Experimental and theoretical study of the buckling of narrow thin plates on an elastic foundation under compression

The paper describes the processes of elastic deformation of thin films under mechanical loading. The film is modeled longitudinally by a compressed plate arranged on an elastic foundation. A computer model of the buckling of the narrow thin plate with a delamination portion located on an elastic foundation is constructed. This paper also touches upon the supercritical behavior of the plate–substrate system. The experiments on the axial compression of a metal strip adhered to a rubber plate are performed, and 2 to 7 buckling modes are obtained therein. The critical loads and buckling modes obtained in the numerical calculations are compared with the experimental data. It is shown that there is the possibility of progressive delamination of the metal plate from the foundation if the critical load is exceeded. It is found that the use of the proposed approach, which, in contrast to other approaches, accounts for the elastic deformation of the substrate, causes the dependence between the critical bending stress and the stiffness of the foundation.     

Modelling of progressive interface failure under combined normal compression and shear stress

The present work is concerned with an analysis of progressive interface failure under normal compressive stress and varying shear stress using the cohesive crack model. The softening model is assumed and frictional linear stress at contact is accounted for. A monotonic loading in anti-plane shear of an elastic plate bonded to a rigid substrate is considered. An analytical solution is obtained by neglecting the effect of minor shear stress component in the plate. The elastic and plate interface compliances are included into the analysis. Three types of solutions are distinguished in the progressive delamination analysis, namely short, medium and long plate solutions. The analysis of quasi-static progressive delamination process clarifies the character of critical points and post-critical response of the plate. The analytical solution provides a reference benchmark for numerical algorithms of analysis of progressive interface delamination. The case of a rigid–softening interface was treated in a companion paper, where also cyclic loading was considered.     

Delamination mechanism maps for a strong elastic coating on an elastic–plastic substrate subjected to contact loading

Hard wear resistant coatings that are subjected to contact loading sometimes fail because the coating delaminates from the substrate. In this report, systematic finite element computations are used to model coating delamination under contact loading. The coating and substrate are idealized as elastic and elastic–plastic solids, respectively. The interface between coating and substrate is represented using a cohesive zone law, which can be characterized by its strength and fracture toughness. The system is loaded by an axisymmetric, frictionless spherical indenter. We observe two failure modes: shear cracks may nucleate just outside the contact area if the indentation depth or load exceeds a critical value; in addition, tensile cracks may nucleate at the center of the contact when the indenter is subsequently removed from the surface. Delamination mechanism maps are constructed which show the critical indentation depth and force required to initiate both shear and tensile cracks, as functions of relevant material properties. The fictitious viscosity technique for avoiding convergence problems in finite element simulations of crack nucleation and growth on cohesive interfaces allows us to explore a wider parametric space that a conventional cohesive model cannot handle. Numerical results have also been compared to analytical analyses of asymptotic limits using plate bending and membrane stretching theories, thus providing guidelines for interpreting the simulation results.     

Quasilinear Theory of Generation of Hydroelastic Waves in a Turbulent Boundary Layer over an Elastic Coating

Wave generation in an incompressible turbulent boundary layer over an elastic coating is investigated. A deviation of the coating surface is represented in the form of a superposition of random-phase harmonics. The response of the flow to wavy surface deflection is determined in the quasilinear approximation. Calculations are carried out on the basis of the local turbulent boundary layer model and a numerical solution of the Prandtl equation. The competition equations for fast waves excited on a low-loss coating are obtained. The results of solving these equations are compared with the known experimental data.     

Axisymmetric contact problem for an elastic half-space and a circular cover plate

The contact interaction problem for a thin circular rigid cover plate and an elastic half-space loaded at infinity by a tensile force directed in parallel to the boundary of the half-space is considered. It is assumed that the cover plate is not resistant to bending deformations. The problem can be reduced to an integral equation of the first kind whose kernel has a logarithmic singularity. The equation is solved approximately by the Multhopp-Kalandia method. The resulting approximate solution is compared with the previously obtained asymptotic solution.     

Modified equations of finite-size layered plates made of orthotropic material. Comparison of the results of numerical calculations with analytical solutions

This paper describes the modified bending equations of layered orthotropic plates in the first approximation. The approximation of the solution of the equation of the three-dimensional theory of elasticity by the Legendre polynomial segments is used to obtain differential equations of the elastic layer. For the approximation of equilibrium equations and boundary conditions of three-dimensional theory of elasticity, several approximations of each desired function (stresses and displacements) are used. The stresses at the internal points of the plate are determined from the defining equations for the orthotropic material, averaged with respect to the plate thickness. The construction of the bending equations of layered plates for each layer is carried out with the help of the elastic layer equations and the conjugation conditions on the boundaries between layers, which are conditions for the continuity of normal stresses and displacements. The numerical solution of the problem of bending of the rectangular layered plate obtained with the help of modified equations is compared with an analytical solution. It is determined that the maximum error in determining the stresses does not exceed 3 %.     

Asymptotic solutions for buckling delamination induced crack propagation in the thin film-compliant substrate system

In a thin film-substrate system in-plane compressive stress is commonly generated in the film due to thermal mismatch in operation or fabrication process. If the stress exceeds a critical value, part of the film may buckle out of plane along the defective interface. After buckling delamination, the interface crack at the ends may propagate. In the whole process, the compliance of the substrate compared with the film plays an important role. In this work, we study a circular film subject to compressive stress on an infinitely thick substrate. We study the effects of compliance of the substrate by modeling the system as a plate on an elastic foundation. The critical buckling condition is formulated. The asymptotic solutions of post-buckling deformation and the corresponding energy release rate of the interface crack are obtained with perturbation methods. The results show that the more compliant the substrate is, the easier for the film to buckle and easier for the interface crack to propagate after buckling.     

Buckling of a stiff film bound to a compliant substrate—Part III:: Herringbone solutions at large buckling parameter

We study the buckling of a compressed thin elastic film bonded to a compliant substrate. An asymptotic solution of the equations for a plate on an elastic foundation is obtained in the limit of large residual stress in the film. In this limit, the film's shape is given by a popular origami folding, the Miura-ori, and is composed of parallelograms connected by dihedral folds. This asymptotic solution corresponds to the herringbone patterns reported previously in experiments: the crests and valleys of the pattern define a set of parallel, sawtooth-like curves. The kink angle obtained when observing these crests and valleys from above are shown to be right angles under equi-biaxial loading, in agreement with the experiments. The absolute minimum of energy corresponds to a pattern with very slender parallelograms; in the experiments, the wavelength is instead selected by the history of applied load.     

正交各向异性平板开孔弹性波的衍射与动应力集中

采用各向异性平板弯曲波动方程及摄动方法，对正交各向异性无限平板开孔弹性波的衍射及动应力集中的问题进行了分析研究，得到了在稳态波作用下此种忆边界条件波动问题的渐近形式分析解。同时采用复变函数方法及保角映射技术，为求解正交各向异性无限平板开孔弹性波的衍射及动力集中问题提供了一种统一规范的分析方法。     

Thin elastic coatings and surface enthalpy

We obtain explicit formulas for the first few terms in the asymptotics of the solution of the problem on the deformation of a body with a thin elastic coating and propose several methods, based on these formulas, for modeling the problem with the use of the Wentzel boundary conditions or a regular perturbation of the boundary. We present estimates of the asymptotic remainders, including those in formulas obtained by modeling.     

各向异性平板开孔动应力集中问题的研究

采用各向异性平板弯曲波动理论及摄动方法，对正交各向异性平板开孔弯曲波的散射及动应力集中问题进行了分析研究，得到了此种平板稳态弯曲波动问题的渐近形式的分析解。同时采用保角映射技术，为求解正交各向异性平板开孔弹性波的散射及动应力集中问题提供了一种统一规范的方法。     

Progressive Frictional Delamination of an Infinite Elastic Film on a Rigid Substrate Due to In-Plane Point Loading

The paper presents a solution to a delamination problem of an infinite elastic film resting on a rigid substrate and loaded by a monotonically increasing in-plane point force. A?rigid-slip contact is assumed between the film and the substrate, leading to the development of two regions at the interface: a damaged zone with a relative slip between the materials, and a region where the interface remains intact. Both film natural and essential boundary conditions are zero on the boundary between these two interfacial zones with the shape of the boundary being a part of the solution. Problem??s self-similarity enables us to obtain an approximate distribution of interfacial traction within the delaminated zone and a shape of the zone itself. For film??s Poisson??s ratio ??=?1 the approximate solution becomes exact. It is argued that this can be treated as a special case of a rigid film sliding on a rigid substrate. The presented approach can be used to obtain approximate closed-form solutions to similar delamination problems.     

An analysis of the imperfection sensitivity of square elastic-plastic plates under axial compression

For a simply supported elastic-plastic square plate under axial compression the post-bifurcation behaviour and the sensitivity to initial imperfections are investigated. An exact asymptotic expansion is given for the initial post-bifurcation behaviour of a perfect plate compressed into the plastic range. The imperfection sensitivity is studied through an asymptotic analysis of the behaviour of the hypoelastic plate that results from neglecting the effect of elastic unloading. The results of the asymptotic analyses are compared with results of a numerical incremental solution by means of a combined finite element—Rayleigh Ritz method. The paper considers the effect of different in-plane boundary conditions and the effect of various degrees of strain hardening.     

Frictionless contact of two parallel congruent rigid cylindrical surfaces coated with thin elastic transversely isotropic incompressible layers: an analytic solution

The paper deals with a frictionless contact problem of two parallel rigid cylindrical surfaces, one encased in the other, coated with thin elastic transversely isotropic and incompressible layers. The coatings of the two circular cylinders may differ. A simplifying approximation for the displacement in the coating enables the problem to be formulated using stress and strain averaged through the coating thickness (for the method, see [Matthewson, M.J., 1981. Axi-symmetric contact on thin compliant coatings. J. Mech. Phys. Solids 29, 89–113]). Analytical results are obtained for the contact width and contact stress distribution. Given this contact stress distribution, an asymptotic analytical solution for the displacement in the coating is then obtained. The results are applied to the human ankle joint and generalized for articular cartilage with depth-dependent properties.     

Cracking of an orthotropic substrate reinforced by an orthotropic plate

A solution method is derived to determine the stress intensity factors for both an internal crack and an edge crack in an orthotropic substrate that is reinforced on its boundary by a finite-length orthotropic plate. The method utilizes the Green’s functions for a pair of dislocations and a concentrated force on the boundary while invoking the concept of superposition. Enforcing the traction-free boundary condition along the crack surfaces and the continuity of displacement gradients along the plate/substrate interface results in a coupled system of singular integral equations. An asymptotic analysis of the kernels in these equations for the region of the junction point between the plate corner and the substrate boundary reveals the strength of the singularity in the case of an edge crack. The numerical solution of the integral equations provides results for the stress intensity factors for both an internal crack and an edge crack perpendicular to the substrate boundary and aligned with one of the corners of the plate. The present results have been validated against previously published stress intensity factors for an internal crack and an edge crack in an isotropic substrate.     

Plate-Like Smart Structures: Reduced Models and Numerical Simulations

In this paper asymptotic models describing the mechanical and electric equilibrium state of two types of smart structures are presented and justified. The first structure consists of an anisotropic elastic thin plate with two surface bonded anisotropic piezoelectric patches and the second one is an anisotropic elastic sandwich thin plate with an inserted anisotropic piezoelectric patch. The two unknowns of the corresponding asymptotic models, the mechanical displacements of the structures and the electric potentials of the patches, are partially decoupled. The former are the solution of modified Kirchhoff-Love plate models, while the latter can be derived as explicit functions of the mechanical displacements. Moreover, different formulas for the electric potential arise as a consequence of diverse electric boundary conditions. We report numerical simulations with these asymptotic models.     

THE FIRST ORDER APPROXIMATION OF NON-KIRCHHOFF-LOVE THEORY FOR ELASTIC CIRCULAR PLATE WITH FIXED BOUNDARY UNDER UNIFORM SURFACE LOADING (Ⅰ)

Based on the approximation theory adopting non-kirchhoff-Love assumption for three dimensional elastic plates with arbitrary shapes^[1],[2], the author derives a functional of generalized variation for three dimensional elastic circular plates, thereby obtains a set of differential equations and the relate boundary conditions to establish a first order approximation theory for elastic circular plate with fixed boundary and under uniform loading on one of its surface. The analytical solution of this problem will present in another paper.     

THE FIRST ORDER APPROXIMATION OF NON-KIRCHHOFF-LOVE THEORYFOR ELASTIC CIRCULAR PLATE WITH FIXED BOUNDARY UNDERUNIFORM SURFACE LOADING (Ⅰ)

I.1ntroductionTheaxisymmetricproblemofthreedimensionalelasticcircularplatecanbetreatedasthreedimensionalaxisymmetricproblemofelasticity.Weconsideracircularplatewithauniformthicknessh,andsetupacircumferentialcoordinates(r,o)onitsmidd1esurfacewithabscissazp…     

Analytical solution for buckling of asymmetrically delaminated Reissner’s elastic columns including transverse shear

The exact analytical solution of buckling in delaminated columns is presented. In order to investigate analytically the influence of axial and shear strains on buckling loads the geometrically exact beam theory is employed with no simplification of the governing equations. The critical forces are then obtained by the linearized stability theory. In the paper, we limit the studies to linear elastic columns with a single delamination, but with arbitrary longitudinal and vertical asymmetry of delamination and arbitrary boundary conditions. The studies of quantitative and qualitative influence of transverse shear are shown in detail and extensive results for buckling loads with respect to delamination length, thickness and longitudinal position are presented.