Sergey A. Lurie

Symmetry conditions in strain gradient elasticity

We study the variational significance of the “order-of-differentiation” symmetry condition of strain gradient elasticity. This symmetry condition stems from the fact that in strain gradient elasticity, one can interchange the order of differentiation in the components of the second displacement gradient tensor. We demonstrate that this symmetry condition is essential for the validity of free variational formulations commonly employed for deriving the field equations of strain gradient elasticity. We show that relying on this additional symmetry condition, one can restrict consideration to strain gradient constitutive equations with a considerably reduced number of independent material coefficients. We explicitly derive a symmetry unified theory of isotropic strain gradient elasticity with only two independent strain gradient material coefficients. The presented theory has simple stability criteria and its factorized displacement form equations of equilibrium allow for expedient identification of the fundamental solutions operative in specific theoretical and application studies.

Multiscale Modeling in the Mechanics of Materials: Cohesion, Interfacial Interactions, Inclusions and Defects

The models of media with multiscale effects are constructed based on the variational formalism. The internal interactions are determined by kinematic constraints of general character. These models take into account the superficial effects and internal interactions similar to interatomic forces of coupling. The models of media with a continuous field of defects of various types are considered. Descriptions of the media with defects, cohesion field models as a special case of the Cosserat models are considered. This paper is devoted to the consideration of disperse composite materials reinforced by nanoparticles. For numerical simulation, the new efficient block analytical-numerical method is developed oriented to the general case of solving problems in complex-shaped 2D and 3D domains. As an example of numerical investigation of the interfacial interactions in heterogeneous media, the medium with inclusions is considered.

Bending problems in the theory of elastic materials with voids and surface effects

In the present study, a comparison of pure, three-point, four-point and cantilever beam bending problems in the frame of the theory of elastic materials with voids (micro-dilatational elasticity) has been provided via analytical modelling and three-dimensional finite-element analysis. We consider the extended variant of the theory with surface effects using a variational approach. At first, we compare the known approximate semi-inverse analytical solution of the pure bending problem with a corresponding three-dimensional finite-element solution in the frame of micro-dilatational theory. It is demonstrated that in the numerical solution – unlike the analytical one – all boundary conditions are satisfied accurately, and there exist distortions of the cross-sections and lateral faces of the beam. The generalized analytical solution of the beam pure bending problem with surface effects is also established and compared with numerical simulations. The effective elastic properties of the beam with micro-dilatations are introduced by comparing its displacements and corresponding classical beam displacements. The influence of scale, coupling and surface parameters on the effective elastic moduli in different bending and simple tension tests is studied. It is shown that all considered types of bending experiments provide the determination of close values of the effective flexural modulus of the beams with different thickness. This means that it is possible to use any bending test with beams of different thickness for reliable identification of the material constants of the theory. It is also possible to use the simple analytical expression for an effective flexural modulus that follows from the analytical solution of the pure bending problem. It is also shown that stiffness of the beam with micro-dilatation in any bending tests should always be higher as compared with the stiffness of such a beam in simple tension. For thin beams, there are no scale effects, and its effective flexural modulus is equal to the material’s Young’s modulus without micro-dilatation. Possible rise of the negative size effects in the model with surface effects is also discussed.

Mechanical behavior of porous Si 3 N 4 ceramics manufactured with 3D printing technology

The paper focuses on experimental measurement and analytical and numerical modeling of the elastic moduli of porous Si3N4 ceramics obtained by 3D printing and pressureless sintering. The pores in such a material have complex irregular shape and porosity varies over a wide range (up to 50%), depending on the technological parameters used. For analytical modeling, we use effective field methods (Mori–Tanaka–Benveniste and Maxwell homogenization schemes) recently developed for pores of superspherical shape. For FEM simulation, we used microstructures generated by overlapping solid spheres and overlapping spherical pores. It is shown that elastic properties of ceramics are largely determined by the granular structure and the concave pore shape, which have been observed in the ceramics microstructure after sintering of the 3D-printed powder green bodies.

On Remarkable Loss Amplification Mechanism in Fiber Reinforced Laminated Composite Materials

In this present work, we investigate damping behavior of filled and layered composite material that has its inclusions coated by viscoelastic coating material. To analyze its behavior, we use generalized self-consistent Eshelby method with correspondence principle approach. The viscous coating layer is assumed to possess properties at its glass transition temperature. This analytical study reveals that at ultra thin coating layer, the composite exhibits very high loss characteristics where its effective loss moduli significantly exceed the loss moduli of both coating and matrix materials. High shearing dissipation mechanism in ultra thin layer of viscoelastic coating material is found to be responsible for this peculiar behavior. This remarkable loss amplification effect is technologically appealing as such composites with high damping and high stiffness properties might be attainable.

Intermediate layer formation between inclusion and matrix during synthesis of unidirectional fibrous composite

The problem of transient layer formation in given composition between inclusions and matrix can be solved using mathematical modeling. This paper suggests a suitable model taking into account multi staging of chemical conversion. Numerical realization of this model allows studying the dynamics of transient layer formation at varying temperature and pressure. As a result, the phase structure and thickness of a transient zone depending on synthesis conditions are obtained.

WAVE-RELAXATION DUALITY OF HEAT PROPAGATION IN FERMI–PASTA–ULAM CHAINS

In our recent work [Gusev and Lurie, Int. J. Mod. Phys. B26 (2012) 125003], we have presented the theory of spacetime elasticity. The theory advocates its own law of heat conduction in solids, involving both wave and relaxation (diffusion) modes of heat propagation. Here we use that heat conduction law — together with the classical Fouriers and Maxwell–Cattaneo laws — to analyze the temperature dynamics occurring in Fermi–Pasta–Ulam (FPU-β) chains. We focus on the acoustic limit and use a combination of the Langevin and molecular dynamics to study the corresponding temperature decay functions. We have found that all the three laws suit well for describing the high-temperature, exponential-form temperature decay functions. However, at low temperatures the wave dynamics becomes prevailing and it is only the spacetime-elasticity heat conduction law that provides the appropriate functional forms. At all the temperatures and wavelengths studied, the observed temperature dynamics is anomalous, and we discuss both theoretical and practical implications of such anomalous behavior

Theory of spacetime elasticity

We present the theory of spacetime elasticity and demonstrate that it involves traditional thermoelasticity. Assuming linear-elastic constitutive behavior and using spacetime transversely-isotropic elastic constants, we derive all principal thermodynamic relations of classical thermoelasticity. We introduce the spacetime principle of virtual work, and use it to derive the equations of motion for both reversible and dissipative thermoelastic dynamics. We show that spacetime elasticity directly implies the Fourier and the Maxwell–Cattaneo laws of heat conduction. However, spacetime elasticity is richer than classical thermoelasticity, and it advocates its own equations of motion for coupled thermoelasticity, complemented by the spectrum of boundary and interface conditions. We argue that the presented framework of spacetime elasticity should prove adequate for describing the thermoelastic phenomena occurring at low temperatures, for interpreting the results of molecular simulations of heat conduction in solids, and also for the optimal heat and stress management in the microelectronic components and the thermoelectric devices.

Gradient Theory of Media with Conserved Dislocations: Application to Microstructured Materials

In the paper, a rigorous continuous media model with conserved dislocations is developed. The important feature of the newly developed classification is a new kinematic interpretation of dislocations, which reflects the connection of dislocations with distortion, change in volume (porosity), and free forming. Our model generalizes those previously derived by Mindlin, Cosserat, Toupin, Aero–Kuvshinskii and so on, and refines some assertions of these models from the point of view of the account of adhesive interactions.

Classification of Gradient Adhesion Theories Across Length Scale

The sequence of continuum theories of adhesion is discussed. We give a brief analysis of the existing theories of adhesion and present a continuum theory of adhesion as a natural generalization of appropriate options for the theory of elasticity and gradient theories of elasticity. We offer a sequence of variational formulations of theories of adhesion and constitutive equations. In addition, the analysis of the structures of the tensors of adhesive elastic modules is presented. As a result, we propose a classification of theories of adhesion and gradient theories of elasticity in terms accounting for scale effects. The classification is based on the qualitative analysis of scale effects of different orders depending on the physical properties of the continuum.