Problems of the Flamant–Boussinesq and Kelvin Type in Dipolar Gradient Elasticity

This work studies the response of bodies governed by dipolar gradient elasticity to concentrated loads. Two-dimensional configurations in the form of either a half-space (Flamant–Boussinesq type problem) or a full-space (Kelvin type problem) are treated and the concentrated loads are taken as line forces. Our main concern is to determine possible deviations from the predictions of plane-strain/plane-stress classical linear elastostatics when a more refined theory is employed to attack the problems. Of special importance is the behavior of the new solutions near to the point of application of the loads where pathological singularities and discontinuities exist in the classical solutions. The use of the theory of gradient elasticity is intended here to model material microstructure and incorporate size effects into stress analysis in a manner that the classical theory cannot afford. A simple but yet rigorous version of the generalized elasticity theories of Toupin (Arch. Ration. Mech. Anal. 11:385–414, 1962) and Mindlin (Arch. Ration. Mech. Anal. 16:51–78, 1964) is employed that involves an isotropic linear response and only one material constant (the so-called gradient coefficient) additional to the standard Lamé constants (Georgiadis et al., J. Elast. 74:17–45, 2004). This theory, which can be viewed as a first-step extension of the classical elasticity theory, assumes a strain-energy density function, which besides its dependence upon the standard strain terms, depends also on strain gradients. The solution method is based on integral transforms and is exact. The present results show departure from the ones of the classical elasticity solutions (Flamant–Boussinesq and Kelvin plane-strain solutions). Indeed, continuous and bounded displacements are predicted at the points of application of the loads. Such a behavior of the displacement fields is, of course, more natural than the singular behavior present in the classical solutions.      

A Generalized Fourier Approximation in Anti-plane Cosserat Elasticity

In the present paper we use the modification of Kupradze’s method of generalized Fourier series for the treatment of interior and exterior Dirichlet and Neumann boundary-value problems arising in a linear theory of anti-plane elasticity which includes the effects of material microstructure.     

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.     

New simple exact penalty function for constrained minimization

By adding one variable to the equality-or inequality-constrained minimization problems, a new simple penalty function is proposed. It is proved to be exact in the sense that under mild assumptions, the local minimizers of this penalty function are precisely the local minimizers of the original problem, when the penalty parameter is sufficiently large.     

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.     

Reduced projection augmented Lagrange bi-conjugate gradient method for contact and impact problems

Based on the numerical governing formulation and non-linear complementary conditions of contact and impact problems,a reduced projection augmented Lagrange bi- conjugate gradient method is proposed for contact and impact problems by translating non-linear complementary conditions into equivalent formulation of non-linear program- ming.For contact-impact problems,a larger time-step can be adopted arriving at numer- ical convergence compared with penalty method.By establishment of the impact-contact formulations which are equivalent with original non-linear complementary conditions,a reduced projection augmented Lagrange bi-conjugate gradient method is deduced to im- prove precision and efficiency of numerical solutions.A numerical example shows that the algorithm we suggested is valid and exact.     

A modulus gradient model for an axially loaded inhomogeneous elastic rod

A new gradient elasticity formulation is proposed for a one-dimensional linear elastic inhomogeneous rod. In the new formulation, similar to the differential relation between the local strain and the gradient enhanced strain in the classical models of gradient elasticity, a differential relation is proposed for the Young’s modulus. Analytical and finite element solutions of the proposed formulation are derived. Results of the proposed model are compared with a classical model of gradient elasticity for a model problem of carbon nanotube reinforced polymer composite.     

Fully explicit Agmon's condition for general states of a special incompressible elastic material

Agmon's condition arises as a necessary condition at the boundary for minimizers in compressible and incompressible elasticity. It is commonly formulated as a statement concerning the solution set of a family of ODEs with constant coefficients. As such, it is algebraic “in principle”.In both the compressible and incompressible cases, Agmon's condition may be recast in a more overtly algebraic form, namely the requirement that a certain family of algebraic Riccati equations (parametrized over the tangent plane) should possess positive solutions. In order to reduce Agmon's condition to a fully explicit set of inequalities involving the components of the incremental elasticity tensor, one must be able to solve the algebraic Riccati equation explicitly. Known situations where this can be done tend to involve highly symmetric states of isotropic materials. It is therefore noteworthy that Agmon's condition may be rendered explicit for any boundary-point of an arbitrarily deformed incompressible neo-Hookean body.     

哈密顿体系与弹性楔体问题

将哈密体系引入到级坐标下的弹性力学楔体问题,利用该体系辛空间的性质,将问题化为本征值和本征向量求解上,得到了完备的解空间,从而改变了弹性力学传统的拉格朗日体系以应力函数为特征的半逆法的讨论去解决该类问题的思路,给出了一条求解该类问题的直接法。     

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

A General Formulation and Solution Scheme for Fractional Optimal Control Problems

Accurate modeling of many dynamic systems leads to a set of Fractional Differential Equations (FDEs). This paper presents a general formulation and a solution scheme for a class of Fractional Optimal Control Problems (FOCPs) for those systems. The fractional derivative is described in the Riemann–Liouville sense. The performance index of a FOCP is considered as a function of both the state and the control variables, and the dynamic constraints are expressed by a set of FDEs. The Calculus of Variations, the Lagrange multiplier, and the formula for fractional integration by parts are used to obtain Euler–Lagrange equations for the FOCP. The formulation presented and the resulting equations are very similar to those that appear in the classical optimal control theory. Thus, the present formulation essentially extends the classical control theory to fractional dynamic system. The formulation is used to derive the control equations for a quadratic linear fractional control problem. An approach similar to a variational virtual work coupled with the Lagrange multiplier technique is presented to find the approximate numerical solution of the resulting equations. Numerical solutions for two fractional systems, a time-invariant and a time-varying, are presented to demonstrate the feasibility of the method. It is shown that (1) the solutions converge as the number of approximating terms increase, and (2) the solutions approach to classical solutions as the order of the fractional derivatives approach to 1. The formulation presented is simple and can be extended to other FOCPs. It is hoped that the simplicity of this formulation will initiate a new interest in the area of optimal control of fractional systems.     

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.     

A family of solutions describing plane strain cylindrical inflation in finite compressible elasticity

Within the context of finite, compressible, isotropic elasticity, a family of solutions describing plane strain cylindrical inflation of cylindrical shells is obtained for a class of materials that includes both the harmonic and Varga materials. Additionally it is shown that the class of materials chsen is the largest class of materials for which the family of solutions is possible.     

Anti-plane shear Lamb''s problem treated by gradient elasticity with surface energy

The consideration of higher-order gradient effects in a classical elastodynamic problem is explored in this paper. The problem is the anti-plane shear analogue of the well-known Lamb's problem. It involves the time-harmonic loading of a half-space by a single concentrated anti-plane shear line force applied on the half-space surface. The classical solution of this problem based on standard linear elasticity was first given by J.D. Achenbach and predicts a logarithmically unbounded displacement at the point of application of the load. The latter formulation involves a Helmholtz equation for the out-of-plane displacement subjected to a traction boundary condition. Here, the generalized continuum theory of gradient elasticity with surface energy leads to a fourth-order PDE under traction and double-traction boundary conditions. This theory assumes a form of the strain-energy density containing, in addition to the standard linear-elasticity terms, strain-gradient and surface-energy terms. The present solution, in some contrast with the classical one, predicts bounded displacements everywhere. This may have important implications for more general contact problems and the Boundary-Integral-Equation Method.     

On the Canonical Elastic Moduli of Linear Plane Anisotropic Elasticity

The linear, planar, anisotropic elastic equilibrium equations are transformed to canonical form, through linear transformations of both coordinates and unknown displacement functions, together with a linear combination of equations. Correspondingly, the six original material moduli are replaced by two canonical elastic moduli. Similar results have been reached by Olver in 1988. However, the method demonstrated in this paper is more concise and direct. As an example, the general solution to the canonical equations is obtained in the case of a pair of double roots.     

Bifurcation Instability in Linear Elasticity with the Constraint of Local Injectivity

There are problems in linear elasticity theory whose corresponding deformations, usually associated with singular stress fields, are open to question because they are not one-to-one and predict self-intersection. Recently, a theory has been advanced to handle such situations, which consists in minimizing the quadratic energy functional of linear elasticity subject to the constraint of local injectivity. In particular, the Jacobian of the deformation gradient is required to be not less than an arbitrarily small positive quantity, and, thus, the local orientation is preserved. Here, this theory is applied to the classical Lekhnitskii problem of an elastic aelotropic circular disk which is loaded on its boundary by a uniform radial pressure. Without the injectivity constraint, this classical linear problem has a unique solution. This example, with the injectivity constraint, already has been considered in previous works, but radial symmetry was assumed in order to reduce the problem from 2D to 1D. Here, making use of an interior penalty formulation, a numerical scheme is implemented that solves a full 2D problem. Remarkably, it is shown that there are values of the material moduli for which the minimal potential energy solution is no longer symmetric, producing a strong azimuthal shear and nominally a 180° rotation of an internal central core of the disk. Although the elastic strain energy is quadratic and convex, the strongly nonlinear character of the constraint allows for bifurcation instabilities. We gratefully acknowledge the partial support of the Minnesota Supercomputing Institute and the Italian “Ministero per l’Università e la Ricerca Scientifica” under the program PRIN 2005 “Affidabilità di elementi in vetro strutturale: indagini teoriche e sperimentali sulla risposta termo-meccanica del materiale e di strutture trasparenti di tipo innovativo”. R.F. gratefully acknowledges the Department of Civil and Environmental Engineering at the Politecnico di Bari, Italy, for their kind hospitality and support during his visit of 2006. We appreciate the helpful comments and suggestions of Paolo Podio-Guidugli on an earlier draft of this work.     

The classical pressure vessel problem for void damage materials

l.IntroductionPressurevesselbearsver}'greatpressureontheinterna1surface.Theremaybedisastrousconsequencesasaresultofthevoiddamage,andtheresearchinthepressurevesseldamageisveryactiveallthetime.IUTAMconvenedaspecia1izedconferenceonnonlineafelasticityanalysis…     

Marcus Integration Method for Shearable Plates

A method proposed by Marcus [5] to integrate the classical biharmonic equation of simply supported, unshearable plates with polygonal contour is extended to apply to shearable plates as well, provided the supporting device is of the ‘hard’ type.     

Plane-strain crack problems in microstructured solids governed by dipolar gradient elasticity

The present study aims at determining the elastic stress and displacement fields around the tips of a finite-length crack in a microstructured solid under remotely applied plane-strain loading (mode I and II cases). The material microstructure is modeled through the Toupin-Mindlin generalized continuum theory of dipolar gradient elasticity. According to this theory, the strain-energy density assumes the form of a positive-definite function of the strain tensor (as in classical elasticity) and the gradient of the strain tensor (additional term). A simple but yet rigorous version of the theory is employed here by considering an isotropic linear expression of the elastic strain-energy density that involves only three material constants (the two Lamé constants and the so-called gradient coefficient). First, a near-tip asymptotic solution is obtained by the Knein-Williams technique. Then, we attack the complete boundary value problem in an effort to obtain a full-field solution. Hypersingular integral equations with a cubic singularity are formulated with the aid of the Fourier transform. These equations are solved by analytical considerations on Hadamard finite-part integrals and a numerical treatment. The results show significant departure from the predictions of standard fracture mechanics. In view of these results, it seems that the classical theory of elasticity is inadequate to analyze crack problems in microstructured materials. Indeed, the present results indicate that the stress distribution ahead of the crack tip exhibits a local maximum that is bounded. Therefore, this maximum value may serve as a measure of the critical stress level at which further advancement of the crack may occur. Also, in the vicinity of the crack tip, the crack-face displacement closes more smoothly as compared to the standard result and the strain field is bounded. Finally, the J-integral (energy release rate) in gradient elasticity was evaluated. A decrease of its value is noticed in comparison with the classical theory. This shows that the gradient theory predicts a strengthening effect since a reduction of crack driving force takes place as the material microstructure becomes more pronounced.