A method to derive approximate explicit solutions for structural mechanics problems

The availability of explicit solutions, i.e. analytical relationships between the structural response and the design variables, allows a more direct and plain treatment of several structural problems. This paper is devoted to derive approximate explicit solutions in the framework of linear static analysis of finite element modeled structures with a given layout (fixed node positions). The proposed procedure is based on a factorization of the element stiffness matrix following the unimodal components concept, which allows a non-conventional assembly of the global stiffness matrix. The exact inversion of that matrix is a trivial task for the case of statically determinate structures, structures with few redundancies or few design variables. An approximate inverse of the stiffness matrix is herein derived for more general structural problems by resorting to the Sherman–Morrison–Woodbury formula.     

On uniqueness of solutions for secondary creep problems

Uniqueness results are established for solutions of secondary creep problems, including the effect of elastic strains, for a large class of domains subject to mixed boundary conditions. Two theorems are proved, one for quasistatic creep and one for dynamic.     

On transient creep bounds and approximate solutions for the torsion of thin strips

Integral equations are derived which govern transient primary and secondary creep in thin rectangular strips subject to torsion. Formal similarity between these equations and others arising in previous work are exploited to obtain bounds, monotonicity and convexity of the stress profile as well as uniform approximations.     

A conformal decomposition finite element method for modeling stationary fluid interface problems

A method is developed for modeling fluid transport in domains that do not conform to the finite element mesh. One or more level set functions are used to describe the fluid domain. A background, non‐conformal mesh is decomposed into elements that conform to the level set interfaces. Enrichment takes place by adding nodes that lie on the interfaces. Unlike other enriched finite element methods, the proposed technique requires no changes to the underlying element assembly, element interpolation, or element quadrature. The complexity is entirely contained within the element decomposition routines. It is argued that the accuracy of the method is no less than that for eXtended Finite Element Methods (XFEM) with Heaviside enrichment. The accuracy is demonstrated using multiple numerical tests. In all cases, optimal rates of convergence are obtained for both volume and surface quantities. Jacobi preconditioning is shown to remove the ill‐conditioning that may result from the nearly degenerate conformal elements. Copyright © 2009 John Wiley & Sons, Ltd.     

Construction of approximate solutions of boundary value impact strain problems

The method for constructing approximate solutions of boundary value problems of impact strain dynamics in the form of ray expansions behind the strain discontinuity fronts is generalized to the case of curvilinear and diverging rays. This proposed generalization is illustrated by an example of dynamics of an antiplane motion of an elastic medium. The ray method is one of the methods for constructing approximate solutions of nonstationary boundary value problems of strain dynamics. It was proposed in [1, 2] and then widely used in nonstationary problems of mathematical physics involving surfaces on which the desired function or its derivatives have discontinuities [3–7]. A complete, qualified survey of papers in this direction can be found in [8]. This method is based on the expansion of the solution in a Taylor-type series behind the moving discontinuity surface rather than in a neighborhood of a stationary point. The coefficients of this series are the jumps of the derivatives of the unknown functions, for which, as a consequence of the compatibility conditions, one can obtain ordinary differential equations, i.e., discontinuity damping equations. In the case where the problem with velocity discontinuity surfaces is considered in a nonlinear medium, this method cannot be used directly, because one cannot obtain the damping equation. A modification of this method for the purpose of using it to solve problems of that type was proposed in [9–11], where, as an example, the solutions of several one-dimensional problems were considered. In the present paper, we show how this method can be transferred to the case of multidimensional impact strain problems in which the geometry of the ray is not known in advance and the rays become curvilinear and diverging. By way of example, we consider a simple problem on the antiplane motion of a nonlinearly elastic incompressible medium.     

Construction of approximate solutions to two-dimensional potential problems of electrohydrodynamics

After transformation to new variables the system of equations describing planar potential electrohydrodynamic flows with a small interaction parameter is converted to a single equation. The particular solution of this equation, which is the electrohydrodynamic analog of Hamel's solution in the dynamics of a viscous liquid, is found. Two types of flows, described by simplified equations, can be distinguished when certain constraints are imposed on the manner in which the electrical parameters vary along the coordinate lines and the terms of the equation correspondingly estimated. These flows are the jet and quasione-dimensional flows of the charged component in a curvilinear electrostatic field and a supper-posed two-dimensional potential flow of the carrier medium. Solutions to the approximate equations are obtained for certain particular cases.Kiev. Transklated from Izvestiya Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 4, pp. 140–147, July–August, 1972.     

On uniqueness of solutions for nonlinear creep problems in symmetric pressure vessels

Uniqueness is established for positive solutions of a nonlinear integral equation which governs the effective stress in internally loaded spherical and incompressible cylindrical pressure vessels subject to primary or secondary transient creep.     

Some approximate solutions of filtration problems in a fissured-porous medium

The equations for the filtration of a fluid in a fissured-porous medium [1] under the assumption that the permeability of the porous blocks is negligible in comparison with the permeability of the cracks and that the porosity of the cracks is negligible in comparison with the porosity of the blocks may be written in the form Here p₁ is the pressure in the cracks, p₂ is the pressure in the porous blocks, is the characteristic lag time, , is the piezoconductivity coefficient. We shall consider the approximate solutions of this system of equations in the case of filtration to a well which penetrates a fissured-porous stratum of thickness h and begins to operate at the moment t=0 with the flow rate Q.The author wishes to tank V. N. Nikolaevskii for discussions of the study.     

On an approximate method of solution of problems in plasticity

An approximate method of solving the variational equation, constructed on the basis of the principle of virtual variation of the deformed state, with a given equation relating the strains (strain rates) is presented. The stress-strain state is then determined from the solution of the above variational equation. The method is demonstrated on an example of the problem of a strip with a rectangular cross section resting on plane-parallel plates.     

Some basic solutions for nonlinear creep

The aim of the paper is to derive the exact analytical expressions for torsion and bending creep of rods that obey the Norton–Bailey, Prandtl–Garofalo and Naumenko–Altenbach–Gorash constitutive models. The common secondary creep constitutive model is the Norton–Bailey law which gives a power law relationship between creep rate and stress. The closed form solutions for fractional Norton–Bailey creep law are derived. The analytical formulas express the torque and bending moment as functions of the time for the period of relaxation. Other formulas express the twist rate and curvature as functions of the time for the duration of engineering creep experiment. The derived formulas are suitable for the practically important problems of machinery. Namely, the formulas are relevant for calculation of hereditary effects for helical, leaf and disk springs and twisted shafts.     

Approximate method for solving relaxation problems in terms of material's damagability under creep

Novosibirsk. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, No. 5, pp. 136–142, September–October, 1994.