Elastostatic Analysis Research Articles

Genericity Analysis for Path‐Following Methods. Application in Unilateral Contact Elastostatics

AbstractGenericity results for sparse parametric optimization problems with equality and inequality constraints are obtained in this paper. Their impact on path‐following (e.g. load incrementation) methods for discretized elastostatic analysis problems with unilateral contact relations are considered in this paper. For the sparse problems which arise after a finite element method discretization, generically appearing singularities along the path of solutions are classified. Perturbations involving only a minimal number of parameters are shown to be sufficient to guarantee genericity. Stability and uniqueness questions for the solution along the path are examined in this way. These results can be used for the solution of inequality mechanics (unilateral) problems by means of load or displacement incrementation techniques; in particular, the choice of appropriate approximation (e.g. order of finite elements) and of the solution algorithm can be based on the results presented here.

A boundary integral equation method for nonlinear heat conduction problems with temperature-dependent material properties

A boundary integral equation formulation for transient heat conduction problems with temperature-dependent material properties is proposed. Direct regular method, a technique used in elastostatic boundary element analysis, is applied for numerical solution. In order to verify the validity of the present formulation, two problems are solved. Present results show good agreement with the exact solution and the results obtained by finite difference methods and finite element method. To evaluate the coefficients of discretized boundary integral equations, some integral formulae are used instead of numerical quadrature. Application of these formulae can reduce the computation time to a large extent and achieve highly accurate evaluation.

Evaluation of Fatigue Strength of Adehsive-Bonded Box Section Beams.

Fatigue tests on adhesive-bonded box section beams under torsion, which are typical members of an automobile body structure, were carried out. Using the finite element three-dimensional elastostatic analysis, the stress intensity factor for interface crack was analyzed and the results of fatigue tests were tried to be arranged by the stress intensity factor range ΔKIII. Judging from the above, we conclude that it is possible unifomly evaluate the torsional fatigue strength of this beam by using ΔKIII.

A Crack Very Close to a Bimaterial Interface

This paper presents the plane elastostatics analysis of a semi-infinite crack perpendicular to a perfectly bonded bimaterial interface. Both cases of the crack approaching the interface and penetrating the interface are addressed. The distance from the tip of the crack to the interface is δ. A singular integral equation approach is used to calculate the stress intensity factor, KI, and the crack-opening displacement at the interface, η, as functions of δ, the Dundurs parameters α and β, and the stress intensity factor kI associated with the same crack terminating at the interface (the case δ = 0). The results are presented as KI=kIδ1/2−λf(α,β) and η=CkIδ1−λη(˜α,β) where λ is the strength of the stress singularity associated with δ = 0, f and η˜ are functions calculated numerically and C is a material constant. These results can be used to determine the stress intensity factor and crack opening displacement of cracks of finite length 2a with one tip at a distance δ from the interface for δ/a≪1. The selected results presented for a crack loaded by a uniform far-field tension in each half-plane show that the stress intensity factors approach their limits at a relatively slow rate.

Path-following energy optimization in unilateral contact problems

Path-following (load incrementation) methods are studied in this paper for elastostatic analysis problems with unilateral contact relations in the framework of a large displacement theory by means of the parametric optimization techniques. Finite element discretization yields sparse polynomial optimization problems with equality and inequality constraints. For such sparse problems generically appearing singularities along the path of solutions are completely classified. Perturbations involving only a minimal number of parameters are shown to be sufficient to guarantee these generic situations. This clarifies stability and uniqueness questions for the solution along the examined path.

Quasidifferentiability in nonsmooth, nonconvex mechanics

Nonconvex and nonsmooth optimization problems arise in advanced engineering analysis and structural analysis applications. In fact the set of inequality and complementarity relations that describe the structural analysis problem are generated as optimality conditions by the quasidifferential potential energy optimization problem. Thus new kind of variational expressions arise for these problems, which generalize the classical variational equations of smooth mechanics, the variational inequalities of convex, nonsmooth mechanics and give a solid, computationally efficient explication of hemivariational inequalities of nonconvex, nonsmooth mechanics. Moreover quasidifferential calculus and optimization software make this approach applicable for a large number of problems. The connection of quasidifferential optimization and nonsmooth, nonconvex mechanics is discussed in this paper. A number of representative examples from elastostatic analysis applications are treated in details. Numerical examples illustrate the theory.

Advanced implementation of the boundary element method in elastostatics

A boundary element program for the elastostatic analysis of two-dimensional problems is introduced. The program's theoretical basis is developed and its computational aspects are highlighted. An efficient algorithm for implementation of symmetry is proposed. Finally, through the solution of three typical examples the validity of the program and its theoretical basis is verified.

A certain mixed boundary value problem for a bimaterial interface

This paper presents the solution of a certain mixed problem for a bimaterial interfacecontaining a cut. In the plane, elastostatic analysis displacements are prescribed on the lower edge of the cut and external stresses are applied on its upper edge. Using complex variable techniques, analytical expressions are derived for all physical quantities, including the compliance of the anchor, the stress field, and the stress singularities at its edges. Selected numerical values are presented for these quantities as functions of elastic mismatch. The developed solution can be used to model a vertically loaded rigid anchor, unbonded on one side, embedded along the interface between two elastic materials.

2-D elastostatic analysis by a symmetric BEM/FEM scheme

An accurate and efficient BEM/FEM scheme for elastostatic structural analysis is developed. The BEM component is based on DeFigueiredo's hybrid displacement boundary element model which leads to a symmetric stiffness matrix. Linear as well as quadratic boundary elements are employed. The FEM component is a displacement based finite element model employing linear as well as quadratic interpolation functions, or the MSC/NASTRAN general purpose finite element program. The resulting BEM/FEM scheme is characterized by symmetric matrices and perfect matching of nodal forces and displacements at the interface of the BEM and FEM domains. This leads to accurate results in an efficient way. Numerical examples dealing with elastostatic stress analysis and fracture mechanics under plane stress conditions are presented in order to illustrate the proposed methodology and demonstrate its merits.

Hybrid finite element and modified mixed finite element solutions of axisymmetric circular plates

Two hybrid finite element formulations and a variant of a mixed finite element technique are outlined for the elastostatic analysis ofthin circular plates under conditions of axial symmetry. The hybrid procedures are the first-, second- and third-order stress, and first- and second-order strain finite element approaches, while the modified mixed finite element technique combines the best features of the standard finite element and stiffness routines, and directly utilizes the basic constitutive bending moment-curvature relationships. The numerical performance of the discussed methods is thoroughly examined and tested against exact solutions. For comparison, the corresponding results from a displacement-based finite element method are shown. The work is intended to assist one in the selection of an appropriate technique for elastic solutions for such problems as axisymmetric plates, rafts and tanks.

Convex multilevel decomposition algorithms for non‐monotone problems

AbstractA convex, multilevel decomposition algorithm is proposed in this paper for the solution of static analysis problems involving non‐monotone, possibly multivalued laws. The theory is developed here for a model structure with non‐monotone interface or boundary conditions. First the non‐monotone laws are written in the form of a difference of two monotone functions. Under this decomposition, the non‐linear elastostatic analysis problem is equivalent to a system of convex variational inequalities and to non‐convex min‐min problems for appropriately defined Lagrangian functions. The solution(s) of each one of the aforementioned problems describe the position(s) of static equilibrium of the considered structure. In this paper a multilevel optimization scheme, due to Auchmuty,1 is used for the numerical solution of the problem. The most interesting feature of this method, from the computational mechanics' standpoint, is the fact that each one of the subproblems involved in the multilevel algorithm is a convex optimization problem, or, in terms of mechanics, an appropriately modified monotone ‘unilateral’ problem. Thus, existing algorithms and software can be used for the numerical solution with minor modifications. Numerical results concerning the calculation of elastic and rigid stamp problems and of material inclusion problems with delamination and non‐monotone stick‐slip frictional effects illustrate the theory.

An improved plate theory of {1,2}-order for thick composite laminates

A new two-dimensional laminate plate theory is developed for the linear elastostatic analysis of thick composite plates. The theory employs equivalent single-layer assumptions for the displacements, transverse shear strains, and transverse normal stress. The inplane and transverse displacements are respectively linear and quadratic expansions through the laminate thickness, where the low-order expansion coefficients correspond to the variables of Reissner's first-order shear-deformable theory. The transverse shear strains and transverse normal stress are assumed to be quadratic and cubic respectively through the thickness ; they are expressed in terms of the kinematic variables of the theory by means of a least-squares compatibility requirement for the transverse strains and explicit enforcement of exact traction boundary conditions on the top and bottom plate surfaces. Application of the virtual work principle results in the loth-order equations of equilibrium and associated Poisson boundary conditions. A major advantage of this theory over other higher-order theories lies in its perfect suitability for finite element approximation : simple C0 continuous interpolations for the kinematic variables of the first-order theory (and, optionally, C−1 interpolations for the two higher-order displacements) are needed to formulate effective and robust two-dimensional plate elements capable of full three-dimensional ply-by-ply strain and stress recovery within the framework of general-purpose finite element codes. In assessing the predictive capability of the theory, analytic solutions for the problem of cylindrical bending are derived and compared with exact three-dimensional elasticity results and those of the earlier version of the {1,2}-order plate theory.

接着継手の疲労強度の界面破壊力学による評価

Fatigue tests of the single lap adhesive joints of thin mild steel plates were carried out and the effect of the thickness of the adhesive and plate on the fatigue strength was investigated. The boundary element two-dimensional elastostatic analysis was also carried out and the effect of the adhesive and plate thickness on the stress distribution at the interface was investigated. Moreover, to evaluate the fatigue strength of adhesive joints quantitatively, the stress intensity factors for interface cracks in the adhesive joints were analyzed by using the BEM. The extrapolation method proposed by the authors was used to determine the stress intensity factors. It was found that the stress intensity factor range ΔKi=√ΔK21+ΔK22 can well characterize the fatigue strengthes of SLJ type adhesive joints with different adhesive and plate thicknesses. It is expected that the interfacial fracture mechanics is very useful for evaluating the fatigue strength of the adhesive joints.

Hybrid stress analysis of boundary and finite elements by a super-element method

This paper is concerned with the hybrid method of boundary element and finite element techniques by means of an “external-super-element” function of the commercial finite element method code msc/nastran. The proposed super-element method preserves the modelling simplicity of the boundary element method and the generality of msc/nastran. Two- and three-dimensional elastostatic analyses are performed to demonstrate the accuracy of this method as well as its applicability to practical problems.

Highly Accurate BEM Elastostatic Analysis of Thin Plates.

A very slender 3D body is accurately solved by using boundary element elastostatic analysis in which accurate evaluation of the quasi-singular and the singular integral for a slender element should be considered. This study also shows the validity of the direct regular method (DRM), particularly for the problem of cantilever plates. From this study, it becomes possible to analyze the thin plate as well as the thin cylindrical shell even if the aspect ratio is in the hundreds.

Alternate BEM formulations for 2- and 3-D anisotropic thermoelasticity

Direct boundary element formulations are presented for the elastostatic analysis of two- and three-dimensional general anisotropic media subject to known temperature distributions. The thermal effects are incorporated in the analysis through a volume integral or alternatively, using particular integrals. A multi-region boundary element approach is adopted, in which particular integrals are derived assuming a quadratic temperature distribution in each sub-region. A multiple regression scheme is used for the best fit quadratic temperature distribution in each sub-region for temperature data given at discrete points. The multi-region analysis capability of the numerical implementation enhances its capacity to model discontinuous temperature fields as well as piecewise homogeneous media, such as laminated composites. The formulations presented are incorporated in a general purpose system and are illustrated with examples.

Development of BEM Elastostatic Analysis System for Composite Laminates.

A BEM analysis system (BEM-LAM) of composite laminates with emphasis on accuracy and efficiency is developed. This system can analyze two-dimensional laminates for both isotropic and orthotropic layers under a plane strain state. Accuracy studies show that BEM-LAM using a subelement division scheme in quasi-singular integral gives very accurate solutions even if the layer aspect ratio is equal to thousands. In this study, the BEM-LAM is applied to analyze the problems of shear deformation, free-edge effects and multilayer structures. Accurate solutions are obtained which satisfy exactly the boundary conditions as well as continuity of traction and displacement on the interface between layers. This system could be applied to optimal design problems of laminate materials.

Boundary Element Elastostatic Analysis of Dissimilar Material Joint and the Interface Crack by a Personal Computer

To analyze the interface crack problems in dissimilar materials easily and efficiently, a boundary element elastostatic analysis program is developed on a personal computer, using Hetenyi's solution instead of Kelvin's as the fundamental solution. By this program, we can analyze the dissimilar material problems without the discretization of the interface, and calculate stress intensity factors of an interface crack. Moreover, by this program, we can deal with the interface crack which is partially closed at a crack tip under shear loadings. This paper presents a simple discriminant to judge the good or bad pair that the dissimilar materials would make from the viewpoint of the stress singularity around the edge point, and also shows the numerical analysis results of the stress distributions on the interface and the stress intensity factors for an interface crack in such paired materials.

Elasto-static analysis by infinite elements : Goel, D C Indian Geotech JV18, N3, July 1988, P266–278

Elasto-static analysis by infinite elements : Goel, D C Indian Geotech JV18, N3, July 1988, P266–278

Duality and symmetry in mixed integral methods of elastostatics

AbstractMixed formulations for integral methods of elastostatic analysis of structures are presented. Both finite and boundary element approaches are considered. A governing system exhibiting symmetry is obtained in both cases by preserving duality in the descriptions of equilibrium and compatibility and reciprocity in the elastic constitutive relations. The theorems of virtual forces and displacements are interpreted as the energetic representation of static–kinematic duality. The quadratic programs associated with the governing system are identified as the theorems of minimum potential energy and co‐energy. Mathematical programming theory is used to establish conditions for the existence and uniqueness of solutions.