Showing papers in "Zamm-zeitschrift Fur Angewandte Mathematik Und Mechanik in 2015"

Band gaps in the relaxed linear micromorphic continuum

Abstract: In this note we show that the relaxed linear micromorphic model recently proposed by the authors can be suitably used to describe the presence of band-gaps in metamaterials with microstructures in which strong contrasts of the mechanical properties are present (e.g. phononic crystals and lattice structures). This relaxed micromorphic model only has 6 constitutive parameters instead of 18 parameters needed in Mindlin- and Eringen-type classical micromorphic models. We show that the onset of band-gaps is related to a unique constitutive parameter, the Cosserat couple modulus μc which starts to account for band-gaps when reaching a suitable threshold value. The limited number of parameters of our model, as well as the specific effect of some of them on wave propagation can be seen as an important step towards indirect measurement campaigns.

On the use of the first order shear deformation plate theory for the analysis of three‐layer plates with thin soft core layer

Abstract: Three-layer laminates with thin soft core layer can be found in many engineering applications. Examples include laminated glasses and photovoltaic panels. For such structures high contrast in the mechanical properties of faces and core requires the use of advanced methods to determine effective material properties of the laminate. In this paper we address the application of the first order shear deformation plate theory to the analysis of laminates with thin and soft core layer. In particular, transverse shear stiffness parameters for three-layered plates with different symmetric configurations are analyzed. For classical sandwiches with thick core layer the result coincides with the Reissner's formula. For the case of thin and compliant core layer the new expression for the effective shear stiffness is derived.

Convergence and stability in balanced norms of finite element methods on Shishkin meshes for reaction-diffusion problems

Abstract: Error estimates of finite element methods for reaction-diffusion problems are often realized in the related energy norm. In the singularly perturbed case, however, this norm is not adequate. A different scaling of the H1 seminorm leads to a balanced norm which reflects the layer behavior correctly. We prove error estimates in balanced norms and investigate also stability questions. Especially, we propose a new C0 interior penalty method with improved stability properties in comparison with the Galerkin FEM.

Efficient time integration for discontinuous Galerkin approximations of linear wave equations

Abstract: We consider the combination of discontinuous Galerkin discretizations in space with various time integration methods for linear acoustic, elastic, and electro-magnetic wave equations. For the discontinuous Galerkin method we derive explicit formulas for the full upwind flux for heterogeneous materials by solving the Riemann problems for the corresponding first-order systems. In a framework of bounded semigroups we prove convergence of the spatial discretization. For the time integration we discuss advantages and disadvantages of explicit and implicit Runge–Kutta methods compared to polynomial and rational Krylov subspace methods for the approximation of the matrix exponential function. Finally, the efficiency of the different time integrators is illustrated by several examples in 2D and 3D for electro-magnetic and elastic waves.

Predictor/corrector co-simulation approaches for solver coupling with algebraic constraints

Abstract: In the paper at hand, co-simulation approaches are analyzed for coupling two solvers. The solvers are assumed to be coupled by algebraic constraint equations. We discuss 2 different coupling methods. Both methods are semi-implicit, i.e. they are based on a predictor/corrector approach. Method 1 makes use of the well-known Baumgarte-stabilization technique. Method 2 is based on a weighted multiplier approach. For both methods, we investigate formulations on index-3, index-2 and index-1 level and analyze the convergence, the numerical stability and the numerical error. The presented approaches require Jacobian matrices. Since only partial derivatives with respect to the coupling variables are needed, calculation of the Jacobian matrices may very easily be calculated numerically and in parallel with the predictor step. For that reason, the presented methods can in a straightforward manner be applied to couple commercial simulation tools without full solver access. The only requirement on the subsystem solvers is that the macro-time step can be repeated once in order to accomplish the corrector step. Within the paper, we introduce methods for coupling mechanical systems. The presented approaches can, however, also be applied to couple arbitrary non-mechanical dynamical systems.

Regularity criteria for the incompressible Hall‐MHD system

Abstract: We study the convergence of the statistical solution of the wave equation. More precisely we consider the Cauchy problem with random initial data. At first we suppose that initial data are homogeneous random field with mixing (in the sense of M.Rosenblatt). We prove weak convergence of the distributions of the solutions to the Gaussian measure. Also we consider the Cauchy problem for the wave equation with random coefficient. An asymptotical behaviour of solutions is discussed, as t → ∞. The answer depends on the initial date. In the cases of summable and periodic initial data the asymptotics is precisely described. The limiting distribution of the solution is non-Gaussian.

Modeling of passive and active external confinement of RC columns with elastic material

Abstract: One of the most attractive applications of composite materials is their use as confining devices for concrete columns, which may result in remarkable increases of strength and ductility. The current investigation focuses on the modeling of reinforced concrete columns passively or actively confined by composites, under axial load. Furthermore, the research highlights the effectiveness and modeling of ultra high extension capacity fiber ropes implemented as external confining reinforcement so as to upgrade ductility and strength of concrete columns. It concerns columns of square cross-section with plain or steel reinforced concrete. The novel confining technique uses composite rope made of polypropylene fibers as passive or pretensioned reinforcement while it presents linear elastic behavior up to failure and is applied by hand. It may enable confined columns to dissipate enormous amounts of earthquake induced energy through concrete deformation. The proposed constitutive model is compared against available experimental results involving circular or square columns with passive or active FRP confinement.

Defects in Nonlinear Elastic Crystals:Differential Geometry, Finite Kinematics, and Second-Order Analytical Solutions

Abstract: : A differential geometric description of crystals with continuous distributions of lattice defects and undergoing potentially large deformations is presented. This description is specialized to describe discrete defects, i.e., singular defect distributions. Three isolated defects are considered in detail: the screw dislocation, the wedge disclination, and the point defect. New analytical solutions are obtained for elastic fields of these defects in isotropic solids of finite extent, whereby terms up to second order in strain, involving elastic constants up to third order, are retained in the stress components. The strain measure used in the nonlinear elastic potential a symmetric function, expressed in material coordinates, of the inverse deformation gradient differs from that used in previous solutions for crystal defects, and is thought to provide a more realistic depiction of mechanics of large deformation than previous theory involving third-order Lagrangian elastic constants and the Green strain tensor. For the screw dislocation and wedge disclination, effects of core pressure and/or possible contraction along the defect line are considered, and radial displacement contributions arise that are absent in the linear elastic solution, affecting dilatation. Stress components are shown to differ from those of linear elastic solutions near defect cores. Volume change from point defects is strongly affected by elastic nonlinearity.

The effect of noise on the response of a vertical cantilever beam energy harvester

Abstract: An energy harvesting concept has been proposed comprising a piezoelectric patch on a vertical cantilever beam with a tip mass. The cantilever beam is excited in the transverse direction at its base. This device is highly nonlinear with two potential wells for large tip masses, when the beam is buckled. For the pre-buckled case considered here, the stiffness is low and hence the displacement response is large, leading to multiple solutions to harmonic excitation that are exploited in the harvesting device. To maximise the energy harvested in systems with multiple solutions the higher amplitude response should be preferred. This paper investigates the amplitude of random noise excitation where the harvester is unable to sustain the high amplitude solution, and at some point will jump to the low amplitude solution. The investigation is performed on a validated model of the harvester and the effect is demonstrated experimentally. c

Buckling and postbuckling of single‐walled carbon nanotubes based on a nonlocal Timoshenko beam model

Abstract: In this research, the axial buckling and postbuckling configurations of single-walled carbon nanotubes (SWCNTs) under different types of end conditions are investigated based on an efficient numerical approach. The effects of transverse shear deformation and rotary inertia are taken into account using the Timoshenko beam theory. The nonlinear governing equations and associated boundary conditions are derived by the virtual displacements principle and then discretized via the generalized differential quadrature method. The small scale effect is incorporated into the model through Eringen's nonlocal elasticity. To obtain the critical buckling loads, the set of linear discretized equations are solved as an eigenvalue problem. Also, to address the postbuckling problem, the pseudo arc-length continuation method is applied to the set of nonlinear parameterized equations. The effects of nonlocal parameter, boundary conditions, aspect ratio and buckling mode on the critical buckling load and postbuckling behavior are studied. Moreover, a comparison is made between the results of Timoshenko beam model and those of its Euler-Bernoulli counterpart for various magnitudes of nonlocal parameter.

Delta wave formation and vacuum state in vanishing pressure limit for system of conservation laws to relativistic fluid dynamics

Abstract: The Riemann solutions to the system of conservation laws to relativistic fluid dynamics with a scaled pressure are shown. Using the method of vanishing pressure limit, it is rigorously proved that the Riemann solution involving two shocks and possibly one contact discontinuity to the system of relativistic fluid dynamics tends to a delta wave solution of pressureless relativistic Euler equations. The intermediate density between the two shocks tends to a weighted δ-measure which forms the delta wave. While the Riemann solution containing two rarefaction waves and perhaps one contact discontinuity tends to a double-contact-discontinuity solution to pressureless relativistic Euler equations. The intermediate state between the two contact discontinuities is a vacuum state. c

Micro-poromechanics model of fluid-saturated chemically active fibrous media

Abstract: We have developed a micromechanics based model for chemically active saturated fibrous media that incorporates fiber network microstructure, chemical potential driven fluid flow, and micro-poromechanics. The stress-strain relationship of the dry fibrous media is first obtained by considering the fiber behavior. The constitutive relationships applicable to saturated media are then derived in the poromechanics framework using Hill's volume averaging. The advantage of this approach is that the resultant continuum model accounts for the discrete nature of the individual fibers while retaining a form suitable for porous materials. As a result, the model is able to predict the influence of micro-scale phenomena, such as the fiber pre-strain caused by osmotic effects and evolution of fiber network structure with loading, on the overall behavior and in particular, on the poromechanics parameters. Additionally, the model can describe fluid-flow related rate-dependent behavior under confined and unconfined conditions and varying chemical environments. The significance of the approach is demonstrated by simulating unconfined drained monotonic uniaxial compression under different surrounding fluid bath molarity, and fluid-flow related creep and relaxation at different loading-levels and different surrounding fluid bath molarity. The model predictions conform to the experimental observations for saturated soft fibrous materials. The method can potentially be extended to other porous materials such as bone, clays, foams and concrete.

Fundamental thermo-electro-elastic solutions for 1D hexagonal QC

Abstract: Static general solutions, fundamental solutions, piezoelectric effect, 1D hexagonal quasicrystal,infinite/half-infinite spaces.This paper is concerned with the fundamental solutions, in the framework of thermo-electro-elasticity, for an infinite/half-infinite space of 1D hexagonal quasicrystals (QCs). To this end, three-dimensional static general solutions, in terms of 5quasi-harmonic functions, are derived with the help of rigorous operator theory and generalized Almansi’s theorem. Foran infinite/half-infinite space subjected to an external thermal load, corresponding problem is formulated by boundaryvalue problems. Appropriate potential functions are set by a trail-and-error technique. Green functions for the problemsin question are obtained explicitly in the closed forms. The present fundamental solutions can be employed to construct3D analysis for crack, indentation and dislocation problems. Furthermore, these solutions also serve as benchmarks forvarious numerical simulations.

Thermo-mechanical couplings in elastomers – experiments and modelling

Abstract: Elastomers take an important role in many industrial applications. In the automotive industries for example, elastomers are used in various bearings, where they inhibit vibration propagation and thereby significantly enhance driving performance and comfort. Several models have been developed to simulate the material's mechanical response to various stresses and strains a component may undergo during its lifetime. So far, these models are commonly developed under isothermal conditions. In this contribution it is shown that the mechanical properties significantly depend on the temperature and that the material heats up under large dynamic deformations. Therefore, an elastomer's behaviour is not described sufficiently with an isothermal approach, a detailed thermo-viscoelastic modelling is required. In this contribution, the behaviour of elastomers is experimentally investigated in order to gain informations about the time- and temperature-dependent mechanical properties. We perform different tests on a natural rubber to emphasize the temperature dependence of the equilibrium stress-strain relation as well as the time-dependent behaviour in relaxation tests. As it is necessary for parameterising a material model, thermal tests are carried out to determine the specific heat capacity, the thermal expansion coefficient and the thermal conductivity. In a second step, we introduce a material model which is able to represent the temperature-dependent viscoelastic material behaviour including large deformations, as well as the self-heating of the material. The model's mechanical parameters are identified on tension tests. In first FE calculations, the applicability of the introduced model is proven by depicting the experimental results of several tension tests at different temperatures. Besides these validations, the self-heating under dynamic load, depending on the loads amplitude and frequency as well as the surrounding temperature is calculated.

Discretization and numerical realization of contact problems for elastic-perfectly plastic bodies. PART II – numerical realization, limit analysis

Abstract: The paper deals with a static case of discretized contact problems for bodies made of materials obeying Hencky's law of perfect plasticity. The main interest is focused on the analysis of the formulation in terms of displacements. This covers the study of: i) a structure of the solution set in the case when the problem has more than one solution ii) the dependence of the solution set on the loading parameter ζ. The latter is used to give a rigorous justification of the limit load approach based on work of external forces as a function of ζ. A model example illustrates the efficiency of the method.

Vertical stress distribution in isotropic half-spaces due to surface vertical loadings acting over polygonal domains

Abstract: By integrating the classical Boussinesq expression we derive analytically the vertical stress distribution induced by pressures distributed with arbitrary laws, up to the third order, over polygonal domains. Thus, one can evaluate in closed form either the vertical stress produced by shell elements, modelling raft foundations by finite elements, acting over a Winkler soil or those induced by a linear pressure distribution simulating axial force and biaxial bending moments over a pad foundation. To this end we include charts and tables, both for rectangular and circular domains, which allow the designer to evaluate the vertical stresses induced by linear load distributions by hand calculations. The effectiveness of the proposed approach is witnessed by the comparison between the analytical results obtained with the proposed formulas and the numerical ones of a FEM discretization of the soil associated with the loading distribution induced by a foundation modeled by plate elements resting on a Winkler soil.

Monitoring and compartmentalized structures

Abstract: The damage acting on a structure can lead to disproportionate consequences, i.e., the global collapse. This extreme situation has to be avoided and, thus, structural monitoring is requested in those structures where human losses are possible and large economic consequences are expected. Static measurement devices are the most economic instrumental set-ups able to highlight the presence of progressive damages. However, this monitoring system suffers from the structural behaviour under the external loads. In many situations, alternate load paths shown the non-effectiveness of the measurement system since the instrumentation are installed on elements not relevant for the response under the given loads. The same problems occur when the structure is compartmentalized, i.e. the structural responses of the single parts dependent on the loads acting almost only on the same single component. In order to measure the degree of compartmentalization, different novel metrics based on stiffness matrix properties are proposed and their effectiveness discussed. The new idea of this paper is to connect compartmentalization of structures with a sort of distance of the stiffness matrix from the set of diagonal matrices. Few examples are illustrated

Some recent results on MDGKN-systems

Abstract: The linearized equations of motion of finite dimensional autonomous mechanical systems are normally written as a second order system and are of the MDGKN type, where the different n × n matrices have certain characteristic properties. These matrix properties have consequences for the underlying eigenvalue problem. Engineers have developed a good intuitive understanding of such systems, particularly for systems without gyroscopic terms (G-matrix) and circulatory terms (N-matrix, which may lead to self-excited vibrations). A number of important engineering problems in the linearized form are described by this type of equations. It has been known for a long time, that damping (D-matrix) in such systems may either stabilize or destabilize the system depending on the structure of the matrices. Here we present some new results (using a variety of methods of proof) on the influence of the damping terms, which are quite general. Starting from a number of conjectures, they were jointly developed by the authors during recent months.

Fractional model of thermoelasticity for a half‐space overlaid by a thick layer

Abstract: In this work, we apply the fractional order theory of thermoelasticity to a 1D problem for a half-space overlaid by a thick layer of a different material. The upper surface of the layer is taken to be traction free and is subjected to a constant thermal shock. There are no body forces or heat sources affecting the medium. Laplace transform techniques are used to eliminate the time variable t. The solution in the transformed domain is obtained by using a direct approach. The inverse Laplace transforms are obtained by using a numerical method based on Fourier expansion techniques. The predictions of the fractional order theory are discussed and compared with those for the generalized theory of thermoelasticity. We also study the effect of the fractional derivative parameters of the two media on the behavior of the solution. Numerical results are computed and represented graphically for the temperature, displacement and stress distributions.

Variational formulation of rate- and state-dependent friction problems

Abstract: We propose a variational formulation of rate- and state-dependent models for the dynamic sliding of a linearly elastic block on a rigid surface in terms of two coupled variational inequalities. Classical Dieterich–Ruina models are covered as special cases. We show existence and uniqueness of solutions for the two spatial subproblems arising from time discretisation. Existence of solutions to the coupled spatial problems is established for Dieterich's state equation through a fixed point argument. We conclude with some numerical experiments that suggest mesh independent convergence of the underlying fixed point iteration, and illustrate quasiperiodic occurrence of stick/slip events.

Effect of scale parameter on the deflection of a nonlocal beam and application to energy release rate of a crack

Abstract: This article studies the influence of the nonlocal scale parameter on the deflection of a nonlocal nanobeam and crack growth Using the Timoshenko hypothesis, a single governing equation is derived and its exact solution can be determined through appropriate end-support conditions Numerical calculations are carried out for a cantilever microtubule in solution at a given flow speed The effects of nonlocal scale parameter on the deflection are discussed Based on the obtained solutions, the double cantilever beam model is utilized to determine energy release rate near a crack tip for an edge crack and a central crack, respectively It is found that the scale parameter plays different roles in determining stress intensity factors and energy release rates, depending on crack constraints When neglecting shear deformation, the results for nonlocal Euler-Bernoulli beams can be directly obtained

Analysis and optimal control for free melt flow boundaries in laser cutting with distributed radiation

Abstract: This work introduces a mathematical model for laser cutting taking account of spatially distributed laser radiation. The model involves two coupled nonlinear partial differential equations describing the interacting dynamical behaviors of the free boundaries of the melt during the process. The model will be investigated by linear stability analysis to study the occurence of ripple formations at the cutting surface. We define a measurement for the roughness of the cutting surface and introduce an optimal control problem for minimizing the roughness with respect to the laser beam intensity along the free melt surface. Necessary optimality conditions will be deduced. Finally, a numerical solution will be presented and discussed by means of the necessary conditions. physical considerations.

Compact closed form solution of the incremental plane states in a pre‐stressed elastic composite with an elliptical hole

Abstract: We consider an unbounded, homogeneous, pre-stressed orthotropic elastic composite containing an elliptical hole and subject to uniform remote tensile and uniform remote tangential shear loads (Mode I and respectively Mode II of fracture). Using the conformal mapping technique and the representation of the stress and displacement fields by complex potentials, we determine the solution of the problem in a compact and elementary form. When the smaller semiaxis of the elliptical hole tends to zero, i.e. the hole becomes a crack, the potentials obtained reduce to a form similar to that of the crack problem, obtained by solving the corresponding Riemann-Hilbert problem.

On some isoperimetric inequalities involving eigenvalues of symmetric free membranes

Abstract: This paper deals with the eigenvalues of the Neumann Laplacian on simply-connected Lipschitz planar domains with some rotational symmetry. Our aim is to continue the investigations from Enache and Philippin [7] and derive new isoperimetric estimates for eigenvalues of higher order.

SH-wave propagation in a continuously inhomogeneous half-plane with free-surface relief by BIEM

Abstract: The anti-plane strain elastodynamic problem for a continuously inhomogeneous half-plane with free-surface relief subjected to time-harmonic SH-wave is studied. The computational tool is a boundary integral equation method (BIEM) based on analytically derived Green’s function for a quadratically inhomogeneous in depth half-plane. To show the versatility of the proposed BIE method, it is considered SH-wave propagation in an inhomogeneous half-plane with free surface relief presented by a semi-circle, semi-elliptic and triangle canyon. The inhomogeneous in depth half-plane is modeled in two different ways: (i) the material properties vary continuously in depth and BIEM based on Green’s function is used; (ii) the material properties vary in a discrete way and the half-plane is presented by a set of homogeneous layers with horizontal interfaces and a hybrid technique based on wave number integration method (WNIM) and BIEM is applied. The equivalence of these two different models is shown. The simulations reveal a marked dependence of the wave field on the material inhomogeneity and the potential of the BIEM based on the Green’s function for half-plane to produce highly accurate results by using strongly reduced discretization mesh in comparison with the conventional boundary element technique using fundamental solution for the full plane. c

Closed-form solution of elastic circular membrane with initial stress under uniformly-distributed loads: Extended Hencky solution

Abstract: The well-known Hencky solution is only applicable to the problem of deformation of the elastic circular membrane without initial stress under transverse uniformly-distributed loads. The problem considered here is a more general case: an initial tensile or compressive stress has been present in the initially flat circular membrane before the membrane is subjected to the transverse loads. The closed-form solution of the considered problem was presented and all the expressions obtained here for displacements, strains and stresses have the same form as those in the well-known Hencky solution. The initial stress plays an important role in the determination of numerical value of the integral constant controlling membrane equation. The solution obtained here can be regressed into the well-known Hencky solution when the initial stress is equal to zero, and it is therefore called extended Hencky solution.

Strategies for stiffness analysis of laminates with microdamage: combining average stress and crack face displacement based methods

Abstract: Many approximate analytical models have been developed to calculate stress state between intralaminar cracks with the aim to predict the degradation of certain elastic property (most often axial modulus or shear modulus) of cross-ply laminate. Often they are plane stress solutions and laminate constants like Poisson's ratios cannot be considered. On the other hand the so called GLOB-LOC approach, presented in WWFE III, allows calculation of any thermo-elastic property of a general symmetric laminate with an arbitrary number of cracks in each layer provided that two local parameters – average and normalized crack opening displacement (COD) and crack face sliding displacement (CSD) are known. In this paper relationships are derived expressing these two parameters (COD and CSD) with average value of transverse stress and in-plane shear stress perturbation between cracks. Expressions are exact and independent on the approximations in the stress model. As examples, average perturbation functions for two shear lag models and Hashin's variational model are used to calculate damaged laminate properties that would not be available in original formulation: Poisson's ratio and thermal expansion coefficients. Predictions are compared with test data for GF/EP laminates and with more accurate predictions based on FEM calculations.

Transient response of a thermoelastic half‐space to mechanical and thermal buried sources

Abstract: With the aid of a complete set of two scalar potential functions, the transient responses of an isotropic thermoelastic half-space subjected to time dependent tractions and heat flux applied to a finite patch at an arbitrary depth below a free surface are derived. Using the displacements- and temperature-potential function relationships, the coupled equations of motion and energy equation are uncoupled, resulting in two (6th and 2nd order) partial differential equations in the cylindrical coordinate system, which are solved with the aid of Fourier series expansion and joint Hankel-Laplace integral transforms. The solutions are also investigated in details for tractions varying with time in terms of a Heaviside step function and heat flux as a Dirac delta function, which may be used as a kernel in any integral based method for more complicated thermoelastodynamic initial-boundary value problems. Due to the complexity of the integrands involved in the general case, the integrals cannot be resolved analytically and thus an appropriate numerical algorithm is used for the inversion of the Laplace and Hankel integral transforms. To demonstrate the pattern of deformations as well as the distribution of change of temperature at the free surface of the half-space, numerical evaluations for these functions are presented for an isotropic material.

Maxwell's stress tensor and the forces in magnetic liquids

Abstract: Maxwell's stress tensor is well known from electromagnetic theory. But correct application of it to practical problems is by no means general knowledge even among experts. In this article we present a survey of the electromagnetic stress tensor and of the electromagnetic forces in strongly polarizable materials. We relate the observed ponderomotoric phenomena to the stress tensor and we present a number of applications in modern devices and processes using ferromagnetic colloids, so-called ferrofluids. We emphazise the correct applications of the stress tensor to these examples in contrast to common popular usage of a so-called “electromagnetic pressure”. We predict some new effects, e.g. density variations and pressure measurements in ferrofluids. Our work is based on two preceding articles by the present authors [4, 33] wherein the electromagnetic stress tensor is deduced from conservation laws together with Maxwell's equations and with thermodynamic relations.

The transverse crack problem for elastic bodies stiffened by thin elastic coating

Abstract: For two types of large-scale bodies, a half-plane and a strip, each of which is weakened by a straight transverse crack, the static problem of elasticity theory is considered. The upper boundary of each body is reinforced by a thin flexible coating. The coating is modeled by special boundary conditions on the upper faces of considered bodies. Three different cases of boundary conditions on the lower face of the strip were studied. By application of generalized integral transforms to the equilibrium equations in displacements the problems were reduced to the solutions of singular integral equations of first kind with Cauchy kernel to the respect of derivative of the crack opening function. In all considered cases the integral equations consists of a singular term, corresponding to crack behavior in an infinite plate, and a regular term, reflecting the influence of various geometric and physical parameters. For various sets of model parameters the solutions of the integral equations were built by small parameter and collocation methods; their structure was analyzed. The values of stress intensity factor in the vicinity of the tips of the crack were obtained and analyzed for different coating materials and geometric parameters of the crack. From the analysis of the numerical results of the problem, it can be concluded that thin flexible coatings significantly reduce stress intensity at a crack tip and therewith significantly increase a reliability of considered elastic bodies.