Contact of a plate with an elastic body: an interface model

The authors study the problem of contact between a plate and an elastic body, using the interface model as introduced by Oden and Martins. The problem is formulated as a variational inequality. The authors use the theory of P-coercive variational inequalities, developed by the authors and K. Schmitt, to examine the existence and uniqueness of the solutions.Work completed with financial support from the National Program of Basic Research in the Natural Sciences of VietNam.     

A new Kirchhoff plate model based on a modified couple stress theory

In this paper a new Kirchhoff plate model is developed for the static analysis of isotropic micro-plates with arbitrary shape based on a modified couple stress theory containing only one material length scale parameter which can capture the size effect. The proposed model is capable of handling plates with complex geometries and boundary conditions. From a detailed variational procedure the governing equilibrium equation of the micro-plate and the most general boundary conditions are derived, in terms of the deflection, using the principle of minimum potential energy. The resulting boundary value problem is of the fourth order (instead of existing gradient theories which is of the sixth order) and it is solved using the Method of Fundamental Solutions (MFS) which is a boundary-type meshless method. Several plates of various shapes, aspect and Poisson’s ratios are analyzed to illustrate the applicability of the developed micro-plate model and to reveal the differences between the current model and the classical plate model. Moreover, useful conclusions are drawn from the micron-scale response of this new Kirchhoff plate model.     

Reflection and transmission of elastic waves at an interface with consideration of couple stress and thermal wave effects

The reflection and transmission of the thermo-elastic coupled waves at an interface of two different couple stress elastic solids are studied in this paper. Based on the Green-Lindsay theory, the governing equations and the constitutive equations are derived. Different from the classic elastic solid, the interface conditions include the surface couple, the rotation angle, the heat flux and the temperature change. The interface conditions are used to obtain the linear algebraic equations set from which the amplitude ratios of reflection and transmission waves to the incident wave can be determined. Then, the normal energy flux conservation is used to validate the numerical results. At last, the influences of two characteristic relaxation times and the five kinds of thermally and micromechanically interface conditions are discussed based on the numerical results. It is found that the thermal wave effects affect only the longitudinal wave while the couple stress effects affect only the transverse waves. The thermo-elastic coupling makes the longitudinal wave and the thermal wave not only dispersive but also attenuated.     

Transient response of an elastic plate to loads with finite characteristic dimensions

The present paper uses a recently developed technique to obtain solutions for an elastic plate acted upon by loads with finite characteristic dimensions. Exact two dimensional transform solutions are derived for a plate suddenly loaded on one of its bounding surfaces. The transform solutions for a general load are rearranged into infinite sums representing multiply-reflected shear and dilatational waves. The interpretation of the infinite sums is verified by ray tracing. The final expressions are used to deduce the Green's function for the velocity. Numerical results are presented for a step function input.     

Some effects of the biaxial tension of an elastic plate with a crack

Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 6, pp. 142–147, November–December, 1991.     

Interaction of an elastic wave with a plate

Within the framework of the dynamic theory of elasticity a numerical solution is set up for an axisymmetric problem which arises in connection with the problem of measuring stresses on the boundary between a solid medium and a rigid wall. In the cylindrical r, z coordinates the medium fills the cylinder z > 0, r < R, the case R being possible when the medium occupies the half-space z > 0. The elastic medium borders on a rigid wall which in the plane z = 0 has a deformable part in the form of a circular elastic plate clamped along the edges. A plane longitudinal wave in the form of a semiinfinite step falls from infinity. The interaction of this wave with the plate is investigated, with the main attention given to the study of the effect of the problem parameters on the deflection of the plate subjected to the wave.Deceased.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 74–85, March–April, 1972.Concluding, the author thanks the participants of the Seminar on Dynamics of Solids, at the Institute of Problems of Mechanics, for the discussion of the paper.     

Flutter of an elastic plate in a channel flow: Confinement and finite-size effects

When a cantilevered plate lies in an axial flow, it is known to exhibit self-sustained oscillations once a critical flow velocity is reached. This flutter instability has been investigated theoretically, numerically and experimentally by different authors, showing that the critical velocity is always underestimated by two-dimensional models. However, it is generally admitted that, if the plate is confined in the spanwise direction by walls, three-dimensionality of the flow is reduced and the two-dimensional models can apply. The aim of this article is to quantify this phenomenon by analysing the effect of the clearance between the plate and the side walls on the flutter instability. To do so, the pressure distribution around an infinite-length plate is first solved in the Fourier space, which allows to develop an analytical model for the pressure jump. This model is then used in real space to compute instability thresholds as a function of the channel clearance, the plate aspect ratio and mass ratio. Our main result shows that, as the value of the clearance is reduced, the convergence towards the two-dimensional limit is so slow that this limit is unattainable experimentally.     

Effects of characteristic material lengths on mode III crack propagation in couple stress elastic–plastic materials

The asymptotic fields near the tip of a crack steadily propagating in a ductile material under Mode III loading conditions are investigated by adopting an incremental version of the indeterminate theory of couple stress plasticity displaying linear and isotropic strain hardening. The adopted constitutive model is able to account for the microstructure of the material by incorporating two distinct material characteristic lengths. It can also capture the strong size effects arising at small scales, which results from the underlying microstructures. According to the asymptotic crack tip fields for a stationary crack provided by the indeterminate theory of couple stress elasticity, the effects of microstructure mainly consist in a switch in the sign of tractions and displacement and in a substantial increase in the singularity of tractions ahead of the crack-tip, with respect to the classical solution of LEFM and EPFM. The increase in the stress singularity also occurs for small values of the strain hardening coefficient and is essentially due to the skew-symmetric stress field, since the symmetric stress field turns out to be non-singular. Moreover, the obtained results show that the ratio η introduced by Koiter has a limited effect on the strength of the stress singularity. However, it displays a strong influence on the angular distribution of the asymptotic crack tip fields.     

A size-dependent composite laminated skew plate model based on a new modified couple stress theory

In this study, a size-dependent composite laminated skew Mindlin plate model is proposed based on a new modified couple stress theory. This plate model can be viewed as a simplified couple stress theory in engineering mechanics. Governing equations and related boundary conditions are derived based on the principle of minimum potential energy. The Rayleigh–Ritz method is employed to obtain the numerical solutions of the center deflections of simply supported plates with different ply orientations. Numerical results show that the normalized center deflections obtained by the proposed model are always smaller than those obtained by the classical one, i.e. the present model can capture the scale effects of microstructures. Moreover, a phenomenon reveals that the ply orientation would make a significant influence on the magnitude of scale effects of composite laminated plates at micro scale. Additionally, the present model of thick skew plate can be degenerated to the model of Kirchhoff plate based on the modified couple stress theory by adopting the assumptions in Bernoulli–Euler beam and material isotropy.     

Relection of elastic waves at the elastically supported boundary of a couple stress elastic half-space

The relection elastic waves at the elastically supported boundary of a couple stress elastic half-space are studied in this paper. Different from the classical elastic solid, there are three kinds of elastic waves in the couple stress elastic solid, and two of them are dispersive. The boundary conditions of a couple stress elastic half-space include the couple stress vector and the rotation vector which disappear in the classical elastic solids. These boundary conditions are used to obtain a linear algebraic equation set, from which the amplitude ratios of relection waves to the incident wave can be determined. Then, the relection coeficients in terms of energy lux ratios are calculated numerically, and the normal energy lux conservation is used to validate the numerical results. Based on these numerical results,the inluences of the boundary parameters, which relect the mechanical behavior of elastic support, on the relection energy partition are discussed. Both the incident longitudinal wave(the P wave) and incident transverse wave(the SV wave) are considered.     

Nonlinear couple stress theory of elastic dielectrics with applications to dynamic deformations

A general nonlinear couple stress theory of elastic dielectrics in the presence of an electric field is formulated and applied to the problem of nonlinear oscillations of the surface of a spherical cavity in an infinite medium. It is observed that the uniform pressure increases while the azimuthal stress decreases considerably from their corresponding values in the elastic dielectric theory without couples stresses.     

Abnormal long wave dispersion phenomena in a slightly compressible elastic plate with non-classical boundary conditions

A two parameter asymptotic analysis is employed to investigate some unusual long wave dispersion phenomena in respect of symmetric motion in a nearly incompressible elastic plate. The plate is not subject to the usual classical traction free boundary conditions, but rather has its faces fixed, precluding any displacement on the boundary. The abnormal long wave behaviour results in the derivation of non-local approximations for symmetric motion, giving frequency as a function of wave number. Motivated by these approximations, the asymptotic forms of displacement components established and long wave asymptotic integration is carried out.     

Surface roughness effects in a short porous journal bearing with a couple stress fluid

In this paper, a theoretical study of the effect of surface roughness in hydrodynamic lubrication of a porous journal bearing with couplestress fluid as lubricant is made. The modified Reynolds equations accounting for the couple stresses and randomized surface roughness structure are mathematically derived. The Christensen stochastic theory of hydrodynamic lubrication of rough surfaces is used to study the effects of surface roughness on the static characteristics of a short porous journal bearing with couplestress fluid as lubricant. Further, it is assumed that, the roughness asperity heights are small compared to the film thickness. It is observed that, the effects of surface roughness on the bearing characteristics are more pronounced for couplestress fluids as compared to the Newtonian fluids.     

负压爆炸载荷和数据采集间隔对弹性开孔板动应力集中的影响

基于ANSYS 7.0/LS-DYNA程序,对3 m3 m四边简支,厚度0.025 m,中心开有0.3 m0.3 m方孔的弹性板,在正压和负压三角爆炸载荷作用下的应力响应进行了分析,利用能量密度时间分布函数（TDFED）确定了动应力集中因子,并给出其计算步骤。计算结果表明计算总时间和数据采集时间间隔对动应力集中因子影响较大,而负压荷载影响较小。     

Antiplane shear crack in a prestressed elastic medium based on the couple stress theory

A prestressed elastic medium containing a mode-Ⅲ crack is studied by means of the couple stress theory(CST). Based on the CST under initial stresses, a governing differential equation along with a mixed boundary value problem is established. The singularities of the couple stress and force stress near the crack tips are analyzed through the asymptotic crack-tip fields resulting from the characteristic expansion method. To determine their intensity, a hypersingular integral equation is derived an...     

Motion of a thin plate in an elastic medium

A solution to the problem of motion of a thin rigid plate in an elastic medium is obtained using the Smirnov-Sobolev method for solving a two-dimensional wave equation.     

Numerical-analytical method to determine the stress state of an elastic rectangular plate

A numerical-analytical method for solving the plane problem of elasticity is proposed. Systems of nonorthogonal functions are used. The method involves the minimization of a quadratic form that is equal to the integral of the sum of squared residuals of the solution and given forces. An explicit expression for stresses is derived. Bessel's inequality and the convergence of the solution are proved. The accuracy of the boundary conditions is estimated. The stress and strain distribution in the plate depending on the maximum magnitude of distributed forces and the size of their localization area is analyzed numerically. New quantitative and qualitative features of the stress distribution in the plate are established __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 90–98, January 2008.     

An analysis on three-dimensional effects near the tip of a crack in an elastic plate

An infinite plate containing a finite through crack under tensile loading is analysed by Fourier transform based on the Kane-Mindlin kinematic assumptions for the quasi-three-dimensional deformation of plates in extension. The asymptotic expressions of stress and displacement fields near the crack tip, the variation of the stress intensity factor with the plate-thickness and the three-dimensional deformation zone near the crack tip are investigated. The results of the analysis show that, (a) the crack-tip stress and displacement fields accounting for the plate-thickness effects are different from the plane stress solutions and this is true even for extremely small parameter (=1–vh/6 a). In a very small region near the crack tip, plane strain solutions prevail; (b) the ratio of the stress intensity factor K_I to the corresponding plane stress one K_I, K_I/K _I ^o , approaches 1/(1–v²) as tends to zero; (c) plane stress solutions can give satisfactory results for points a distance from the crack tip greater than about three-fourths of the plate-thickness; (d) the linear elastic result for the zone of three-dimensional effects is approximately valid for an elasto-plastic material with linear strain-hardening when the plastic tangential mudulus E_t is not very small.The Project Supported by National Natural Science Foundation of China.