A second strain gradient elasticity theory with second velocity gradient inertia – Part I: Constitutive equations and quasi-static behavior

A multi-cell homogenization procedure with four geometrically different groups of cell elements (respectively for the bulk, the boundary surface, the edge lines and the corner points of a body) is envisioned, which is able not only to extract the effective constitutive properties of a material, but also to assess the “surface effects” produced by the boundary surface on the near bulk material. Applied to an unbounded material in combination with the thermodynamics energy balance principles, this procedure leads to an equivalent continuum constitutively characterized by (ordinary, double and triple) generalized stresses and momenta. Also, applying this procedure to a (finite) body suitably modelled as a simple material cell system, in association with the principle of the virtual power (PVP) for quasi-static actions, an equivalent structural system is derived, featured by a (macro-scale) PVP having the typical format as for a second strain gradient material model. Due to the surface effects, the latter model does work as a combination of two subsystems, i.e. the bulk material behaving as a Cauchy continuum, and the boundary surface operating as a membrane-like boundary layer, each subsystem being in (local and global) equilibrium by its own. Further, the applied (ordinary) boundary traction splits into two (response-dependent) parts, i.e. the “Cauchy traction” transmitted to the bulk material and the “Gurtin–Murdoch traction” acting, together with all other boundary tractions, upon the boundary layer. The role of the boundary layer as a two-dimensional manifold enclosing a Cauchy continuum is elucidated, also with the aid of a discrete model. A strain gradient elasticity theory is proposed which includes a minimum total potential energy principle featuring the relevant boundary-value problem for quasi-static loads and its (unique) solution. A simple application is presented. Two appendices are included, one reports the proof of the global equilibrium of the boundary layer, the other is concerned with double and triple stresses. The paper is complemented by a companion Part II one on dynamics. Previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137] are improved and extended.     

A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior

This paper is the sequel of a companion Part I paper devoted to the constitutive equations and to the quasi-static behavior of a second strain gradient material model with second velocity gradient inertia. In the present Part II paper, a multi-cell homogenization procedure (developed in the Part I paper) is applied to a nonhomogeneous body modelled as a simple material cell system, in conjunction with the principle of virtual work (PVW) for inertial actions (i.e. momenta and inertia forces), which at the macro-scale level takes on the typical format as for a second velocity gradient inertia material model. The latter (macro-scale) PVW is used to determine the equilibrium equations relating the (ordinary, double and triple) generalized momenta to the inertia forces. As a consequence of the surface effects, the latter inertia forces include (ordinary) inertia body forces within the bulk material, as well as (ordinary and double) inertia surface tractions on the boundary layer and (ordinary) inertia line tractions on the edge line rod; they all depend on the acceleration in a nonstandard way, but the classical laws are recovered in the case of no higher order inertia. The classical linear and angular momentum theorems are extended to the present context of second velocity gradient inertia, showing that the extended theorems—used in conjunction with the Cauchy traction theorem—lead to the local force and moment (stress symmetry) motion equations, just like for a classical continuum. A gradient elasticity theory is proposed, whereby the dynamic evolution problem for assigned initial and boundary conditions is shown to admit a Hamilton-type variational principle; the uniqueness of the solution is also discussed. A few simple applications to wave propagation and dispersion problems are presented. The paper indicates the correct way to describe the inertia forces in the presence of higher order inertia; it extends and improves previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137]. Overall conclusions are drawn at the end of the paper.     

Multiple interacting circular nano-inhomogeneities with surface/interface effects

A two-dimensional problem of multiple interacting circular nano-inhomogeneities or/and nano-pores is considered. The analysis is based on the Gurtin and Murdoch model [Gurtin, M.E., Murdoch, A.I., 1975. A continuum theory of elastic material surfaces. Arch. Ration. Mech. Anal. 57, 291–323.] in which the interfaces between the nano-inhomogeneities and the matrix are regarded as material surfaces that possess their own mechanical properties and surface tension. The precise component forms of Gurtin and Murdoch's three-dimensional equations are derived for interfaces of arbitrary shape to provide a basis for critical review of various modifications used in the literature. The two-dimensional specification of these equations is considered and their representation in terms of complex variables is provided. A semi-analytical method is proposed to solve the problem. Solutions to several example problems are presented to: (i) examine the difference between the results obtained with the original and modified Gurtin and Murdoch's equations, (ii) compare the results obtained using Gurtin and Murdoch's model and those for a problem of nano-inhomogeneities with thin membrane-type interphase layers, and (iii) demonstrate the effectiveness of the approach in solving problems with multiple nano-inhomogeneities.     

A modified beam model based on Gurtin–Murdoch surface elasticity theory

In this work, a modified surface-effect incorporated beam model based on Gurtin and Murdoch (GM) surface elasticity theory is established by satisfying the required balance equations on surfaces, which is often overlooked by researchers in this field. With the refinement, the proposed model is more rigorous in mathematics and mechanics compared with GM theory-based beam models in literature. To demonstrate the model, the problem for static bending of simply supported beam considering surface effects is solved by applying the general equations derived, and numerical results are obtained and discussed.

     

Surface free energy effects on the postbuckling behavior of cylindrical shear deformable nanoshells under combined axial and radial compressions

In the traditional continuum mechanics, the effects of surface free energy are generally ignored. However, this cannot be the case for nanostructures because of their high surface to volume ratio; surface energy plays an important role in the mechanical responses. In the present study, the nonlinear buckling and postbuckling characteristics of cylindrical nanoshells subjected to combined axial and radial compressions are investigated in the presence of surface energy effects. To this end, Gurtin–Murdoch elasticity theory is implemented into the classical first-order shear deformation shell theory to develop an efficient size-dependent shell model incorporating surface free energy effects. Subsequently, a boundary layer theory is employed including surface effects in conjunction with the nonlinear prebuckling deformations, the large postbuckling deflections and the initial geometric imperfection. Finally, a solution methodology based on a two-stepped singular perturbation technique is utilized to obtain the size-dependent critical buckling loads and equilibrium postbuckling paths corresponding to the both axial dominated and radial dominated loading cases. It is observed that for the both axial dominated and radial dominated loading cases, surface free energy effects cause to increase the both critical buckling load and critical end-shortening of shear deformable nanoshell made of silicon.     

Mathematical study of boundary-value problems within the framework of Steigmann–Ogden model of surface elasticity

Mathematical questions pertaining to linear problems of equilibrium dynamics and vibrations of elastic bodies with surface stresses are studied. We extend our earlier results on existence of weak solutions within the Gurtin–Murdoch model to the Steigmann–Ogden model of surface elasticity using techniques from the theory of Sobolev’s spaces and methods of functional analysis. The Steigmann–Ogden model accounts for the bending stiffness of the surface film; it is a generalization of the Gurtin–Murdoch model. Weak setups of the problems, based on variational principles formulated, are employed. Some uniqueness-existence theorems for weak solutions of static and dynamic problems are proved in energy spaces via functional analytic methods. On the boundary surface, solutions to the problems under consideration are smoother than those for the corresponding problems of classical linear elasticity and those described by the Gurtin–Murdoch model. The weak setups of eigenvalue problems for elastic bodies with surface stresses are based on the Rayleigh and Courant variational principles. For the problems based on the Steigmann–Ogden model, certain spectral properties are established. In particular, bounds are placed on the eigenfrequencies of an elastic body with surface stresses; these demonstrate the increase in the body rigidity and the eigenfrequencies compared with the situation where the surface stresses are neglected.     

A generalized continuum theory with internal corner and surface contact interactions

We consider a classical derivation of a continuum theory, based on the fundamental balance laws of mass and momenta, for a body with internal corner and surface contact interactions. The balances of mass and linear and angular momentum are applied to the arbitrary parts of a continuum which supports non-classical internal corner and surface contact interactions. The form of the specific corner contact interaction force measured per unit length of the corner is derived. A generalized form of Cauchy’s stress theorem is obtained, which shows that the surface traction on an oriented surface depends in a specific way on both the oriented unit normal as well as the curvature of the surface. An explicit form of the surface-couple traction which acts on every oriented surface is obtained. Two fields in the continuum, which are denoted as stress and hyperstress fields, are shown to exist, and their role in representing the surface traction and the surface-couple traction is identified. Finally, the field equations for this theory are determined, and a fundamental power theorem is derived. In the absence of internal corner and surface-couple traction interactions, the equations of classical continuum mechanics are recovered. There is no appeal to any ‘principle of virtual power’ in this work.     

On uniqueness for the traction problem in finite elasticity

In many problems of interest the (Cauchy) surface traction is given as a function of position on the deformed surface. A class of loadings sufficiently general to include these problems is considered, and within the context of the fraction problem in finite elasticity, a number of uniqueness results are established. This work extends results obtained for the mixed problem by Gurtin and Spector.

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Resumé Dans plusieurs proble\`mes interessants la traction surfacique de Cauchy est donnée comme une fonction de la position dans la surface deformée. Une telle classe des charges, suffisamment géerale, est considerée et un nombre des résultats d'unicité est établit dans le cadre du probleme de traction de l'elasticité non linéaire. Cet ouure prolonge des résultats pour le proble\`me mixte obtenus par Gurtin et Spector.

     

Transient thermal stresses in a medium with a circular cavity with surface effects

A two-dimensional, transient, uncoupled thermoelastic problem of an infinite medium with a circular nano-scale cavity is considered. The analysis is based on the generalized Gurtin and Murdoch model [Murdoch, A.I., 2005. Some fundamental aspects of surface modelling. Journal of Elasticity 80, 33–52.] where the surface of the cavity possesses its own surface tension and thermomechanical properties. A semi-analytical solution for the problem is obtained using a complex variable boundary integral equation method and the Laplace transform. Several examples are presented to study the significance of surface thermomechanical properties and surface tension, and to compare the results obtained using the generalized Gurtin and Murdoch model and a thin interphase layer model.     

On uniqueness in finite elasticity with general loading

In many problems of interest the (Cauchy) surface traction is given as a function of position on the deformed surface. A class of loadings sufficiently general to include these problems is considered and within the context of finite elasticity a number of uniqueness results are established. A key ingredient is the result of Gurtin and Spector that uniqueness holds in any convex, stable set of deformations.

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Résumé Dans plusieurs problémes interessants la traction surfacique de Cauchy est donnée comme une fonction de la position dans la surface deformée. Une telle classe des charges, suffisamment générale, est considerée et un nombre des résultats d'unicité et établit dans le cadre d'élasticité non linéaire. Le résultat de Gurtin et Spector (unicité est valable sur un ensemble stable et convexe queleonque des déformations) est utilisé.

     

Nonlinear resonance responses of geometrically imperfect shear deformable nanobeams including surface stress effects

In this work, a thorough investigation is presented into the nonlinear resonant dynamics of geometrically imperfect shear deformable nanobeams subjected to harmonic external excitation force in the transverse direction. To this end, the Gurtin–Murdoch surface elasticity theory together with Reddy’s third-order shear deformation beam theory is utilized to take into account the size-dependent behavior of nanobeams and the effects of transverse shear deformation and rotary inertia, respectively. The kinematic nonlinearity is considered using the von Kármán kinematic hypothesis. The geometric imperfection as a slight curvature is assumed as the mode shape associated with the first vibration mode. The weak form of geometrically nonlinear governing equations of motion is derived using the variational differential quadrature (VDQ) technique and Lagrange equations. Then, a multistep numerical scheme is employed to solve the obtained governing equations in order to study the nonlinear frequency–response and force–response curves of nanobeams. Comprehensive studies into the effects of initial imperfection and boundary condition as well as geometric parameters on the nonlinear dynamic characteristics of imperfect shear deformable nanobeams are carried out through numerical results. Finally, the importance of incorporating the surface stress effects via the Gurtin–Murdoch elasticity theory, is emphasized by comparing the nonlinear dynamic responses of the nanobeams with different thicknesses.     

Statics and kinematics of discrete Cosserat-type granular materials

A theoretical framework is presented for the statics and kinematics of discrete Cosserat-type granular materials. In analogy to the force and moment equilibrium equations for particles, compatibility equations for closed loops are formulated in the two-dimensional case for relative displacements and relative rotations at contacts. By taking moments of the equilibrium equations, micromechanical expressions are obtained for the static quantities average Cauchy stress tensor and average couple stress tensor. In analogy, by taking moments of the compatibility equations, micromechanical expressions are obtained for the (infinitesimal) kinematic quantities average rotation gradient tensor and average Cosserat strain tensor in the two-dimensional case. Alternatively, these expressions for the average Cauchy stress tensor and the average couple stress tensor are obtained from considerations of the equivalence of the continuum force and couple traction vectors acting on a plane and the resultant of the discrete forces and couples acting on this plane. In analogy, the expressions for the average rotation gradient tensor and the average Cosserat strain tensor are obtained from considerations of the change of length and change of rotation of a line element in the two-dimensional case. It is shown that the average particle stress tensor is always symmetrical, contrary to the average stress tensor of an equivalent homogenized continuum. Finally, discrete analogues of the virtual work and complementary virtual work principles from continuum mechanics are derived.     

Singular integral equations in elastic dielectrics

Galerkin representations for the displacement vector, polarization vector and the potential field are obtained by elementary matrix inversions of the equations of equilibrium. Matrices of fundamental solutions of an infinite elastic dielectric continuum subjected to a concentrated body force, an electric force, and a charge density, are constructed. Theorems are proved on the discontinuity of double layer potentials and R, M, M operators of single layer potentials. By means of these theorems, the solution of the two basic boundary value problems has been reduced to the solution of a system of seven singular integral equations.     

Exact resultant equilibrium conditions in the non-linear theory of branching and self-intersecting shells

We formulate the exact, resultant equilibrium conditions for the non-linear theory of branching and self-intersecting shells. The conditions are derived by performing direct through-the-thickness integration in the global equilibrium conditions of continuum mechanics. At each regular internal and boundary point of the base surface our exact, local equilibrium equations and dynamic boundary conditions are equivalent, as expected, to the ones known in the literature. As the new equilibrium relations we derive the exact, resultant dynamic continuity conditions along the singular surface curve modelling the branching and self-intersection as well as the dynamic conditions at singular points of the surface boundary. All the results do not depend on the size of shell thicknesses, internal through-the-thickness shell structure, material properties, and are valid for an arbitrary deformation of the shell material elements.     

An admissibility condition for equilibrium shocks in finite elasticity

The equations governing the equilibrium of a finitely deformed elastic solid are derived from the Principle of Minimum Potential Energy. The possibility of the deformation gradient and the stresses being discontinuous across certain surfaces in the body — “equilibrium shocks” — is allowed for. In addition to the equilibrium equations, natural boundary conditions and traction continuity condition, a supplementary jump condition which is to hold across the surface of discontinuity is derived. This condition is shown to imply that a stable equilibrium shock must necessarily be dissipation-free.     

Closed-form formulas for the effective properties of random particulate nanocomposites with complete Gurtin–Murdoch model of material surfaces

The objective of this work is to present an approach allowing for inclusion of the complete Gurtin–Murdoch material surface equations in methods leading to closed-form formulas defining effective properties of particle-reinforced nanocomposites. Considering that all previous developments of the closed-form formulas for effective properties employ only some parts of the Gurtin–Murdoch model, its complete inclusion constitutes the main focus of this work. To this end, the recently introduced new notion of the energy-equivalent inhomogeneity is generalized to precisely include all terms of the model. The crucial aspect of that generalization is the identification of the energy associated with the last term of the Gurtin–Murdoch equation, i.e., with the surface gradient of displacements. With the help of that definition, the real nanoparticle and its surface possessing its own distinct elastic properties and residual stresses are replaced by an energy-equivalent inhomogeneity with properties incorporating all surface effects. Such equivalent inhomogeneity can then be used in combination with any existing homogenization method. In this work, the method of conditional moments is used to analyze composites with randomly dispersed spherical nanoparticles. Closed-form expressions for effective moduli are derived for both bulk and shear moduli. As numerical examples, nanoporous aluminum is investigated. The normalized bulk and shear moduli of nanoporous aluminum as a function of residual stresses are analyzed and evaluated in the context of other theoretical predictions.     

Anti-plane shear Green’s functions for an isotropic elastic half-space with a material surface

This paper presents analytical Green’s function solutions for an isotropic elastic half-space subject to anti-plane shear deformation. The boundary of the half-space is modeled as a material surface, for which the Gurtin–Murdoch theory for surface elasticity is employed. By using Fourier cosine transform, analytical solutions for a point force applied both in the interior or on the boundary of the half-space are derived in terms of two particular integrals. Through simple numerical examples, it is shown that the surface elasticity has an important influence on the elastic field in the half-space. The present Green’s functions can be used in boundary element method analysis of more complicated problems.     

On boundary potential energies in deformational and configurational mechanics

This contribution deals with the implications of boundary potential energies, i.e. in short surface, curve and point potentials, on deformational and configurational mechanics. Within the realm of deformational mechanics the surface/curve potentials are allowed in the most general case to depend on the deformation, the surface/curve deformation gradient and the spatial surface normal/curve tangent and are parametrised in the material placement and the material surface normal/curve tangent. The point potentials depend on the deformation and are parametrised in the material placement. From the configurational mechanics perspective the roles of fields and parametrisations are reversed. By considering variational arguments based on the kinematics of deforming surfaces/curves, in particular the relevant surface/curve stresses and distributed forces contributing to (localized) deformational and configurational force balances at surfaces/curves/points, which extend the common traction boundary conditions, are derived. Thereby, dissipative distributed configurational forces that are energetically conjugate to configurational changes are introduced as definitions. The (localized) force balances at surfaces/curves/points together with the contributing stresses and distributed forces within deformational and configurational mechanics display an intriguing duality. The resulting dissipative configurational tractions at the boundary are exemplified for some illustrative cases of boundary potentials.