Problems Of Linear Elasticity Theory Research Articles

Geosynthetic materials properties analysis suitable for controlling reflective cracks in asphalt concrete

Reflective cracks appearing in the new asphalt concrete layer laid during reconstruction over the existing concrete or asphalt concrete pavement is the main cause of premature highways failure. Cracks of this type appear over defects in the old pavement and grow through the new pavement under the influence of vehicle loads and fluctuations in the temperature field. One of the common ways to combat reflective cracks is the use of an intermediate reinforcing layer made of special geosynthetic material, redistributing part of the loads. The paper proposes a multilayer pavement structural model and a temperature field change model based on the design documentation of the M-11 «Neva» highway and data from the temperature monitoring station. Temperature stresses сalculation arising in reinforced asphalt concrete layers in the existing crack vicinity of the old pavement is performed by the finite element method using the approximate method of Hills and Brien, reducing the calculation to the sequence of solving problems of linear elasticity theory. The dependencies of asphalt concrete rigidity modulus on temperature and time of load action are calculated on the basis of penetrative index and the binder softening temperature, taking into account the inevitable aging of the binder during paving and pavement operation. It is shown that significant reduction of dangerous tensile stresses in asphalt concrete is possible at reinforcement with geogrids with maximum tensile strength not lower than 100 kN/m, made on the basis of high-modulus glass or carbon fibers, under condition of joint work of asphalt concrete and reinforcing layers in the absence of mutual slippage.

Solution to Force Problem of Linear Elasticity Theory for Quarter Space with Edgeuniform Forces

Solution to Force Problem of Linear Elasticity Theory for Quarter Space with Edgeuniform Forces

Integro-differential approach to solving problems of linear elasticity theory

In order to construct efficient algorithms for solving problems of linear elasticity theory, the approach is used based on the introduction of integral relations between components of the stress and strain tensors. In the framework of the model proposed, the integro-differential boundary value problem is reduced to a variational problem to which well-developed methods of numerical analysis are applicable [1, 2]. To demonstrate the potentialities of the proposed approach, we employ the numerical-analytical method of finding approximations for desired stress functions and displacement functions. 1. We consider an elastic body that occupies a certain region Ω with boundaries γ . We assume that the displacement and stresses are given at the parts γ u and γ σ of the boundary, respectively, ( γ u  γ σ = 0, γ u ← γ σ = γ ). The stress‐strain state of the body is described by the set of differential equations of the linear elasticity theory:

A differential operator approach to multidimensional optimal control

We use recent improvements in the parametrizations of controllable linear multidimensional systems to show how to transform the study of a linear quadratic optimal problem into that of a variational problem without constraints. We give formal conditions on the differential module defined by the linear control system to pass from the Pontryagin approach to a purely Euler–Lagrange variational problem. This formal approach uses the cost function in order to link the locally exact sequence formed by the controllable system and its parametrizations with the sequence formed by their formal adjoint operators. In the case of partial differential equations, this scheme is typical for any problem of linear elasticity theory and electromagnetism.

Measurement of cellular forces at focal adhesions using elastic micro-patterned substrates

Mechanical force is known to play an important role in the regulation of cellular behaviour, including adhesion, motility, differentiation and proliferation. For stationary, mechanically active cells like fibroblasts, adhesion to flat substrates occurs mainly at sites of focal adhesions, which are micron-sized protein aggregates at the plasma membrane, which on the cytoplasmic side connect to the actin cytoskeleton. In recent years, evidence has been growing that focal adhesions act as mechanosensors which convert mechanical force into biochemical signalling. We have investigated the relationship between force and aggregation at focal adhesions by a new method which combines elastic micro-patterned substrates (to record substrate deformation), fluorescence labelling of focal adhesion proteins (to monitor aggregation) and numerical solution of the inverse problem of linear elasticity theory (to calculate forces at focal adhesions). We have found that force correlates linearly with lateral size of aggregation, with a stress constant of 5.5 nN/μm 2. This finding indicates that mechanosensing involves regulation of aggregation.

Calculation of Forces at Focal Adhesions from Elastic Substrate Data: The Effect of Localized Force and the Need for Regularization

Forces exerted by stationary cells have been investigated on the level of single focal adhesions by combining elastic substrates, fluorescence labeling of focal adhesions, and the assumption of localized force when solving the inverse problem of linear elasticity theory. Data simulation confirms that the inverse problem is ill-posed in the presence of noise and shows that in general a regularization scheme is needed to arrive at a reliable force estimate. Spatial and force resolution are restricted by the smoothing action of the elastic kernel, depend on the details of the force and displacement patterns, and are estimated by data simulation. Corrections arising from the spatial distribution of force and from finite substrate size are treated in the framework of a force multipolar expansion. Our method is computationally cheap and could be used to study mechanical activity of cells in real time.

Accuracy of the Richardson Extrapolation Method in Eigenvalue Problems

The accuracy of the Richardson extrapolation method is investigated for the eigenvalue problem of linear elasticity theory with zero Dirichlet boundary conditions in a three-dimensional beam. O( h^4 )-estimates of eigenvalue and eigenfunction vector errors are obtained with the constraint that the eigenfunction belongs to the space {\bf W}_2^4 ( \Omega ).

On the improvement of the strength of fibrous material using mechanical or thermal action

We consider an elastic isotropic material containing a fibrous inclusion or a void subject to loads. We study the possibility of increasing the strength of the material using “control action.” As examples of such actions we consider an additional external load, application of pressure in the cavities (voids), and variation of the temperature of uniform heating of the medium with the inclusions. Studies are carried out for two-dimensional problems of linear elasticity theory. From the class of all possible actions we single out a subclass leading to states of equal strength along the phase interface. We study the efficiency of these types of action and show that in certain cases they lead to a manifold increase in the strength of the material. We give the results of numerical studies.

The problem of the justification of variational formulations of the boundary element method

An investigation is made of the convergence of boundary element approximations of variational solutions of the boundary-value problems of linear elasticity theory, using double- and simple-layer potentials (DLP, SLP) /1, 2/. Auxiliary propositions are proved, concerning the basis property of the approximating sequence of potentials. The results may also be used to justify variational formulations of the boundary element method (BEM) is applied to the solution of other second-order elliptic boundary-value problems.

On certain formulations of the boundary-elements method

Variational formulations are proposed for the boundary-element method (BEM) to solve linear problems of elasticity theory with a known Green's tensor. Unlike existing BEM formulations utilizing the method of weighted residuals /1/ or boundary integral equations /2/, the formulations to be considered below use a variational formulation of the problems for boundary functionals /3/ in a set of allowable functions in the form of double-layer potentials whose density is given in the form of BEM basis functions. Also examined is a BEM variational formulation on the basis of minimization problems for Trefftz generalized functionals of the fundamental boundary value problems of linear elasticity theory /4/. A basis for the formulations is presented. Utilization of the proposed BEM formulations is effective in solving boundary-contact problems; consequently, a numerical realization is examined with an example of a unilateral variational problem (of the generalized Signorini problem type /5/) corresponding to the classical contact problem of inserting a stamp into an elastic half-plane with approximation of the possible contact boundary by isoparametric boundary elements.

On an effective algorithm for minimizing generalized Trefftz functionals of linear elasticity theory

The problem of minimizing the generalized Trefftz functionals of three-dimensional elasticity theory results in a minimax problem for the Lagrangian. An algorithm is proposed for searching for the saddle point in coordinate functions not subjected to any constraints in the domain and on the boundary (this is the efficiency of the algorithm). The convergence of the approximate solution is investigated. The Trefftz variational method /1/ is convenient for solving boundary value problems of mathematical physics in that the dimensionality of the problem being solved is reduced because of its reduction to the solution of equations defined on the domain boundary. At the same time, when constructing the solution using the Ritz process, say, the coordinate functions should be selected so that they satisfy the differential equation of the boundary value problem in the domain, which is a serious constraint. An approach is proposed below that uses Lagrange multipliers to reduce this constraint when minimizing the generalized Trefftz functionals of the fundamental boundary value problems of linear elasticity theory. The results obtained can also be used to minimize the classical Trefftz functionals of the boundary value problems of mathematical physics /1/. Generalized Trefftz functionals were constructed in /2, 3/ for the fundamental problems of linear elasticity theory with continuous and discontinuous elasticity coefficients. The functionals are minimized in solutions (ordinary or generalized) for the linear equilibrium equation for an elastic medium in displacements. Assuming the existence of a coordinate system of functions satisfying the equilibrium equation (in the generalized sense) in /4/, the Ritz process was investigated for solving problems to minimize the generalized Trefftz functionals in an example of the second boundary value problem of three-dimensional elasticity theory. The practical construction of the above-mentioned coordinate system is a fairly complex problem. At the same time, the differential equation of the boundary value problem in whose solutions the minimum of the functionals is sought, can be considered as a linear constraint in the problem of minimizing the Trefftz functionals. Then such a minimization problem with linear constraints can be reduced to the minimax problem of a certain Lagrangian (by using reciprocity theory).

Bending, extension, and torsion of naturally twisted rods

Saint-Venant /1/ established that the spatial problem of linear elasticity theory of the deformation of straight rods with a load-free side surface allows of practically complete investigation: the extension problem is solved exactly (if the boundary layer is ignored), and the bending and torsion problems reduce to Neumann problems for the Laplace equation in the region of the rod cross-section (see /2, 3/). It is shown below that an analogous situation holds for a naturally twisted rod: the spatial problem is successfully reduced to a Neumann-type problem for a certain system of second-order elliptic equations in the cross-section. It is essential that this can be done for an arbitrary value of the rod twist. For zero twist the problem in the section reduces to the Saint-Venant problem. In the case of centrally-symmetric sections, the problem decomposes into two independent problems, on bending and on extensiontorsion. Variational principles and certain bilateral estimates of the extension and torsion stiffness are constructed for the latter, and the case of oblong sections is investigated. The extension-torsion problem for naturally twisted rods was examined earlier in /4/. The difference from this research is discussed in Sect.4.

A new boundary element technique for solving plane problems of linear elasticity: improved theory and an application to fracture mechanics

An improved boundary element method for solving plane problems of linear elasticity theory is described. The method is based on the Muskhelishvili complex variable representation for the displacement and stress fields. The paper shows how to take account of symmetry about the x and/or y axes. The potential accuracy of the method is illustrated by its application to the calculation of stress intensity factors associated with cracks in both a square and a circular plate. The crack problem is solved using a Gauss-Chebyshev representation of a singular integral equation by a set of linear algebraic equations. The integral equation involves an analytic function which takes account of the presence of the external boundary. This function is determined directly using the boundary element method. Numerical results are believed to be more accurate than the existing published values which are quoted to four significant figures.

A new boundary element technique for solving plane problems of linear elasticity: 1. Theory

A new boundary elements technique for solving plane problems of linear elasticity theory is described. The method is based upon the Muskhelishvili complex variable representation of the displacement and stress fields involving two independent complex functions. These functions are represented by complex Cauchy integrals where the path of integration is taken around the external boundary of the solid. Two complex density functions appearing in the integrands of the Cauchy integrals are represented by spline functions and these are determined by the application of appropriate boundary conditions. The theory presented is suitable only for bounded simply-connected regions.

On an approach to the investigation of the signorini problem using the idea of duality

Reciprocal variational problems for the boundary functionals of linear elasticity theory, defined on convex closed sets of functions, are formulated in an example of the Signorini problem. Certain unilateral boundary value problems of linear elasticity theory result in variational problems for such functionals. The reciprocity relationship is proved, and error estimates are presented of the approximate solutions of unilateral boundary value problems which can be used, for instance, in solving contact problems of linear elasticity theory.

On the convergence of the method of homogeneous linear approximations in problems of the theory of plasticity of inhomogeneous bodies: PMM vol. 36, n≗3, 1972, pp. 519–527

A formulation of the mathematical problem of equilibrium is given for a deformable solid whose elastic and plastic characteristics are functions of the coordinates. Extraction of the nonlinear terms and terms taking account of the deviation of the elastic moduli from some constant values results in the method of linear successive approximations, which is analogous to the method of elastic solutions. A problem of linear elasticity theory for a homogeneous solid subjected to fictitious mass and surface forces governed by the preceding approximation is solved in each approximation. Using the apparatus of functional analysis, the convergence of the method in application to the first and second boundary value problems is proved. In many cases the solid and the medium are inhomogeneous relative to the mechanical properties. The inhomogeneity originates either during shaping of the material (solidification from the melt, tempering, ageing, etc.) or is the result of the presence of inhomogeneous temperature fields, racioactive exposure, and other physical and chemical fields. The elastic and plastic characteristics of the material are hence functions of the coordinates, which are introduced either explicitly (for example, for naturally inhomogeneous media), or by using field functions (the exposure dose [1], the degree of austenite transformation in tempering of steel [2], etc.).