Second-order Displacement Research Articles

A Volterra edge dislocation problem in second-order elasticity theory

A displacement function technique has been developed to study plane strain problems in second-order elasticity theory for incompressible elastic materials. The formulation culminates in the development of a single inhomogeneous biharmonic equation of the form [Formula: see text] for the second-order displacement function [Formula: see text]. In the second-order, the isotropic stress associated with constitutive behaviour is governed by an inhomogeneous harmonic equation of the form [Formula: see text]. Both functions [Formula: see text] and [Formula: see text] depend only on the solution to the analogous problem in classical elasticity. While there are similarities to the Airy stress function, the displacement function approach enables the evaluation of the first- and second-order displacement fields purely through its derivatives, thus eliminating the introduction of arbitrary rigid body terms normally associated with formulations where the strains derived from a stress function approach need to be integrated. The displacement function approach is applied to develop a second-order elasticity solution to the problem of a Volterra edge dislocation.

Fractional-order velocity-dilatation-rotation viscoelastic wave equation and numerical solution based on a constant-Q model

Viscoelastic wave equations based on the constant- Q (CQ) model can accurately describe the amplitude dissipation and phase distortion of waves in anelastic media. However, only three velocity or displacement components can be obtained directly by solving such equations. Starting from the time-domain second-order displacement viscoelastic wave equation, we derived the decoupled P- and S-wave displacement vector viscoelastic wave equation by using the polarization difference of P- and S-wave propagation in isotropic media. The equation can be transformed into the velocity-dilatation-rotation viscoelastic wave equation containing the first-order temporal derivative and fractional Laplacian operators, which can be solved directly by using the staggered-grid finite-difference and pseudospectral methods. We use the low-rank decomposition method to approximate the derived mixed space-wavenumber domain fractional Laplacian operators for modeling wave propagation in heterogeneous attenuating media. We also demonstrated the precision of our equation by comparing the numerical solutions with the analytical solutions. Furthermore, compared with the conventional velocity-stress viscoelastic wave equation, experimental results demonstrate that our equation can separate the pure P and S waves from the mixed wavefield during wavefield continuation. In addition, it can be separated into an equation containing predominantly an amplitude attenuation or a phase distortion term.

An overset mesh-free finite-difference method for seismic modeling including surface topography

An overset mesh-free finite-difference method (FDM) is developed for 2D time-domain seismic wave modeling using second-order elastic displacement wave equations in a generally isotropic medium that exhibits complex surface topography and subsurface structures. The complex surface topography is discretized using an overset mesh-free node generation method, thus naturally arranging surface nodes on the surface topography and outperforming the Cartesian grid in terms of geometric flexibility. A Cartesian grid is used to discretize the entire computational region in terms of computational efficiency. The QR-decomposition radial-basis-function FDM (QR-RBF-FDM) and FDM are adopted to solve the equation independently in the mesh-free node and Cartesian grid. The data transfers between them are conducted through QR-RBF interpolation to facilitate mesh-free calculation, thus eliminating the requirement for node-to-grid matching. The improved free-surface boundary condition is implemented in the QR-RBF-FDM. Furthermore, a more appropriate formula for the outward normal direction is developed. The numerical solutions of our method are compared with those of the curvilinear grid FDM and standard FDM to verify the effectiveness of our method. The results indicate that our method avoids numerical scattering induced by staircase approximations of Cartesian grids while saving considerable computational costs. Therefore, this method is a better candidate than the mesh-free FDM for seismic numerical simulation with free surface topography.

Elastic wave pre-stack reverse-time migration based on the second-order P- and S-wave decoupling equation

Abstract The problems of large computation, large storage and low-frequency noises have been the key factors limiting the development of elastic wave pre-stack reverse-time migration (RTM). Most research has centered on the wave equation of first-order velocity stress and using Helmholtz separation to get pure primary and secondary waves. However, the cost is enormous and the amplitude and phase of wavefields will be changed. So, we use the second-order P- and S-wave decoupling equations to construct seismic wavefields that reduce the number of variables, and the storage space required for boundary wavefields is reduced by ∼78%. Then we bring the boundary-saving approach into our study during the wavefield extrapolation process so that we can reconstruct the source wavefields completely. Finally, for the wave equations of second-order displacement used in this study, the Poynting vector expressed by the traditional method was not applicable. Therefore, we derived the formula of the energy flux density vector represented by displacements and used it in directional traveling wave decomposition to suppress low-frequency noise in the imaging profile. Subsequently, based on the inner product imaging conditions, the RTM of the 2-D salt dome model was calculated. The results illustrate the effectiveness and practicability of the proposed solution.

Generalized quasiharmonic approximation via space group irreducible derivatives

The quasiharmonic approximation (QHA) is the simplest nontrivial approximation for interacting phonons under constant pressure, bringing the effects of anharmonicity into temperature-dependent observables. Nonetheless, the QHA is often implemented with additional approximations due to the complexity of computing phonons under arbitrary strains, and the generalized QHA, which employs constant stress boundary conditions, has not been completely developed. Here we formulate the generalized QHA, providing a practical algorithm for computing the strain state and other observables as a function of temperature and true stress. We circumvent the complexity of computing phonons under arbitrary strains by employing irreducible second-order displacement derivatives of the Born-Oppenheimer potential and their strain dependence, which are efficiently and precisely computed using the lone irreducible derivative approach. We formulate two complementary strain parametrizations: a discretized strain grid interpolation and a Taylor series expansion in symmetrized strain. We illustrate our approach by evaluating the temperature and pressure dependence of select elastic constants and the thermal expansion in thoria (${\mathrm{ThO}}_{2}$) using density functional theory with three exchange-correlation functionals. The QHA results are compared to our measurements of the elastic constant tensor using time-domain Brillouin scattering and inelastic neutron scattering. Our irreducible derivative approach simplifies the implementation of the generalized QHA, which will facilitate reproducible, data-driven applications.

Locating Neighborhood Displacement Risks to Climate Gentrification Pressures in Three Coastal Counties in Florida

Residential displacement as a result of climate change impacts will manifest on varied temporal and spatial scales affecting both coastal and inland communities. As coastal residents contend with more frequent shocks and stress from sea-level rise, many will eventually need to abandon their homes and property. Inland communities consisting of socioeconomically vulnerable populations with less exposure to climate change impacts will presumably be at risk for residential displacement. This study uses three counties in Florida—Duval, Pinellas, and Miami-Dade—as cases, and employs publicly available data. The well-known principal components analysis generated representative components of housing, socioeconomic, and demographic characteristics known to predispose neighborhoods to the risk of secondary residential displacement. An additive model fit to the individual principal components analysis retained six components for Duval and Miami-Dade counties and seven for Pinellas county. The subsequent displacement risk categorization captures interrelationships between neighborhood socioeconomic and biophysical risk and allows for the identification of areas with higher secondary displacement risks. High potential second-order displacement was found in 30 percent, 25 percent, and 20 percent of the block groups in Duval, Miami-Dade, and Pinellas counties, respectively. This article serves as a framework to operationalize factors that predispose a neighborhood to climate gentrification driven by sea-level rise.

On Spencer’s displacement function approach for problems in second-order elasticity theory

The paper describes the displacement function approach first proposed by AJM Spencer for the formulation and solution of problems in second-order elasticity theory. The displacement function approach for the second-order problem results in a single inhomogeneous partial differential equation of the form , where is Stokes’ operator and depends only on the first-order or the classical elasticity solution. The second-order isotropic stress is governed by an inhomogeneous partial differential equation of the form , where is Laplace’s operator and depends only on the first-order or classical elasticity solution. The introduction of the displacement function enables the evaluation of the second-order displacement field purely through its derivatives and avoids the introduction of arbitrary rigid body terms normally associated with formulations where the strains need to be integrated. In principle, the displacement function approach can be systematically applied to examine higher-order effects, but such formulations entail considerable algebraic manipulations, which can be facilitated through the use of computer-aided symbolic mathematical operations. The paper describes the advances that have been made in the application of Spencer’s fundamental contribution and applies it to the solution of Kelvin’s concentrated force, Love’s doublet, and Boussinesq’s problems in second-order elasticity theory.

Analysis of P wave induced second-order harmonic field at solid–fluid interface with potential matrix algorithm

Based on the second-order harmonic potential theory, the characteristics of the second-order harmonic field generated at the solid–liquid interface induced by P wave incidence are analyzed. A planar model of the solid–liquid interface is established to study the variation of the second-order displacement field versus the incident angles. The homogeneous solution coefficient matrix, refraction and reflection coefficient matrix are introduced. According to the boundary conditions and Lagrange's various parameters method, the second-order displacement field is obtained, and its dependence on the solid–liquid interface is investigated. The different effects of boundary on the tangential displacement and normal displacement are demonstrated. Numerical simulation shows that the complete solution varies slightly at the incident angle, and the tangential displacement and the normal displacement change sharply at a mutation angle θω due to the boundary effect.

Second-Order Ultrasound Elastography With L1-Norm Spatial Regularization.

Time delay estimation (TDE) between two radio-frequency (RF) frames is one of the major steps of quasi-static ultrasound elastography, which detects tissue pathology by estimating its mechanical properties. Regularized optimization-based techniques, a prominent class of TDE algorithms, optimize a nonlinear energy functional consisting of data constancy and spatial continuity constraints to obtain the displacement and strain maps between the time-series frames under consideration. The existing optimization-based TDE methods often consider the L2 -norm of displacement derivatives to construct the regularizer. However, such a formulation over-penalizes the displacement irregularity and poses two major issues to the estimated strain field. First, the boundaries between different tissues are blurred. Second, the visual contrast between the target and the background is suboptimal. To resolve these issues, herein, we propose a novel TDE algorithm where instead of L2 -, L1 -norms of both first- and second-order displacement derivatives are taken into account to devise the continuity functional. We handle the non-differentiability of L1 -norm by smoothing the absolute value function's sharp corner and optimize the resulting cost function in an iterative manner. We call our technique Second-Order Ultrasound eLastography (SOUL) with the L1 -norm spatial regularization ( L1 -SOUL). In terms of both sharpness and visual contrast, L1 -SOUL substantially outperforms GLobal Ultrasound Elastography (GLUE), tOtal Variation rEgulaRization and WINDow-based time delay estimation (OVERWIND), and SOUL, three recently published TDE algorithms in all validation experiments performed in this study. In cases of simulated, phantom, and in vivo datasets, respectively, L1 -SOUL achieves 67.8%, 46.81%, and 117.35% improvements of contrast-to-noise ratio (CNR) over SOUL. The L1 -SOUL code can be downloaded from http://code.sonography.ai.

The singularity in the second-order solution of flexural-gravity waves and the influence of plate length on the second-order response of a finite elastic plate

Based on potential flow theory, the second-order interaction between nonlinear waves with infinite and finite elastic plates is examined by using the eigenfunction expansion method and Higher Order Boundary Element Method (HOBEM) coupling with the plate's Green function in the frequency domain. For infinite elastic plates, the singular phenomenon of second-order flexural-gravity waves is examined, and for finite elastic plates, the effects of plate length on the maximum second-order displacement and occurring frequency are studied. The maximum second-order displacement of the finite elastic plate increases with increasing plate length, and the frequency of the maximum displacement decreases and approaches the singular frequency of the corresponding infinite elastic plate with increasing plate length. In addition, the effects of the plate bending stiffness and water depth on the maximum second-order displacement are also investigated.

Second–order models for the bending analysis of thin and moderately thick circular cylindrical shells

The suitability of employing the second-order shear deformation theory to static bending problems of thin and moderately thick isotropic circular cylindrical shells was investigated. Two variant forms of the polynomial second-order displacement models were considered. Both models account for quadratic expansions of the surface displacements along the shell thickness, although the second model (SSODM) was augmented by the initial curvature term. The equilibrium equations were derived by use of the principle of virtual work. Navier analytical solutions were obtained under simply supported boundary conditions. The results of the displacements and stresses revealed that the theory formulated on the SSODM provides a good depiction of the bending response of thin and moderately thick shells and are in close agreement with those of the first and higher-order shear deformation theories (FSDT; HSDT). The ability of the theory formulated on the first model (FSODM) to predict adequate values of displacements and stresses in thin shells was found to be significantly affected by changes in length to radius of curvature (l/a) ratios.

Analytical study on the effectiveness of hybrid FRP strengthening on behaviour of slender reinforced concrete square columns

Strengthening of reinforced concrete (RC) columns using external bonded (EB) fibre-reinforced polymer (FRP) composites has become popular since last three decades. The effectiveness of EB confinement reduces for slender columns due to the secondary moment arising from applied axial loads. The effectiveness of the hybrid FRP strengthening system for enhancing the performance of RC slender square columns is analytically investigated. Hybrid FRP strengthening uses a combination of EB FRP wrapping for confinement and longitudinal near-surface mounted (NSM) FRP laminates for bending. Hybrid FRP strengthening aims to reduce the second-order lateral displacement of the slender RC column and increase its axial capacity and flexural stiffness. A numerical modelling approach is presented to capture second-order effects in hybrid FRP strengthened slender RC square column based on Newmark's central difference scheme. Both material and geometric nonlinearities are accounted for in the calculations. The predictions are validated with test results available in the literature. After validation, a parametric study is carried out to evaluate the influence of slenderness ratio, the number of EB FRP layers, NSM laminate ratio, modulus of NSM laminate, and buckling of FRP composites. Results show that hybrid FRP strengthening is effective in improving the performance of slender RC columns.

Technical Note: Internal Second-Order Stiffness: A Refined Approach to the Rm Coefficient to Account for the Influence of P-? on P-?

One component of the B2 amplifier method of addressing second-order effects is the RM coefficient, which represents the influence of P-? on P-? effects. This paper presents the background for RM based on LeMessurier’s paper, “A Practical Method of Second-Order Analysis: Part 2—Rigid Frames,” (1977), and makes explicit the simplifications entailed in the AISC Specification for Structural Steel Buildings (AISC, 2016b) formulation for this coefficient. These simplifications, while providing for reliable strength design, can overestimate the P-? effect for typical building applications, especially if applied to drift. A simple formula for RM based on the work of LeMessurier permits a more precise estimate, which can be used as a component of both force and displacement amplifiers presented in this paper. This explicit approach to the RM coefficient provides the basis for clear presentation of the relationships first-order and second-order stiffness (both internal and external), including the distinct effects of P-? and P-? stiffness reductions on equilibrium at the second-order displacement.

Micromechanical modeling of a cracked elliptically orthotropic medium

In this paper, we derive the second-order crack opening displacement tensor for an arbitrarily oriented elliptical crack in an elliptically orthotropic (EO) matrix. This result is obtained in explicit closed form. The approach is based on the Saint-Venant’s idea of linear transformation between boundary value problems for elliptically orthotropic and isotropic bodies. The solution utilizes the classical representation of an ellipsoid crack where the smallest aspect ratio approaches zero and the transformation of the Taylor expansion of the corresponding Hill tensor. It is shown, in particular, that transformed cracks have neither the same in-plane aspect ratio nor the same vanishing aspect ratio. It requires a correction factor in the crack opening displacement tensor. Some specific relative orientations of the crack with respect to the symmetry planes of the EO matrix are considered in detail and effective properties are calculated in the case of randomly distributed cracks. The result is also extended to the case of a cylindrical (plane strain) crack.

Large amplitude vibrations of a deepwater riser conveying oscillatory internal fluid flow

A marine riser operated in deepwater could encounter harsh environmental conditions and exhibit complex vibration phenomena which require thorough investigation, especially using nonlinear geometric diagnosis. The aim of this paper was to propose a theoretical formulation for undertaking large amplitude vibration analysis of a deepwater marine riser conveying oscillatory internal fluid flow. The work-energy principle based on virtual displacement was utilized to formulate the riser mathematical model, accounting for internal strain energy and external work done. The geometric nonlinearity of the riser regarding extensible elastica theory was considered, which accounted for axial stretching and the large deformed curvature of the riser. The finite element method was utilized to form the equation of the motion system for solving numerical solutions. Based on Taylor's series expansion, the stiffness matrices with second-order nonlinear dynamic displacements of the riser were regarded in the equation of motion. The hydrodynamic ocean force modeled in terms of the squared riser velocity also yielded the nonlinear hydrodynamic damping matrix. The inertial force initiated by the pulsatile flow of transported fluid was taken into account as a harmonic oscillation representing the pump operation. Performing the Newmark time integration incorporated direct iteration on the equation of motion provided the nonlinear vibration response. A thorough parametric study was accomplished which investigated the influences of geometric nonlinearity, the forcing frequencies of a hydrodynamic wave, and an internal pulsatile flow on the large amplitude dynamic response characteristics of a deepwater riser.

On a consistent rod theory for a linearized anisotropic elastic material: I. Asymptotic reduction method

An asymptotic reduction method is introduced to construct a rod theory for a linearized general anisotropic elastic material for space deformation. The starting point is Taylor expansions about the central line in rectangular coordinates, and the goal is to eliminate the two cross-section spatial variables in order to obtain a closed system for displacement coefficients. This is first achieved, in an ‘asymptotically inconsistent’ way, by deducing the relations between stress coefficients from a Fourier series for the lateral traction condition and the three-dimensional (3D) field equation in a pointwise manner. The closed system consists of 10 vector unknowns, and further refinements through elaborated calculations are performed to extract bending and torsion terms and to obtain recursive relations for the first- and second-order displacement coefficients. Eventually, a system of four asymptotically consistent rod equations for four unknowns (the three components of the central-line displacement and the twist angle) are obtained. Six physically meaningful boundary conditions at each edge are obtained from the edge term in the 3D virtual work principle, and a one-dimensional rod virtual work principle is also deduced from the weak forms of the rod equations.

A Re-Examination of Wave Dispersion and on Equivalent Spatial Gradient of the Integral in Bond-Based Peridynamics

In this paper, wave dispersion properties in bond-based peridynamics (BBPD) are re-examined. By BBPD formulation, wave dispersion (frequency (ω) - wavenumber (k)) relation is known to be transcendental in nature. In fact, for uniform micromodulus function (C), there exists a frequency bandwidth with multiple k. Further, there is a ω within this bandwidth at which propagating k are infinite in number. Question that needs an answer is can all k propagate in the BBPD? In literature, an agreement between certain continuum gradient models (CM) and BBPD has been reported. This agreement is established in the sense that, for a judicious choice of C, wave dispersion properties are the same between BBPD and CM. This equivalence between a finite-ordered (CM) and an infinite-ordered (BBPD) displacement gradient model motivates to an associated converse question: given a C, does BBPD integral represent an effective displacement gradient of a finite order? In this paper, an attempt has been made to fix an order to the BBPD integral with the help of a Fourier frequency-based spectral study, under a one-dimensional rod setting. Both kinetically local and nonlocal formulations of BBPD are considered. It is argued that only one wave mode may propagate in a BBPD rod. This implies, Neumann force condition at a boundary should read as a spatial gradient of order one. This further implies, the BBPD spatial integral effectively represents a second-order displacement gradient in space. The wave motion study of this paper corroborates with the literature that applies local boundary conditions to nonlocal boundary value problems.

Fast numerical method to generate halo catalogues in modified gravity (part I): second-order Lagrangian perturbation theory

ABSTRACT We present a new numerical method to determine second-order Lagrangian displacement fields in presence of modified gravity (MG). We start from the extension of Lagrangian perturbation theory (LPT) to a class of MG models, which can be described by a parametrized Poisson equation. We exploit Fast Fourier transforms to compute the full source term of the differential equation for the second-order Lagrangian displacement field. We compare its mean to the source term computed for specific configurations, for which a k-dependent solution can be found numerically. We choose the configuration that best matches the full source term, thus obtaining an approximate factorization of the second-order displacement field as the space term valid for Λ Cold Dark Matter (ΛCDM) times a k-dependent, second-order growth factor. Such approximation is used to compute second-order displacements for particles. The method is tested against N-body simulations run with standard and f(R) gravity: we rely on the results of a friends-of-friends code run on the N-body snapshots to assign particles to haloes, then compute the halo power spectrum. We find very consistent results for the two gravity theories: second-order LPT (2LPT) allows to recover the N-body halo power spectrum within ∼10 per cent precision to k ∼ 0.2–0.4 h Mpc−1, as well as halo positions. We show that the performance of 2LPT with MG is the same (within 1 per cent) as the one obtained for standard ΛCDM case. This formulation of 2LPT can quickly generate dark matter distributions with f(R) gravity, and can easily be extended to other MG theories.

Deformation of beams in the grade consistent theory of microstretch elastic solids

A microstretch continuum is a material with microstructure in which the microelements can stretch and contract independently of their translations and rotations. This paper is concerned with the grade consistent theory of microstretch elastic solids, where the second-order displacement is added to the classical set of independent constitutive variables. We study the equilibrium of a homogeneous and isotropic elastic beam loaded by tractions distributed over its plane ends. First, the problem of bending and extension is investigated. It is shown that the solution of the problem can be expressed in terms of solutions of three plane strain problems. Then, we study the problem of torsion in the framework of the grade consistent theory of microstretch elastic solids. This problem is solved with the help of three torsion functions. The results are used to investigate the torsion of a right circular cylinder.

Method of manufactured solutions code verification of elastostatic solid mechanics problems in a commercial finite element solver

Much progress has been made in advancing and standardizing verification, validation, and uncertainty quantification practices for computational modeling in recent decades. However, examples of rigorous code verification for solid mechanics problems in literature remain scarce, particularly for commercial software and for the non-trivial large-deformation analyses and nonlinear materials typically needed to simulate medical devices. Here, we apply the method of manufactured solutions (MMS) to verify a commercial finite element code for elastostatic solid mechanics analyses using linear-elastic, hyperelastic (neo-Hookean), and quasi-hyperelastic (Hencky) constitutive models. Analytical source terms are generated using Python/SymPy and are implemented in ABAQUS/Standard without modification to solver source code. Source terms for the three constitutive models are found to vary nearly six orders of magnitude in the number of mathematical operations they contain. Refinement studies reveal second-order displacement and first-order (or higher) stress and strain convergence in response to mesh refinement for all constitutive models and first-order displacement convergence in response to pseudo-time increment refinement for the Hencky-elastic case. We also investigate the sensitivity of convergence order to quantitatively minor changes to the underlying mathematical model using an exploratory case. Code used to generate the MMS source terms and simulation input files are provided as Supplemental Material.