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We present a framework for the inverse design of spatially graded phononic materials based on ray tracing. Spatial grading of phononic materials allows the unit cell to vary smoothly in space. Compared to periodic architectures, spatial grading opens up a vast design space that promises new capabilities for manipulating elastic and acoustic waves. However, the use of spatial grading to control wave propagation has been limited to simple gradings, largely due to the bottleneck of modeling efficiency, and to the long-wavelength limit of low frequencies. In this work, ray tracing is exploited as an efficient alternative, around which we develop an optimization framework based on the adjoint state method toward the flexible inverse design of graded phononic materials. We demonstrate the design of graded mass-spring networks for objectives including focusing all directions of a point source, broadband focusing of a plane wave, and frequency sorting. These objectives are out of reach of periodic phononic materials, highlighting the high potential of spatially graded phononic materials. Moreover, our results demonstrate the importance of wave dispersion, which is classically neglected in the long-wavelength limit of elastodynamics. [ABSTRACT FROM AUTHOR]

The effective inclusion of a priori knowledge when embedding known data in physics‐based models of dynamical systems can ensure that the reconstructed model respects physical principles, while simultaneously improving the accuracy of the solution in the previously unseen regions of state space. This paper presents a physics‐constrained data‐driven discrepancy modeling method that variationally embeds known data in the modeling framework. The hierarchical structure of the method yields fine scale variational equations that facilitate the derivation of residuals which are comprised of the first‐principles theory and sensor‐based data from the dynamical system. The embedding of the sensor data via residual terms leads to discrepancy‐informed closure models that yield a method which is driven not only by boundary and initial conditions, but also by measurements that are taken at only a few observation points in the target system. Specifically, the data‐embedding term serves as residual‐based least‐squares loss function, thus retaining variational consistency. Another important relation arises from the interpretation of the stabilization tensor as a kernel function, thereby incorporating a priori knowledge of the problem and adding computational intelligence to the modeling framework. Numerical test cases show that when known data is taken into account, the data driven variational (DDV) method can correctly predict the system response in the presence of several types of discrepancies. Specifically, the damped solution and correct energy time histories are recovered by including known data in the undamped situation. Morlet wavelet analyses reveal that the surrogate problem with embedded data recovers the fundamental frequency band of the target system. The enhanced stability and accuracy of the DDV method is manifested via reconstructed displacement and velocity fields that yield time histories of strain and kinetic energies which match the target systems. The proposed DDV method also serves as a procedure for restoring eigenvalues and eigenvectors of a deficient dynamical system when known data is taken into account, as shown in the numerical test cases presented here. [ABSTRACT FROM AUTHOR]

Least‐squares reverse time migration is an increasingly popular technique for subsurface imaging, especially in the presence of complex geological structures. However, elastic least‐squares reverse time migration algorithms face significant practical and numerical challenges when migrating multi‐component seismic data acquired from irregular topography. Many associated issues can be avoided by abandoning the Cartesian coordinate system and migrating the data to a generalized topographic coordinate system conformal to surface topology. We introduce a generalized anisotropic elastic least‐squares reverse time migration methodology that uses the numerical solutions of tensorial elastodynamics for propagating wavefields in computational domains influenced by free‐surface topography. We define a coordinate mapping assuming unstretched vertically translated meshes that transform an irregular physical domain to a regular computational domain on which calculating numerical elastodynamics solutions is straightforward. This allows us to obtain numerical solutions of forward and adjoint elastodynamics and generate subsurface images directly in topographic coordinates using a tensorial energy‐norm imaging condition. Numerical examples demonstrate that the proposed generalized elastic least‐squares reverse time migration algorithm is suitable for generating high‐quality images with reduced artefacts and better balanced reflectivity that can accurately explain observed data acquired from topography in a medium characterized by arbitrary heterogeneity and anisotropy. Finally, the computational cost of our method is comparable to that of an equivalent Cartesian elastic least‐squares reverse time migration numerical implementation. [ABSTRACT FROM AUTHOR]

Time discontinuous Galerkin space-time finite element method (ST/FEM) can be used for developing arbitrary high-order accurate and unconditionally stable time integration schemes for elastodynamics problems. The existing ST/FEMs can be classified as the single-field and two-field ST/FEM: in the former method, either displacement or velocity, is independent and discontinuous in time. In contrast, in the latter method, both displacement and velocity fields are independent and discontinuous in time. Both methods have third-order accuracy for linear interpolation in time, higher than typical time integration schemes used in semi-discretized. However, these methods currently lack a unified computational framework, so each method requires a separate implementation. Therefore, the main goal of the present study is to develop a generalized computational framework that can facilitate the derivation and implementation of the existing linear-in-time ST/FEMs in a unified manner. This framework is developed by realizing that existing methods differ through the treatments of displacement-velocity relationships, which can be unified through displacement functions. In addition, by employing this framework, a new ST/FEM, which is designated as LC v-ST/FEM, is derived from the linear combination of displacement functions of single-field and two-field ST/FEMs. LC v-ST/FEM contains a user-defined parameter α ∈ [ 0 , 1 ] , which can be used for controlling the high-frequency dissipation characteristics. From finite difference analysis and numerical solutions of benchmark problems, it is demonstrated that the proposed method is the third order accurate in time, unconditionally stable, and contains negligible numerical dispersion error for all 0 ≤ α ≤ 1 . Moreover, for α ≠ 0 , the method can attenuate the spurious high-frequency components from the velocity and displacement fields. [ABSTRACT FROM AUTHOR]

A technique for developing convex dual variational principles for the governing PDE of nonlinear elastostatics and elastodynamics is presented. This allows the definition of notions of a variational dual solution and a dual solution corresponding to the PDEs of nonlinear elasticity, even when the latter arise as formal Euler–Lagrange equations corresponding to non-quasiconvex elastic energy functionals whose energy minimizers do not exist. This is demonstrated rigorously in the case of elastostatics for the Saint-Venant Kirchhoff material (in all dimensions), where the existence of variational dual solutions is also proven. The existence of a variational dual solution for the incompressible neo-Hookean material in 2-d is also shown. Stressed and unstressed elastostatic and elastodynamic solutions in 1 space dimension corresponding to a non-convex, double-well energy are computed using the dual methodology. In particular, we show the stability of a dual elastodynamic equilibrium solution for which there are regions of non-vanishing length with negative elastic stiffness, i.e. non-hyperbolic regions, for which the corresponding primal problem is ill-posed and demonstrates an explosive 'Hadamard instability;' this appears to have implications for the modeling of physically observed softening behavior in macroscopic mechanical response. [ABSTRACT FROM AUTHOR]

For a one dimensional 2-lattice we write down the momentum equation and the equation ruling the shift vector for the dynamic case as a first order system and characterize it in terms of its hyperbolicity. We use similar arguments for problems of increasing difficulty and tackle: one dimensional 3-lattices, three dimensional 2-lattices, three dimensional 3-lattices and finally the most general case of three dimensional (n + 1) -lattices. Our approach is confined to the geometrically and materially linear elastodynamic theory and is valid for generic anisotropic materials that have the multilattice structure. The main finding is that, with the assumptions adopted, the presence of each shift vector is related with zero eigenvalues when the system is written as a first order system. [ABSTRACT FROM AUTHOR]

The concept of metamaterial recently emerged as a new frontier of scientific research, encompassing physics, materials science and engineering. In a broad sense, a metamaterial indicates an engineered material with exotic properties not found in nature, obtained by appropriate architecture either at macro-scale or at micro-/nano-scales. The architecture of metamaterials can be tailored to open unforeseen opportunities for mechanical and acoustic applications, as demonstrated by an impressive and increasing number of studies. Building on this knowledge, this theme issue aims to gather cutting-edge theoretical, computational and experimental studies on elastic and acoustic metamaterials, with the purpose of offering a wide perspective on recent achievements and future challenges. This article is part of the theme issue, 'Current developments in elastic and acoustic metamaterials science (Part 2)'. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to address a class of hybrid optimal control problems constrained with hyperelasticity and constant global volume. This type of problems can intervene for example in the mechanical aspects of cardiac activity. The time deformation of the heart tissue is modeled with the elastodynamics equations dealing with the displacement field as main unknown. These equations are coupled with a pressure whose time variations are aimed to be maximized. This pressure variable corresponds to a Lagrange multiplier associated with the so-called global injectivity condition, translating the fact that the total volume of the domain remains constant. We develop an optimal control approach in a general framework that covers in particular the maximization of the variations of this pressure, and also the time the maximum is reached, defining what we call a hybrid optimal control problem. Mathematical analysis based on the $ \mathrm {L}^p $ Lp-parabolic maximal regularity is provided for the state equations and the rigorous derivation of optimality conditions. Numerical simulations for a toy-model illustrate the capacity of the approach. [ABSTRACT FROM AUTHOR]

Motivated from standard procedures in linear wave equations, we write the equations of classical elastodynamics as a linear symmetric hyperbolic system in terms of the displacement gradient ( u x ) and the velocity ( u t ); this is in contrast with common practice, where the stress tensor and the velocity are used as the basic variables. We accomplish our goal by a judicious use of the compatibility equations. The approach using the stress tensor and the velocity requires use of the time differentiated constitutive law as a field equation; the present approach is devoid of this need. The symmetric form presented here is based on a Cartesian decomposition of the variables and the differential operators that does not alter the Hamiltonian structure of classical elastodynamics. We comment on the differences of our approach with that using the stress tensor in terms of the differentiability of the coefficients and the differentiability of the solution. Our analysis is confined to classical elastodynamics, namely geometrically and materially linear anisotropic elasticity which we treat as a linear theory per se and not as the linearization of the nonlinear theory. We, nevertheless, comment on the symmetrization processes of the nonlinear theories and the potential relation of them with the present approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider a dynamic model of fracture for viscoelastic materials, in which the constitutive relation, involving the Cauchy stress and the strain tensors, is given in an implicit nonlinear form. We prove the existence of a solution to the associated viscoelastic dynamic system on a prescribed time-dependent cracked domain via a discretization-in-time argument. Moreover, we show that such a solution satisfies an energy-dissipation balance in which the energy used to increase the crack does not appear. As a consequence, in analogy to the linear case this nonlinear model exhibits the so-called viscoelastic paradox. [ABSTRACT FROM AUTHOR]

We show the existence of a weak solution for a system of partial differential equations describing the motion of a flexible solid inside a fluid: A nonlinear, viscoelastic, n-dimensional bulk solid governed by a PDE including inertia is interacting with an incompressible fluid governed by the (n-dimensional) Navier-Stokes equation for n ≥ 2. The result is the first allowing for large bulk deformations in the regime of long time existence for fluid-structure interactions. The existence is achieved by introducing a novel variational scheme involving two time-scales that allows us to extend the method of minimizing movements to hyperbolic problems involving nonconvex and degenerate energies. [ABSTRACT FROM AUTHOR]

A recent mixed formulation of the Virtual Element Method in 2D elastostatics, based on the Hu-Washizu variational principle, is here extended to 2D elastodynamics. The independent modeling of the strain field, allowed by the mixed formulation, is exploited to derive first order quadrilateral Virtual Elements (VEs) not requiring a stabilization (namely, self-stabilized VEs), in contrast to the standard VEs, where an artificial stabilization is always required for first order quads. Lumped mass matrices are derived using a novel approach, based on an integration scheme that makes use of nodal values only, preserving the correct mass in the case of rigid-body modes. In the case of implicit time integration, it is shown how the combination of a self-stabilized stiffness matrix with a self-stabilized lumped mass matrix can produce excellent performances both in the compressible and quasi-incompressible regimes with almost negligible sensitivity to element distortion. Finally, in the case of explicit dynamics, the performances of the different types of derived VEs are analyzed in terms of their critical time-step size. [ABSTRACT FROM AUTHOR]

Various spatial-gradient extensions of standard viscoelastic rheologies of the Kelvin–Voigt, Maxwell's, and Jeffreys' types are analysed in linear one-dimensional situations as far as the propagation of waves and their dispersion and attenuation. These gradient extensions are then presented in the large-strain nonlinear variants where they are sometimes used rather for purely analytical reasons either in the Lagrangian or the Eulerian formulations without realizing this wave propagation context. The interconnection between these two modelling aspects is thus revealed in particular selected cases. [ABSTRACT FROM AUTHOR]

This paper presents an analytical study on the in-plane vibrations of a rectangular elastic lattice plate. The plane lattice is modelled considering central and angular interactions. The lattice difference equations are shown to coincide with a spatial finite difference scheme of the corresponding continuous plate. The considered lattice converges to a 2D linear isotropic elastic continuum at the asymptotic limit for a sufficiently small lattice spacing. This continuum has a free Poisson's ratio, which must be lower than that foreseen by the rare-constant theory, to preserve the definite positiveness of the associated discrete energy. Exact solutions for the in-plane eigenfrequencies and modes are analytically derived for the discrete plate. The stiffness characterising the lattice interactions at the boundary is corrected to preserve the symmetry properties of the discrete displacement field. Two classes of constraints are considered, i.e., sliding supports at the nodes, one normal and the other parallel to the boundary. For both boundary conditions, a single equation for the eigenfrequency spectrum is derived, with two families of eigenmodes. Such behaviour of the lattice plate is like that of the continuous plate, the eigenfrequency spectrum of which has been given by Rayleigh. The convergence of the spectrum of the lattice plate towards the spectrum of the continuous plate from below is confirmed. Two continuous size-dependent plate models, considering the strain gradient elasticity and non-local elasticity, respectively, are built from the lattice difference equations and are shown to approximate the plane lattice accurately. [ABSTRACT FROM AUTHOR]

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Copyright of Transactions of Nanjing University of Aeronautics & Astronautics is the property of Editorial Department of Journal of Nanjing University of Aeronautics & Astronautics and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

In this paper we consider the vanishing viscosity limit of solutions to the initial boundary value problem for the compressible viscoelastic equations in the half space. When the initial deformation gradient does not degenerate and there is no vacuum initially, we establish the uniform regularity estimates of solutions to the initial-boundary value problem for the three-dimensional compressible viscoelastic equations in the Sobolev spaces. Then we justify the vanishing viscosity limit of solutions of the compressible viscoelastic equations based on the uniform regularity estimates and the compactness arguments. Both the no-slip boundary condition and the Navier-slip type boundary condition on velocity are addressed in this paper. On the one hand, for the corresponding vanishing viscosity limit of the compressible Navier-Stokes equations with the no-slip boundary condition, it is impossible to derive such uniform energy estimates of solutions due to the appearance of strong boundary layers. Consequently, our results show that the deformation gradient can prevent the formation of strong boundary layers. On the other hand, our results also provide two different kinds of boundary conditions suitable for the well-posedness of the initial-boundary value problem of the elastodynamic equations via the method of vanishing viscosity. Finally, it is worth noting that we take advantage of the Lagrangian coordinates to study the vanishing viscosity limit for the fixed boundary problem in this paper. [ABSTRACT FROM AUTHOR]

Metamaterials (MMs), which include phononic crystals (PCs) as a particular type, exhibit anomalous wave propagation properties through artificial design of topologies or lattice forms of unit‐cells. Recent advancements in MMs signify an ascendant research trend, providing promising design ideas and means for unprecedented wave propagation properties. The imperative for on‐demand, real‐time active control of wave propagation underscores the significance of tunable manipulation of acoustic/elastic waves, promoting the design and development of tunable MMs. Furthermore, the versatility of intelligent materials and their ongoing development and innovation contribute significantly to the emergence of diverse intelligent MMs. This comprehensive survey provides an overview of recent advancements and current research trends in the interdisciplinary field of intelligent MMs with electro‐/magneto‐mechanical couplings. The primary objective of the review is to emphasize significant progress in agile manipulation of acoustic/elastic waves in electro‐/magneto‐mechanical coupled MMs, followed by an in‐depth exploration of intelligent metasurfaces, topological MMs, non‐Hermitian parity‐time symmetric wave systems, odd elastic MMs, and spatiotemporally modulated MMs. Special emphasis is given to multi‐field coupling effects. The review concludes with a summary and outlines potential prospects, offering a timely and informative guide for future studies on actively tunable PCs and MMs in practical engineering applications. [ABSTRACT FROM AUTHOR]

The classical Lagrange problem for dynamical systems introduces a Lagrangian action functional defined for any dynamical process that is envisioned to take place over a fixed interval of time with its state at each time lying on an unknown, but prescribed, configuration between two given end points in an n -dimensional state space R n . It is proposed that the fundamental dynamical field equation that characterizes the dynamical process and determines the precise motion between the two given end points is the Euler–Lagrange equation related to the stationarity of the Lagrangian action functional, expressed as the integral of a particularly formulated action density over the fixed time interval, among all admissible configurations that span the two given end points. Thus stated, this variational calculus problem introduces variations of a configuration that carries a dynamical process, and emphasizes the novelty and need to express explicitly how the configuration influences the state of that process. At each time during a dynamical process the state is subjected to an extrinsic force (classically taken to be conservative) which must be transmitted to the configuration that carries the process and, by action-reaction the configuration responds with a configuration contact force on the state of equal magnitude but opposite direction. This allows the Lagrangian action functional for a dynamical process to be interpreted as the difference between the average kinetic energy of the dynamical process that is carried by that configuration and the average configurational work done by the configuration contact force on the moving state as the state traverses that configuration during the fixed time interval. The aim in the Problem of Lagrange is to extremize this difference over all admissible configurations. The implication is that given a time interval and initial and final end points in the space of all states, the dynamical process of physical interest must follow a configuration that optimizes the gap between the average expended kinetic energy and the average expended configurational work. When the optimal condition is met and the dynamical process is so restricted, the difference between these average expenditures of energy and work will be at a local maximum, a local minimum, or a saddle point known as a condition of "least action". Herein, we investigate the optimization implications of this novel interpretation of the action functional for the Problem of Lagrange for dynamical systems for a general, possibly non-conservative, state-dependent extrinsic force field. We show that only a conservative state-dependent extrinsic force field is allowable within the statement of the problem and, thus, reaffirm the predominant classical hypothesis of restricting attention to conservative extrinsic force fields. We close with a section which covers an analogous investigation of the Problem of Lagrange for nonlinear elastodynamics. [ABSTRACT FROM AUTHOR]

Mapping corrosion depths along pipeline sections using guided-wave-based tomographic methods is a challenging task. Accurate defect sizing depends heavily on the precision of the forward model in guided wave tomography. This model is fitted to measured data using inversion techniques. This study evaluates the effectiveness of a recursive extrapolation scheme for tomography applications and full waveform inversion. It employs a table-driven approach, with precomputed extrapolation operators stored across a spectrum of wavenumbers. This enables fast modelling for extensive pipe sections, approaching the speed of ray tracing while accurately handling complex velocity models within the full frequency band. This ensures an accurate representation of diffraction phenomena. The study examines the assumptions underlying the extrapolation approach, namely, the negligible reflection and conversion of modes at defects. In our tomography approach, we intend to use multiple wave modes— A 0 , S 0 , and S H 1 —and helical paths. The acoustic extrapolation method is validated through numerical studies for different wave modes, solving the 3D elastodynamic wave equation. Comparison with an experimentally measured single-mode wavefield from an aluminium plate with an artificial defect reveals good agreement. [ABSTRACT FROM AUTHOR]

The optimal local truncation error method (OLTEM) with unfitted Cartesian meshes recently developed for the scalar wave and heat equations for heterogeneous materials is extended to a more complex case of a system of the elastodynamics PDEs. Compact 9-point stencils (similar to those for linear finite elements) are used for OLTEM. Compared to our previous results, a new approach is used for the calculation of the right-hand side of the stencil equations due to body forces. It significantly simplifies the analytical derivations of OLTEM for time-dependent problems. There are no unknowns on interfaces between different materials; the structure of the global semi-discrete equations for OLTEM is the same for homogeneous and heterogeneous materials. For the first time we have also developed OLTEM with the diagonal mass matrix. In contrast to many known approaches with some ad-hoc calculations of the diagonal mass matrix, OLTEM offers a rigorous approach which is a particular case of OLTEM with the non-diagonal mass matrix. Another novelty of the article is a new post-processing procedure for the accurate calculations of stresses. It includes the same compact 9-point stencils as those in basic computations and uses the accelerations and the displacements at the grid points along with the PDEs for the stress calculations. OLTEM yields accurate numerical results for heterogeneous materials with big contrasts in the material properties of different components. Numerical experiments for elastic heterogeneous materials show: a) at the same number of degrees of freedom (dof), OLTEM with unfitted Cartesian meshes is more accurate than linear finite elements with similar stencils and conformed meshes; at the engineering accuracy of 0.1 % for the displacements, OLTEM reduces the number of dof by more than 20 times; at the engineering accuracy of 0.1 % for the stresses, OLTEM with the new post-processing procedure reduces the number of dof by more than 104 times compared to linear finite elements; b) at the same number of dof, OLTEM with unfitted Cartesian meshes is even more computationally efficient than high-order finite elements with much wider stencils and conformed meshes. This will lead to a huge reduction in the computation time for elastodynamics problems solved by OLTEM and will allow the direct computations of some complex wave propagation and structural dynamics problems for heterogeneous materials without the scale separation. [ABSTRACT FROM AUTHOR]

Equations of motion are derived for (visco)elastic, self-gravitating and variably rotating planets. The equations are written using a decomposition of the elastic motion that separates the body's elastic deformation from its net translational and rotational motion as far as possible. This separation is achieved by introducing degrees of freedom that represent the body's rigid motions; it is made precise by imposing constraints that are physically motivated and that should be practically useful. In essence, a Tisserand frame is introduced exactly into the equations of solid mechanics. The necessary concepts are first introduced in the context of a solid body, motivated by symmetries and conservation laws, and the corresponding equations of motion are derived. Next, it is shown how those ideas and equations of motion can readily be extended to describe a layered fluid–solid body. A possibly new conservation law concerning inviscid fluids is then stated. The equilibria and linearization of the fluid–solid equations of motion are discussed thereafter, along with new equations for use within normal-mode coupling calculations and other Galerkin methods. Finally, the extension of these ideas to the description of multiple, interacting fluid–solid planets is qualitatively discussed. [ABSTRACT FROM AUTHOR]

We introduce a general differentiable solver for time-dependent deformation problems with contact and friction. Our approach uses a finite element discretization with a high-order time integrator coupled with the recently proposed incremental potential contact method for handling contact and friction forces to solve ODE- and PDE-constrained optimization problems on scenes with complex geometry. It supports static and dynamic problems and differentiation with respect to all physical parameters involved in the physical problem description, which include shape, material parameters, friction parameters, and initial conditions. Our analytically derived adjoint formulation is efficient, with a small overhead (typically less than 10% for nonlinear problems) over the forward simulation, and shares many similarities with the forward problem, allowing the reuse of large parts of existing forward simulator code. We implement our approach on top of the open-source PolyFEM library and demonstrate the applicability of our solver to shape design, initial condition optimization, and material estimation on both simulated results and physical validations. [ABSTRACT FROM AUTHOR]

Engineered piezoelectric metamaterials can possess electro-momentum coupling between the macroscopic momentum and electric stimuli. This indicates the applicability of such metamaterials for wave scattering with an extra design degree of freedom, in the same way as Willis materials. To fully utilize this novel bianisotropy, we derive for the first-time theoretical bounds on electro-momentum coupling in wave scattering via energy conservation. As this coupling acts on both elastodynamics and electromagnetics, the polarizability tensor is generalized to fill their link in the bounds. Our derived bounds are verified via analytical scattering solutions. Results show that the bianisotropic scattering can be of the same order as the non-bianisotropic terms via the aid of electro-momentum coupling even for small Willis coupling. We further reveal the possibility of using electro-momentum coupling for tunable cloaking. This sheds light on the promising potential of piezoelectric metamaterials for tunable scattering devices whose bianisotropy can be modulated by external electric stimuli. [ABSTRACT FROM AUTHOR]

This work focuses on the numerical performance of HHT- α and TR-BDF2 schemes for dynamic frictionless unilateral contact problems between an elastic body and a rigid obstacle. Nitsche's method, the penalty method, and the augmented Lagrangian method are considered to handle unilateral contact conditions. Analysis of the convergence of an opposed value of the parameter α ~ for the HHT- α method is achieved. The mass redistribution method has also been tested and compared with the standard mass matrix. Numerical results for 1D and 3D benchmarks show the functionality of the combinations of schemes and methods used. [ABSTRACT FROM AUTHOR]

The thermodynamic model of viscoelastic deformable solids at finite strains is formulated in a fully Eulerian way in rates. Also, effects of thermal expansion or buoyancy due to evolving mass density in a gravity field are covered. The Kelvin–Voigt rheology with a higher-order viscosity (exploiting the concept of multipolar materials) is used, allowing for physically relevant frame-indifferent stored energies and for local invertibility of deformation. The model complies with energy conservation and Clausius–Duhem entropy inequality. Existence and a certain regularity of weak solutions are proved by a Faedo–Galerkin semi-discretization and a suitable regularization. Subtle physical limitations of the model are illustrated on thermally expanding neo-Hookean materials or materials with phase transitions. [ABSTRACT FROM AUTHOR]

We propose a new implicit–explicit scheme to address the challenge of modeling wave propagation within thin structures using the time‐domain finite element method. Compared to standard explicit schemes, our approach renders a time marching algorithm with a time step independent of the plate thickness and its associated discretization parameters (mesh step and order of approximation). Relying on the standard three dimensional elastodynamics equations, our strategy can be applied to any type of material, either isotropic or anisotropic, with or without discontinuities in the thickness direction. Upon the assumption of an extruded mesh of the plate‐like geometry, we show that the linear system to be solved at each time step is partially lumped thus efficiently treated. We provide numerical evidence of an adequate convergence behavior, similar to a reference solution obtained using the well‐known leapfrog scheme. Further numerical investigations show significant speed up factors compared to the same reference scheme, proving the efficiency of our approach for the configurations of interest. [ABSTRACT FROM AUTHOR]

Since Euler's original gyro-dynamic analysis nearly two and a half centuries ago, the use of multi-body dynamics (MBD) has spread widely in application scope from large displacement rigid body dynamics to infinitesimal amplitude elastodynamics. In some cases, MBD has become a multi-physics multi-scale analysis, comprising contact mechanics, tribo-dynamics, terramechanics, thermodynamics, biomechanics, etc. It is an essential part of all analyses in many engineering disciplines, including vehicle engineering. This paper provides an overview of historical developments with emphasis on vehicle development and investigation of observed phenomena, including noise, vibration and harshness. The approach undertaken is comprehensive and provides a uniquely focused perspective, one which has not hitherto been reported in the literature. [ABSTRACT FROM AUTHOR]

This article focuses on computational studies evaluating the influence of crystallinity, residual stresses, and out-of-plane (OOP) deterministic switching on Terfenol-D nano/microstructures. The computational models use both coupled and uncoupled Landau–Liftshitz–Gilbert equations with elastodynamics to study strain-induced magnetization reorientation. A Voronoi tessellation approach models the crystal distribution in the microstructures subjected to residual stresses with good agreement to experimental data including large changes in coercivity values, i.e., from 100 to 3000 Oe. Parametric studies show how the coercivity is manipulated with residual stresses, including a magnetoelastically induced perpendicular-magnetic-anisotropy (PMA), important for memory applications. Additional parametric studies focus on epitaxially deposited micro-disks, revealing that residual stresses can create magnetoelastically dominant easy axes along the ⟨ 110 ⟩ directions, which are energetically favorable relative to the intrinsic ⟨ 111 ⟩ magnetocrystalline easy axes. Modification of the global easy axis is used to design a strain-mediated multiferroic composite consisting of a 20 nm epitaxially deposited Terfenol-D memory bit with PMA grown on a PZT substrate. The multiferroic disk achieves OOP deterministic clocking with an applied voltage. [ABSTRACT FROM AUTHOR]

We extend the work of Marazzato et al. [Int. J. Numer. Methods Eng, 121:5295‐5319] on elastodynamics to the treatment of contact. To that end, we propose adequate handling of boundary conditions, either through the resolution of local problems on each of the face displacement unknowns or through interpolation from nearest neighbours. Adapting the time‐integration strategy adopted in Marazzato et al. [Comput Methods Appl Mech Eng, 347:906‐927], it is possible to conserve both momentum and a pseudo‐energy exactly. Numerical results are presented to illustrate the accuracy of contact treatment and the energy conservation of the system. [ABSTRACT FROM AUTHOR]

Founded on a novel analytical formulation that led to a rigorous yet compact path-integral representation of the time-domain elastodynamic half-space Green's function, a unified analysis of the possible occurrence of different singular wavefront behaviour in the response under arbitrary impulsive internal or surface point loads at arbitrary source-receiver locations is presented. With the decomposition of the general solution into distinct initiating and reflected wave group integrals that share a common factored format and simple contour definitions, the mathematical framework is shown to allow a straightforward identification of the specific conditions and the particular wave groups that are responsible for the singular wavefront phenomena without resorting to advanced analytic function theories or asymptotic methods. Analytic characterizations of the nature, strength and direction of all intrinsic singular wavefront behaviours of the three-dimensional Green's function in three canonical cases of source-receiver configurations are given in a dual integral-closed form format to facilitate their theoretical understanding as well as computational applications. Graphical illustrations of their variation with the source-receiver configuration and the medium's Poisson's ratio together with relevant comparison and clarifications of some classical treatments are included. [ABSTRACT FROM AUTHOR]

Recently, Murthy et al. (Commun Comput Phys 2:23, 2017. http://dx.doi.org/10.4208/cicp.OA-2016-0259) and Escande et al. (Lattice Boltzmann method for wave propagation in elastic solids with a regular lattice: theoretical analysis and validation, 2020. arXiv.doi:1048550/ARXIV.2009.06404. arXiv:2009.06404) adopted the Lattice Boltzmann Method (LBM) to model the linear elastodynamic behaviour of isotropic solids. The LBM is attractive as an elastodynamic solver because it can be parallelised readily and lends itself to finely discretised simulations of dynamic effects in continua, allowing transient phenomena such as wave propagation to be modeled efficiently. This work proposes simple local boundary rules which approximate the behaviour of Dirichlet and Neumann boundary conditions with an LBM for elastic solids. The boundary rules are shown to be consistent with the target boundary values in the first order. An empirical convergence study is performed for the transient tension loading of a rectangular plate, with a Finite Element (FE) simulation being used as a reference. Additionally, we compare results produced by the LBM for the sudden loading of a stationary crack with an analytical solution from Freund (Dynamic fracture mechanics. Cambridge Monographs on Mechanics. Cambridge University Press, Cambridge, 1990. https://doi.org/10.1017/CBO9780511546761). [ABSTRACT FROM AUTHOR]

The thermodynamic model of viscoelastic deformable magnetic materials at finite strains is formulated in a fully Eulerian way in rates. The Landau theory applied to a ferro-to-paramagnetic phase transition, the gradient theory (due to an exchange energy) for magnetization with general mechanically dependent coefficient, hysteresis in magnetization evolution by the Gilbert equation involving an objective corotational time derivative of magnetization, and the demagnetizing field are considered in the model. The Kelvin–Voigt viscoelastic rheology with a higher-order viscosity (exploiting the concept of multipolar materials) is used, allowing for physically relevant frame-indifferent stored energies and for local invertibility of deformation. The model complies with energy conservation and Clausius–Duhem entropy inequality. An existence and a certain regularity of weak solutions are proved by a Faedo–Galerkin semi-discretization and a suitable regularization. [ABSTRACT FROM AUTHOR]

The inverse problem of identifying a small unknown inclusion or cavity in an elastic medium is considered. Important applications are structural damage identification, medical imaging and geophysical exploration; the latter is the focus of the present investigation. It is assumed that the (isotropic but possibly heterogeneous) material properties of the medium are known, except for the presence of a small inclusion of a different material or of a small cavity. The goal is to find the location, size and shape of the inclusion or cavity. Identification is performed using full waveform inversion (FWI) and the adjoint method for the efficient calculation of the gradient of the misfit function. The identification is accomplished by considering a single unknown material‐property field. The inclusion manifests itself in the inverse solution as a local region where the unknown material property becomes significantly different than the known background property. The limiting case of cavity identification requires special treatment in the minimization process to avoid failure, as Bürchner et al. showed empirically for the scalar wave equation. Their ρ$$ \rho $$‐scaling approach, whose success is explained mathematically here for the first time, is extended to elastodynamics. The performance of the proposed method is demonstrated via numerical examples, involving two geophysical models: a homogeneous model and the heterogeneous Marmousi model, which is a standard testing model in geophysical research. [ABSTRACT FROM AUTHOR]

This manuscript presents an efficient solver for the linear system that arises from the hierarchical Poincar\'e--Steklov (HPS) discretization of three dimensional variable coefficient Helmholtz problems. Previous work on the HPS method has tied it with a direct solver. This work is the first efficient iterative solver for the linear system that results from the HPS discretization. The solution technique utilizes GMRES coupled with a locally homogenized block-Jacobi preconditioner. The local nature of the discretization and preconditioner naturally yield the matrix-free application of the linear system. Numerical results illustrate the performance of the solution technique. This includes an experiment where a problem approximately 100 wavelengths in each direction that requires more than a billion unknowns to achieve approximately 4 digits of accuracy takes less than 20 minutes to solve. [ABSTRACT FROM AUTHOR]

To evaluate the mechanical behavior and the load transfer mechanisms of a geosynthetic-reinforced pile-supported (GRPS) embankment under traffic load, an analytical model is developed based on elastodynamic theory, and its solution is derived rigorously. Since the present model is continuum based, the displacement and stress distributions in the embankment can be obtained analytically. Additionally, the present model takes the anisotropic property of the reinforced layer into consideration. The governing equations of the model are solved by introducing Fourier expansions of the field variables and making use of the theory of differential equations. With the boundary and interface conditions, the solution of the system is determined. The influences of some critical parameters are investigated. Here we show the load velocity has a significant influence on the response of the system, especially when it approaches the critical point. The present results suggest that the maximum efficiency of a single pile Max{EPi} at v = 100 m/s is almost 2 times higher than its value at v = 1 m/s. Some other parameters, such as pile cap spacing, embankment height, and the stiffness ratio between subsoil and piles, also affect the distributions of EPi , while the total pile efficiency EP only depends on the supporting properties. Some practical suggestions are also presented based on the parametric studies. [ABSTRACT FROM AUTHOR]

In our previous study (Part I), the anti-plane steady-state hyperbolic mode III fracture of a magneto-flexoelectric material was solved for the displacement, the polarization, and the magnetic fields. The solution, however, was based on the assumption of the development of strain discontinuities, and the propagation of the crack-tip was related to a critical shear strain. However, in the current study, the asymptotic details of the fields close to the crack-tip were investigated. The asymptotic analysis assumes strain continuity at the crack-tip (discontinuity in the strain gradients) and reveals the existence of a positive dynamic J-integral. The asymptotic analysis was performed not only for hyperbolic but also for elliptic conditions, and the energy release rate was calculated as a function of the crack-tip velocity in both regimes. These results are very different from those predicted by classical singular elastodynamics, where the dynamic J-integral is zero when super-shear is attained and there can be only an elliptic solution. Moreover, the results are very useful for couple-stress elastodynamics where equivalent length scales are present due to the analogy with flexoelectricity. [ABSTRACT FROM AUTHOR]

The stress propagation for a three-dimensional elastic sphere under a diametric loading condition in the framework of the linear elastodynamics is revisited. By describing displacements in terms of scalar and vector potentials using the Helmholtz theorem, the Navier-Cauchy equation, the time evolution equation for displacements, is converted to the wave equation. The wave equation is Laplace transformed and further solved in spherical coordinates to develop the tabular expressions for displacement and stress in terms of modified spherical Bessel functions, and the coefficients are determined to satisfy the initial and boundary conditions. The obtained solutions are given in the form of an inverse Laplace transform. For the steady solution, the long-time limit of the obtained solutions is derived by using the final value theorem of the Laplace transform. The unsteady solutions are obtained by applying the residue theorem of complex analysis by adding a path with zero contribution in the complex plane. The obtained solution includes longitudinal and transverse waves, and also Rayleigh waves propagating on the surface. The origin of the von Schmidt wave is discussed as a reflected wave produced by the longitudinal wave travelling on the surface. It is also discussed that the wave is bent, unlike in the case of semi-infinite systems.

A high-order accurate implicit-mesh discontinuous Galerkin framework for wave propagation in single-phase and bi-phase solids is presented. The framework belongs to the embedded-boundary techniques and its novelty regards the spatial discretization, which enables boundary and interface conditions to be enforced with high-order accuracy on curved embedded geometries. High-order accuracy is achieved via high-order quadrature rules for implicitly-defined domains and boundaries, whilst a cell-merging strategy addresses the presence of small cut cells. The framework is used to discretize the governing equations of elastodynamics, written using a first-order hyperbolic momentum-strain formulation, and an exact Riemann solver is employed to compute the numerical flux at the interface between dissimilar materials with general anisotropic properties. The space-discretized equations are then advanced in time using explicit high-order Runge–Kutta algorithms. Several two- and three-dimensional numerical tests including dynamic adaptive mesh refinement are presented to demonstrate the high-order accuracy and the capability of the method in the elastodynamic analysis of single- and bi-phases solids containing complex geometries.

This paper presents an analytical study on the in-plane vibrations of a rectangular elastic lattice plate. The plane lattice is modelled considering central and angular interactions. The lattice difference equations are shown to coincide with a spatial finite difference scheme of the corresponding continuous plate. The considered lattice converges to a 2D linear isotropic elastic continuum at the asymptotic limit for a sufficiently small lattice spacing. This continuum has a free Poisson’s ratio, which must be lower than that foreseen by the rare-constant theory, to preserve the definite positiveness of the associated discrete energy. Exact solutions for the in-plane eigenfrequencies and modes are analytically derived for the discrete plate. The stiffness characterising the lattice interactions at the boundary is corrected to preserve the symmetry properties of the discrete displacement field. Two classes of constraints are considered, i.e., sliding supports at the nodes, one normal and the other parallel to the boundary. For both boundary conditions, a single equation for the eigenfrequency spectrum is derived, with two families of eigenmodes. Such behaviour of the lattice plate is like that of the continuous plate, the eigenfrequency spectrum of which has been given by Rayleigh. The convergence of the spectrum of the lattice plate towards the spectrum of the continuous plate from below is confirmed. Two continuous size-dependent plate models, considering the strain gradient elasticity and non-local elasticity, respectively, are built from the lattice difference equations and are shown to approximate the plane lattice accurately.

We consider the retarded potential boundary integral equation, arising from the 3D elastic (vector) wave equation problem, endowed with a Dirichlet condition on the boundary and null initial conditions. For its numerical solution, we employ a weak formulation related to the energy of the system and we discretize it by a Galerkin-type boundary element method (BEM). This approach, called energetic BEM, has been already applied in the context of time-domain acoustic (scalar) wave propagation and it has revealed accurate and stable even on large time intervals of analysis. In particular, when standard (constant) shape functions for time discretization are employed, the double integration in time can be performed analytically. Then, one is left with the task of evaluating double space integrals, whose integration domains are generally delimited by the wave fronts of the primary and the secondary waves. Since the accurate computation of the integrals involved in the numerical scheme is a key issue for the stability of the method, we propose an efficient evaluation strategy, based on the exact detection of the integration domain. The presented numerical tests show the effectiveness of the proposed approach. [ABSTRACT FROM AUTHOR]

In recent years, the development of lattice Boltzmann methods (LBMs) for solids has gained attention. Fracture mechanics as a viable application for these methods has been presented before, albeit for mode III cracks in the context of an LBM for antiplane shear deformation. The performance of the LBM itself is promising, while the usage of a regular lattice simplifies the modelling of fractures significantly. Recent advancements in LBMs for solids, especially the description of Dirichlet‐ and Neumann‐type boundary conditions, now make it possible to extend the LBM simulation of crack propagation to the plane strain case with modes I and II crack opening, including growth with non‐uniform speed in arbitrary directions. For this, the configurational force acting on a crack tip is utilised. The definition of the moments of the LBM, which are based on the balance laws of continuum mechanics, render the evaluation of macroscopic fields in the configuration straightforward. In this work, the general in‐plane case of dynamic crack propagation is shown and necessary considerations for the implementation are discussed. Lastly, numerical examples showcase the capabilities of the proposed method to model dynamic fractures and establish a proof‐of‐concept. [ABSTRACT FROM AUTHOR]

We provide a fully nonlinear port‐Hamiltonian formulation for discrete elastodynamical systems as well as a structure‐preserving time discretization. The governing equations are obtained in a variational manner and represent index‐1 differential algebraic equations. Performing an index reduction, one obtains the port‐Hamiltonian state space model, which features the nonlinear strains as an independent state next to position and velocity. Moreover, hyperelastic material behavior is captured in terms of a nonlinear stored energy function. The model exhibits passivity and losslessness and has an underlying symmetry yielding the conservation of angular momentum. We perform temporal discretization using the midpoint discrete gradient, such that the beneficial properties are inherited by the developed time stepping scheme in a discrete sense. The numerical results obtained in a representative example are demonstrated to validate the findings. [ABSTRACT FROM AUTHOR]