Boussinesq–Flamant problem in gradient elasticity with surface energy

The strain gradient elasticity theory with surface energy is applied to Boussinesq–Flamant problem. The solution for the vertical displacements at the surface of half space due to the surface normal line load is presented. The theory includes both volumetric and surface energy terms. Boussinesq–Flamant problem in the strain gradient elasticity is solved by means of Fourier transform. The results obtained show that the vertical displacements of half space in the gradient elasticity are some different from that in the classical elasticity and the effects of the strain gradient parameters (material characteristic lengths) on the vertical displacements do exist.     

Gradient elasticity with surface energy: mode-III crack problem

In this paper an anisotropic strain-gradient dependent theory of elasticity is exploited, which contains both volumetric and surface energy gradient dependent terms. The theory is applied to the solution of the mode-III crack problem and is extending previous results by Aifantis and co-workers. The two boundary value problems corresponding to the “unclamped” and “clamped” crack tips, respectively, are solved analytically. It turns out that the first problem is physically questionable for some values of the surface energy parameter, whereas the second boundary value problem is leading to a cusping crack, which is consistent with Barenblatt's theory without the incorporation of artificial assumptions.     

Anti-plane shear crack in a functionally gradient piezoelectric material

The main objective of this paper is to study the singular nature of the crack-tip stress and electric displacement field in a functionally gradient piezoelectric medium having material coefficients with a discontinuous derivative. The problem is considered for the simplest possible loading and geometry, namely, the anti-plane shear stress and electric displacement in-plane of two bonded half spaces in which the crack is parallel to the interface. It is shown that the square-root singularity of the crack-tip stress field and electric displacement field is unaffected by the discontinuity in the derivative of the material coefficients. The problem is solved for the case of a finite crack and extensive results are given for the stress intensity factors, electric displacement intensity factors, and the energy release rate. Project supported by the National Natural Science Foundation of China (No. 10072041), the National Excellent Young Scholar Fund, of China (No. 10125209) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P. R. C..     

A nonlinear microbeam model based on strain gradient elasticity theory with surface energy

A microscale nonlinear Bernoulli–Euler beam model on the basis of strain gradient elasticity with surface energy is presented. The von Karman strain tensor is used to capture the effect of geometric nonlinearity. Governing equations of motion and boundary conditions are obtained using Hamilton’s principle. In particular, the developed beam model is applicable for the nonlinear vibration analysis of microbeams. By employing a global Galerkin procedure, the ordinary differential equation corresponding to the first mode of nonlinear vibration for a simply supported microbeam is obtained. Numerical investigations show that in a microbeam having a thickness comparable with its material length scale parameter, the strain gradient effect on increasing the beam natural frequency is higher than that of the geometric nonlinearity. By increasing the beam thickness, the strain gradient effect decreases or even diminishes. In this case, geometric nonlinearity plays the main role on increasing the natural frequency of vibration. In addition, it is shown that for beams with some specific thickness-to-length parameter ratios, both geometric nonlinearity and size effect have significant role on increasing the frequency of nonlinear vibration.     

The Boussinesq problem in dipolar gradient elasticity

The three-dimensional axisymmetric Boussinesq problem of an isotropic half-space subjected to a concentrated normal quasi-static load is studied within the framework of dipolar gradient elasticity involving linear constitutive relations and small strains. Our main concern is to determine possible deviations from the predictions of classical linear elastostatics when a more refined theory is employed to attack the problem. Of special importance is the behavior of the new solution near to the point of application of the load where pathological singularities exist in the classical solution. The use of the theory of gradient elasticity is intended here to model the response of materials with microstructure in a manner that the classical theory cannot afford. A linear version of this theory (as regards both kinematics and constitutive response) results by considering a linear isotropic expression for the strain-energy density that depends on strain gradient terms, in addition to the standard strain terms appearing in classical elasticity and by considering small strains. Through this formulation, a microstructural material constant is introduced, in addition to the standard Lamé constants. The solution method is based on integral transforms and is exact. The present results show significant departure from the predictions of classical elasticity. Indeed, continuous and bounded displacements are predicted at the points of application of the concentrated load. Such a behavior of the displacement field is, of course, more natural than the singular behavior exhibited in the classical solution.     

Anti-plane shear of rectangular region with two cracks

An approach for obtaining the solution of multiple cracks in a rectangular region is presented. A pair of concentrated shear force applied to the crack surface is used in conjunction with the method of superposition for generating the results for cracks whose surfaces are free from tractions. A system of Fredholm integral equations can always be obtained to yield numerical results.     

Cavitation in finite elasticity with surface energy effects

Cavity formation in incompressible as well as compressible isotropic hyperelastic materials under spherically symmetric loading is examined by accounting for the effect of surface energy. Equilibrium solutions describing cavity formation in an initially intact sphere are obtained explicitly for incompressible as well as slightly compressible neo-Hookean solids. The cavitating response is shown to depend on the asymptotic value of surface energy at unbounded cavity surface stretch. The energetically favorable equilibrium is identified for an incompressible neo-Hookean sphere in the case of prescribed dead-load traction, and for a slightly compressible neo-Hookean sphere in the case of prescribed surface displacement as well as prescribed dead-load traction. In the presence of surface energy effects, it becomes possible that the energetically favorable equilibrium jumps from an intact state to a cavitated state with a finite cavity radius, as the prescribed loading parameter passes a critical level. Such discontinuous cavitation characteristics are found to be highly dependent on the relative magnitude of the surface energy to the bulk strain energy.     

Anti-plane dynamic problem of an electroelastic cylinder with thin rigid inclusions excited by a system of surface electrodes

Summary The anti-plane mixed boundary problem of electroelasticity for vibrations of an infinite piezoceramic cylinder with a thin rigid inclusion is considered. Using the developed integral representation of the solution, the boundary problem is reduced to a system of singular integro-differential equations of the second kind with resolvent kernels. Calculations yeild the amplitude-frequency characteristics of the piecewise homogeneous cylinder. The behaviour of electroelastic fields, both within the cylinder and on its boundary, is given.     

A bound on the strain energy for the traction problem in finite elasticity with localized non-zero surface data

For the boundary value problem in finite elasticity in which nonzero tractions are given on a connected subdomain of the boundary, the rest of the boundary is stress-free, and there are no body forces, a bound is obtained for the strain energy in terms of the L ₂ integral norm of the surface tractions with the constant involved depending only upon and the material constants.The result is obtained in the context of finite elasticity under the assumptions that the unstressed body occupies a convex domain and the displacement gradients are sufficiently small. In the context of the linear theory, the same result is obtained without these assumptions.

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Zusammenfassung Wir betrachten ein Randwertproblem in der nichtlinearen Elastizität in dem nur ein zusammenhängendes Teilgebiet der Randfläche belastet ist, sonst aber keine Randbelastung oder Körperkräfte vorhanden sind. Eine Schranke für die Verzerrungsenergie wird mittels der L ₂ Integralnorme der Randbelastung hergeleitet, wobei die auftretende Konstante nur von dem Teilgebiet und von den Eigenschaften des Materials abhängig ist.Das Ergebnis gilt für die nichtlinearen Elastizität, unter den Vorraussetzungen dass das unbelastete Material ein konvexes Gebiet besetzt und dass die Verschiebungsgradiente hinreichend klein sind. Das gleiche Ergebnis gilt in der linearen Elastizität ohne diese Vorraussetzungen.

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The second author was a visitor at Georgia Institute of Technology, School of Mathematics, at the time that the revised version was prepared.     

Anti-plane shear crack growth rate of piezoelectric ceramic body with finite width

The theory of linear piezoelectricity is applied to develop an anti-plane crack growth rate equation of a finite crack in a piezoelectric ceramic body with finite width. Plastic zone is assumed to be confined to a sheet ahead of both crack edges similar to the strip model for in-plane loading. The procedure consists of reducing a system of dual integral equations to a Fredholm integral equation of the second kind. The accumulated plastic displacement criterion is used for developing a solution for the crack growth rate. Numerical values of crack growth rate are obtained and the results are displayed graphically to exhibit the electroelastic interactions.     

Anti-plane shear waves in a fibre-reinforced composite with a non-linear imperfect interface

The propagation of non-linear elastic anti-plane shear waves in a unidirectional fibre-reinforced composite material is studied. A model of structural non-linearity is considered, for which the non-linear behaviour of the composite solid is caused by imperfect bonding at the “fibre–matrix” interface. A macroscopic wave equation accounting for the effects of non-linearity and dispersion is derived using the higher-order asymptotic homogenisation method. Explicit analytical solutions for stationary non-linear strain waves are obtained. This type of non-linearity has a crucial influence on the wave propagation mode: for soft non-linearity, localised shock (kink) waves are developed, while for hard non-linearity localised bell-shaped waves appear. Numerical results are presented and the areas of practical applicability of linear and non-linear, long- and short-wave approaches are discussed.     

Solution of the Eshelby problem in gradient elasticity for multilayer spherical inclusions

We consider gradient models of elasticity which permit taking into account the characteristic scale parameters of the material. We prove the Papkovich–Neuber theorems, which determine the general form of the gradient solution and the structure of scale effects. We derive the Eshelby integral formula for the gradient moduli of elasticity, which plays the role of the closing equation in the self-consistent three-phase method. In the gradient theory of deformations, we consider the fundamental Eshelby–Christensen problem of determining the effective elastic properties of dispersed composites with spherical inclusions; the exact solution of this problem for classical models was obtained in 1976.     

Singularity-free dislocation dynamics with strain gradient elasticity

The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.     

ANTI-PLANE FRACTURE ANALYSIS OF FUNCTIONALLY GRADIENT MATERIAL INFINITE STRIP WITH FINITE WIDTH

The special case of a crack under mode III conditions was treated, lying parallel to the edges of an infinite strip with finite width and with the shear modulus varying exponentially perpendicular to the edges. By using Fourier transforms the problem was formulated in terms of a singular integral equation. It was numerically solved by representing the unknown dislocation density by a truncated series of Chebyshev polynomials leading to a linear system of equations. The stress intensity factor (SIF) results were discussed with respect to the influences of different geometric parameters and the strength of the non-homogeneity. It was indicated that the SIF increases with the increase of the crack length and decreases with the increase of the rigidity of the material in the vicinity of crack. The SIF of narrow strip is very sensitive to the change of the non-homogeneity parameter and its variation is complicated. With the increase of the non-homogeneity parameter, the stress intensity factor may increase, decrease or keep constant, which is mainly determined by the strip width and the relative crack location. If the crack is located at the midline of the strip or if the strip is wide, the stress intensity factor is not sensitive to the material non-homogeneity parameter.     

Strain gradient elasticity theory for antiplane shear cracks. Part I: Oscillatory displacements

Solutions of crack problems for a strain gradient elastic theory whose crack opening displacements are monotonic rather than oscillatory in profile are examined in this (portion of a two-part) paper. A new analytical solution having a simple form is also obtained through the introduction of parabolic cylindrical coordinates and expansion of the imposed displacement along the crack surfaces in terms of orthogonal polynomials. Uniqueness of solution is also discussed, as well as finite-difference approximations of the governing equations for use in numerical solutions of boundary value problems.     

Balance laws and energy release rates for cracks in dipolar gradient elasticity

It is the purpose of this work to derive the balance laws (in the Günther–Knowles–Sternberg sense) pertaining to dipolar gradient elasticity. The theory of dipolar gradient (or grade 2) elasticity derives from considerations of microstructure in elastic continua [Mindlin, R.D., 1964. Microstructure in linear elasticity. Arch. Rational 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 gradient of both strain and rotation (additional terms). The balance laws are derived here through a more straightforward procedure than the one usually employed in classical elasticity (i.e. Noether’s theorem). Indeed, the pertinent balance laws are obtained through the action of the standard operators of vector calculus (grad, curl and div) on appropriate forms of the Hamiltonian of the system under consideration. These laws are directly related to the energy release rates in the processes of crack translation, rotation and self-similar expansion. Under certain conditions, they are identified with conservation laws and path-independent integrals are obtained.     

Microstructures with finite surface energy: the two-well problem

We study solutions of the two-well problem, i.e., maps which satisfy uSO(n)ASO(n)B a.c. in ⁿ , where A and B are n×n matrices with positive determinants. This problem arises in the study of microstructure in solid-solid phase transitions. Under the additional hypothesis that the set E where the gradient lies in SO(n) A has finite perimeter, we show that u is locally only a function of one variable and that the boundary of E consists of (subsets of) hyperplanes which extend to and which do not intersect in . This may not be the case if the assumption on E is dropped. We also discuss applications of this result to magnetostrictive materials.     

Distinguishing shear banding from shear thinning in flows with a shear stress gradient

Shear banding occurs in complex fluids that exhibit a non-monotonic constitutive instability, such as wormlike micelles, and potentially also in polymeric fluids with presumably monotonic constitutive behavior. However, velocity profiles for shear thinning fluids in geometries possessing a stress gradient, such as Taylor-Couette flow, could be misidentified as shear banding. To address this, we present a model-free experimental procedure to distinguish shear banding from strong shear thinning using high-resolution velocimetry. The approach is developed and validated using simulations using the d-Giesekus model and is based upon the behavior of the width of the apparent interface between the high and low shear rate regions. It is then tested using experimental data for model wormlike micellar solutions. The method allows shear banding to be distinguished from shear thinning in cases where this difference is otherwise indistinguishable. As a by-product, it also provides an estimate of the stress diffusivities for shear banding fluids.