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The formalism of the internal variable theory is applied to extend Navier-Stokes equations. The internal variable theory provides a thermodynamically consistent derivation of constitutive relations and equations of motion without a priori specifying the nature of internal variables. Both single and dual internal variables cases are thoroughly examined. The similarities and differences of the approaches are emphasized. In the single internal variable framework, the elimination of the internal variable results in Maxwell-type constitutive relations and hyperbolic equations of motion. The dual internal variable technique enables us to create even more sophisticated fluid flow models with coupled equations for fluid motion and internal variable evolution.

The influence of a microstructure on heat conduction in solids is studied using the internal variable approach. Two variants of the internal variable treatment are compared by means of the numerical simulation of two-dimensional heat conduction in a plate under a localised thermal pulse loading. Computation of the same problem by the different internal variable descriptions produces qualitatively dissimilar results. The single internal variable approach leads to a diffusional type of the internal variable evolution. In contrast, the dual internal variable technique provides a wave-like evolution of the internal variables and, as the consequence, the corresponding wave-like heat transfer. The results are obtained in the dimensionless form, and parameters of models are chosen to emphasise the features of each model.

Two-dimensional wave propagation in an aluminium strip is examined experimentally, analytically, and numerically in this paper. The high-frequency pulse is generated by a piezoelectric transducer, and the velocity component normal to the strip edge is measured by the non-contact vibrometer system. Inplane waves are then investigated under plane stress conditions using the analytical method based on Laplace and Fourier transforms. For numerical solution, explicit variants of finite element and finite volume solvers are utilized. It is shown that the used numerical algorithms confirm the analytical results and that they are suitable to represent the transient wavefield observed in the experiment including several wave reflections from upper and bottom strip boundaries.

Mathematical structure of generalized heat equations in rigid solids is analyzed from the point of view of internal variables theory. Internal variables are used for accounting for the influence of inner microstructure on heat conduction. It is shown that all the known extensions of the classical heat equation can be recovered in the framework of the internal variables theory. The introduction of a single internal variable does not change the parabolic type of the heat conduction equations. Hyperbolic generalized heat conduction equations can be derived only by means of dual internal variables.

In the paper, the finite element method and the finite volume method are used in parallel for the simulation of a pulse propagation in periodically layered composites beyond the validity of homogenization methods. The direct numerical integration of a pulse propagation demonstrates dispersion effects and dynamic stress redistribution in physical space on the example of a one-dimensional layered bar. Results of numerical simulations are compared with the analytical solution constructed specifically for the considered problem. The analytical solution as well as numerical computations show the strong influence of the composition of constituents on the dispersion of a pulse in a heterogeneous bar and the equivalence of results obtained by two numerical methods.

Application of internal variables to the description of the influence of a microstructure on the overall behavior of solids is presented and discussed. It is demonstrated on the one-dimensional example that capabilities of strain gradient models are much less than those provided by internal variables.

In this paper, elastic wave propagation in a one-dimensional micromorphic medium characterized by two internal variables is investigated. The evolution equations are deduced following two different approaches, namely using: (i) the balance of linear momentum and the Clausius–Duhem inequality, and (ii) an assumed Lagrangian functional (including a gyroscopic coupling) together with a variational principle. The dispersion relation is obtained and the possibility of the emerging band gaps is shown in such microstructured materials. Some numerical simulations are also performed in order to highlight the dispersive nature of the material under study.

The paper is devoted to evolving discontinuities in elastic solids. A discontinuity is represented as a singular set of material points. Evolution of a discontinuity is driven by the configurational force acting at such a set. The main attention is paid to the determination of the velocity of a propagating discontinuity. Martensitic phase transition fronts and brittle cracks are considered as representative examples.

The B-spline variant of the finite element method (FEM) is tested in one-dimensional discontinuous elastic wave propagation. The B-spline based FEM (called Isogeometric analysis IGA) uses spline functions as testing and shape functions in the Galerkin continuous content. Here, the accuracy of stress distribution and spurious oscillations of the B-spline based FEM are studied in numerical modeling of one-dimensional propagation of stress discontinuities in a bar, where the analytical solution is known. For time integration, the Newmark method, implicit form of the generalized-α method, the central difference method and the predictor/multi-corrector method are tested and compared. The use of lumped and consistent mass matrices in explicit time integration is discussed. Due to accuracy, the consistent mass matrix is preferred in explicit time integration in IGA.

The description of heat conduction in microstructured solids is presented in the framework of the dual internal variable approach. One of the internal variables is identified with microtemperature, i.e. the fluctuation of macroscopic temperature due to the inhomogeneity of the body. It is shown that the microstructure influence may result in a hyperbolic heat propagation for the microtemperature. The macroscale heat conduction is described by a parabolic equation which is coupled with the hyperbolic equation for the microtemperature.

The construction of a two-dimensional finite volume numerical scheme based on the representation of computational cells as thermodynamic systems is presented explicitly. The main advantage of the scheme is an accurate implementation of conditions at interfaces and boundaries. It is demonstrated that boundary conditions influence the wave motion even in the simple case of a homogeneous waveguide.

Mindlin's microelasticity theory is reduced to the one-dimensional setting to examine its prediction of the response of microstructured materials on a pulse loading. The one-dimensional setting requires a minimal number of additional material parameters. Since analytical solutions may be found only in highly simplified asymptotic cases, numerical simulations are performed by means of a finite-volume numerical scheme modifying the wave-propagation algorithm. The calculated response of a material described by the Mindlin microelasticity is compared then with the corresponding response of an “equivalent” periodic laminate.

The comparison of numerical simulations of the classical problem of the single slit diffraction in optical, acoustic, and elastic cases is presented in the paper in the plane strain setting. It is shown that wave fields downstream the slit are similar in optical and acoustic cases, as expected. Corresponding wave fields in the single slit diffraction using elastic materials become essentially different from optical and acoustic cases. This is an effect of elastic waves propagating inside the plate forming the slit.

A special mesh adaptation technique and a precise discontinuity tracking are presented for an accurate, efficient, and robust adaptive-mesh computational procedure for one-dimensional hyperbolic systems of conservation laws, with particular reference to problems with evolving discontinuities in solids. The main advantage of the adaptive technique is its ability to preserve the modified mesh as close to the original fixed mesh as possible. The constructed method is applied to the martensitic phase-transition front propagation in solids.

In nonlinear theories the axiom of equipresence requires all the effects of the same order to be taken account. In this paper the mathematical modelling of deformation waves in media is analysed involving nonlinear and dispersive effects together with accompanying phenomena caused by thermal or electrical fields. The modelling is based on principles of generalized continuum mechanics developed by G.A. Maugin. The analysis demonstrates the richness of models in describing the physical effects in media with complex properties.

Dynamic materials are artificially constructed in such a way that they may vary their characteristic properties in space or in time, or both, by an appropriate arrangement or control. These controlled changes in time can be provided by the application of an external (non-mechanical) field, or through a phase transition. In principle, all materials change their properties with time, but very slowly and smoothly. Changes in properties of dynamic materials should be realized in a short or quasi-nil time lapse and over a sufficiently large material region. Wave propagation is a characteristic feature for dynamic materials because it is also space and time dependent. As a simple example of the complex behavior of dynamic materials, the one-dimensional elastic wave propagation is studied numerically in periodic structures whose properties (mass density, elasticity) can be switched suddenly in space and in time. It is shown that dynamic materials have the ability to dynamically amplify, tune, and compress initial signals. The thermodynamically consistent high-resolution finite-volume numerical method is applied to the study of the wave propagation in dynamic materials. The extended analysis of the influence of inner reflections on the energy localization in the dynamic materials is presented.

Elastic wave propagation through diffraction gratings is studied numerically in the plane strain setting. The interaction of the waves with periodically ordered elastic inclusions leads to a self-imaging Talbot effect for the wavelength equal or close to the grating size. The energy localization is observed at the vicinity of inclusions in the case of elastic gratings. Such a localization is absent in the case of rigid gratings.

Results of heat pulse experiments in various artificial and natural materials are reported in the paper. The experiments are performed at room temperature with macroscopic samples. It is shown that temperature evolution does not follow the Fourier's law but well explained by the Guyer-Krumhansl equation. The observations confirm the ability of non-equilibrium thermodynamics to formulate universal constitutive relations for thermomechanical processes., 4 pages and 11 figures

Wave dispersion is a characteristic feature of solids with microstructure. The development of models of linear wave dispersion effects in isotropic solids is used as an example of the construction of more and more sophisticated dispersive wave equations. The considered models are distinct in foundations and derivations.

It is demonstrated that the dual internal variable approach is able to predict a hyperbolic character of heat conduction at the microscale. One of the internal variables is identified with microtemperature, i.e., the fluctuation of macroscopic temperature due to the inhomogeneity of the body. The macroscopic heat conduction equation remains parabolic, but coupled with the hyperbolic evolution equation for the microtemperature.

The overall description of thermomechanical processes in microstructured solids includes both direct and indirect couplings of equations of motion and heat conduction at the macrolevel. In addition to the conventional direct coupling, there exists the coupling between macromotion and microtemperature evolution. This means that the macrodeformation can induce microtemperature perturbations due to the heterogeneity in the presence of a microstructure. These perturbations, propagating with finite speed, can induce, in turn, corresponding changes in macrotemperature.

The introduction of double dual internal variables provides the complete extension of the classical thermoelasticity theory onto the case of microstructured solids. This extension keeps the structure of canonical balances of momentum and energy and provides the thermodynamically consistent evolution equations for microdeformation and microtemperature. Evolution equations in the case of dual internal variables are hyperbolic and coupled with the equations of macromotion.

The theory of a single internal variable of state is well established including the weak nonlocality and the enrichment by extra entropy flux. The theory is based on the consideration of the internal variable of state as a tool for taking into account the internal dissipation. Inertial effects are absent in this theory by definition, which leads to parabolic evolution equations for the internal variables of state. Due to the simplicity, the single internal variable theory is the necessary first step in the construction of more sophisticated material models.

The introduction of nonlinear terms at both macro- and micro-levels does not change the formalism of the internal variable approach. In the non-dissipative case, the nonlinear terms can be balanced with dispersion providing the well-known models of solitonic behavior. The interplay between micro-and macro- nonlinearities allows to achieve more sophisticated models of nonlinear dispersive wave propagation than those in the homogeneous solids.

Several well-known dispersive wave propagation models together with generalization of the Mindlin-type models are derived by using internal variables. The adopted phenomenological approach is based on the material formulation of continuum mechanics and provides the full thermodynamic consistency due to the dual internal variables concept.

It is shown that inertial terms appear naturally in the thermodynamic theory with dual internal variables and the conditions of their appearance is well understandable in terms of mechanical notions. This demonstrates the difference between the standard single internal variable theory and the dual internal variables concept.

It is shown how dual weakly non-local internal variables and extra entropy fluxes can be introduced in the framework of canonical thermomechanics on the material manifold. This extension of the single internal variable formalism allows one to derive a hyperbolic evolution equation for internal variables in the non-dissipative case. Since the dissipation inequality is the basis of the derivation, it ensures the thermodynamic consistency of the obtained evolution equations.

It is demonstrated on the example of one-dimensional elastic pulse propagation that predictions of the Mindlin micromorphic theory may be inefficient if the size of inhomogeneity is comparable with the pulse length. This means that the Mindlin microelasticity is still an approximation of the description of the behavior of materials, which is satisfactory only for long wavelengths or for small size of inhomogeneities.

The numerical simulation of elastic wave propagation through a grating in a homogeneous medium shows the characteristic Talbot self-imaging effect. The size of the imaging and patterns can be controlled by changing the periodicity of grating and wavelength, respectively. It provides a novel route to modulate elastic wave distributions in solids with inclusions. Time dependent simulation suggests a long-distance elastic-wave-based interconnection in solids. An interferometer based on the elastic wave Talbot effect is discussed.

Prediction of the thermoelastic behavior of microstructured materials suggests a more general description of thermal processes in addition to the generalized continuum description extending the conventional continuum mechanics for incorporating intrinsic microstructural effects. Double dual internal variables are introduced in order to couple inertial microstructural effects like microdeformation and diffusive microstructural effects like microtemperature. The full coupled system of governing equations provides a complete extension of the classical thermoelasticity theory onto the case of microstructured solids.

Governing equations of dissipative generalized solid mechanics are derived by thermodynamic methods in the Piola-Kirchhoff framework using the Liu procedure. The isotropic small strain case is investigated in more detail. The connection to the Ginzburg-Landau type evolution, dual internal variables, and a thermodynamic generalization of the standard linear solid model of rheology is demonstrated. Specific examples are chosen to emphasize experimental confirmations and predictions beyond less general approaches., Comment: 23 pages, sign error revision

Numerical simulations of the thermoelastic response of a microstructured material on a thermal loading are performed in the one-dimensional setting to examine the influence of temperature gradient effects at the microstructure level predicted by the thermoelastic description of microstructured solids (Berezovski et al. in J. Therm. Stress. 34:413–430, 2011). The system of equations consisting of a hyperbolic equation of motion, a parabolic macroscopic heat conduction equation, and a hyperbolic evolution equation for the microtemperature is solved by a finite-volume numerical scheme. Effects of microtemperature gradients exhibit themselves on the macrolevel due to the coupling of equations of the macromotion and evolution equations for macro- and microtemperatures.

International audience; Reflection and transmission of elastic waves at the interface between two distinct micromorphic media are considered in the one-dimensional setting. Dual internal variable approach is used for the description of the microstructure influence on the global motion. It is shown that reflection and transition coefficients for plane waves depend on the coupling between macro-and micro-motions as well as on the choice of the microstructural interaction at the interface. Numerical simulations exhibiting results with promising technological implications are shown.

Thermoelastic wave propagation suggests a coupling between elastic deformation and heat conduction in a body. Microstructure of the body influences the both processes. Since energy is conserved in elastic deformation and heat conduction is always dissipative, the generalization of classical elasticity theory and classical heat conduction is performed differently. It is shown in the paper that a hyperbolic evolution equation for microtemperature can be obtained in the framework of the dual internal variables approach keeping the parabolic equation for the macrotemperature. The microtemperature is considered as a macrotemperature fluctuation. Numerical simulations demonstrate the formation and propagation of thermoelastic waves in microstructured solids under thermal loading.

The asymptotic stability of solutions of the Mindlin-type microstructure model for solids is analyzed in the paper. It is shown that short waves are asymptotically stable even in the case of a weakly non-convex free energy dependence on microdeformation.

Wave propagation in materials with embedded two different microstructures is considered. Each microstructure is characterized by its own length scale. The dual internal variables approach is adopted yielding in a Mindlin-type model including both microstructures. Equations of motion for microstructures are coupled with the balance of linear momentum for the macromotion, but not coupled with each other. Corresponding dispersion curves are provided and scale separation is pointed out.

The extension of the thermoelasticity theory by weakly non-local dual internal variables enriched by an extra entropy flux for the thermomechanical description of the behavior of microstructured solids is presented. The internal variables take into account the distributed effect of microdeformations or microtemperatures (and their gradients) on the overall macroscopic behavior. The evolution equations for microtemperatures can be hyperbolic, which can induce wave-like propagation for macrotemperature due to the coupling of equations.