Some optimization problems for elastic bodies

Two optimization problems for finite deformations of an elastic body are formulated. In the first problem, the motion that gives minimum value (maximum value) to a certain functional is chosen from a single family of controllable motions. The necessary conditions for optimality are stated for one specific problem of this type. In the second problem, the minimizing (maximizing) motion is chosen from the class of all possible motions connecting two configurations of the body. The necessary conditions for the optimality of singular solutions are obtained for one specific example of this type.This work was supported by the Pokrajinska Zajednica za Naucni Rad, Novi Sad, Yugoslavia.     

Saint-venant principle for paraboloidal elastic bodies

We study the asymptotic behavior of solutions to the linear problem of elasticity in a domain Ω with paraboloidal exit at infinity. Properties of solutions and the condition of the existence of solutions depend on a parameter γ∈[0,1] characterizing the velocity of extending the paraboloid (a cylinder and a cone correspond to the cases γ=0¤γ=1 respectively). Asymptotic formulas are deduced for displacement fields generating forces and moments “applied at infinity”. The Saint-Venant principle is verified for “oblong” bodies such as paraboloids, cylinders, and narrow cones. The following question turns out to be a key one: What rigid displacements belong to the energy space obtained by completion of $C_0^\infty (\bar \Omega )^3 $ by the energy norm? The dimension d_γ of the lineal R_γ of rigid energy displacements is computed (in this case, d₀=6, d₁=0, and the function γ?d_γ has jumps at the points γ=1/4, 1/2, 3/4). We also clarify the reasons why it is necessary to distinguish the notions “energy solution” and “solution with finite energy”. We also discuss the phenomenon of a boundary layer that appears near the endpoints of spindle-like rods and is described by energy solutions in paraboloids. As is shown, in order to have the well-posed formulation of the boundary conditions in one-dimensional models of such rods, it is necessary to use the weakened Saint-Venant principle, i.e., replace R₀ with R_γ: for γ>1/4. If we apply the strong principle, we arrive at an overdetermined limit one-dimensional problem. Bibliography: 71 titles.     

Junction problem for elastic and rigid inclusions in elastic bodies

An equilibrium problem for an elastic body is considered. It is assumed that the body has a thin elastic inclusion and a thin rigid inclusion. We analyze a junction problem assuming that the inclusions have a joint point. Different equivalent problem formulations are discussed, and existence of solutions is proved. A set of junction conditions is found. We investigate a convergence to infinity and to zero of a rigidity parameter of the elastic inclusion. A delamination of the elastic inclusion is also investigated. In this case, inequality‐type boundary conditions are imposed at the crack faces to prevent a mutual penetration between crack faces. Copyright © 2015 John Wiley & Sons, Ltd.     

Mixed interface problems for anisotropic elastic bodies

Three-dimensional mathematical problems of the elasticity theory of anisotropic piecewise homogeneous bodies are discussed. A mixed type boundary contact problem is considered where, on one part of the interface, rigid contact conditions are give (jumps of the displacement and the stress vectors are known), while on the remaining part screen or crack type boundary conditions are imposed. The investigation is carried out by means of the potential method and the theory of pseudodifferential equations on manifolds with boundary.     

Some least work principles for elastic bodies

Zusammenfassung Es werden Prinzipien hergeleitet, welche für gewisse Klassen von Deformationen eines elastischen Körpers die Verformungen mit kleinster Formänderungsenergie bestimmen. Diese Prinzipien werden auf Zylinder mit Endlasten und auf den belasteten Halbraum angewandt.     

A compact finite element method for elastic bodies

A nonconforming finite element method is described for treating linear equilibrium problems, and a convergence proof showing second order accuracy is given. The close relationship to a related compact finite difference scheme due to Phillips and Rose [1] is examined. A Condensation technique is shown to preserve the compactness property and suggests an approach to a certain type of homogenization.     

Celestial mechanics of elastic bodies

We construct time independent configurations of two gravitating elastic bodies. These configurations either correspond to the two bodies moving in a circular orbit around their center of mass or strictly static configurations.     

General relations for waves in one-dimensional elastic systems

Common features inherent in waves propagating in one-dimensional elastic systems are pointed out. Local laws of energy and wave momentum transfer when the Lagrangian of an elastic system depends on the generalized coordinates and their derivatives up to the second order inclusive are presented. It is shown that in a reference system moving with the phase velocity, the ratio of the energy flux density to the wave momentum flux density is equal to the phase velocity. It is established that for systems, the behaviour of which is described by linear equations or by nonlinear equations in the unknown function, the ratio of the mean values of the energy flux density to the wave momentum density is equal to the product of the phase and group velocities of the waves.     

Non-classical interface problems for piecewise homogeneous anisotropic elastic bodies

Two non-classical model interface problems for piecewise homogeneous anisotropic bodies are studied. In both problems on the contact surface jumps of the normal components of displacement and stress vectors are given. In addition, in the first problem (Problem H) the tangent components of the displacement vectors are given from both sides of the contact surface, while in the second one (Problem G) the tangent components of the stress vectors are prescribed on the same surface. The existence and uniqueness theorems are proved by means of the boundary integral equation method, and representations of solutions by single layer potentials are established. In the investigation the general approach of regularization of the first kind of integral equations is worked out for the case of two-dimensional closed smooth manifolds. An equivalent global regularizer operator is constructed explicitly in the form of a singular integro-differential operator.     

Mixed impedance transmission problems for vibrating layered elastic bodies

Direct scattering problems for partially coated piecewise homogenous and inhomogeneous layered obstacles in linear elasticity lead to mixed impedance transmission problems for the steady‐state elastic oscillation equations. For a piecewise homogenous isotropic composite body, we employ the potential method and reduce the mixed impedance transmission problem to an equivalent system of boundary pseudodifferential equations. We give a detailed analysis of the corresponding pseudodifferential operators, which live on the interface between the layers and on a proper submanifold of the boundary of the composite elastic body, and establish uniqueness and existence results for the original mixed impedance transmission problem for arbitrary values of the oscillation frequency parameter; this is crucial in the study of inverse elastic scattering problems for partially coated layered obstacles. We also investigate regularity properties of solutions near the collision curves, where the different boundary conditions collide, and establish almost best Hölder smoothness results. Further, we analyze the asymptotic behavior of the stress vector near the collision curve and derive explicit formulas for the stress singularity exponents. The case of Lipschitz surfaces is briefly treated separately. In the case of a composite body containing homogeneous or inhomogeneous finite anisotropic inclusions, we develop an alternative hybrid method based on the so‐called nonlocal approach and reduce the mixed transmission problem to an equivalent functional‐variational equation with a sesquilinear form that ‘lives’ on a bounded part of the layered composite body and its boundary. We show that this sesquilinear form is coercive and that the corresponding variational equation is uniquely solvable. Copyright © 2015 John Wiley & Sons, Ltd.     

Rotating elastic bodies in Einstein gravity

We prove that, given a stress‐free, axially symmetric elastic body, there exists, for sufficiently small values of the gravitational constant and of the angular frequency, a unique stationary, axisymmetric solution to the Einstein equations coupled to the equations of relativistic elasticity with the body performing rigid rotations around the symmetry axis at the given angular frequency. © 2009 Wiley Periodicals, Inc.     

On crack propagation shapes in elastic bodies

We consider a two dimensional elastic isotropic body with a curvilinear crack. The formula for the derivative of the energy functional with respect to the crack length is discussed. It is proved that this derivative is independent of the crack path provided that we consider quite smooth crack propagation shapes. An estimate for the derivative of the energy functional being uniform with respect to the crack propagation shape is derived.     

On crack propagation shapes in elastic bodies

We consider a two dimensional elastic isotropic body with a curvilinear crack. The formula for the derivative of the energy functional with respect to the crack length is discussed. It is proved that this derivative is independent of the crack path provided that we consider quite smooth crack propagation shapes. An estimate for the derivative of the energy functional being uniform with respect to the crack propagation shape is derived.     

Uncertainty quantification for linear elastic bodies with two fluctuating input parameters

In this work, we present a numerical method for solving partial differential equations (PDEs) with stochastic coefficients for a linear elastic body. To this end, a stochastic finite element method is applied. We distinguish two different cases for an isotropic material with two fluctuating input parameters in order to analyse the optimal choice of input parameters. Using the GALERKIN projection, the final stochastic equation system is reduced to a system of deterministic PDEs. Subsequently, the solution is determined iteratively. Finally, a numerical example for a plate with a ring hole is presented. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Torsion of prismatic elastic bodies containing screw dislocations

The problem of the stressed state of a prismatic anisotropic rod containing screw dislocations, the axes of which are parallel to the rod axis, is considered. Such defects may arise during the growth of filamentary crystals (metal “whiskers”), and may also exist in multiply connected cylindrical structures. The torsion of an anisotropic elastic bar with a multiply connected cross-section is investigated initially, assuming that the stresses and strains are single-valued but dispensing with the requirement that the warping function should be single-valued. The boundary-value problem is formulated in terms of the Prandtl stress function, which, unlike the warping function, is single-valued in a multiply connected region. A variational formulation of the boundary-value problem for the stress function is given. From the variational principle obtained a torsion boundary-value problem is formulated when there are lumped or continuously distributed dislocations. A modification of the membrane analogy for the torsion problem is proposed which takes into account the presence of dislocations. General theorems of the theory of the torsion of a rod containing dislocations are formulated. An effective formula is derived for the angle of torsion of a bar due to a specified dislocation distribution. Problems on dislocations in a thin-walled rod and a rectangular anisotropic bar are solved.     

Internal and boundary calculations of thin elastic bodies

A method for the separate construction of the main stress-strain state (the internal calculation) and the boundary corrections (the boundary calculations) are discussed in the case of a linear static problem in the theory of shells and plates. It is assumed that the internal calculation is carried out using an iterative process based on the Kirchhoff-Love theory. The boundary calculation involves the construction of antiplane and plane boundary layers, that is, in the initial approximation they reduce to the solution of antiplane and plane problems in the theory of elasticity.

Investigation of the asymptotic behaviour of the boundary corrections shows that near a weakly clamped edge only the correction from the antiplane boundary layer is important and that near a fairly rigidly clamped edge only the correction from the plane boundary layer is important.

The advisability of the use of the shear theory of the bending of plates for investigating boundary elastic phenomena is discussed from the point of view of the results obtained. It is shown that, close to the free edge, its use is justified and is adequate for the method described in the paper both with regard to the numerical results and with regard to the nature of the mathematical apparatus. As a method for investigating boundary elastic phenomena, shear theories lose their meaning close to a fairly rigidly clamped edge since they only enable one to construct the minor part of the correction asymptotically.