On the (u, p) problem in the theory of elasticity

A problem of the theory of elasticity is considered for a body with vectors of displacements u and loads p simultaneously defined on one part of the body and with undefined conditions on the remaining part of the body. For a doubly connected domain, where the vectors u and p are set on one of its boundaries (inner or outer), an iterative method based on reduction of the initial problem to a sequence of mixed problems is justified. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 3, pp. 100–103, May–June, 2006.     

On Aerodynamic Forces for Viscous Compressible Flow

The paper is aimed at reviewing and adding some new results to our recent work on a force theory for viscous compressible flows around a finite body. It has been proposed to analyze aerodynamic forces directly in terms of fluid elements of nonzero vorticity and density gradient. Let ρ denote the density, u the velocity, and ω the vorticity. It is demonstrated that for largely separated flows about bluff bodies, there are two major source elements: R _e(x) =−?u ²∇ρ·∇ϕ and V _e(x) =−u×ω·∇ϕ, where ϕ is an acyclic potential, generated by the solid body moving with unit velocity in the negative direction of the force considered. In particular, under mild conditions, the (unique) choice of ϕ enforces that the elements R _e(x) and V _e(x) decay rapidly away from the body. Four kinds of finite body are considered: a circular cylinder, a sphere, a hemi sphere-cylinder, and a delta wing of elliptic section—all in transonic-to-supersonic regimes. From an extensive numerical study carried out for these bodies, it is found that these two elements contribute to 95% or more of the total drag or lift for all the cases under consideration. Moreover, R _e(x) due to density gradient becomes progressively important relative to V _e(x) due to vorticity as the Mach number increases. The present method of force analysis enables effective analysis and assessment of relative importance of aerodynamics forces, contributed from individual flow structures. The analysis could therefore be very much useful in view of the rapid growth in numerical fluid dynamics; detailed (either local or global) flow information is often available. The paper is dedicated to Sir James Lighthill in honor of his great contributions to aeronautics on the occasion of the publication of his collected works. Received 3 January 1997 and accepted 11 April 1997     

Interaction between elliptical and ellipsoidal inclusions under bending stress fields

Summary  This paper deals with interaction problems of elliptical and ellipsoidal inclusions under bending, using singular integral equations of the body force method. The problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are densities of body forces distributed in the x,y and r,θ,z directions in infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical and the ellipsoidal boundaries, the unknown functions are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield the exact solutions for a single elliptical or spherical inclusion under a bending stress field. It yields rapidly converging numerical results for interface stresses in the interaction of inclusions. Received 9 September 1999; accepted for publication 15 January 2000     

Radiation from a sphere in compressible plasma

Summary The Hansen's vector wave functions have been modified so it could apply directly for the solution of general exterior boundary value problems in compressible plasma for a spherical geometry. The vector wave functionL has been included to represent an acoustic wave and the three angular orthogonal functions have been defined using complex Fourier series in the azymuthal direction. Using this modified method of vector wave functions, the exterior spherical boundary value problem in compressible isotropic plasma has been solved. The boundary conditions prescribed over the surface of the sphere have been the tangential electric field (or the tangential magnetic field) and the radial component of the velocity vector (or the pressure). From those four basic boundary value problems the coefficients have been derived and several particular cases has been discussed.The research reported in this paper was supported in part by the National Science Foundation, U.S.A.     

Numerical solution of singular integral equations in stress concentration problems under longitudinal shear loading

Summary The paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy-type singularities, where unknown functions are densities of body forces distributed in the longitudinal direction of an infinite body. First, four kinds of fundamental density functions are introduced to satisfy completely the boundary conditions for an elliptical boundary in the range 0≤φ _k ≤2π. To explain the idea of the fundamental densities, four kinds of equivalent auxiliary body force densities are defined in the range 0≤φ _k ≤π/2, and necessary conditions that the densities must satisfy are described. Then, four kinds of fundamental density functions are explained as sample functions to satisfy the necessary conditions. Next, the unknown functions of the body force densities are approximated by a linear combination of the fundamental density functions and weight functions, which are unknown. Calculations are carried out for several arrangements of elliptical holes. It is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of the present solutions. Received 26 May 1998; accepted for publication 27 November 1998     

Corner Singularities and Regularity Results for the Reissner/Mindlin Plate Model

The theory for elliptic boundary value problems for general elliptic systems is used in order to investigate systematically corner singularities and regularity for weak solutions to a broad class of boundary value problems for the Reissner/Mindlin plate model in polygonal domains. The regularity results for the deflection of the midplane and for the rotation of fibers normal to the midplane are formulated in Sobolev spaces H ^s , where s>1 is a real number. The number s depends on the geometry, the material parameters and the boundary conditions in general and is related to a decomposition of the fields in a singular and a regular part. The leading singular terms are calculated for a wide class of boundary conditions (36 combinations). The results are critically compared with those known from a stress potential approach.     

A hybrid immersed boundary and material point method for simulating 3D fluid–structure interaction problems

A numerical method is developed for solving the 3D, unsteady, incompressible Navier–Stokes equations in curvilinear coordinates containing immersed boundaries (IBs) of arbitrary geometrical complexity moving and deforming under forces acting on the body. Since simulations of flow in complex geometries with deformable surfaces require special treatment, the present approach combines a hybrid immersed boundary method (HIBM) for handling complex moving boundaries and a material point method (MPM) for resolving structural stresses and movement. This combined HIBM & MPM approach is presented as an effective approach for solving fluid–structure interaction (FSI) problems. In the HIBM, a curvilinear grid is defined and the variable values at grid points adjacent to a boundary are forced or interpolated to satisfy the boundary conditions. The MPM is used for solving the equations of solid structure and communicates with the fluid through appropriate interface‐boundary conditions. The governing flow equations are discretized on a non‐staggered grid layout using second‐order accurate finite‐difference formulas. The discrete equations are integrated in time via a second‐order accurate dual time stepping, artificial compressibility scheme. Unstructured, triangular meshes are employed to discretize the complex surface of the IBs. The nodes of the surface mesh constitute a set of Lagrangian control points used for tracking the motion of the flexible body. The equations of the solid body are integrated in time via the MPM. At every instant in time, the influence of the body on the flow is accounted for by applying boundary conditions at stationary curvilinear grid nodes located in the exterior but in the immediate vicinity of the body by reconstructing the solution along the local normal to the body surface. The influence of the fluid on the body is defined through pressure and shear stresses acting on the surface of the body. The HIBM & MPM approach is validated for FSI problems by solving for a falling rigid and flexible sphere in a fluid‐filled channel. The behavior of a capsule in a shear flow was also examined. Agreement with the published results is excellent. Copyright © 2007 John Wiley & Sons, Ltd.     

Some problems in the Z-C-X space

The new concepts of the Z-C-X space and excellent cone are introduced. Some problems of random semiclosed 1-set-contractive operator are investigated in the Z-C-X space. At first, an important inequality is proved. Secondly, several new conclusions are proved by means of random fixed point index in the theory of random topological degree. A random solution of a class of random operator equations under conditions of imitating the parallelogram law is obtained, famous Altman’s theorem is generalized in partially ordered Z-C-X space. Therefore, some new results are obtained.     

Swirling turbulent wake behind a self-propelled body

The development of the swirling turbulent axisymmetric wake of a self-propelled body is modeled numerically. The flow pattern is calculated within the framework of the thin shear layer approach for nonclosed system of the motion and continuity equations. The closed system of equations is written for two different formulations of the closure relations. The numerical solution of the problem is performed with the use of the finite-difference algorithm realised on moving grids. The algorithm is conservative with respect to the laws of conservation of the momentum and the angular momentum. The experimentally measured distributions are used as the initial conditions. Both the models described agree well with the experimental data of Gavrilov N., Demenkov A., Kostomakha V., Chernykh G. (2000), Experimental and numerical modelling of turbulent wake behind self-propelled body, J. Appl. Mech. Tech. Phys., 41 (4), 619–627. It is demonstrated that at the large distances downstream from the body the solution of the problem approaches the self-similar one.     

Sensitivity analysis of the non-linear dynamic response of beams using space-time perturbation modes

The focus of the present work is directed towards the development of an effective reduced basis technique for calculating the sensitivity of the non-linear dynamic structural response of mechanical systems with respect to variations in the design variables.The proposed methodology is formulated within the context of a mixed space-time finite element method, which naturally allows the treatment of initial and boundary value problems. The time dependency of the solutions is implied in the assumed space-time modal shapes, and hence the partial differential equations of motion are directly reduced to a set of non-linear simultaneous equations of a purely algebraic nature.The independent field variables are approximated in terms of perturbations modes or path derivatives with respect to a load control parameter. These modes, extracting information about the kinematic and dynamic behavior of the structural system through the higher order derivatives of the strain and kinetic energies, are appropriate bases for non-linear dynamic problems. The sensitivity derivatives of the field variables are then approximated using a combination of perturbation modes and of their sensitivity derivatives.The resulting computational procedure offers high potential for the effective and numerically efficient sensitivity analysis of dynamic systems exhibiting periodic-in-time response. The proposed methodology is illustrated addressing non-linear beam problems subjected to harmonic loading and the results obtained are compared with those of a full finite-element model.Nomenclature (O, I _i), (is1, 2, 3) Inertial frame of origin O - (P, s _i), (is1, 2, 3) Local frame in the undeformed configuration - (Q, s _i ^*), (is1, 2, 3) Local frame in the deformed configuration - t Time - l Abscissa along the beam reference line - L Beam length - (·)s(·)/t Partial derivative with respect to time - (·)s(·)/l Partial derivative with respect to space - u Position vector of the beam reference line - r Rotation parameters - ds(u, r) Generalized displacement vector - R(r) Rotation tensor associated with r - (r) Tensor defined in equations (4) and (5) - s· Finite rotation vector - as(a _s, a _v) Conformal rotation vector - Angular velocity - ws(u, ) Generalized velocity vector - k Curvature - e Generalized strains - ps(h, l) Generalized momenta - fs(s, m) Generalized sectional stress resultants - f _es(S _e, m _e) Applied external loads - M Inertia tensor     

The Potential Scope of the Ultrasonic Surface Reflection Method Towards Mechanical Characterisation of Isotropic Materials. Part 2. Experimental Results

Background

This paper is Part 2 of a study on the scope of the ultrasonic Surface Reflection Method (SRM). Part 1 deals with the theoretical conditions for a satisfactory usage of this method.

Objective

This second part validates the practical feasibility and reliability of the SRM method by comparison with the conventional Transmission Method (TM) in cases where the latter is applicable.

Methods

Two experimental devices (one for SRM and one for TM) are developed and measurements of shear and bulk moduli are carried out at ultrasonic frequency (610 kHz) and at room temperature.

Results

The experimental conditions in terms of sample geometry, pulse characteristics and interfacial transmission required to obtain a given accuracy on the measurement are stated. The SRM is then validated against other experimental methods and is used to determine the shear modulus of a carbon black filled neoprene at ambient temperature (T?=?21 °C) and ultrasonic frequency.

Conclusions

The benefit brought by this method is well demonstrated: a unique measurement allows the determination of all the moduli of a highly damping isotropic material (carbon black filled neoprene) not achievable by other methods.

     

Boundary element method for buckling eigenvalue problem and its convergence analysis

The conditions for determining solution of buckling eigenvalue problem are discussed. The corresponding system of integral equations with constraint conditions and boundary variational equations with Lagrange multiplier are established. The theorems on the existence and uniqueness of the solution for these problems are given. The corresponding boundary element method is constructed and the error estimation for the approximation solution is obtained. Finally the numerical example is given. Foundation item: the National Natural Science Foundation Pre-research Project (T4107015) Biography: Ding Rui (1969-)     

Discussion of Antiplane Singularities of Piezoelectric–Dielectric and Piezoelectric–Conductor Wedges

This technical note deals with two special topics from our previous paper (Chue and Chen in Arch Appl Mech 72 673–685, 2003) in Archive of Applied Mechanics: the effects of electrical conditions imposed on the edges and bonded interfaces of piezoelectric–dielectric and piezoelectric–conductor wedges on antiplane problems. After employing relatively realistic electrical conditions, we found that stress and electric displacement singularities are altered when boundary conditions and/or continuity conditions are changed, and we compared the results with those of previous studies.     

Two notes on nonlocal field theory

In this paper, two fundamental problems completely unsolved in nonlocal field theory are studied. The first is the dependence of nonlocal residuals. By studying this problem, a theorem concerning the relationship between the residuals of nonlocal body force and nonlocal moment of momentum is given and proven. The other problem is how to give the stress boundary conditions in the linear theory of nonlocal elasticity. The stress boundary conditions obtained in this paper can not only answer why the nonlocal stress solution satisfying the boundary conditionst _ji ^(s) n _j ¦O ₂ =p _i (p _i is a constant) on the surface of crack does not exist but also give a model of the molecular cohesive stress on the crack tip.     

On Sanders' energy-release rate integral and conservation laws in finite elastostatics

Sanders showed in 1960, within the framework of two-dimensional elasticity, that in any body a certain integral I around a closed curve containing a crack is path-independent. I is equal to the rate of release of potential energy of the body with respect to crack length. Here we first derive, in a simple way, Sanders' integral I for a loaded elastic body undergoing finite deformations and containing an arbitrary void. The strain energy density need not be homogeneous nor isotropic and there may be body forces. In the absence of body forces, for flat continua, and for special forms of the strain energy density, it is shown that I reduces to the well-known vector and scalar path-independent integrals often denoted by J, L, and M.     

On the reduction of the normality conditions in equality-constrained optimization problems in mechanics

A large class of problems in mechanics leads to the minimization of an objective function under equality constraints. In fact, inequality constraints can always be transformed into equality constraints by means of slack variables. The classical approach to solve equality-constrained problems relies on Lagrange multipliers, whose first-order normality conditions (FONC) lead to a system of nonlinear algebraic equations. This system of equations involves as many equations as unknowns, composed of the design variables and Lagrange multipliers, and hence, is amenable to a host of solution methods. In this paper, two methods to eliminate the Lagrange multipliers are reported, by which a reduced system of normality conditions is obtained. Reduction is conducted here either symbolically or numerically using an isotropic orthogonal complement L of the Jacobian matrix of the equality constraints. The relations thus resulting are cast into what is termed the dual form of the FONC. When the problem allows for symbolic calculations, a semi-graphical approach is applied, which leads to the global optimum of the problem at hand. However, the main novelty of the paper lies in an algorithm that returns the stationary points of a constrained optimization problem without requiring the closed-form expressions of the dual form of the FONC. Moreover, numerically efficient and stable procedures are given for the intermediate computational steps. The application of this algorithm is demonstrated with three examples from mechanics.     

Unsteady Flux-Vector-Based Green Element Method

This article extends the mathematical formulation and solution procedure of the modified ‘q-based’ GEM to unsteady situations, namely to the modified unsteady ‘q-based’ GEM. Solutions that provide information on the evolution of the pressure and the flux over long time intervals are available by incorporating the additional dimension of time into steady problems. This approach is first tested by solving an example for which an analytical solution is available. The numerical results for this example is found to be in excellent agreement with the analytical solution. Several problems involving geological features, such as wells and faults, are then investigated, with different properties applying to the faults. A strong influence of the low permeability faults is in evidence in these problems.     

Function theoretic solutions to problems of orthotropic elasticity

The plane strain problem for a two dimensional orthotropic elastic body is investigated. In particular analytic representations for the solution of the displacement boundary value problem and the stress boundary value problems are found. To this end, the Navier equations are reduced by means of composite transformations to normal form. These are the so-called equations for bianalytic function of the type (k). The generalized Cauchy integral formula for this function theory is used to obtain representation formulae. A simplified method to solve these problems by bianalytic function theory is given for certain situations of plane strain for an orthotropic elastic body. AMS (MOS): 35A20, 35CO5, 35G15, 35J55.Applied Mathematics Institute Technical Report No. 140A, July 1983.The work of this author was supported in part by grant no. DE-AC01-81ER-10967 from the Department of Energy.