On the boundary layer phenomenon in bending of thick annular sector plates using third-order shear deformation theory

In this article, the bending equations of thick annular sector plates are extracted based on the third-order shear deformation theory. Using a function, called boundary layer function, the coupled system of equations is converted into two decoupled equations. These equations are used to find a closed form solution for bending of thick transversely isotropic annular sector plates. It is shown that the solution of one of the decoupled equations has a boundary layer behavior like that of Mindlin plate theory. It is seen that the value of the boundary layer function for third order shear deformation theory is higher than that of the Mindlin theory. Thus, variations of stress components in the edge zone of the plate are more significant. Also, as in the Mindlin plate theory, there exist no boundary layer, a weak boundary layer, and a strong boundary layer effect for simply supported, clamped, and free edges, respectively.     

Role of diffusional coherency strain theory in the discontinuous precipitation in Mg-Al alloy

Discontinuous precipitation (DP) occurs in many alloy systems under certain conditions. It is called discontinuous precipitation because precipitation occurs on prior matrix grain boundaries followed by grain boundary movement. The DP nodule consists of alternate lamellae of the precipitate and the matrix respectively. The chemical driving force for DP is one of solute supersaturation. Although solute supersaturation is responsible for precipitation, it has to be coupled with another driving force to explain grain boundary migration. This coupling driving force has been identified to be diffusional coherency strain which has been verified to be active in diffusion induced grain boundary migration and liquid film migration. To test diffusional coherency strain theory for discontinuous precipitation Mg-7Al and Mg-7Al-1Pb alloys were studied. While the fraction transformed was high at 6% in Mg-7Al alloy, dit was significantly low at 2% in Mg-7Al-1Pb alloy. The velocity of DP nodules decreased by half in alloy with Pb as compared to the alloy without Pb. Theoretical calculations also predict that the misfit parameter δ_th decreases with the addition of Pb. These observations are an evidence to the fact that diffusional coherency strain is the most active driving force for the movement of the grain boundaries of the DP nodules during discontinuous precipitation in Mg-Al alloy.     

The one-temperature field thermo-mechanical theory of fluid-filled porous materials

Summary The one-temperature field thermo-mechanical theory of a fluid-filled porous material with a linearly elastic solid and a Newtonian viscous fluid is established. The author's previous work, phenomenological constitutive theory for fluid-filled porous materials with solid/fluid outer boundaries, is introduced into a conventional mixture theory. Kinetical and kinematical quantities defined for a sample with a solid/fluid outer boundary, including interaction terms, correspond to those in the mixture theory directly from their definitions. The approximations involved in one to one correspondence between the constitutive equations in the mixture theory and those in the phenomenological theory are made clear.     

On boundary conditions for a certain class of problems in mixture theory

An additional boundary condition is proposed for solid-fluid mixtures for the situation in which a mixture boundary is in a saturated state. This condition on the boundary is derived from a thermodynamic characterization of the state and takes the form of a relationship between the total stress tensor, the stretch tensor and the volume fraction of the solid. This additional condition is sufficient to make several boundary value problems involving mixtures, determinate.     

A uniqueness theorem for the general theory of heat conduction with finite wave speeds

Uniqueness of solutions in nonlinear heat conduction is considered. It is established that for a certain class of heat conductors with memory, at most one solution can exist with perscribed temperature on the boundary.     

On the boundary integral formulation of thermo-viscoelasticity theory

A formulation of the boundary integral equation method for generalized thermo-viscoelasticity is given. Fundamental solutions of the corresponding differential equations are obtained. An outline of the implementation of the boundary element method is discussed for the solution of the above boundary equations. Special emphasis is given to the representation of the primary fields, namely, temperature and displacement.     

Computational mechanics based on the theory of boundary eigensolutions

The theory of boundary eigensolutions for boundary value problems is applied to the development of computational mechanics formulations. The boundary element and finite element methods that result are consistent with the mathematical theory of boundary value problems. Although the approach is quite general, this paper focuses on potential problems. For these problems, both methods employ potential and boundary flux as primary variables. Convergence characteristics of the new flux‐oriented finite element method are also developed. By utilizing suitable boundary weight functions, the formulations are written exclusively in terms of bounded quantities, even for non‐smooth problems involving notches, cracks and mixed boundary conditions. The results of numerical experiments indicate that the algorithms perform in concert with the underlying theory and thus provide an attractive alternative to existing approaches. Beyond this, the approach developed here provides a new perspective from which to view computational mechanics, and can be used to obtain a better understanding of boundary element and finite element methods. Comparisons with closed‐form boundary eigensolutions are also presented in order to provide a means for assessing the numerical methods. Copyright © 2001 John Wiley & Sons, Ltd.     

A numerical technique in the hydrodynamic theory of foil bearings

The hydrodynamic theory of foil bearings is reviewed. The relationship between fluid pressure and film thickness is discussed. The compressibility of gas is included in the analysis. It is shown that the basic equation for determination of pressure distribution becomes a third order boundary value problem in terms of film thickness. A simple numerical scheme for solution of the nonlinear boundary value problem is developed and some examples are considered and discussed.     

Approximate solution of an internal problem in laminar boundary layer theory

Laminar flow of a liquid in a channel of arbitrary shape is examined with account for interaction of the boundary layer with the core. A simple approximate method is described for calculating the boundary layer in a channel with arbitrary generator.     

An advanced boundary element method for solving 2D and 3D static problems in Mindlin's strain‐gradient theory of elasticity

An advanced boundary element method (BEM) for solving two‐ (2D) and three‐dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin's Form‐II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative, is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems, respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems is illustrated with the solution of simple 2D problems. Copyright © 2010 John Wiley & Sons, Ltd.     

The simple boundary element method for multiple cracks in functionally graded media governed by potential theory: a three‐dimensional Galerkin approach

The simple boundary element method consists of recycling existing codes for homogeneous media to solve problems in non‐homogeneous media while maintaining a purely boundary‐only formulation. Within this scope, this paper presents a ‘simple’ Galerkin boundary element method for multiple cracks in problems governed by potential theory in functionally graded media. Steady‐state heat conduction is investigated for thermal conductivity varying either parabolically, exponentially, or trigonometrically in one or more co‐ordinates. A three‐dimensional implementation which merges the dual boundary integral equation technique with the Galerkin approach is presented. Special emphasis is given to the treatment of crack surfaces and boundary conditions. The test examples simulated with the present method are verified with finite element results using graded finite elements. The numerical examples demonstrate the accuracy and efficiency of the present method especially when multiple interacting cracks are involved. Copyright © 2005 John Wiley & Sons, Ltd.     

On complex potentials in elasticity theory

Summary In this paper a complex representation is found for the solution of the traction and displacement boundary value problems for the equations of bending of thin elastic plates established on the basis of Kirchhoff's kinematic assumption.With 1 Figure     

A higher-order shear deformation theory of laminated elastic shells

A higher-order shear deformation theory of elastic shells is developed for shells laminated of orthotropic layers. The theory is a modification of the Sanders' theory and accounts for parabolic distribution of the transverse shear strains through thickness of the shell and tangential stress-free boundary conditions on the boundary surfaces of the shell. The Navier-type exact solutions for bending and natural vibration are presented for cylindrical and spherical shells under simply supported boundary conditions.     

Further study on the refined theory of rectangle deep beams

Based on the refined theory for narrow rectangular deep beams, two different displacement boundary conditions of the fixed end of a cantilever beam are used to study the deformation of the beam. One is the conventional simplified displacement boundary condition, and the other is a new boundary condition determined by the least squares method. Three load cases are investigated, which are a transverse shear force at the free end of the beam, a uniformly distributed load at the top surface, and a linearly distributed load at the top surface, respectively. Solutions are given for both the refined theory and the Timoshenko beam theory and are compared with the known solutions from the elastic theory and results by the finite element method. It is shown that the solutions of the refined theory coincide with those of the elastic theory; the solutions from the Timoshenko theory by using the two different displacement boundary conditions are the same; the refined theory by using the new boundary condition provides better results than using the conventional boundary condition and also better than those of the Timoshenko beam theory.     

Product fragility and damage boundary theory

The theoretical basis for the widely accepted Damage Boundary Curve approach to product fragility and the applicability of its associated ASTM fragility assessment test procedure are first discussed. The basic assumption that the product can be modelled as a collection of linear spring/mass systems is shown to lead to specific characteristics of the product's damage boundary curve which are not borne out by the results of shock tests performed on some common consumer products. Discrepancies between theoretical predictions and experimental results are attributed to the inability of the model to account for progressive failure due to plastic deformation accumulated as a result of repeated shocks. An alternative damage boundary curve for a rigid/perfectly-plastic model is then developed and shown to be a better estimate of fragility for a broader class of products based on the results of shock tests. Modifications to the standard ASTM test procedure required to obtain the damage boundary curve for this type of product are suggested. Finally, the damage boundary theory for a generalized model which incorporates both elastic and plastic properties is developed. The feature of the damage boundary curve which are shown to survive regardless of the choice of model for the product are the critical velocity change and critical acceleration parameters. Those which do not are the shape of the damage region and the associated test procedures.     

破损边界理论局限性的讨论

传统的破损边界理论是用冲击实验机求取冲击加速度,然后绘制破损边界曲线,从而确定产品的破损条件.根据智能三维加速度测试仪测得的跌落曲线,提出了破损边界理论的局限性.     

On thermoelastic transients in a general theory of heat conduction with finite wave speeds

Summary The linear Chen-Gurtin-Pipkin theory of heat conduction in a deformable material is employed to study the one dimensional problem of a homogeneous thermoelastic half space subjected to thermal and mechanical disturbances at its boundary. A ray series approach is used to generate asymptotic wavefront expansions for the temperature, strain, and stress response of the medium to the disturbances. General properties of the propagation process are obtained simply and directly. We specialize the solution to the case for which this theory reduces to that of Lord and Shulman and demonstrate that our results for this example agree with asymptotic results obtained previously by other investigators.With 6 Figures     

Wrinkle-free solutions in the theory of curved circular membranes

Rotationally symmetric deformations of a curved circular elastic membrane under a vertical surface load are studied, with prescribed radial stresses or radial displacements at the edge. Considering Reissner's theory of thin shells of revolution suffering small strains but arbitrarily large deflections and rotations, the determination of the principal stresses in the membrane is shown to be equivalent to the solution of a single, second-order ODE, expressed in terms of a geodesic variable. Analytical techniques are applied in order to determine a limit curve of those boundary data, which subdivide the parameter range into complementary domains of existence and non-existence of tensile solutions. Finally, the more restricted subdomain of those boundary parameters is determined which admit wrinkle-free solutions, i.e. solutions governed by a nonnegative radial and circumferential stress component.     

A free boundary problem in the theory of lubrication

The theory of variational inequalities is applied to a free boundary problem arising in the hydrodynamic lubrication. We deal mainly with the properties of the pressure distribution in the clearance space and with the shape of the cavitated region.     

A new approach to the theory of grain boundaries

It is shown that the classical picture of a grain boundary in terms of a single array of lattice dislocations is incomplete. In addition, it is also necessary to incorporate into the boundary a second array of continuously distributed surface dislocations of infinitesimal strength and of opposite sign to those of the lattice dislocations, but with the same total magnitude. Furthermore, the lattice dislocations can also dissociate into a continuous distribution by the formation of cores. As the angular misorientation of the grain boundary increases, more and more of the surface dislocations combine with the lattice dislocations, in turn resulting in the formation of a larger and larger stress-free ledge, with a consequent overall reduction in the strength of the remaining lattice dislocation.