Two-dimensional nonstationary problem of elastic diffusion for an isotropic one-component layer

A locally equilibrium model of mechanodiffusion which comprises a coupled system of motion equations for an elastic body and a mass transfer equation is used to solve the two-dimensional nonstationary problem of elastic diffusion for an isotropic one-component layer. The solution is constructed using Fourier series, Laplace time transforms, and Fourier transforms for the spatial coordinate. The Laplace transform originals are found analytically, and the Fourier transforms are inverted by quadrature formulas.     

Elastic nonlinear stability analysis of thin rectangular plates through a semi-analytical approach

A semi-analytical approach to the elastic nonlinear stability analysis of rectangular plates is developed. Arbitrary boundary conditions and general out-of-plane and in-plane loads are considered. The geometrically nonlinear formulation for the elastic rectangular plate is derived using the thin plate theory with the nonlinear von Kármán strains and the variational multi-term extended Kantorovich method. Emphasis is placed on the effect of destabilizing loads and on the derivation of the solution methodologies required for tracking a highly nonlinear equilibrium path, namely: parameter continuation and arc-length continuation procedures. These procedures, which are commonly used for the solution of discretized structural systems governed by nonlinear algebraic equations, are augmented and generalized for the direct application to the PDE. The boundary value problem that results from the arc-length continuation scheme and consists of coupled differential, integral, and algebraic equations is re-formulated in a form that allows the use of standard numerical BVP solvers. The performance of the continuation procedures and the convergence of the multi-term extended Kantorovich method are examined through the solution of the two-dimensional Bratu–Gelfand benchmark problem. The applicability of the proposed approach to the tracking of the nonlinear equilibrium path in the post-buckling range is demonstrated through numerical examples of rectangular plates with various boundary conditions.     

Asymptotic stability in nonlinear elastic materials with dissipation

This paper is concerned with the effects of dissipation on classical solutions of the displacement and mixed initial-boundary value problems for a nonlinear elastic material. A proof of the asymptotic stability of the null solution in three dimensions is presented and the necessary assumptions are carefully discussed. The derivation of an analogous result for non-zero equilibrium solutions of the one-dimensional problem completes this work.     

风与柔性结构流固耦合作用的预处理方法研究

针对强耦合方法求解风与柔性结构流固耦合作用时,大量计算资源都耗费在对强耦合方程求解中这一弊端,本文研究了强耦合方程的预处理求解方法。在风与柔性结构流固耦合作用的强耦合整体方程的基础上,将时空离散和线性化后的类似结构方程看成是鞍点问题,首先推导得到了类似结构方程的预处理矩阵;再基于此推导出了强耦合整体方程的预处理矩阵。首先采用预处理方法对经典二维流固耦合问题进行了计算,验证了提出的预处理矩阵的正确性;然后对风与三维膜结构的流固耦合作用进行了分析,评估了所提出预处理方法的相关计算参数。计算结果表明,所提出的预处理方法可使强耦合整体方程的求解在计算精度和计算效率上都得到较大提升,证明本文提出的预处理方法适用于风与柔性结构的流固耦合分析。     

Interaction of a crack with a pair of thin, elastic inclusions

The influence of a pair of thin, elastic inclusions on the stress intensity at a crack tip is considered. The interaction problem is formulated as a set of coupled singular integral equations and the solution is obtained by the Gauss-Chebyshev numerical technique. The stress intensity is computed as a function of certain geometric and elastic parameters of the inclusions, and the toughening effect of these parameters is evaluated.     

Solution of the Three-Dimensional Thermoelastic Problem in Stresses

A method is proposed for constructing a general solution to the three-dimensional thermoelastic equations in stresses (compatibility conditions and equilibrium equations). This general solution is valid for both an elastic half-space and an elastic layer. The second thermoelastic boundary-value problem for a half-space is studied in detail as an example     

Analytic postbuckling solution of a pre-stressed infinite beam bonded to a linear elastic foundation

The postbuckling deflection of an infinite beam that is bonded to a linear elastic foundation and is subjected to an internal compressive stress is analyzed. The nonlinear equilibrium equation that governs the problem considers extensional deformation of the beam. An analytic solution of the nonlinear equilibrium equation is presented and is found to be in good agreement with numerical simulations of the problem. The numerical simulations confirm that for a linear elastic foundation the postbuckling deflection is periodic. The analytic solution shows that the postbuckling wavelength is unaffected by the level of internal stress, and is equal to the wavelength at the critical state.     

A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids

A variational formulation of the coupled thermo-mechanical boundary-value problem for general dissipative solids is presented. The coupled thermo-mechanical boundary-value problem under consideration consists of the equilibrium problem for a deformable, inelastic and dissipative solid with the heat conduction problem appended in addition. The variational formulation allows for general dissipative solids, including finite elastic and plastic deformations, non-Newtonian viscosity, rate sensitivity, arbitrary flow and hardening rules, as well as heat conduction. We show that a joint potential function exists such that both the conservation of energy and the balance of linear momentum equations follow as Euler-Lagrange equations. The identification of the joint potential requires a careful distinction between equilibrium and external temperatures, which are equal at equilibrium. The variational framework predicts the fraction of dissipated energy that is converted to heat. A comparison of this prediction and experimental data suggests that α-titanium and Al2024-T conform to the variational framework.     

Domain Decomposition Methods for Sensitivity Analysis of a Nonlinear Aeroelasticity Problem

Abstract

We consider the nonlinear aeroelasticity problem of the interaction between a viscous, incompressible fluid and Lin elastic solid undergoing large displacement. The non-linearities of the problem formulation include the solid and fluid governing equations. as well as thc dependence of the How geometry on the solid deformation. The resulting coupling is thus two-way. We develop domain-decomposition methods for solution and sensitivity analysis of the coupled problem. The domain decomposition is in the form of a block-Gauss-Seidel-like prcconditioncr that decomposes ihc coupled-domain problem into distinct nonovcrlapping fluid and solid subdotnain problems. The preconditioner thus enables exploitation or single-domain algorithms for solid and fluid mechanics discretization and solution. On the other hand, two-way fluid-solid coupling is retained within the residuals, which is essential for correct sensitivities. Sensitivities of field quantities can be found with little additional work beyond that required for solving the coupled fluid-solid system. The methodology developed here is illustrated by the solution of a problem of viscous incompressible flow about an infinite clastic cylinder. Sensitivities of the resulting velocity and displacement fields with respect to elastic modulus and fluid viscosity are computed.     

The effective thermoelectroelastic properties of microinhomogeneous materials

The paper is concerned with composite materials which consist of a homogeneous matrix phase with a set of inclusions uniformly distributed in the matrix. The components of these materials are considered to be ideally elastic and exhibit piezoelectric properties. One of the variants of the self-consistent scheme, the Effective Field Method (EFM) is applied to calculate effective dielectric, piezoelectric and thermoelastic properties of such materials, taking into account the coupled electroelastic effects. At first the coupled thermoelectroelastic problem for a homogeneous medium with an isolated inclusion is solved. For an ellipsoidal inclusion and constant external field the solution of this problem is found in a closed analytic form. This solution is then used in the EFM to derive the effective thermoelectroelastic operator for the composite containing a random array of ellipsoidal inclusions. Explicit formulae for the electrothermoelastic constants are given for composites, reinforced by spheroidal inclusions.     

The principle of correspondence between elastic and piezoelectric problems

Summary  A correspondence principle is established between elastic and piezoelectric problems for transversely isotropic materials, in such a way that the knowledge of an elastic solution yields fully coupled electro–elastic fields for the corresponding piezoelectric problem, provided the elastic solution is written in a certain form. The implementation of this principle is illustrated by constructing, in a routine way, several piezoelectric solutions involving crack and punch problems (one of them has not been solved previously). Received 12 Feburary 2002; accepted for publication 29 April 2002     

Phase mixtures in dynamics of pseudoelasticity

We give a numerical treatment of phase mixtures in pseudoelasticity from a purely mathematical point of view. It is based on a surprising result that the approximate solution may consist of persistent oscillations in strain which resemble the experimentally observed interface patterns. Such a solution is obtained from a sequence of solutions for a rate-type viscoelastic problem with a non-monotone equilibrium stress-strain relation, for which in the limit as the viscosity tends to infinity the viscoelastic problem reduces to the rate-independent elastic problem describing phase transitions. In this manner, it seems to give yet another perspective for the phase mixture from dynamic point of view as the evolution of an unstable state, in contrast to the traditional treatment from stability analysis for phase equilibrium.     

Equilibrium morphology of misfit particles in elastically stressed solids under chemo-mechanical equilibrium conditions

This paper studies the effects of chemical, elastic and interfacial energies on the equilibrium morphology of misfit particles due to phase separation in binary alloys under chemo-mechanical equilibrium conditions. A continuum framework that governs the chemo-mechanical equilibrium of the system is first developed using a variational approach by treating the phase interface as a sharp interface endowed with interfacial excess energy. An extended finite element method (XFEM) in conjunction with the level set method is then developed to simulate the behaviors of the coupled chemo-mechanical system. The coupled chemo-mechanics model together with the numerical techniques developed here provides an efficient simulation tool to predict the equilibrium morphologies of precipitates in phase separate alloys.     

Analytical solution to a mixed boundary value elastic problem of a roller-guided panel of laminated composite

This study presents an analytical solution to elastic field in a roller-guided panel of symmetric cross-ply laminated composite material. The mixed boundary value two-dimensional plane stress elasticity problem is formulated in terms of a single displacement potential function. This reduces the problem to the solution of a single fourth order partial differential equation of equilibrium as the other equilibrium equation is satisfied automatically. The solution is obtained in terms of an infinite Fourier series. To present some numerical results, a panel of glass/epoxy laminated composite is considered and different components of stress and displacement at different sections of the panel are presented graphically. To justify the present analytical solution, it is compared with the finite element solution obtained by using the commercial software ANSYS. It is found that the two solutions agree well with each other. This ensures that the formulation developed in this study based on the displacement potential approach can be used to obtain analytical solution of an elastic field in structural elements of laminated composite under any mode of boundary conditions prescribed in terms of either stress, displacement or any combination of these.     

The elastic response of a cohesive aggregate—a discrete element model with coupled particle interaction

A model is presented for the deformation of a cohesive aggregate of elastic particles that incorporates two important effects of large-sized inter-particle junctions. A finite element model is used to derive a particle response rule, for both normal and tangential relative deformations between pairs of particles. This model agrees with the Hertzian contact theory for small junctions, and is valid for junctions as large as half the nominal particle size. Further, the aggregate model uses elastic superposition to account for the coupled force–displacement response due to the simultaneous displacement of all of the neighbors of each particle in the aggregate. A particle stiffness matrix is developed, relating the forces at each junction to the three displacement degrees of freedom at all of the neighboring-particle junctions. The particle response satisfies force and moment equilibrium, so that the model is properly posed to allow for rigid rotation of the particle without introducing rotational degrees of freedom. A computer-simulated sintering algorithm is used to generate a random particle packing, and the stiffness matrix is derived for each particle. The effective elastic response is then estimated using a mean field or affine displacement calculation, and is also found exactly by a discrete element model, solving for the equilibrium response of the aggregate to uniform-strain boundary conditions. Both the estimate and the exact solution compare favorably with experimental data for the bulk modulus of sintered alumina, whereas Hertzian contact-based models underestimate the modulus significantly. Poisson's ratio is, however, accurately determined only by the full equilibrium discrete element solution, and shown to depend significantly on whether or not rigid particle rotation is permitted in the model. Moreover, this discrete element model is sufficiently robust, so it can be applied to problems involving non-homogeneous deformations in such cohesive aggregates.     

The Action of an Arbitrary Load on the Surface of an Orthotropic Half-Space

The exact solution to the first boundary-value problem for a half-space is constructed on the basis of the general solution of the equilibrium equations for an orthotropic medium (nine elastic constants). The stress–strain state of an orthotropic half-space whose surface is under an arbitrarily applied concentrated force is described as an example. The well-known solution for the isotropic case is obtained by the same scheme, which confirms the reliability of the result.     

运用Hoek-Brown经验准则分析圆形硐室围岩弹塑性应力和位移

Hoek-Brown经验准则能够较容易地用来估计节理岩体的强度,在岩石工程界已得到广泛的应用和认可。只要用定量指标合理地描述岩体质量,就可确定岩体的强度,因此Hoek-Brown经验准则为通过岩体质量评分评价岩体稳定性架起了一座桥梁,并对需要采取的加固支护措施提出建议。地下硐室开挖后改变了岩体的初始应力状态,围岩应力产生应力重分布现象。当硐室周边围岩应力状态超过岩体弹性极限状态而进入塑性状态时,塑性区内岩体的应力满足极限平衡条件。本文简单回顾了当侧压力系数为1时,圆形硐室围岩的弹性应力和位移,在此基础上以Hoek-Brown经验准则为极限平衡条件,得到圆形硐室轴对称平面应变问题的围岩弹塑性应力和位移的分析解。     

A free rectangular plate on elastic foundation

In the theory of elastic thin plates, the bending of a rectangular plate on the elastic foundation is also a difficult problem. This paper provides a rigorous solution by the method of superposition. It satisfies the differential equation, the boundary conditions of the edges and the free corners. Thus we are led to a system of infinite simultaneous equations. The problem solved is for a plate with a concentrated load at its center. The reactive forces from the foundation should be made to be in equilibrium with the concentrated force to see whether our calculation is correct or not.     

Conditions of stability and instability for a pair of arbitrarily stratified compressible fluids in an arbitrary non-uniform gravity field

The work continues and develops authors’ previous investigation of stability in the small for a two-layer system of inhomogeneous compressible fluids in the uniform gravity field. Here we present a solution of a similar problem in the case of arbitrary non-uniform potential gravity field. The equilibrium stratification of both density and elastic properties of the fluids is supposed arbitrary, as well as the shape of open on top reservoir filled by the fluids. The problem of stability of equilibrium is analyzed as the corresponding problem for the non-linearly elastic bodies, basing on the static energy criterion with regard for the boundary conditions at all parts of the boundary. The crucial element of the analysis is conversion of the quadratic functional of second variation of total potential energy of the system into a “canonical” form that enables to determine its sign. Making use of this canonical form, we obtain almost coinciding with each other necessary and sufficient conditions for stability (those being valid also for an arbitrary number of layers).     

Navier solution for the elastic equilibrium problems of anisotropic skew thin plate wite variable thickness in nonlinear theories

This paper discusses the elastic equilibrium problems of anisotropic skew thin plate ofvariable thickness simply supported on all four sides in nonlinear theories,and uses theNavier method to seek an approach to the problem,and to illustrate the solution with theexamples.In conclusion,the mention is made of the scope of application and theconvergency of the solution.