Bounds to internal forces for elastic-plastic solids subjected to variable loads

Summary Considering an elastic-plastic workhardening solid with piecewise linear yield surfaces and a piecewise linear workhardening law, we give a method for constructing bounds to the internal forces and to the (hardened) yield stresses produced by the action of variable loads at any point of the body and at any time. The loading history is supposed to be unknown, but the loads range within a given domain.

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Sommario Considerando un solido elasto-plastico incrudente con superfici di plasticizzazione lineari a tratti e legge di incrudimento lineare a tratti, si fornisce un metodo per la costruzione di maggiorazioni sulle forze interne nonché sulle tensioni limite (incrudite) provocate dai carichi in un punto qualunque del solido ed in un qualunque istante. La storia di carico è incognita, ma i carichi variano all'interno di un dato dominio.

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The results presented in this paper were obtained in the course of a research project sponsored by the National (Italian) Research Council, C.N.R., PAdIS Committee.     

A finite element solution to turbulent diffusion in a convective boundary layer

A second-order closure turbulence model is used to simulate the plume behaviour of a passive contaminant dispersed in a convective boundary layer. A time-splitting finite element scheme is used to solve the set of partial differential equations. It is shown that the second-order closure model compares favourably with recent findings from laboratories, wind-tunnel experiments and large-eddy simulations. We also compare the second-order closure model with the commonly used K-diffusion model for the same meteorological conditions. Case studies also show the effects of model parameters and turbulence variables on the plume behaviour.     

Finite expansion of a hole in an elastic-plastic thin plate of finite thickness

A finite deformation theory for axially symmetric thin elastic-plastic plates is obtained by a consistent approximation from the corresponding three-dimensional theory. This theory can be applied to plates of finite thickness, and is different from the usual plane stress theory. The problem of the expansion of a circular hole in an infinite plate is investigated and the effect of the transversely variable stress components is studied.     

A decomposed Crouzeix-Raviart element for the finite element solution of the Navier-Stokes equation

A methodology for the decomposition of the Crouzeix-Raviart finite element into six linear subelements is described. The resulting element is shown to satisfy the Brezzi-Babu?ka compatibility condition. The error bounds are also established. A comparison in accuracy between this and the standard Crouzeix-Raviart element is presented for driven cavity flows. Other results include the execution time for the DCR element and the Crouzeix-Raviart element along with both analytical and numerical integration. It is shown that the decomposed element results in shorter execution times with only marginal changes in accuracy.     

A transient solution method for the finite element incompressible Navier-Stokes equations

A numerical procedure for solving the time-dependent, incompressible Navier-Stokes equations is presented. The present method is based on a set of finite element equations of the primitive variable formulation, and a direct time integration method which has unique features in its formulation as well as in its evaluation of the contribution of external functions. Particular processes regarding the continuity conditions and the boundary conditions lead to a set of non-linear recurrence equations which represent evolution of the velocities and the pressures under the incompressibility constraint. An iteration process as to the non-linear convective terms is performed until the convergence is achieved in every integration step. Excessively artificial techniques are not introduced into the present solution procedure. Numerical examples with vortex shedding behind a rectangular cylinder are presented to illustrate the features of the proposed method. The calculated results are compared with experimental data and visualized flow fields in literature.     

Three dimensional analytical solution for finite circular cylinders subjected to indirect tensile test

This paper derives a new three-dimensional (3-D) analytical solution for the indirect tensile tests standardized by ISRM (International Society for Rock Mechanics) for testing rocks, and by ASTM (American Society for Testing and Materials) for testing concretes. The present solution for solid circular cylinders of finite length can be considered as a 3-D counterpart of the classical two dimensional (2-D) solutions by Hertz in 1883 and by Hondros in 1959. The contacts between the two steel diametral loading platens and the curved surfaces of a cylindrical specimen of length H and diameter D are modeled as circular-to-circular Hertz contact and straight-to-circular Hertz contact for ISRM and ASTM standards respectively. The equilibrium equations of the linear elastic circular cylinder of finite length are first uncoupled by using displacement functions, which are then expressed in infinite series of some combinations of Bessel functions, hyperbolic functions, and trigonometric functions. The applied tractions are expanded in Fourier–Bessel series and boundary conditions are used to yield a system of simultaneous equations. For typical rock cylinders of 54 mm diameter subjected to ISRM indirect tensile tests, the contact width is in the order of 2 mm (or a contact angle of 4°) whereas for typical asphalt cylinders of 101.6 mm diameter subjected to ASTM indirect tensile tests the contact width is about 10 mm (or a contact angle of 12°). For such contact conditions, 50 terms in both Fourier and Fourier–Bessel series expansions are found sufficient in yielding converged solutions. The maximum hoop stress is always observed within the central portion on a circular section close to the flat end surfaces. The difference in the maximum hoop stress between the 2-D Hondros solution and the present 3-D solution increases with the aspect ratio H/D as well as Poisson’s ratio ν. When contact friction is neglected, the effect of loading platen stiffness on tensile stress in cylinders is found negligible. For the aspect ratio of H/D = 0.5 recommended by ISRM and ASTM, the error in tensile strength may be up to 15% for both typical rocks and asphalts, whereas for longer cylinders with H/D up to 2 the error ranges from 15% for highly compressible materials, and to 60% for nearly incompressible materials. The difference in compressive radial stress between the 2-D Hertz solution or 2-D Hondros solution and the present 3-D solution also increases with Poisson’s ratio and aspect ratio H/D. In summary, the 2-D solution, in general, underestimates the maximum tensile stress and cannot predict the location of the maximum hoop stress which typically locates close to the end surfaces of the cylinder.     

Determination of displacement of hole profile in cylindrical shells subjected to torsion using holographic interferometry

Moscow Engineering and Physics Institute. Translated from Prikladnaya Mekhanika, Vol. 24, No. 7, pp. 63–69, July, 1988.     

Effective properties of a composite beam subjected to torsion

The problem of finding the effective characteristics of a rectilinear beam under pure torsion is considered. The problem can be reduced to determining the torsional stress function from the solution of a boundary-value problem in a cross section of the beam for a partial differential equation with variable coefficients. Two special boundary-value problems are formulated to find the effective characteristics. It is shown that the effective coefficients are reciprocal in the case of torsion of a layer with nonuniform thickness. In the two-dimensional case, the problem is solved by a finite element method. The cases of a square beam with single and multiple inclusions are discussed. The dependence of the effective characteristics on the inclusion volume fraction is analyzed.     

Bursting pressure of thick-walled cylinders subjected to internal and external pressures,axial load and torsion

A finite-total-strain, incompressible, analytical solution is presented to predict load-deformation relations for loads from zero to failure for thick-walled cylinders subjected to internal pressure, external pressure, axial load and torsion. The solution assumes that the material is an isotropic hardening material that obeys the von Mises yield condition. The flow law incorporates the prandtl-Reuss stressstrain relations and a loading function represented by the tension true-stress vs. true-strain diagram. Poisson's ratio is assumed to be equal to one-half for both elastic and plastic strains. The difference between the strains given by the incompressible solution and the correct strains are calculated for one set of elastic loads; the strains given by the incompressible solution are then corrected based on the assumption that each correction is proportional to the increase in the given component of load. Good agreement is indicated between the corrected incompressible solution and data obtained from cylinders made of either SAE 1045 steel, OFHC copper, or aluminum alloy 1100. Paper was presented at 1975 SESA Spring Meeting held in Chicago, IL on May 11–16. This investigation was funded by Rock Island Arsenal and was conducted at Research Directorate, GEN Thomas J. Rodman Laboratory, under the Laboratory Research Cooperative Program of ARO-D.     

A directed rod subjected to internal constraints

In this paper, the field equations for a directed rod are given. The form of the reaction forces with respect to a general internal constraint imposed on the rod is also derived, based on the assumption that the reaction forces do no work in any deformation compatible with the constraint. Finally, the rod subjected to both special and fundamental internal constraints is treated.     

A finite element solution of unsteady two-dimensional flow in cascades

A theory is presented for unsteady two-dimensional potential transonic flow in cascades of compressor and turbine blades using a mesh of triangular finite elements. The theory leads to a computer program, FINSUP, which is fast and has moderate storage requirements, so that it can be run on a personal computer. Comparisons with other theories in special cases show that the program is accurate in subsonic flow, and that in supersonic flow, although the wave effects are smeared by the numerical process, the results for overall blade force and moment have acceptable accuracy. The program is useful for engineering assessment of unstalled flutter of actual compressor and turbine blades.     

Stress intensity factors for circular hole and inclusion using finite element alternating method

A centre cracked plate subjected to remote tensile and shear loading is considered for the analysis. Effect of circular hole and influence of shrunk fit inclusion on stress intensity factors are studied. Multiply connected domain boundary value problem is solved using finite element alternating method (FEAM). Parametric studies involving drilled hole/inclusion sizes and locations are investigated. Energy release rates evaluated using the stress field obtained by FEAM are in good agreement with other methods. The optimum location in reducing the stress intensity factor with hole/inclusion is obtained and located at a distance 20% of semi-crack length from crack tip on the side opposite the ligament for Mode-I loading and it is also observed that the location is almost invariant of hole sizes. For Mode-II loading, the optimum location for the hole is located at a distance about 23% of semi-crack length from the middle of the crack along the transverse direction.     

Finite deformation solution of a dynamic problem of combined compressive and shear loading for an elastic-plastic half-space

A self-similar problem involving combined compressive and shear loading of a non work-hardening elastic-plastic half-space is solved, for large deformations, within the framework proposed earlier by J.R. Willis (1969). The thermodynamic system, which was left open in the general framework, is taken to be that proposed recently by G.W. Swan and C.K. Thornhill (1974). Results are presented for oblique loading of a block of low carbon steel. For an imposed normal velocity of the order of 1,100 m s^?1, a smooth shear wave behind the shock that is formed is possible only for small transverse velocities, perhaps no greater than 14 m s^?1, depending upon the value assumed for the yield stress. Larger transverse velocities would give rise to a thin layer of intense shear near the surface of the block, whose study would require allowance for both work-hardening and temperature-dependent yield stress.     

A finite element method for the solution of two-dimensional transonic flows in cascades

Steady two-dimensional transonic flow is calculated in cascades of compressor and turbine blades using a mesh of triangular finite elements. A velocity potential is used, the equations being solved by the Newton-Raphson technique. The resulting computer program is fast, and is shown to give good accuracy. Shock waves are well represented, provided they are not too strong.     

A combined programming and iteration algorithm for finite element analysis of three-dimensional contact problems

Comparing with two-dimensional contact problems, three-dimensional frictional contact problems are more difficult to deal with, because of the unknown slip direction of the tangential force and enormous computing time. In order to overcome these difficulties, a combined PQP (Parametric Quadratic Programming) and iteration method is derived in this paper. The iteration algorithm, which alleviates the difficulty of unknown slip direction, is used along with the PQP method to cut down computing costs. Numerical example is given to demonstrate the validity of the present algorithm. The project supported by the Machinary and Electronics Ministry of China     

Uncoupled finite element solution of biharmonic problems for vector potentials

A method for the uncoupled solution of three-dimensional biharmonic problems for the vector potential in viscous incompressible flow is presented. The strategy applied in a previous work on vector Poisson equations is employed to reduce the vector fourth-order problem to a sequence of scalar biharmonic problems. A finite element aimed at the implementation of the method in a discrete version is considered. A conjugate gradient algorithm which is particularly efficient for the uncoupled solution method is also described.     

A fatigue-testing machine for combined bending and torsion

A testing machine is designed and developed in such a way as to be capable of exerting simultaneous cyclic bending and twisting moments, such moments being uniformly distributed along the entire gage length of the test specimen. The ratio of bending to twisting moments can be adjusted within the range 0 to 0.5, the angular deflection of specimen being controlled in the machine. Experimental results first obtained for reversed torsion show that the fatigue life is best expressed in terms of the shearstrain range or the shear-stress amplitude in potential form. Further wark on the subject is proceeding.