Surface interaction potentials in elasticity

Dedicated to Bernard D. Coleman on his sixtieth birthday     

Surface effects of adsorption-induced resonance analysis on micro/nanobeams via nonlocal elasticity

The governing differential equation of micro/nanbeams with atom/molecule adsorption is derived in the presence of surface effects using the nonlocal elasticity.The effects of the nonlocal parameter,the adsorption density,and the surface parameter on the resonant frequency of the micro/nanobeams are investigated.It is found that,in addition to the nonlocal parameter and the surface parameter,the bending rigidity and the adsorption-induced mass exhibit different behaviors with the increase in the adsorption density depending on the adatom category and the substrate material.     

Surface effect and non-local elasticity in wave propagation of functionally graded piezoelectric nano-rod excited to applied voltage

A non-local solution for a functionally graded piezoelectric nano-rod is presented by accounting the surface effect. This solution is used to evaluate the characteristics of the wave propagation in the rod structure. The model is loaded under a two-dimensional (2D) electric potential and an initially applied voltage at the top of the rod. The mechanical and electrical properties are assumed to be variable along the thickness direction of the rod according to the power law. The Hamilton principle is used to derive the governing differential equations of the electromechanical system. The effects of some important parameters such as the applied voltage and gradation of the material properties on the wave characteristics of the rod are studied.     

Acoustic axes in elasticity

New results are presented for the degeneracy condition of elastic waves in anisotropic materials. The condition for the existence of acoustic axes involves a traceless symmetric third order tensor that must vanish identically. It is shown that all previous representations of the degeneracy condition follow from this acoustic axis tensor. The conditions for existence of acoustic axes in elastic crystals of orthorhombic, tetragonal, hexagonal and cubic (RTHC) symmetry are reinterpreted using the geometrical methods developed here. Application to weakly anisotropic solids is discussed, and it is shown that the satisfaction of the acoustic axes conditions to first order in anisotropy does not in general coincide with true acoustic axes.     

Conservation laws in elasticity

In this paper the basic results involved in the application of Noether's theorem relating symmetry groups and conservation laws to the variational problems of homogeneous elastostatics are outlined. General methods and conditions for the existence of variational and generalized symmetries are presented. Applications will be considered in subsequent papers in this series.The research reported here was supported in part by the U. S. National Science Foundation, Grant NSF MCS 81-00786.     

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

The present work deals with the uniqueness theorem for plane crack problems in solids characterized by dipolar gradient elasticity. The theory of gradient elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Micro-structure in linear elasticity. Arch. Ration. Mech. Anal. 16, 51–78] and is appropriate to model materials with periodic structure. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain (as in classical elasticity) and the second gradient of the displacement (additional term). Specific cases of the general theory employed here are the well-known theory of couple-stress elasticity and the recently popularized theory of strain-gradient elasticity. These cases are also treated in the present study. We consider an anisotropic material response of the cracked plane body, within the linear version of gradient elasticity, and conditions of plane-strain or anti-plane strain. It is emphasized that, for crack problems in general, a uniqueness theorem more extended than the standard Kirchhoff theorem is needed because of the singular behavior of the solutions at the crack tips. Such a theorem will necessarily impose certain restrictions on the behavior of the fields in the vicinity of crack tips. In standard elasticity, a theorem was indeed established by Knowles and Pucik [Knowles, J.K., Pucik, T.A., 1973. Uniqueness for plane crack problems in linear elastostatics. J. Elast. 3, 155–160], who showed that the necessary conditions for solution uniqueness are a bounded displacement field and a bounded body-force field. In our study, we show that the additional (to the two previous conditions) requirement of a bounded displacement-gradient field in the vicinity of the crack tips guarantees uniqueness within the general form of the theory of dipolar gradient elasticity. In the specific cases of couple-stress elasticity and pure strain-gradient elasticity, the additional requirement is less stringent. This only involves a bounded rotation field for the first case and a bounded strain field for the second case.     

Internal constraints in linear elasticity

Sufficient conditions are obtained for continuous dependence of solutions of boundary value problems of linear elasticity on internal constraints. Arbitrary hyperelastic materials with arbitrary (linear) internal constraints are included. In particular the results of Bramble and Payne, Kobelkov, Mikhlin for homogeneous, isotropic, incompressible materials are obtained as a special case. In the case of boundary value problem of place, a compatibility condition is obtained between the internal constraints and the boundary data which is necessary for the existence of solutions. With a further coercivity assumption on the compliance tensor, it is shown that the compatibility condition is also sufficient for existence. An orthogonal decomposition theorem for second order tensor fields modeled after Weyl's decomposition of solenoidal and gradient fields leads to the variational formulation of the problem and existence theorems.Almost all the results here apply to materials both with or without internal constraints. For internally constrained materials however, the verification of certain hypothesis is surprisingly non-trivial as indicated by the computation in the appendix.     

Two-plate problems in linear elasticity

Summary Some necessary and/or sufficient conditions for the existence of an equilibrium configuration of a couple of thin, noninterpenetrating plates are given. Various types of boundary data are considered, and some unstable cases discussed in detail.

---

Sommario Si danno alcune condizioni necessarie e/o sufficienti per l'esistenza di una configurazione di equilibrio senza interpenetrazione di una coppia di piastre sottili. Si considerano vari tipi di dati al bordo e si discutono in dettaglio alcuni casi instabili.

---

---

This work was supported in party by Istituto di Analisi Numerica del C.N.R., by Gruppo Nazionale di Analisi Funzionale e Applicazioni del C.N.R. and by Ministero della Pubblica Istruzione, Italy.     

Two-sided estimates in unilateral elasticity

The paper presents an extension to unilateral problems of the classical method of bounding (above and below) the solutions of linear self-adjoint boundary value problems. Using this extension the solution of the general unilateral problem in linear elasticity is bounded in energy by two suitable defined admissible states belonging to two complementary convex sets.     

Diffuse fields in dynamic elasticity

In this article the problem of Green function retrieval from correlations is approached from a theoretical point of view and for this purpose an integral identity is considered: a representation theorem of the correlation type for an inhomogeneous region embedded in a homogeneous space. The full homogeneous case is studied with the theorem and it is concluded that, in the resulting field, the energy is equipartitioned. In infinite space this means that the ratio of P and S energy densities stabilizes to a constant value. That equipartition is reached in the classical sense is also demonstrated. Thus, in infinite space the energy densities associated with the possible degrees of freedom tend to share in equal parts the available energy.

The representation theorem permits the verification of the well known result that by averaging correlations of motions from diffuse, equipartitioned fields within an inhomogeneous, anisotropic, elastic medium it is possible to retrieve its Green function. As a result of this it is shown that the average autocorrelation of the diffuse displacement field at a point is proportional to the imaginary part of the Green function at the source precisely at this point. As a consequence, the energy density of the diffuse field is proportional to the trace of the imaginary part of the Green tensor at the source. Thus, the analytical form of the Green function permits the establishment, in and around an inhomogeneous region, of the theoretical energy density of a diffuse field.

In both homogeneous and inhomogeneous cases (i.e. localized elastic inclusions or cavities) the equipartition of the background illumination (the so called incident field in scattering theory) is a necessary and sufficient condition to retrieve the exact Green function from correlations. Local effects lead to energy ratios that fluctuate in space and frequency. The boundary of a half-space produces in its interior fluctuations of energy densities that are local effects of the diffuse field. These results may be useful to assess the diffuse nature of seismic ground motion from a limited set of observation points and to detect the presence of a target by its signature in the distribution of diffuse energy.     

Energetic bounds in finite elasticity

Summary We study the conditions under which the internal work of deformation in an elastic isotropic body in finite deformations may be bounded by results obtained from a suitably defined linear infinitesimal problem. The values of the constants appearing in the principal inequalities are calculated and discussed for a certain class of extensional deformations.     

Crack-tip problem in non-local elasticity

Solutions are presented for the one- and two-dimensional Griffith crack problems in non-local elasticity. The displacements and stresses are determined in an elastic plate, weakened by a sharpedged line crack. The plate is loaded by a uniform tension perpendicular to the line of the crack at infinity. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis.     

Radiation conditions in asymmetric elasticity

In the present paper the radiation conditions of the Sommerfield type for a linear homogeneous and isotropic micropolar elasticity are discussed. A regular solution (u, ) of the fundamental system of field equations in an infinite domain has been defined using the radiation conditions for % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% WbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGWbGaaiykaaqabaGccaGGPaaaaa!4834!\[(u_{(p)} , + \varphi _{(p)} )\] and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% ZbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGPaaaaa!483A!\[(u_{(s)} , + \varphi _{(s)} )\], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaadwhacqGH9aqpcaWG1bWaaSbaaSqaaiaa% cIcacaWGWbGaaiykaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGSaGaaeiiaiabgEGirlabgEna0kaa% dwhadaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9iaaic% dacaGGSaGaaeiiaiabgEGirhrbmv3yPrwyGm0BUn3BSvgaiyGacaWF% 1bWaaSbaaSqaaGqaciaa+HcacaGFZbGaa4xkaaqabaGccqGH9aqpca% aIWaGaaiilaiaabccacqaHvpGAcqGH9aqpcqaHvpGAdaWgaaWcbaGa% aiikaiaadchacaGGPaaabeaakiabgUcaRiabew9aQnaaBaaaleaaca% GGOaGaam4CaiaacMcaaeqaaOGaaiilaiaabccacqGHhis0cqGHxdaT% cqaHvpGAdaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9i% aaicdacaGGSaGaaeiiaiabgEGirlabew9aQnaaBaaaleaacaqGOaGa% ae4CaiaabMcaaeqaaOGaeyypa0JaaGimaaaa!809B!\[u = u_{(p)} + u_{(s)} ,{\text{ }}\nabla \times u_{(p)} = 0,{\text{ }}\nabla u_{(s)} = 0,{\text{ }}\varphi = \varphi _{(p)} + \varphi _{(s)} ,{\text{ }}\nabla \times \varphi _{(p)} = 0,{\text{ }}\nabla \varphi _{{\text{(s)}}} = 0\], and formulae of the Betti type for an infinite domain with a cavity have been derived.

---

Zusammenfassung Betrachtet werden die Ausstrahlungsbedingungen der Sommerfeldschen Art für lineare homogene mikropolare Elastizitätstheorie. Die reguläre Lösung (u, ) der grundlegenden Gleichungen für einen unendlichen Raum wird mit Hilfe der Bedingungen für Paare % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% WbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGWbGaaiykaaqabaGccaGGPaaaaa!4834!\[(u_{(p)} , + \varphi _{(p)} )\] und % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaacIcacaWG1bWaaSbaaSqaaiaacIcacaWG% ZbGaaiykaaqabaGccaGGSaGaey4kaSIaeqy1dO2aaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGPaaaaa!483A!\[(u_{(s)} , + \varphi _{(s)} )\] definiert, wobei % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbnL2yY9% 2CVzgDGmvyUnhitvMCPzgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqe% fqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYlf9irVeeu0d% Xdh9vqqj-hEeeu0xXdbba9frFf0-OqFfea0dXdd9vqaq-JfrVkFHe9% pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGaciaaca% qabeaadaabauaaaOqaaiaadwhacqGH9aqpcaWG1bWaaSbaaSqaaiaa% cIcacaWGWbGaaiykaaqabaGccqGHRaWkcaWG1bWaaSbaaSqaaiaacI% cacaWGZbGaaiykaaqabaGccaGGSaGaaeiiaiabgEGirlabgEna0kaa% dwhadaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9iaaic% dacaGGSaGaaeiiaiabgEGirhrbmv3yPrwyGm0BUn3BSvgaiyGacaWF% 1bWaaSbaaSqaaGqaciaa+HcacaGFZbGaa4xkaaqabaGccqGH9aqpca% aIWaGaaiilaiaabccacqaHvpGAcqGH9aqpcqaHvpGAdaWgaaWcbaGa% aiikaiaadchacaGGPaaabeaakiabgUcaRiabew9aQnaaBaaaleaaca% GGOaGaam4CaiaacMcaaeqaaOGaaiilaiaabccacqGHhis0cqGHxdaT% cqaHvpGAdaWgaaWcbaGaaiikaiaadchacaGGPaaabeaakiabg2da9i% aaicdacaGGSaGaaeiiaiabgEGirlabew9aQnaaBaaaleaacaqGOaGa% ae4CaiaabMcaaeqaaOGaeyypa0JaaGimaaaa!809B!\[u = u_{(p)} + u_{(s)} ,{\text{ }}\nabla \times u_{(p)} = 0,{\text{ }}\nabla u_{(s)} = 0,{\text{ }}\varphi = \varphi _{(p)} + \varphi _{(s)} ,{\text{ }}\nabla \times \varphi _{(p)} = 0,{\text{ }}\nabla \varphi _{{\text{(s)}}} = 0\] ist. Es wird gezeigt, dass derartige reguläre Lösung mit Hilfe der Formel Bettischer Art für einem unendlichen Raum mit einem Hohlraum dargestellt werden kann.