Showing papers on "Plane stress published in 1982"

Influence of void nucleation on ductile shear fracture at a free surface

Abstract: A n approximate continuum model of a ductile, porous material is used to study the influence of the nucleation and growth of micro-voids on the formation of shear bands and the occurrence of surface shear fracture in a solid subject to plane strain tension. Bifurcation into diffuse modes is analysed for a plane strain tensile specimen described by these constitutive relations, which account for a considerable plastic dilatancy due to void growth and for the possibility of non-normality of the plastic flow law. In particular, bifurcation into surface wave modes and the possible influence of such modes triggering shear bands is investigated. For solids with initial imperfactions such as a surface undulation, a local material inhomogeneity on an inclusion colony, the inception and growth of plastic flow localization is analysed numerically. Both the formation of void-sheets and the final growth of cracks in the shear bands is described numerically. Some special features of shear band development in the solid obeying non-normality are studied by a simple model problem.

Ductile fracture by cavity nucleation between larger voids

Abstract: A mechanism of ductile fracture involving the interaction of relatively large voids with small-scale voids is studied by a computational model. The larger voids are described as circular cylindrical holes arranged in a doubly periodic array in the initial state. In the matrix material between these voids the nucleation and growth of much smaller voids is accounted for by using approximate constitutive equations for a ductile, porous medium. The computations show bands of highly localized straining and void growth, initiating at the surfaces of larger voids and growing into the matrix material, until the bands connect two neighbouring voids. The materials are analysed both under plane strain conditions and under conditions approximating those in a round tensile bar. The failure strains obtained under different principal stress ratios show rather good agreement when plotted against a measure of the stress-triaxiality.

Numerical prediction of collapse loads using finite element methods

Abstract: In this paper, the ability of a displacement-type finite element analysis to predict collapse loads accurately is investigated. For the usual assumptions of ideal plasticity and infinitesimal deformations, attention is focused on undrained geotechnical problems. The theoretical criterion originally developed by Nagtegaal et al is applied to each member of the serendipidity quadrilateral and triangular family of elements, up to and including those with a quartic displacement expansion. This method of assessing the suitability of a particular type of element is shown to be valid for any constitutive law which attempts to enforce the constant volume condition at failure, such as critical state type soil models. The method is also generalized to permit an assessment, a priori, of the suitability of any given mesh which is composed of a finite number of elements of the same type. It is postulated that the 15-noded, cubic strain triangle is theoretically capable of accurate computations in the fully plastic range for undrained geotechnical situations which involve axial symmetry or plane strain. This prediction is verified by a series of numerical experiments on footing problems. Extending the work of Nagtegaal et al, it is established theoretically that if lower order finite elements are employed rigorously for non-trivial undrained problems with axial symmetry, then it is impossible to predict the exact limit load accurately, regardless of how refined the mesh may be. (Author/TRRL)

Void nucleation effects on shear localization in porous plastic solids

Abstract: The influence of void nucleation occurring during the deformation history on shear localization is investigated by employing a constitutive model of a rate independent porous plastic solid. Both plastic strain controlled and stress controlled nucleation processes are simulated. Two deformation histories are considered, one corresponding to uniaxial tension and the other to plane strain tension. The enhanced triaxiality at the center of a neck is simulated by application of Bridgman's solution for the stress and deformation state at the minimum section of a necked bar. The destabilizing effect that arises when void nucleation is stress controlled and nucleation occurs over a narrow range of stress is illustrated. Results are also presented employing parameter values representative of spheroidized carbon steels employed in a recent experimental study carried out by Fisher [23] and the predictions of the model are discussed in light of the experimental observations.

Asymptotic analysis of growing plane strain tensile cracks in elastic-ideally plastic solids

Abstract: An exact asymptotic analysis is presented of the stress and deformation fields near the tip of a quasistatically advancing plane strain tensile crack in an elastic-ideally plastic solid. In contrast to previous approximate analyses, no assumptions which reduce the yield condition, a priori, to the form of constant in-plane principal shear stress near the crack tip are made, and the analysis is valid for general Poisson ratio ν. Specific results are given for ν = 0.3 and 0.5, the latter duplicating solutions in previous work by L.I. Slepyan, Y.-C. Gao and the present authors. The crack tip field is shown to divide into five angular sectors of four different types ; in the order in which these sweep across a point in the vicinity of the advancing crack, they are : two plastic sectors which can be described asymptotically (i.e., as r → 0, where r is distance from the crack tip) in slip-line terminology as ‘constant stress’ and ‘centered fan’ sectors, respectively ; a plastic sector of non-constant stress which cannot be described asymptotically in terms of slip lines; an elastic unloading sector; and a trailing plastic sector of the same type as that directly preceding the elastic sector. Further, these four different sector types constitute the full set of asymptotically possible solutions at the crack tip. As is known from prior work, the plastic strain accumulated by a material point passing through such a moving ‘centered fan’ sector is O(ln r) as r → 0 ; it is proved in the present work that the plastic strain accumulated by a material point passing through the ‘constant stress’ sector ahead of a growing crack must be less singular than In r as r → 0. As suggested also in earlier studies, the rate of increase of opening gap δ at a point currently at a distance r behind, but very near, the crack tip is given for crack advance under contained yielding by δ = αJσ0+β(σ0E)a ln(Rr) where a is crack length, σ0 is tensile yield strength, E is Young's modulus, J is the value of the J-integral taken in surrounding elastic material, and the parameters α and R are undetermined by the asymptotic analysis. The exact solution for ν = 0.3 gives β = 5.462, which agrees very closely with estimates obtained from finite element solutions. An approximate analysis based on use of slip line representations in all plastic sectors is outlined in the Appendix.

Numerical modeling of intraplate deformation: Simple mechanical models of continental collision

Abstract: We study a model of continental collision in which one of the continents acts as a rigid die indenting the other plate which flows as an incompressible viscoplastic medium. We consider two extreme cases of plane deformation: (1) plane strain which corresponds to an infinitely thick lithospheric plate, and (2) plane stress corresponding to a very thin plate. Deformation of the lithosphere, a thick plate, should be intermediate between those extremes. We found that the flow in the plane strain case is quite similar to that obtained by slip line, theory. The plane stress results are quite different, since in this case most of the plate shortening is taken up by the thickening of the lithosphere. We also explored the role of boundary conditions on the flow, in particular, the role of the side walls containing the flow of the lithosphere. In the case of a free lateral boundary the main feature is a flow of matter toward this free wall and a S-like pattern for the horizontal stress field. For a rigid wall, on the other hand, the plane strain and the plane stress results are quite different. In the first case, there is a large return flow on the sides of the punch, the material being extruded along the only free surface available. In the plane stress case, the return flow disappears, and the material displaced by the penetration of the die tends to thicken the plate. The role of a nonlinear constitutive relation is studied for power law creep. As the power of the flow limit increases, the flow retains its general features, but the deformation localizes creating sharper contrasts between high and low strain rate areas; in plane stress, the effect of nonlinearity is to increase the contrasts in vertical motion. Available data for Asia are discussed in the light of the new results.

Deformation of Ideal Granular Materials

Abstract: The formulation of the equations which govern the deformation and flow of granular materials is an outstanding problem in continuum mechanics. This article concentrates on one model for the mechanical behaviour of granular materials, termed the “double-shearing model”, in which deformation is assumed to occur by simultaneous shearing on the two families of characteristic curves of the stress equations. We review the formulation of the equations of the double-shearing model for plane deformations, and describe the known exact solutions of these equations. Further developments of the plane strain theory are then given in which the velocity equations are expressed in a new form from which some general results are derived. In the last part of the article the theory is extended to include problems in three dimensions. The three-dimensional theory is applied to some special cases of nonplanar deformations, and in particular we formulate a new theory of axially symmetric deformations of granular materials, and apply this theory to the solution of some boundary value problems.

The equilibrium field near the tip of a crack for finite plane strain of incompressible elastic materials

Abstract: This investigation is concerned with the deformations and stresses in a slab of all-around infinite extent containing a traction-free plane crack, under conditions of plane strain. The analysis is carried out within the framework of the fully nonlinear equilibrium theory of homogeneous and isotropic incompressible elastic solids. For a fairly wide class of such materials and general loading conditions at infinity, assymptotic estimates appropriate to the various field quantities near the crack-tips are deduced. For a subclass of the materials considered, these results — in contrast to the analogous predictions of the linearized theory — lead to the conclusion that the crack opens up in the neighborhood of its tips even if the applied loading is antisymmetric about the plane of the crack, (e.g., Mode II loading). It is shown further that the non-linear global crack problem corresponding to such a loading in general cannot admit an antisymmetric solution.

Elastic–Plastic Crack Growth

Abstract: Summary The asymptotic structure of near-tip stress and deformation fields is analyzed for cracks growing in elastic–perfectly plastic solids. A general formulation is presented for materials of arbitrary yield condition and associated flow rule, including anisotropic response, although detailed results are presented only for isotropic materials of the Huber–Mises type. Centered fan sectors of singular straining at a crack tip are shown to be general, independent of details of material response, as are also the types of plastic strain singularities associated with stationary and growing cracks. Previously results for the Huber–Mises material are recovered by specialization from the general formulation, some corrections are made, and recent work on using results of such a near-tip analysis as a basis for predicting plane strain stable crack growth is reviewed briefly.

Saint-Venant End Effects In Composites

Abstract: In this paper, we demonstrate that the neglect of elastic end effects, usually justified by appealing to Saint-Venant's principle, cannot be applied routine ly in problems involving composite materials. In particular, for fiber rein forced composites, the characteristic decay length over which end effects are significant is, in general, several times longer than the corresponding length for isotropic materials. For plane strain or generalized plane stress of a highly anisotropic transversely isotropic (or orthotropic) material, modeling a fiber- reinforced composite, the characteristic decay length is of order b(E/G) 1/2, where b is the maximum dimension perpendicular to the fibers and E, G are the longitudinal Young's modulus and shear modulus respectively. Thus when E/G is large, as for fiber-reinforced composites, end effects are transmitted over a distance which is of the order of several specimen widths. This is in marked contrast with the situation for isotropic materials where decay lengths of one ...

Mixed mode plane stress ductile fracture

Abstract: Mixed mode ductile fractures in thin sheets are shown to be possible. The staggered deep edge notch tension specimen enablesJ p , the plane stress propagation value of theJ integral, and dJ/dα, the rate of increase inJ with crack growth to be measured from the specific work of fracture. TheJ integral can also be separated into its two component modesJ 1 andJ n. For the particular low alloy steel testedJ p is virtually independent of the mode of fracture, but for other materialsJ p may be dependent on the fracture mode.

The cantilever beam under tension, bending or flexure at infinity

Abstract: A novel technique, the method of projection, is applied to the plane strain problems of determining the tractions, and stress intensity factors, at the fixed end of a cantilever beam under tension, bending or flexure at infinity. The method represents a useful alternative to the integral equation method of Erdogan, Gupta and Cook, and possesses certain advantages; in particular it is much easier to extend the present method to the more difficult dynamics case. An unusual feature of the method is that the required tractions are expanded as a series whose terms have the natural role of displacements rather than stresses.

Application of the line-spring model to a cylindrical shell containing a circumferential or axial part-through crack

Abstract: An approximate solution was obtained for a cylindrical shell containing a part-through surface crack. It was assumed that the shell contains a circumferential or axial semi-elliptic internal or external surface crack and was subjected to a uniform membrane loading or a uniform bending moment away from the crack region. A Reissner type theory was used to account for the effects of the transverse shear deformations. The stress intensity factor at the deepest penetration point of the crack was tabulated for bending and membrane loading by varying three dimensionless length parameters of the problem formed from the shell radius, the shell thickness, the crack length, and the crack depth. The upper bounds of the stress intensity factors are provided by the results of the elasticity solution obtained from the axisymmetric crack problem for the circumferential crack, and that found from the plane strain problem for a circular ring having a radial crack for the axial crack. The line-spring model gives the expected results in comparison with the elasticity solutions. Results also compare well with the existing finite element solution of the pressurized cylinder containing an internal semi-elliptic surface crack.

Elastic-Plastic Line-Spring Finite Elements for Surface-Cracked Plates and Shells

Abstract: The elastic line-spring finite element of Parks, et al. [1], which was incorporated into the ABAQUS© finite element program, is extended to include elastic-plastic response for approximate evaluation of the J -integral in surface-cracked plates or shells. J -analysis of a long axial crack in a pressurized cylinder is performed both with the virtual crack extension method and with a single elastic-plastic line-spring element attached to a 180-deg ring of shell elements constrained to (axial) plane strain. Agreement of the two models is generally good, both in the elastic range (as was noted earlier by Buchalet and Bamford) and in the plastic range. An axially cracked, internally pressurized cylinder containing a semi-elliptical flaw of aspect ratio a/c = 1/3, and of varying maximum relative depths a/t has also been analyzed.

Predicting Cutting Forces for Oblique Machining Conditions

Abstract: Using orthogonal (plane strain) machining theory together with certain simplifying assumptions based on experimental observations it is shown how the three components of cutting force in oblique machining can be predicted from a knowledge of the work material flow stress and thermal properties and the cutting conditions. A comparison of predicted and experimental cutting force results is given.

Generalized Structural Strength Criteria of Reinforced Plastics Under Plane Stress

Abstract: Strength criteria for reinforced plastics have been presented considering the possibility of fiber, matrix or bond failure

Edge Effects in Laminated Composite Plates

Abstract: The attenuation of self-equilibrated edge stress states into the interior of a laminated plate composed of an arbitrary number of bonded, elastic, anisotropic layers is investigated in the context of Saint-Venant’s principle using the exponential decay results of Toupin, Knowles, and Horgan. To model the plate’s behavior, a semianalytical method is used with finite element interpolations over the thickness and exponential decay into the plate’s interior. The formulation leads to a second-order algebraic eigensystem whose eigenvalues are the characteristic inverse decay lengths, and corresponding right eigenvectors depict the displacement distributions of self-equilibrated traction states. Orthogonality relations between these right and left eigenvectors of the adjoint system are established. An eigenvector expansion for representing arbitrary self-equilibrated edge tractions is then presented. This approach is useful in revealing the interlaminar effects and their decay rates in a laminated composite plate under plane strain. Two examples are provided where the interlaminar phenomena due to eigenstates of self-equilibrated edge stress are illustrated.

Analysis of the Net Tension Failure Mode in Composite Bolted Joints

Abstract: A failure analysis has been developed to predict the net tension strength of composite bolted joints. The analysis involved: 1) the development of a representative stress profile along the net tension failure plane and 2) the in corporation of this stress distribution into the appropriate failure criterion. A two-dimensional, plane stress finite element analysis employing quadrilateral elements with orthotropic material properties was used to determine the state of stress along the failure plane. The stress distribution along this plane was described by a polynomial expression and was generalized to apply to a wide range of geometries. The "point stress" failure criterion was applied to the generalized stress distribution equation resulting in the desired net tension strength prediction. Development of the failure criterion for a particular material system and laminate configuration involved empirically determining two notch sensitivity parameters, m and C. In this study, notch sensitivity parameters were...

Deformation and cracking of thin second-phase layers on deforming metals at elevated temperature

Abstract: Deformation, cracking, and spalling of oxide layers or protective coatings on high-temperature components during creep may have a significant influence on the lifetime of the components. The continuum-mechanical equations for thin second-phase layers on deforming metals are formulated for plane stress taking into account elastic and creep deformation, as well as stress-independent straining such as thermal expansion. Linear or non-linear viscous sliding of the scale on the metal is allowed, and microscopic models for sliding are discussed. Cracks in the scale are assumed to form once a critical tensile stress is attained. The theory is especially applied to the stress fields and crack formation in scales on round creep-test specimens. The dependence of the calculated crack spacing on the applied strain rate compares well with experimental results reported in the literature for oxide scales on ferritic steel.

Direct solution of plasticity problems in soils by the method of characteristics

Abstract: The method of characteristics is well established as a direct method of solving for the stresses in plasticity deforming soils under conditions of plane strain. The application of similar methods to problems of axial symmetry, and to soils with non-homogeneous properties is described and some illustrative examples given. The method of characteristics may also be used for the solution of the plastic displacement equations, although the exact form which these equations should take is still a matter of controversy. Illustrations are given of displacement calculations using the simplifying assumptions either of an associated flow rule (which is unrealistic for a frictional material) or using a fixed rotation term (which, although it has no physical justification, leads to realistic displacement fields). Refs.

Development of preferred orientation in plane strain deformed limestone: Experiment and theory

Abstract: Carbonate rocks deform preferentially by twin gliding on e={01¯18} and slip on r ={10¯14} and f={02¯21} In polycrystalline aggregates strong textures develop We report on experimentally produced textures in triaxial plane strain geometry with orthorhombic symmetry at 200° C and 400° C Pole figure of the experimentally deformed specimens are compared quantitatively with theoretical simulations based on the Taylor theory using both slip and mechanical twinning as mechanisms Agreement at low and high temperature is satisfactory and documents that models developed for fcc metals can be applied to low symmetry minerals provided that deformation mechanisms are known and that mechanical twinning is properly accounted for Comparison with experimental results indicates that strain was nearly homogeneous at the conditions considered and the same may apply to many geological textures Three texture types are described which are differentiated mainly by the relative importance of e twinning