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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]

We present an augmented Lagrangian-based approach for stress-constrained topology optimization of structures subjected to general dynamic loading. The approach renders structures that satisfy the stress constraints locally at every time step. To solve problems with a large number of stress constraints, we normalize the penalty term of the augmented Lagrangian function with respect to the total number of constraints (i.e., the number of elements in the mesh times the number of time steps). Moreover, we solve the stress-constrained problem effectively by penalizing constraints associated with high stress values more severely than those associated with low stress values. We integrate the equations of motion using the HHT-α method and conduct the sensitivity analysis consistently with this method via the "discretize-then-differentiate" approach. We present several numerical examples that elucidate the effectiveness of the approach to solve dynamic, stress-constrained problems under several loading scenarios including loads that change in magnitude and/or direction and loads that change in position as a function of time. [ABSTRACT FROM AUTHOR]

• 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]

Stress-heat flux characterization of linear dynamic coupled thermoelasticity for an inhomogeneous isotropic infinite cylinder under plane strain conditions and zero heat-flux normal to the plane is presented. It is shown that for a bounded cross-section of the cylinder, a 3D stress-heat flux process is generated by a 2D one, and a uniqueness theorem for the associated 2D initial-boundary value problem is established. In addition, an asymptotic approach to the 2D stress-heat flux initial-boundary value problem, in the form of a power series with respect to a small thermoelastic coupling field, is proposed. Also, Green' s formulas for time-periodic complex-valued solutions to 2D stress-heat flux field equations are obtained; and the existence of a globally constrained real-valued periodic and attenuated on the time-axis stress-heat flux mode satisfying homogeneous natural boundary conditions is proved. The stress-heat flux characterization covers a large class of FGM's with physical properties smoothly distributed over the cross-section of the infinite cylinder. The results obtained are complementary to those of linear dynamic coupled thermoelasticity published up to date and should prove useful for a number of researchers in the field. [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]

We deal with a new phase nucleation in a phase-transforming bar caused by a collision of two non-stationary waves. We consider an initial stage of dynamical process in the finite bar before the moment of time when the waves emerged due to new phase nucleation reach the ends of the bar. The model of a phase-transforming bar with trilinear stress-strain relation is used. The problem is formulated as a scale-invariant initial value problem with additional restrictions in the form of several inequalities involving the problem parameters. We consider the particular limiting case where the stiffness of a new phase inclusion is much greater than the stiffness of the initial phase and obtain the asymptotic solution in the explicit form. In particular, the domains of existence of the solution in the parameter space are constructed. [ABSTRACT FROM AUTHOR]

An exact three-dimensional elasticity solution for buckling of a circular cross-section beam with one end fixed and the other end free is obtained. A comparison is made using the approximate solution by Kardomateas and an exact two-dimensional solution by Chattopadhyay and Gu. The buckling loads of both exact solutions are in good agreement with each other, but the buckling load using the approximate solution is slightly higher than the exact solutions. The exact solutions support Engesser's formula rather than Haringx's formula for the buckling load of a beam with transverse shear effect. [ABSTRACT FROM AUTHOR]

• An energy-momentum stable explicit solver hyperbolic PDEs is proposed. • Time-marching formula has a time-error estimation to guarantee method's convergence. • The method stability is analyzed, obtaining parameters to ensure its convergence. • Numerical examples involving finite-strains and large-displacement are presented. • Energy-momentum stable results are obtained when lumped mass matrices are used. We propose a time-adaptive predictor/multi-corrector method to solve hyperbolic partial differential equations, based on the generalized- α scheme that provides user-control on the numerical dissipation and second-order accuracy in time. Our time adaptivity uses an error estimation that exploits the recursive structure of the variable updates. The predictor/multicorrector method explicitly updates the equation system but computes the residual of the system implicitly. We analyze the method's stability and describe how to determine the parameters that ensure high-frequency dissipation and accurate low-frequency approximation. Subsequently, we solve a linear wave equation, followed by non-linear finite strain deformation problems with different boundary conditions. Thus, our method is a straightforward, stable and computationally efficient approach to simulate real-world engineering problems. Finally, to show the performance of our method, we provide several numerical examples in two and three dimensions. These challenging tests demonstrate that our predictor/multicorrector scheme dynamically adapts to sudden energy releases in the system, capturing impacts and boundary shocks. The method efficiently and stably solves dynamic equations with consistent and under-integrated mass matrices conserving the linear and angular momenta as well as the system's energy for long-integration times. [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]

This work presents dynamic analyses of one-dimensional bar and Euler–Bernoulli beam problems with Generalized Finite Element Method (GFEM). Enrichment monomials are trigonometric and exponential functions. A beam free vibration problem is analyzed to assess the element’s robustness and efficiency. Next, an elastodynamic analysis of a bar is performed using several enrichment levels. Finally, a dynamic elastoplastic analysis of a beam problem is carried out. Error measures and nonlinear strains are estimated. Results from GFEM are compared to results from a conventional FE formulation to show GFEM’s level of efficiency in solving Euler–Bernoulli beam elastic as well as elastoplastic dynamic problems. [ABSTRACT FROM AUTHOR]

The positive influence of crystallographic compatibility on the thermal transformation stability has been already investigated extensively in the literature. However, its influence on the stability of the shape memory effect or superelasticity used in actual applications is still unresolved. In this investigation sputtered films of a highly compatible TiNiCuCo composition with a transformation matrix middle eigenvalue of 1 ± 0.01 are exposed to thermal as well as to superelastic cycling. In agreement with previous results the thermal transformation of this alloy is with a temperature shift of less than 0.1K for 40 cycles very stable; on the other hand, superelastic degradation behaviour was found to depend strongly on heat treatment parameters. To reveal the transformation dissimilarities between the differently heat-treated samples, the microstructure has been analysed by transmission electron microscopy, in situ stress polarization microscopy and synchrotron analysis. It is found that good crystallographic stability is not a sufficient criterion to avoid defect generation which guarantees high superelastic stability. For the investigated alloy, a small grain size was identified as the determining factor which increases the yield strength of the composition and decreases the functional degradation during superelastic cycling. [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]

For a subsonically self-similarly expanding spherical inclusion with dilatational transformation strain in a linear elastic solid, the governing system of partial differential equations is shown to be elliptic under scaling of uniform stretching of the variables, and the resulting elliptic equation is solved by satisfying the Hadamard jump conditions on the moving boundary. The solution has the Eshelby constant stress property for the interior domain, and can thus be used for the expanding inhomogeneity with transformation strain according to Eshelby (1957) . The driving force on the moving boundary is also obtained. [ABSTRACT FROM AUTHOR]

We propose a finite element formulation for a coupled elasticity–reaction–diffusion system written in a fully Lagrangian form and governing the spatio-temporal interaction of species inside an elastic, or hyper-elastic body. A primal weak formulation is the baseline model for the reaction–diffusion system written in the deformed domain, and a finite element method with piecewise linear approximations is employed for its spatial discretization. On the other hand, the strain is introduced as mixed variable in the equations of elastodynamics, which in turn acts as coupling field needed to update the diffusion tensor of the modified reaction–diffusion system written in a deformed domain. The discrete mechanical problem yields a mixed finite element scheme based on row-wise Raviart–Thomas elements for stresses, Brezzi–Douglas–Marini elements for displacements, and piecewise constant pressure approximations. The application of the present framework in the study of several coupled biological systems on deforming geometries in two and three spatial dimensions is discussed, and some illustrative examples are provided and extensively analyzed. [ABSTRACT FROM AUTHOR]

This paper presents the investigation of guided wave propagation in prestressed double-cylinder structure. Based on Hertzian contact theory, the interaction between two rods is treated as a plane strain problem and the stress state in the waveguide under the static load is obtained. The stress state is considered as a prestressed configuration for elastic wave propagation analysis in double cylindrical rods. The elastodynamic equation of the prestressed structure is established with the updated Lagrangian formulation and the wave finite element (WFE) method. Firstly, the equation is verified by the application on an aluminum rod compared with the Euler–Bernoulli beam theory. Then, dispersion curves for single rod and double cylindrical rods without prestress are computed. Besides, at the dimensionless frequency 0.25 the propagating modes in double cylindrical rods are identified with mode shapes and displacement vectors. The guided waves in double rods consist of the modes different from single rod. Particularly, there exist two kinds of torsional-like modes, one of which has a twist center and the other has two. The latter changes obviously with the increase of prestress in the waveguide. At low frequencies, torsional-like modes are very sensitive to the variation of prestress; in addition, the prestress configuration has little influence on propagating modes at mid-frequencies but some at high frequencies. [ABSTRACT FROM AUTHOR]

Stress characterization of isothermal elastodynamics for an inhomogeneous transversely isotropic infinite cylinder under plane strain conditions is presented. The cylinder, referred to the Cartesian coordinates x i ( i = 1,2,3) and denoted here by B i ( 3 ) , is identified with an infinite solid cylinder of which the rotational symmetry axis coincides with the x 3 -axis, and the geometrical axis coincides with the x i -axis ( i = 1,2,3); and the plane strain conditions exist in the plane x i = 0, ( i = 1,2,3). A cross-section of the cylinder B i ( 3 ) with the plane x i = 0 is denoted by C i ( 3 ) . It is shown that a 3D stress wave S ij ( i,j = 1,2,3) propagating in the cylinder B i ( 3 ) is generated by a solution to a 2D pure stress initial-boundary value problem for C i ( 3 ) , and a uniqueness theorem for the 2D problem is established. In particular, a pure stress initial boundary value problem for C 3 ( 3 ) involving only two (out of five) elastic moduli: c 11 and c 12 is formulated, and it is shown that the problem accommodates two types of the surface stress wave problems for a transversely isotropic semi-space with a traction free boundary and with an inhomogeneity depending on its depth. The first type is obtained when C 3 ( 3 ) is the semi-space: x 1 < ∞ ,   0 < x 2 < ∞ ; and the density ρ , and moduli c 11 , and c 12 depend on x 2 only, while the second type is obtained when C 3 ( 3 ) is the semi-space: 0 < x 1 < ∞ ,   x 2 < ∞ ; and the density ρ , and moduli c 11 , and c 12 depend on x 1 only. Also, it is shown that the two pure stress initial boundary value problems for the cross-sections C 2 ( 3 ) and C 1 ( 3 ) involve only four (out of five) elastic moduli c 11 , c 33 , c 13 and c 44 ; and since the ( x 1 , x 2 ) is a plane of isotropy, a formulation for C 1 ( 3 ) may be obtained from that of C 2 ( 3 ) by suitable transformation of indices for the strain and stress tensor fields. Also, it is shown that each of the two problems accommodates two types of the surface stress wave problems for a transversely isotropic semi-space with a traction-free boundary and with an inhomogeneity depending on its depth. The results obtained should prove useful for both the analytical and numerical studies of the surface stress waves in an inhomogeneous transversely isotropic elastic semi-space under plane strain conditions. [ABSTRACT FROM AUTHOR]

Homogeneous nucleation of dislocations is the dominant dislocation generation mechanism at strain rates above 108 s-1 at those rates, homogeneous nucleation dominates the plastic relaxation of shock waves in the same way that Frank-Read sources control the onset of plastic flow at low strain rates. This article describes the implementation of homogeneous nucleation in dynamic discrete dislocation plasticity (D3P), a planar method of discrete dislocation dynamics (DDD) that offers a complete elastodynamic treatment of plasticity. The implemented methodology is put to the test by studying four materials--Al, Fe, Ni, and Mo--that are shock loaded with the same intensity and a strain rate of 1010 s-1. It is found that, even for comparable dislocation densities, the lattice shear strength is fundamental in determining the amount of plastic relaxation a material displays when shock loaded. [ABSTRACT FROM AUTHOR]

In this article the multiplicative decomposition of the deformation gradient into one part constrained in the direction of the axis of anisotropy and one part describing the directional deformation is proposed. This leads to a clear division of the deformation and stress states in the direction of anisotropy and a remaining part. The decomposition is explained in detail and a constitutive model of hyper-elasticity is proposed for the case of transversal isotropy, where the behavior of the model is investigated with the aid of simple analytical examples. The model is also investigated for inhomogeneous deformation states using high-order finite elements based on hierarchical shape functions showing the sensitivity of the accuracy of the results in the case of anisotropic media. [ABSTRACT FROM AUTHOR]

The object of analysis is a plane structure reinforced by a system of thin parallel-distributed ribs. It will be assumed that the number of the ribs is very large. The thickness of neighbouring ribs can smoothly change. The aim of contribution is to derive 2D-macroscopic mathematical models describing elastodynamic behaviour of the plate structure in plane-stress state. The consideration will be based on the tolerance averaging technique [9,10]. The general results of the contribution will be illustrated by the analysis of the free vibrations of a structure under consideration. [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]

The symmetric frequency-domain problem of the interaction effects in rectangular lattice system of coplanar penny-shaped cracks located in an infinite elastic solid is numerically investigated. The problem is reduced to a boundary integral equation for the crack-opening-displacement in a unit cell by means of 3D periodic elastodynamic Green's function. This function is adopted for the effective calculation by its representation in the form of exponentially convergent Fourier integrals. A collocation method is used for the solution of the boundary integral equation. Numerical results for the mode-I dynamic stress intensity factor in the crack vicinities are obtained and analyzed depending on the wave number and the lattice size. [ABSTRACT FROM AUTHOR]

The aim of this paper is the design of a new one-step implicit and thermodynamically consistent Energy–Momentum (EM) preserving time integration scheme for the simulation of thermo-electro-elastic processes undergoing large deformations. The time integration scheme takes advantage of the notion of polyconvexity and of a new tensor cross product algebra. These two ingredients are shown to be crucial for the design of so-called discrete derivatives fundamental for the calculation of the second Piola–Kirchhoff stress tensor, the entropy and the electric field. In particular, the exploitation of polyconvexity and the tensor cross product, enable the derivation of comparatively simple formulas for the discrete derivatives. This is in sharp contrast to much more elaborate discrete derivatives which are one of the main downsides of classical EM time integration schemes. The newly proposed scheme inherits the advantages of EM schemes recently published in the context of thermo-elasticity and electro-mechanics, whilst extending to the more generic case of nonlinear thermo-electro-mechanics. Furthermore, the manuscript delves into suitable convexity/concavity restrictions that thermo-electro-mechanical strain energy functions must comply with in order to yield physically and mathematically admissible solutions. Finally, a series of numerical examples will be presented in order to demonstrate robustness and numerical stability properties of the new EM scheme. [ABSTRACT FROM AUTHOR]

Highlights: [•] We developed and accurate and efficient method for stress concentration studies. [•] Any configuration of inclusions in an elastic matrix can be considered. [•] Discretization of only the surfaces and interfaces of the problem is required. [•] Interacting inclusions in a solid matrix has important applications in composite materials. [ABSTRACT FROM AUTHOR]

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]

This paper presents an overview of modeling fractal media by continuum mechanics using the method of dimensional regularization. The basis of this method is to express the balance laws for fractal media in terms of fractional integrals and, then, convert them to integer-order integrals in conventional (Euclidean) space. Following an account of this method, we develop balance laws of fractal media (continuity, linear and angular momenta, energy, and second law) and discuss wave equations in several settings (1d and 3d wave motions, fractal Timoshenko beam, and elastodynamics under finite strains). We then discuss extremum and variational principles, fracture mechanics, and equations of turbulent flow in fractal media. In all the cases, the derived equations for fractal media depend explicitly on fractal dimensions and reduce to conventional forms for continuous media with Euclidean geometries upon setting the dimensions to integers. We also point out relations and potential extensions of dimensional regularization to other models of microscopically heterogeneous physical systems. [ABSTRACT FROM AUTHOR]

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]

In structural mechanics the stresses are of a primary interest. Due to the required regularity of the approximation it is often inconvenient to obtain the stresses from the displacement methods. Mixed and hybrid methods are thus proposed to avoid this obstacle. The compatibility conditions should be fulfilled in order to obtain a unique solution. In this paper the compatibility equations of non‐linear elasticity are considered. Despite its very long history, this issue has not been satisfactorily resolved for non‐simply‐connected bodies until recently [1]. Its well known however, that the compatibility conditions are linearly dependent. The aim of the present contribution is to include the spectral methods [2] in order to improve the finite element solution procedure for the reversible elastodynamics problem [3], formulated for a large strain deformation with the deformation gradient tensor as one of the basic unknowns. In the case of the small strains the components of the deformation gradient tensor are of different magnitude. The main contribution is the derivation of the solution strategy in order to take this information into account. Applying this strategy to the implicit time integration method the exponent decrease of the condition numbers of the iteration matrices was achieved. Consequently the accuracy of the calculations should increase and the convergence for spectral method solutions should considerably improve. The theoretical findings are confirmed by a selected numerical example from structural mechanics. [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]

An orthogonal meshless finite volume method has been presented to solve some elastodynamic crack problems. An orthogonal weighted basis function is used to construct shape function so there is no problem of singularity in this new form. In this work, for three-dimensional dynamic fracture problems, a new displacement function is used at the tip of the crack to give a new OMFVM. When the new OMFVM is used, the singularity of the stresses at the tip of the crack can be shown to be better than that in the primal OMFVM. High computational efficiency and precision are other benefits of the method. Solving some sample crack problems of thin-walled structures show a good performance of this method. [ABSTRACT FROM AUTHOR]

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]