energy conjugate stresses of the higher-order deformation gradients in peridynamics and the stress measures in strain gradient theories. In addition to the horizon, length-scale parameters from strain gradient theories are explicitly incorporated into the higher-order peridynamic material correspondence models, which enables application of peridynamics theory to materials at micron and sub-micron scales where length-scale effects are significant.

Inadequacy of Third-Order Elastic Coefficients for Predicting Nonlinearity in Highly n-Type-Doped Silicon Resonators

Third-order nonlinear elastic coefficients of silicon have been assumed to be the dominant factor in describing the nonlinear behavior of silicon-based resonators in literature. In this article, it is postulated that in spite of the common belief, third-order elastic coefficients may not be adequate to explain the nonlinear elastic behavior of silicon micro-resonators at high n-type doping concentrations. The nonlinear behavior observed in degenerately n-type-doped bulk-extensional mode resonators aligned to $\langle {100}\rangle $ orientation is carefully studied in which a spring-hardening effect is observed at large vibration amplitudes. It is shown that the existing analytical/numerical calculations and a proposed finite element model that are based on the utilization of third-order elastic (TOE) coefficients will all predict spring-softening in such resonators, thus suggesting insufficiency of the nonlinear model.

Nonlinear elasticity of silica nanofiber

Optical nanofibers (ONFs) represent versatile nanophotonic platforms for important photonic applications such as optical sensing and quantum and nonlinear optics. The attractiveness of ONFs arises from the tight optical confinement, their wide evanescent field in the subwavelength limit, their surface acoustic properties, and their high tensile strength. Here we investigate Brillouin light scattering in silica-glass ONFs under high tensile strain and show that the fundamental properties of elastic waves dramatically change due to elastic anisotropy and nonlinear elasticity for strain larger than 2%. This yields to unexpected Brillouin strain coefficients for all Brillouin resonances including surface and hybrid waves, followed by a nonlinear evolution at high tensile strength. We further provide a complete theoretical analysis based on third-order nonlinear elasticity of silica that agrees well with our experimental data. These new regimes open the way to the development of compact tensile strain optical sensors based on nanofibers.

Finite Third-order Gradient Elasticity and Thermoelasticity

A constitutive format for the third-order gradient elasticity is suggested. It includes both isotropic and anisotropic non-linear behavior under finite deformations. Appropriate invariant stress and strain variables are introduced, which allow for reduced forms of the elastic energy law that identically fulfill the objectivity requirement. After working out the transformation behavior under a change of the reference placement, the symmetry transformations for third-order materials can be introduced. After the mechanical third-order theory, an extension to thermoelasticity is given, and necessary and sufficient conditions are derived from the Clausius-Duhem inequality.

Linear and nonlinear elastic properties of isostatic graphite and graphite of MPG-7 brand

The paper presents the results of experimental studies of linear and nonlinear elastic properties of isostatic graphite and graphite of MPG-7 brand. In samples of isotropic structural graphite having the shape of a rectangular parallelepiped, the velocities of longitudinal and shear elastic waves and their dependence on the uniaxial compression of the sample under study were measured. Based on the results of these measurements, independent components of the second-order elasticity tensor in the samples were determined. The acoustoelastic effect in structural graphite was studied. The Thurston-Brugger method was used to calculate the independent components of the third-order elasticity tensor in samples of isostatic graphite and graphite of MPG-7 brand.

Anti-plane crack solutions in higher-order elasticity

The asymptotic solutions for cracks in a linear elastic medium under plane or anti-plane state were obtained by Williams (1957) nearly sixty years ago. However, solutions for cracks in second-order elasticity are unavailable, in contrast to those based on the neo–Hookean models, e.g., Knowles (1997). This paper addresses the formulation and solution of crack problems under finite anti-plane deformation using higher-order elasticity. It is found that: (1) a combined second- and third-order elasticity is necessary to ensure that full equilibrium is satisfied, (2) the equilibrium equations are non-homogeneous partial differential equations (pde's) with variable coefficients, and (3) exact particular solutions can be obtained while the homogeneous pde's reduce to nonlinear eigenvalue problems that can be solved numerically. The results show that: (1) the displacement or stress dependence on the radial coordinate is generally a function of the elastic constants, (2) the Piola–Kirchhoff stress matrix is fully populated, with induced normal stresses, in-plane as well as out-of-plane shear stresses, (3) singular stresses at the crack tip may exist, depending on the eigenvalues, and (4) normal stresses are predicted on the crack faces, which may lead to anomalous mechanical behavior in soft biological materials.

Mode pair selection of circumferential guided waves for cumulative second-harmonic generation in a circular tube

The appropriate mode pairs of primary and double-frequency circumferential guided waves (CGWs) have been investigated and selected for generation of the cumulative second harmonics, which are applicable for quantitative assessment of damage/degradation in a circular tube. The selection criteria follow the requirements: the higher efficiency of cumulative second-harmonic generation (SHG) of primary CGW propagation, and the larger response sensitivity of cumulative SHG to material damage/degradation [characterized by variation in the third-order elastic (TOE) constants]. The acoustic nonlinearity parameter β of CGW propagation and the change rate of normalized β versus the TOE constants of tube material are, respectively, used to describe the efficiency of SHG and its response sensitivity to damage/degradation. Based on the selection criteria proposed, all the possible mode pairs of primary and double-frequency CGWs satisfying the phase velocity matching have been numerically examined. It has been found that there are indeed some mode pairs of CGW propagation with the larger values both of β and the change rate of normalized β versus the TOE constants. The CGW mode pairs found in this paper are of practical significance for quantitative assessment of damage/degradation in the circular tube.

Probabilistic analysis and comparison of stress-dependent rock physics models

A rock physics model attempts to account for the nonlinear stress dependence of seismic velocity by relating changes in stress and strain to changes in seismic velocity and anisotropy. Understanding and being able to model this relationship is crucial for any time-lapse geophysical or geohazard modelling scenario. In this study, we take a number of commonly used rock physics models and assess their behaviour and stability when applied to stress versus velocity measurements of a large (dry) core data set of different lithologies. We invert and calibrate each model and present a database of models for over 400 core samples. The results of which provide a useful tool for setting a priori parameter constraints for future model inversions. We observe that some models assume an increase in VP/VS ratio (hence Poisson’s ratio) with stress. A trait not seen for every sample in our data set. We demonstrate that most model parameters are well constrained. However, third-order elasticity models become ill-posed when their equations are simplified for an isotropic rock. We also find that third-order elasticity models are limited by their approximation of an exponential relationship via functions that lack an exponential term. We also argue that all models are difficult to parametrize without the availability of core data. Therefore, we derive simple relationships between model parameters, core porosity and clay content. We observe that these relationship are suitable for estimating seismic velocities of rock but poor when comes to predicting changes related to effective stress. The findings of this study emphasize the need for improvement to models if quantitatively accurate predictions of time-lapse velocity and anisotropy are to be made. Certain models appear to better fit velocity depth log data than velocity–stress core data. Thus, there is evidence to suggest a limitation in core data as a representation of the stress dependence of the subsurface. The differences in the stress dependence of the subsurface compared to that measured under laboratory conditions could potentially be significant. Although potentially difficult to investigate, its importance is of great significance if we wish to accurately interpret the stress dependence of subsurface seismic velocities.

The Third-Order Elastic Moduli and Debye Temperature of SrFe2As2 and BaFe2As2: a First-Principles Study

The second- and third-order elastic constants of SrFe2As2 and BaFe2As2 with tetragonal structure are investigated by the first-principles methods combined with homogeneous deformation theory. The effective second-order elastic constants are obtained from the second- and third-order elastic constants for SrFe2As2 and BaFe2As2. Based on the effective second-order elastic constants, the phase stabilities and the Debye temperature of SrFe2As2 and BaFe2As2 have been investigated. It is predicted that SrFe2As2 and BaFe2As2 with tetragonal structure are not mechanically stable above 12.70 and 14.28 GPa, respectively, which agree with the available experimental and theoretical values.