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• A newly-developed h p -finite element is extended to linear elastodynamic problems of thin shells of revolution. • The h p -shell-finite element is based on the hybridization of a three-field dual-mixed variational formulation. • The theoretical model does not rely on the standard hypotheses used in Naghdi- and Koiter-type shell models. • The unmodified, inverse three-dimensional constitutive relation is applied to homogeneous and isotropic materials. • The convergence behavior of the locking-free h p -shell element is tested comprehensively through natural frequency analyzes. A newly-developed, dimensionally reduced, h p -type axisymmetric shell finite element model is extended to linear elastodynamic problems of thin shells of revolution. The h p -shell finite element relies on the hybridized version of a three-field dual-mixed variational formulation, the application of which dictates the obligate usage of the inverse three-dimensional constitutive relation for homogeneous and isotropic materials, thereby ensuring the volumetric locking-free characteristic of the shell model at theory level. The fundamental fields are the a priori non-symmetric stress tensor, the displacement vector, the infinitesimal rotation vector and the hybrid variable defined on the element interfaces. Since the dimensional reduction process guided by this hybrid-mixed formulation does not necessitate the use of any classical kinematic assumptions appearing in the scientific literature, the inverse 3D Hooke's law does not have to be modified. The numerical performance of the shell finite element is analyzed comprehensively for natural frequency computations of clamped-free and simply supported, silicone, conoid, spherical and hyperboloid shells of revolution. From their relative error convergence behaviors it follows that the extended hybrid-mixed shell finite element is not sensitive to the decrease of the slenderness ratio, namely providing reliable, uniformly stable numerical results for both h - and p -approximation. From theoretical point of view, the beneficial properties of the hybrid-mixed h p shell finite element model are as follows: (i) this is effectively applicable to modeling not only extremely thin but also moderately thick shell structures with transverse shear deformations, as well as (ii) both the through-the-thickness variation and the membrane stress normal to the shell mid-surface are retained as independent variables, making it much easier to upbuild shell model for contact problems. From numerical point of view, the nice feature of the hybrid-mixed h p shell finite element is that the global flexibility matrix of the system can be inverted block-wise at element level at cheap computational cost during the assembling procedure because of the hybridization technique. [ABSTRACT FROM AUTHOR]

In this paper, we show the global existence of classical solutions to the incompressible elastodynamics equations with a damping mechanism on the stress tensor in dimension three for sufficiently small initial data on periodic boxes, that is, with periodic boundary conditions. The approach is based on a time-weighted energy estimate, under the assumptions that the initial deformation tensor is a small perturbation around an equilibrium state and the initial data have some symmetry. [ABSTRACT FROM AUTHOR]

Highlights • Analytical method is developed to solve dynamic contact problem of graded coatings. • The rigid cylindrical punch moves at a constant speed on the graded coating. • The singular integral equation is solved to reveal the elastodynamic effects. • Results of the present analytical method are verified by comparison studies. • Dynamic effect on contact stresses due to punch speed is found to be significant. Abstract This paper presents an analytical method developed to investigate the dynamic frictional contact mechanics between a functionally graded coating and a rigid moving cylindrical punch. Governing partial differential equations of elastodynamics are solved analytically by applying Galilean and Fourier transformations. Interface continuity and boundary conditions are written and contact problem is then reduced to a singular integral equation of the second kind. The singular integral equation is solved numerically by means of an expansion-collocation method. Developed solution procedures are verified through the comparisons made to the results available in the literature. Presented parametric analyses illustrate the effects of punch speed, coefficient of friction, material inhomogeneity and geometric parameters upon the contact stresses. It is shown that, especially at higher punch speeds, the difference between contact stresses obtained through elastodynamic and elastostatic solutions is rather significant. A formulation based on the elastodynamic theory, as presented in the current study, is required to compute more realistic contact stresses. [ABSTRACT FROM AUTHOR]

Present paper deals with the vibration analysis of nano plate supported by Winkler foundation using Eringen's elasticity theory with Classical Plate theory. Rayleigh-Ritz method has been employed in this study for finding frequencies of plate subjected to different edge conditions. The obtained results are first tested for convergence and then validated with the published literature. Further study is carried out to analyse the effect of various parameters on natural nondimensional frequencies of nano plate resting on Winkler foundation. The study reveals that the non-local effect has significant effect on vibration behaviour of nano plate resting on elastic foundation. Observations shows that on increasing the Winkler foundation modulus and aspect ratio the nondimensional frequency parameters increases. [ABSTRACT FROM AUTHOR]

The paper deals with numerical evaluation of the scattered wave and dynamic stress concentration fields in a finite anisotropic solid containing multiple nano-cavities. 2D plane-strain state and in-plane wave motion are assumed. The proposed mechanical model combines classical elastodynamic theory for the bulk general anisotropic solid and the Gurtin-Murdoch theory of surface elasticity assuming localized constitutive equation for the infinitely thin interface between the cavity and the matrix. The developed computational methodology is based on the following: (a) displacement boundary integral equations along existing boundaries using the analytically derived through Radon transform fundamental solution of the equation of motion of the bulk anisotropic solid; (b) non-classical boundary conditions of the Gurtin-Murdoch model along the interface between the matrix and cavities taking into consideration a jump in the stresses as one moves from the bulk material to the cavity due to the presence of surface elasticity; and (c) elastic-viscoelastic correspondence principle. The accuracy of the developed software is proven by comparisons of the obtained results solved by boundary element method and finite element method. A detailed parametric study reveals the sensitivity of the wave field to different key factors such as size, number and configuration of the cavities, surface and bulk material properties. [ABSTRACT FROM AUTHOR]

The dynamic pull-in instability of a thermoelectromechanically loaded micro-beam is investigated based on the 'symmetric stress' gradient elasticity theory and Euler-Bernoulli beam theory. The beam is subjected to the combined action of an electric voltage, axial static force and a uniform temperature change. By employing Galerkin's method, the nonlinear partial differential governing equation is decoupled into a set of nonlinear ordinary differential equations, which are then solved using Runge-Kutta method. Numerical results show that compared with the size-dependent micro-beam model, the classical elasticity theory in which the size effect is ignored underestimates the pull-in voltage. The effects of size, temperature change, axial force, geometric nonlinearity, fringe effect, initial gap, beam length and width on the pull-in instability of the micro-beam are discussed in detail. [ABSTRACT FROM AUTHOR]

A new type of displacement pile, the X-section cast-in-place concrete (XCC) pile, has recently been developed in China. Extensive field tests and laboratory experiments are undertaken to evaluate its performance and quantify the non-uniform deformation effect (NUDE) of the X-shaped cross section during installation. This paper develops a simplified theoretical model that attempts to capture the NUDE. Based on the theory of complex variable plane elasticity, closed-form solutions of the stress and displacement for the X-shaped cavity boundary value problem are given. Subsequently, the analytical solution is used to evaluate the NUDE, the concrete filling index (CFI), and the perimeter reduction coefficient of the XCC pile cross section. The computed results are compared with field test results, showing reasonable agreement. The present simplified theoretical model reveals the deformation mechanism of the X-shaped cavity and facilitates application of the newly developed XCC pile technique in geotechnical engineering. [ABSTRACT FROM AUTHOR]

In this paper, a mixed Lagrangian multiplier formulation is developed for consistent size-dependent couple stress elastodynamic response. This extends previous work on quasistatic couple stress response that also requires the rotation to be one half of the curl of displacement. A Lagrange multiplier, which turns out to be related to the skew-symmetric part of the force stress, is used to constrain the relation between rotation and displacement, so that rotation becomes an independent variable and $$\hbox {C}^{1}$$ continuity is avoided in the weak form. The finite element method is applied first to obtain the matrix form of the Lagrangian formulation in space, and then, discrete variational time integration is introduced to produce the discrete action. Afterward, the variation of discrete action is applied to derive the governing equations in algebraic form, which are used, along with the initial condition relations, to obtain the solutions at finite time steps. Finally, the problems of uniaxial deformation of a square plate, transverse deformation of a cantilever, and uniform traction on a square plate with a hole are investigated under this formulation, and the results are compared to existing methods where possible. [ABSTRACT FROM AUTHOR]

An atomistically-meaningful pseudocontinuum representation for the nontrivial lattice dynamics of a finite monatomic chain with linear elastic interactions between nearest neighbor atoms is analytically deduced by mean of a dynamic mechanical analysis extending the memory-dependent pseudocontinuum viewpoint suggested in [M. Charlotte and L. Truskinovsky, Lattice dynamics from a continuum viewpoint , J. Mech. Phys. Solids, 60, pages 1508–1544 (2012)]. For a correct description of the lattice dynamics at its interstice length scale, the pseudocontinuum model integrates both the bulk and boundary inertial (heat-vibration) effects of the atomistic medium through specific modifications of the classical elastodynamic Newton’s law model: these modifications involve a generalization of the D’Alembert’s principle of inertial forces and Neumann-Robin’s boundary conditions, without increasing the number of initial and boundary conditions of the generic mechanical evolution problem, unlike all other generalized continuum models proposed in the literature up to this date. Owing to the spatially local and one-dimensional nature of the discrete and pseudocontinuum models, relationships are thus more clearly pinpointed between the elastodynamic normal stress field of that exact generalized continuum representation and the cohesive (or internal) and inertial forces operating at the lattice sites within the bulk of a finite-size monatomic chain and at its boundary. [ABSTRACT FROM AUTHOR]

The size-dependent nonlinear buckling and postbuckling characteristics of circular cylindrical nanoshells subjected to the axial compressive load are investigated with an analytical approach. The surface energy effects are taken into account according to the surface elasticity theory of Gurtin and Murdoch. The developed geometrically nonlinear shell model is based on the classical Donnell shell theory and the von Kármán's hypothesis. With the numerical results, the effect of the surface stress on the nonlinear buckling and postbuckling behaviors of nanoshells made of Si and Al is studied. Moreover, the influence of the surface residual tension and the radius-to-thickness ratio is illustrated. The results indicate that the surface stress has an important effect on prebuckling and postbuckling characteristics of nanoshells with small sizes. [ABSTRACT FROM AUTHOR]

This paper presents a set of analytic solutions for the transient elastodynamic response of orthotropic cylindrical tubes to sequential moving pressures with specific profiles. The general form of the presented formulations and the solution methods are applicable to a number of theoretical and practical problems. However, the final solutions are tailored for sequential gaseous detonations, with direct application in the stress analysis of pulse detonation engines (PDE). The PDE generates trust by high cycling of gaseous detonations and is regarded as a promising candidate for providing very efficient propulsion systems for aviation and electric power generation. The presented analytic solutions are validated with the available experimental data and complementary finite element simulations. Representative analyses are carried out for an experimental detonation tube subjected to different boundary conditions and loading sequences. It is shown that the proper (or improper) combinations of the relevant parameters can result in substantial attenuation (or amplification) of the tube vibration. It is also demonstrated that the realistic analysis of the overall vibrational behavior, which can be highly affected by the specifications of the loadings and boundary conditions, requires the simulation of hundreds of sequential detonations. [ABSTRACT FROM AUTHOR]

The effect of temperature change on dynamic performances of planar 3-RRR flexible parallel robots is studied in this paper. In general, the strain and stress are produced not only by the external exciting force, but also by temperature change. The strain energy that is caused by temperature change should not be ignored. Based on the Hamilton principle and the finite element method, the general equations of motion of 3-RRR systems are determined, in which the effect of temperature change is taken into consideration. The equations of motion of planar 3-RRR flexible parallel robots are solved using Gear’s algorithm. The commercial Ansys 13.0 software is used to confirm the validity of the theory model. The effect of the daily temperature change of different cities on the performance of 3-RRR flexible parallel robots is investigated. Numerical results show that a small temperature change will cause a significant change in the stress of the flexile links of planar 3-RRR flexible parallel robots. The effects of temperature change should not be ignored when analyzing the dynamic performances of planar 3-RRR flexible parallel robots. [ABSTRACT FROM PUBLISHER]

Abstract: A mixed second order stabilised Petrov–Galerkin finite element framework was recently introduced by the authors (Lee et al., 2014) [46]. The new mixed formulation, written as a system of conservation laws for the linear momentum and the deformation gradient, performs extremely well in bending dominated scenarios (even when linear tetrahedral elements are used) yielding equal order of convergence for displacements and stresses. In this paper, this formulation is further enhanced for nearly and truly incompressible deformations with three key novelties. First, a new conservation law for the Jacobian of the deformation is added into the system providing extra flexibility to the scheme. Second, a variationally consistent Petrov–Galerkin stabilisation methodology is derived. Third, an adapted fractional step method is presented for both incompressible and nearly incompressible materials in the context of nonlinear elastodynamics. For completeness and ease of understanding, these three improvements are presented both in small and large strain regimes, studying the eigenstructure of the resulting systems. A series of numerical examples are presented in order to demonstrate the robustness of the enhanced methodology with respect to the work previously published by the authors. [Copyright &y& Elsevier]

In this paper, we analyse three commonly discussed 'flaws' of linearized elasticity theory and attempt to resolve them. The first 'flaw' concerns cylindrically orthotropic material models. Since the work of Lekhnitskii (), there has been a growing body of work that continues to this day, that shows that infinite stresses arise with the use of a cylindrically orthotropic material model even in the case of linearized elasticity. Besides infinite stresses, interpenetration of matter is also shown to occur. These infinite stresses and interpenetration occur when the ratio of the circumferential Young modulus to the radial Young modulus is less than one. If the ratio is greater than one, then the stresses at the center of a spinning disk are found to be zero (recall that for an isotropic material model, the stresses are maximum at the center). Thus, the stresses go abruptly from a maximum value to a value of zero as the ratio is increased to a value even slightly above one! One of the explanations provided for this extremely anomalous behaviour is the failure of linearized elasticity to satisfy material frame-indifference. However, if this is the true cause, then the anomalous behaviour should also occur with the use of an isotropic material model, where, no such anomalies are observed. We show that the real cause of the problem is elsewhere and also show how these anomalies can be resolved. We also discuss how the formulation of linearized elastodynamics in the case of small deformations superposed on a rigid motion can be given in a succinct manner. Finally, we show how the long-standing problem of devising three compatibility relations instead of six can be resolved. [ABSTRACT FROM AUTHOR]

Abstract: An analytical approach is employed to investigate the transient and steady-state stresses in an isotropic, homogeneous half-space subjected to moving concentrated loads with subsonic speeds. Applying the Stokes–Helmholtz resolution to the Navier’s equation of motion for the half-space results in a system of wavetype partial differential equations. Based on the new moving coordinate system, a modified system of partial differential equations is obtained. Applying a concurrent two-sided and one-sided Laplace transformation, this system is modified to a system of ordinary differential equations, the solutions of which are obtained with respect to boundary conditions. The transformed transient stresses can be inverted by the Cagniard–de Hoop method. Special properties of Laplace transformation yield the steady-state stresses through an analytical approach. Numerical examples are presented to illustrate the methodology. Final results revealed the importance of considering the stresses related to the initial stages of the loading. [Copyright &y& Elsevier]

Abstract: The unsymmetrical discharge of a granular solid from a thin-walled cylindrical metal silo is well known to be a potential prelude to catastrophic buckling failure. The mechanics of this structure have only been very slowly unraveled in recent years with the help of powerful nonlinear finite element analyses. The associated buckling collapse is now known to be caused by localized axial membrane compression and occurs under predominantly elastic conditions. However, such high compressive stress concentrations are also known to produce significantly less imperfection sensitivity than uniform compression. For this reason, the search for an appropriately detrimental imperfection form under eccentric discharge has been quite a long one. This study presents and explores a novel form of long-wave imperfection using a superellipse to parametrise the entire shell geometry. This imperfection, termed ‘superelliptical flattening’, is judged to be potentially present in practical silo construction, and is shown to cause significant decreases in the nonlinear buckling strength of an imperfect slender silo under eccentric discharge. [Copyright &y& Elsevier]

Abstract: A robust and efficient computational method for reconstructing the elastodynamic structural response of truss, beam, and frame structures, using measured surface-strain data, is presented. Known as “shape sensing”, this inverse problem has important implications for real-time actuation and control of smart structures, and for monitoring of structural integrity. The present formulation, based on the inverse Finite Element Method (iFEM), uses a least-squares variational principle involving section strains (also known as strain measures) of Timoshenko theory for stretching, torsion, bending, and transverse shear. The present iFEM methodology is based on strain–displacement relations only, without invoking force equilibrium. Consequently, both static and time-varying displacement fields can be reconstructed without the knowledge of material properties, applied loading, or damping characteristics. Two finite elements capable of modeling frame structures are derived using interdependent interpolations, in which interior degrees of freedom are condensed out at the element level. In addition, relationships between the order of kinematic-element interpolations and the number of required strain gauges are established. Several example problems involving cantilevered beams and three-dimensional frame structures undergoing static and dynamic response are discussed. To simulate experimentally measured strains and to establish reference displacements, high-fidelity MSC/NASTRAN finite element analyses are performed. Furthermore, numerically simulated measurement errors, based on Gaussian distribution, are also considered in order to verify the stability and robustness of the methodology. The iFEM solution accuracy is examined with respect to various levels of discretization and the number of strain gauges. [Copyright &y& Elsevier]

The energy-momentum method, a space-time discretization strategy for elastic problems in nonlinear solid, structural, and multibody mechanics relies critically on a discrete derivative operation that defines an approximation of the internal forces that guarantees the discrete conservation of energy and momenta. In the case of nonlinear elastodynamics, the formulation for general hyperelastic materials is due to Simo and Gonzalez, dating back to the mid-nineties. In this work we show that there are actually infinite second order energy-momentum methods for elastodynamics, all of them deriving from a modified midpoint integrator by an appropriate redefinition of the stress tensor at equilibrium. Such stress tensors can be interpreted as the solutions to local convex projections, whose precise definitions lead to different methods. The mathematical requirements of such projections are identified. Based on this geometrical interpretation several conserving methods are examined. [ABSTRACT FROM AUTHOR]

Abstract: This work presents a new technique yielding computable bounds of quantities of interest in the framework of linear visco-elastodynamics. A novel expression for the error representation is introduced, alternative to the previous ones using the Cauchy–Schwarz inequality. The proposed formulation utilizes symmetrized forms of the error equations to derive error bounds in terms of energy error measures. The practical implementation of the method is based on constructing admissible fields for both the original problem and the adjoint problem associated with the quantity of interest. Here, the flux-free technique is considered to compute the admissible stress fields. The proposed methodology yields estimates with better quality than the ones based on the Cauchy–Schwarz inequality. In the studied examples the bound gaps obtained are approximately halved, that is the estimated intervals of confidence are reduced. [Copyright &y& Elsevier]

Abstract: A three-dimensional semi-analytic analysis based on the linear elasticity theory is offered to study the transient vibration characteristics of an arbitrarily thick, simply supported, functionally graded (FGM) rectangular plate, resting on a linear Winkler–Pasternak viscoelastic foundation, and subjected to general distributed driving forces of arbitrary temporal and spatial variations. The problem solution is obtained by adopting a laminate model in conjunction with the powerful state space solution technique involving a global transfer matrix and Durbin’s numerical Laplace inversion algorithm. Numerical calculations are carried out for the transient displacement and stress responses of aluminum-zirconia FGM square plates of selected thickness parameters and compositional gradients, resting on “soft” or “stiff” elastic foundations, under the action of moving transverse forces as well as uniformly distributed blast loads. Also, the response curves for the FGM plates are compared with those of equivalent bilaminate plates containing comparable total volume fractions of constituent materials. It is observed that the material gradient variation is substantially more influential on the dynamic stress concentrations induced across the plate thickness than on the displacement response of the inhomogeneous plates. In particular, the displacement response of the equivalent bilaminate plates can provide an accurate estimate for prediction of the dynamic response of the corresponding FGM plates, especially for thick plates resting on a stiff foundation. Limiting cases are considered and good agreements with the data available in the literature as well as with the computations made by using a commercial finite element package are obtained. [Copyright &y& Elsevier]

Abstract: In this paper, a two-dimensional symmetric-Galerkin boundary integral formulation for elastodynamic fracture analysis in the frequency domain is described. The numerical implementation is carried out with quadratic elements, allowing the use of an improved quarter-point element for accurately determining frequency responses of the dynamic stress intensity factors (DSIFs). To deal with singular and hypersingular integrals, the formulation is decomposed into two parts: the first part is identical to that for elastostatics while the second part contains at most logarithmic singularities. The treatment of the elastostatic singular and hypersingular singular integrals employs an exterior limit to the boundary, while the weakly singular integrals in the second part are handled by Gauss quadrature. Time histories (transient responses) of the DSIFs can be obtained in a post-processing step by applying the standard fast Fourier transform (FFT) and algorithm to the frequency responses of these DSIFs. Several test examples are presented for the calculation of the DSIFs due to two types of impact loading: Heaviside step loading and blast loading. The results suggest that the combination of the symmetric-Galerkin boundary element method and standard FFT algorithms in determining transient responses of the DSIFs is a robust and effective technique. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a mixed stress formulation for linear elastodynamic analysis based on a modified Hellinger–Reissner functional and a consistent approach for selecting finite element approximations are presented. The key idea in this new approach is to choose stress approximations by taking into account for suitable modes to equilibrate inertia forces. Feasibility and effectiveness of the proposed approach are numerically verified through three benchmark tests. [Copyright &y& Elsevier]

I present analytical time-domain expressions for the Green's functions, which represent the transient stress response of an infinite, homogeneous and isotropic medium to a constant slip rate on a triangular fault that continues perpetually after the slip onset. The solution can be utilized in the formulation of boundary element methods, which discretize a fault plane of any arbitrary geometry into an assembly of small triangular elements and assume the slip rate to be piecewise constant within each discrete element. [ABSTRACT FROM AUTHOR]

In this article, it is demonstrated that current methods of modelling plasticity as the collective motion of discrete dislocations, such as two-dimensional discrete dislocation plasticity (DDP), are unsuitable for the simulation of very high strain rate processes (106 s−1 or more) such as plastic relaxation during shock loading. Current DDP models treat dislocations quasi-statically, ignoring the time-dependent nature of the elastic fields of dislocations. It is shown that this assumption introduces unphysical artefacts into the system when simulating plasticity resulting from shock loading. This deficiency can be overcome only by formulating a fully time-dependent elastodynamic description of the elastic fields of discrete dislocations. Building on the work of Markenscoff & Clifton, the fundamental time-dependent solutions for the injection and non-uniform motion of straight edge dislocations are presented. The numerical implementation of these solutions for a single moving dislocation and for two annihilating dislocations in an infinite plane are presented. The application of these solutions in a two-dimensional model of time-dependent plasticity during shock loading is outlined here and will be presented in detail elsewhere. [ABSTRACT FROM AUTHOR]