Uniqueness in Kelvin–Voigt elasticity with higher gradients

This paper deals with the propagation of acceleration waves in constrained linear elastic materials, within the framework of the so-called linearized finite theory of elasticity, as defined by Hoger and Johnson in [12, 13]. In this theory, the constitutive equations are obtained by linearization of the corresponding finite constitutive equations with respect to the displacement gradient and significantly differ from those of the classical linear theory of elasticity. First, following the same procedure used for the constitutive equations, the amplitude condition for a general constraint is obtained. Explicit results for the amplitude condition for incompressible and inextensible materials are also given and compared with those of the classical linear theory of elasticity. In particular, it is shown that for the constraint of incompressibility the classical linear elasticity provides an amplitude condition that, coincidently, is correct, while for the constraint of inextensibility the disagreement is first order in the displacement gradient. Then, the propagation condition for the constraints of incompressibility and inextensibility is studied. For incompressible materials the propagation condition is solved and explicit values for the squares of the speeds of propagation are obtained. For inextensible materials the propagation condition is solved for plane acceleration waves propagating into a homogeneously strained material. For both constraints, it is shown that the squares of the speeds of propagation depend by terms that are first order in the displacement gradient, while in classical linear elasticity they are constant.

Estimate of the error occurring in the linearization of geometrically non-linear problems of the theory of elasticity

The problem of linear elasticity for free harmonic (periodic) and solitary bell-shaped (nonperiodic) waves in an isotropic half-space with stress-free plane boundary is considered. The half-space is made of either conventional (classical structural) or nonconventional (nonclassical auxetic) material. Two cases of wave damping are studied: rapid (surface wave) and periodic (nonsurface wave). The following conclusions on a free harmonic wave are drawn: a surface wave exists in materials of both classes, but the ratio of the wave velocity to the velocity of a transverse plane wave in auxetic materials is somewhat lower than in conventional materials; a nonsurface wave cannot be described by the approach applied to conventional materials, but can theoretically exist in auxetic materials where there are two wave velocities. For a solitary (bell-shaped) wave, the assumption that the wave velocity depends on the wave phase is substantiated and some constraint is imposed on the time of travel of the wave and the way the wave velocity varies with time. The following conclusions are drawn: a rapidly damped bell-shaped wave cannot be described by the approach for both classes of materials, whereas a periodically damped bell-shaped wave can be described

The 21st Winterschool on Electronic Properties of Novel Materials (IWEPNM 2007) took place from 10 March to 17 March 2007. It continued the tradition of winterschools of this type in Kirchberg/Tirol in its well-known atmosphere of collegiality and emphasis on scientific exchange between young and experienced scientists. The sessions at the IWEPNM 2007 focused on quantum information, carbon nanostructure synthesis, purification and separation, the characterization and properties of carbon nanostructures, fullerenes, endohedrals, and fullerides, excitons in carbon nanotubes, the chemical treatment of carbon nanotubes, nanotube filling and double-walled nanotubes, graphite and graphene, non-carbonaceous nanostructures, electron and heat transport in carbon nanostructures, composites with carbon nanotubes, and on theory and applications of these materials. This proceedings volume is divided into according sections. The program of the Winterschool was composed of invited oral presentations, poster presentations and mini-workshops. The oral presentations were scheduled in morning and evening sessions, the mini-workshops took place during the afternoons, and the posters were presented and discussed in two late-evening sessions. In 2007 the Winterschool celebrated an anniversary: It had been of taking place for the 20th time at the Hotel Sonnenalp, a hotel ideally suited for the close interactions of international participants from many countries. We are grateful to the manager of Hotel Sonnenalp, Frau Edith Mayer, and her staff for the local arrangements as well as for her patience with many special requests during the meeting. We owe special thanks to the hotel staff for their careful organization of the celebration. The event benefited substantially from support of the Universitat Wien and the Verein zur Forderung der Winterschulen in Kirchberg, as well as from numerous industrial sponsors. We greatly appreciate their financial contribution without which the meeting would have hardly become possible. Finally, special acknowledgments go to all contributors of the IWEPNM 2007 and to the authors of this volume, the proceedings of the Winterschool. Viera Skakalova was instrumental in compiling and editing the proceedings. In the name of all participants who have enjoyed the Winterschools since 1985 we express special thanks to Hans Kuzmany for his outstanding engagement in organizing the IWEP for so many years. His experience and advice will be needed for future Winterschools. The next Winterschool in this series will take place from 1 March to 8 March 2008, again in Kirchberg/Tirol, and will be organized by Christian Thomsen from Berlin. Wien, Darmstadt, Stuttgart, Berlin, 2007

Mechanical and structure properties of cellular auxetic materials

Constitutive modeling within peridynamic theory considers the collective deformation at each time of all the material within a δ-neighborhood of any point of a peridynamic body. The assignment of the parameter δ, called the horizon, is treated as a material property. The difference displacement quotient field in this neighborhood, rather than the extension scalar field, is used to generate a three-dimensional state-based linearly elastic peridynamic theory. This yields an enhanced interpretation of the kinematics between bonds that includes both length and relative angle changes. A free energy function for a linearly elastic isotropic peridynamic material that contains four material constants is proposed as a model, and it is used to obtain the force vector state and the associated modulus state for this material. These states are analogous to, respectively, the stress field and the fourth-order elasticity tensor in classical linear theory. In the limit of small horizon, we find that only three of the four peridynamic material constants are related to the classical elastic coefficients of an isotropic linear elastic material, with one of the three constants being arbitrary. The fourth peridynamic material constant, which accounts for the coupling effect of both bond length and relative angle change, has no effect on the limit, but remains a part of the peridynamic model. The determination of the two undetermined constants is the subject of future investigation. Peridynamic models proposed elsewhere in the literature depend on the deformation state through its dilatational and deviatoric parts and contain only two peridynamic material constants, in analogy to the classical linear elasticity theory. Observe from above that our model depends on both length and relative angle changes, as in classical linear theory, but, otherwise, is not limited to having only two material constants. In addition, our model corresponds to a nonordinary material, which represents a substantial break with classical models.

Displacements around a tunnel, occurring as a result of excavation, consist of the elastic and plastic parts. In this paper, we discuss the elastic part of displacements as a result of excavation, called net displacement. In general, the previous analytical solutions presented for determining the displacements around a circular tunnel in an elastic medium do not give the net displacements directly. The well-known Kirsch solution is the most widely used method for determining the induced stresses and net displacements around a circular opening in a biaxially-loaded plate of homogeneous, isotropic, continuous, linearly elastic material. However, the complete solution for obtaining the net displacements has not been presented or highlighted in the available literature. Using the linear elasticity, this paper reviews and presents three different analytical methods for determining the net displacements directly as well as induced stresses around a circular tunnel. The three solution methods are the Lame' method, airy stress function method, and complex variable method. The tunnel is assumed to be situated in an elastic, continuum, and isotropic medium in the plane strain condition. The solutions are presented for both the hydrostatic and non-hydrostatic in situ stresses in the 2D biaxial loading condition along with an internal pressure. Loading and unloading in tunneling occurring as a result of excavation and stress differences between the induced and initial ones are considered to evaluate the net displacements directly. Finally, some examples are given to demonstrate the complete solution and show the difference between the net elastic displacements as a result of excavation and total elastic displacements that are not real.

Uniqueness for plane crack problems in dipolar gradient elasticity and in couple-stress elasticity

Pressure vessel problem for chiral elastic tubes

Universal displacements are those displacements that can be maintained, in the absence of body forces, by applying only boundary tractions for any material in a given class of materials. Therefore, equilibrium equations must be satisfied for arbitrary elastic moduli for a given anisotropy class. These conditions can be expressed as a set of partial differential equations for the displacement field that we call universality constraints. The classification of universal displacements in homogeneous linear elasticity has been completed for all the eight anisotropy classes. Here, we extend our previous work by studying universal displacements in inhomogeneous anisotropic linear elasticity assuming that the directions of anisotropy are known. We show that universality constraints of inhomogeneous linear elasticity include those of homogeneous linear elasticity. For each class and for its known universal displacements, we find the most general inhomogeneous elastic moduli that are consistent with the universality constrains. It is known that the larger the symmetry group, the larger the space of universal displacements. We show that the larger the symmetry group, the more severe the universality constraints are on the inhomogeneities of the elastic moduli. In particular, we show that inhomogeneous isotropic and inhomogeneous cubic linear elastic solids do not admit universal displacements and we completely characterize the universal inhomogeneities for the other six anisotropy classes.

: For the traction boundary value problem in nonlinear elastostatics for a body which is convex in its undeformed reference state and with the assumption of sufficiently small strains (but not necessarily small displacement gradients), an upper bound is obtained for the elastic strain energy in terms of the L sub 2-integral norms of the surface tractions and body forces with the constant depending only upon the ratio of the outer and inner diameters and the physical constants of the material. This result extends previous known results in linear elasticity (infinitesimal displacement gradients) and finite elasticity (small but finite displacement gradients) into the small strain theory of nonlinear elasticity. (Author)

An integral bound on the strain energy for the traction problem in non-linear elasticity with sufficiently small strains

Energy theorems and the J-integral in dipolar gradient elasticity

We investigate the properties of an isotropic linear elastic peridynamic material in the context of a three-dimensional state-based peridynamic theory, which considers both length and relative angle changes, and is based on a free energy function proposed in previous work that contains four material constants. To this end, we consider a class of equilibrium problems in mechanics to show that, in interior points of the body where deformations are smooth, the corresponding solutions in classical linear elasticity are also equilibrium solutions in peridynamics. More generally, we show that the equations of equilibrium are satisfied even when two of the four peridynamic constants are arbitrary. Pure torsion of a cylindrical shaft and pure bending of a cylindrical beam are particular cases of this class of problems and are used together with a correspondence argument proposed elsewhere to determine these two constants in terms of the elasticity constants of an isotropic material from the classical linear elasticity. One of the constants has a singularity in the Poisson ratio, which needs further investigation. Two additional experiments concerning bending of cylindrical beam by terminal load and anti-plane shear of a hollow cylinder, which do not belong to the previous class of problems, are used to validate these results.

The rapid advancement of micro-nano acoustic devices has led their core acoustic structures to shrink to the nanoscale level. The influence of surface effects on the mechanical properties of thin-film materials on a nanoscale becomes increasingly prominent, and the classical elasticity theory struggles to accurately describe their mechanical behavior on this scale. In this paper, a mechanical model of nano-SiO<sub>2</sub>/Si heterostructured thin films that considers surface effects is developed using surface elasticity theory. This model incorporates the key parameter of surface energy density. In this paper, a mechanical model of heterostructured nano-SiO<sub>2</sub>/Si films is developed using the surface elasticity theory, incorporating surface effects through the introduction of surface energy density as a key parameter. Using the Fourier integral transform method, analytical expressions for stress and displacement fields under surface traction are systematically derived, revealing the influence of surface effects on the mechanical behavior of materials on a nanoscale by comparing the analytical solution with that from the classical theory. The results show that when the surface stress distribution deviates by 3% from that predicted by the classical theory, the microscopic properties of the material become significant, and the surface effect cannot be ignored in a range of five times the width of the excitation region 2<i>a</i>. As the size of the excitation region decreases, the surface effect is significantly increases and the stress distribution within the excitation region and near the boundary becomes more concentrated than the counterparts in the classical theory. The shear stress is no longer zero, and an extreme value is observed at the boundary, which is significantly different from that predicted by the classical theory of elasticity. The transverse and longitudinal displacements are reduced compared with those from the classical theory, and the surface stiffness and deformation resistance of the material are greatly enhanced. Significant surface effects occur on nano-heterostructure thin films, leading to large deviations in stress and displacement distributions from the results of elasticity theory. Therefore, the classical elasticity assumptions are no longer applicable in the corresponding nanoscale range. The results demonstrate that the propagation of ultrahigh-frequency nano- length acoustic waves in nanoscale solid film surfaces is significantly affected by the scale effect. The failure of the classical elastic wave theory on a nanoscale is of great value for the study of nanoscale acoustic theory. Furthermore, these findings provide a theoretical basis for the subsequent development of more precise models of interfacial effects and a more detailed investigation of the influence of the film-substrate modulus ratio.