Showing papers on "Plane stress published in 1983"

Ground Response Curves for Rock Tunnels

Abstract: Calculations of support pressure—tunnel convergence relationships, or ground response curves are used to improve understanding or rocksupport interaction and to aid in the dimensioning of tunnel support elements. Methods of response curve calculation are presented for a tunnel of circular cross‐section excavated in a rock mass initially subjected to a hydrostatic in situ stress field. Plane strain conditions are assumed. The two solutions presented use nonlinear peak and residual rock mass strength criteria. Particular consideration is given to the influence of plastic volumetric strains in the rock mass surrounding the tunnel. A closed‐form solution is presented for an elastic‐brittle‐plastic material behavior model in which post‐peak dilatancy occurs at a constant rate with major principal strain. A second solution is presented as a stepwise sequence of calculations for an elastic‐strain, softening‐plastic model in which post‐peak dilatancy occurs at a lower rate with major principal strain in the const...

On the role of strain and stress state in ductile failure

Abstract: U sing plane strain and axisymmetric notched tensile specimens in combination with finite deformation stress analysis, the strain to initiate failure in a range of structural steels is shown to be co-related by the state of stress for both axisymmetric and plane states of strain. The implications of this result for ductile failure terminated by flow localisation is discussed in the light of the theoretical work on localised flow.

Localization of deformation in rate sensitive porous plastic solids

Abstract: The effect of material rate sensitivity on the localization of deformation in a porous visco-plastic solid is examined under plane strain tension and axisymmetric tension conditions. The plastic flow rule proposed by Gurson [3], modified to account for material rate sensitivity, is adopted to model the plastic softening behavior that arises due to void nucleation and growth. An initial imperfection in the form of a planar band is assumed and a material instability is sought as the deformation proceeds.

The Boundary Element Method Applied to Inelastic Problems

Abstract: 1 Introduction and Motivation.- 1.1 Introduction.- 1.2 Literature Survey-Nonlinear Applications.- 1.3 Layout of Notes.- 2 Basic Theory.- 2.1 Introduction.- 2.2 Theory of Elasticity.- 2.3 Inelastic Behaviour of Materials.- 2.4 Governing Equations.- 3 Boundary Element Formulation for Elastic Problems.- 3.1 Introduction.- 3.2 Somigliana's Identity.- 3.3 Fundamental Solutions.- 3.4 Stresses at Internal Points.- 3.5 Boundary Integral Equation.- 3.6 Infinite and Semi-Infinite Regions.- 3.7 Numerical Implementation.- 3.8 Examples - Half-Plane Formulation.- 4 Boundary Element Equations for Inelastic Problems.- 4.1 Introduction.- 4.2 Somigliana's Identity for Inelastic Problems.- 4.3 Internal Stresses.- 4.4 Alternative Boundary Element Formulations.- 4.4.1 Initial Strain.- 4.4.2 Initial Stress.- 4.4.3 Fictitious Tractions and Body Forces.- 4.5 Half-Plane Formulations.- 4.6 Spatial Discretization.- 4.7 Internal Cells.- 5 Elastoplastic Boundary Element Analysis.- 5.1 Introduction.- 5.2 Some Simple Elastoplastic Relations.- 5.3 Initial Strain - Numerical Solution Technique.- 5.4 Examples - Initial Strain Formulation.- 5.4.1 Perforated Aluminium Strip.- 5.4.2 Polystyrene Crazing Problem.- 5.4.3 Plane Strain Punch.- 5.4.4 Thick Cylinder.- 5.5 General Elastoplastic Stress-Strain Relations.- 5.6 Initial Stress-Outline of Solution Techniques.- 5.7 Examples - Kelvin Implementation.- 5.7.1 Notched Tensile Specimen.- 5.7.2 Deep Circular Tunnel.- 5.7.3 Rough Punch.- 5.8 Examples - Half-Plane Implementation.- 5.8.1 Strip Footing.- 5.8.2 Shallow Tunnel.- 6 Viscoplasticity and Creep Using Boundary Elements.- 6.1 Introduction.- 6.2 Rate Dependent Constitutive Equations.- 6.3 Solution Technique.- 6.4 Examples.- 6.4.1 Deep Beam.- 6.4.2 Thin Disc.- 6.4.3 Plate Under Thermal Shrinkage.- 7 General Discussion and Conclusions.- References.- Appendix A Indirect Computation of Principal Values.- Appendix B Stress Rates at Boundary Nodes.- Appendix C Displacements Due to Constant Inelastic Strain Fields.- Appendix D Some Particular Expressions for 2-D Inelastic Problems.

Tectonic stresses in the lithosphere

Abstract: Various types of observables (earthquake focal mechanisms, in situ measurements and geological deformations) give information about the large scale lithospheric stress field. The latter has often been explained by postulating appropriate forces acting at the edges and beneath the plates. This approach ignores the role of mass heterogeneities within the lithosphere. Here we analyze the effect of both boundary and internal forces on the stress pattern and show that both contributions are of comparable magnitude. The presence of internal sources makes the problem three-dimensional. We show however that it can be reduced to a two-dimensional plane stress formulation, whereby the edge forces are expressed by the ‘non hydrostatic stresses’ and the basal shear is increased by the addition of a term proportional to the gradient of the mean vertical stress. For the oceanic lithosphere we derive a compression that increases with age. The comparison with geophysical observables yields an upper bound of a few bars on the magniude of the basal drag. For the continents we infer the existence of an underlying upper mantle somewhat denser than under oceans.

Large deformations near a tip of an interface-crack between two Neo-Hookean sheets

Abstract: This paper contains an asymptotic investigation - within the nonlinear theory of elastostatic plane stress - of the deformations and stresses near the tips of a traction-free interface-crack between two dissimilar semi-infinite Neo-Hookean sheets. The results obtained are free of oscillatory singularities of the kind predicted by the linearized theory, which would require the two deformed faces of an interface-crack to overlap in the vicinity of its tips. Instead, the crack is found to open smoothly near its ends, regardless of the specific loading at infinity.

An analysis of thep-version of the finite element method for nearly incompressible materials

Abstract: In this paper we analyze the behavior of the so-calledp-version of the finite element method when applied to the equations of plane strain linear elasticity. We establish optimal rate error estimates that are uniformly valid, independent of the value of the Poisson ratio,v, in the interval ]0, 1/2[. This shows that thep-versiondoes not exhibit the degeneracy phenomenon which has led to the use of various, only partially justified techniques of reduced integration or mixed formulations for more standard finite element schemes and the case of a nearly incompressible material.

Dynamic Behavior of Pile Groups

Abstract: The semi‐analytical solution of the dynamic behavior of vertical pile groups is presented. The solution accounts for the pile‐soil‐pile dynamic interaction and is based on modeling the soil as a plane strain continuum and the piles as a set of finite elements. The method is generally enough to model any pile group geometry in a layered soil. The proposed method is simple since it needs no special soil modeling, accurate since its results compared well with the results of finite elements method, and cheaper to use than the finite element method. Some case studies are presented which show that the dynamic pile‐soil‐pile interaction is important for longer distances between the piles than the static case. It is shown also that correction factors due to dynamic pile‐soil‐pile interaction is frequency dependent.

Modeling of large-scale accretionary wedge deformation

Abstract: Instantaneous stress and velocity fields within an active accretionary wedge can be calculated using plastic slip line theory, given the assumptions of a perfectly plastic rheology and plane strain conditions. After selecting a plastic yield stress and an average density, a stress field (slip line field) can be computed for a wedge of known cross-sectional configuration from known stress boundary conditions on the upper surface. Basal surface tractions are derived without basal boundary conditions. The geometry of the computed slip line field can then be used to constrain the lower limit of the yield stress. The known velocity boundary conditions are insufficient for calculation of a velocity field because the coupling between the wedge and the subducting plate is unknown. Consequently, basal velocity distributions must be assumed to permit the construction of velocity fields. These assumed distributions are constrained by known rates of surface uplift and rates of tilt as well as inferred patterns of instantaneous strain rates within a wedge. The Sunda accretionary wedge west of central Sumatra is used to illustrate the plasticity approach. A minimum average yield stress of 20–30 MPa and a weaker basal layer of variable effective strength are indicated for this example. Surface uplift rates may be affected significantly by regions of high basal strain rate, finite velocity discontinuities, and underplating. The known pattern of uplift rates near Nias Island, located on the outer arc ridge, is consistent with the constraint that 90% of the total incremental shortening of the wedge is concentrated within 20 km of the trench. Given a sufficient number of uplift and tilt rate observations across a particular wedge, the relative influence of underplating, tectonic erosion, and basal strain rate variations could be assessed using the perfectly plastic model.

The reflection of a symmetric Rayleigh-Lamb wave at the fixed or free edge of a plate

Abstract: A semi-infinite plate of homogeneous isotropic, linearly elastic material occupies the region x≥0, |y|≤1, -∞

Generic buckling curves for specially orthotropic rectangular plates

Abstract: Using a double affine transformation, the classical buckling equation for specially orthotropic plates and the corresponding virtual work theorem are presented in a particularly simple fashion. These dual representations are characterized by a single material constant, called the generalized rigidity ratio, whose range is predicted to be the closed interval from 0 to 1 (if this prediction is correct then the numerical results using a ratio greater than 1 in the specially orthotropic plate literature are incorrect); when natural boundary conditions are considered a generalized Poisson's ratio is introduced. Thus the buckling results are valid for any specially orthotropic material; hence the curves presented in the text are generic rather than specific. The solution trends are twofold; the buckling coefficients decrease with decreasing generalized rigidity ratio and, when applicable, they decrease with increasing generalized Poisson's ratio. Since the isotropic plate is one limiting case of the above analysis, it is also true that isotropic buckling coefficients decrease with increasing Poission's ratio.

Plastic behavior of aluminum-killed steel following plane-strain deformation

Abstract: Aluminum-killed steel sheets have been subjected to plane-strain prestrain in three ways: two-pass rolling, multi-pass rolling, and inplane, plane-strain tension. Subsequent uniaxial tensile tests were performed to evaluate the residual work-hardening behavior. The subsequent hardening curves depended primarily on the relative direction between major strain axes in the two deformation stages and very little on the specific prestrain procedure. These curves showed high initial yield stresses followed by a region of low (or negative) work hardening rate. This behavior contrasted with earlier results for 70/30 brass sheet, and a model of subsequent tensile behavior based on a strain-induced stress transient emerged.

Dislocation microstructures in steel during deep drawing

Abstract: Simple and complex loadings are performed on low-carbon steel sheets in order to simulate deep drawing. The loading include uniaxial tension, plane strain, equibiaxial stretching, uniaxial tension followed by equibiaxial stretching, and reverse sequence. As the strain is increased dislocations tend to arrange themselves in walls delimiting cells of material in which the dislocation density is low. The cell shape and its evolution depend on the strain and the strain path. Some geometrical parameters (cell size, wall thickness) and wall orientation measurements allow the microstructure to be characterized. In each grain, dislocation walls appear to be connected with active slip planes. Cell size and wall thickness decrease with increasing strain along any strain path. During complex loadings the dislocation microstructure that develops is typical of the last loading path rather than the previous one.

Instabilities of a Finitely Deformed Fiber-Reinforced Elastic Material

Abstract: The instabilities of a finitely deformed Blatz-Ko material reinforced with fibers in a single direction (an anisotropic, nonlinearly elastic material) are examined. The loading is one of plane strain uniaxial stress with the load axis being inclined with respect to the fiber direction. In general one finds a critical applied stress (or stretch) at which surface instabilities of the body occur. This is then followed by a shear band-type instability. For a certain range of fiber orientations these instabilites appear a maximum stress level has been reached. It has also been found that when the fiber direction approaches the loading axis, the material is stabilized in tension but destabilized in compression.

Preseismic rupture progression and great earthquake instabilities at plate boundaries

Abstract: We present a procedure for modeling the initially quasi-static upward progression of a zone of slip from some depth in the lithosphere toward the earth's surface, along a transform plate margin, culminating in a great crustal earthquake. Stress transmission in the lithosphere is analyzed with a generalized Elsasser model, in which elastic lithospheric plates undergo plane stress deformation and are coupled by an elementary foundation model to a Maxwellian viscoelastic asthenosphere. Upward progression of rupture over a finite length of plate boundary, corresponding to a seismic gap along strike, is analyzed by a method based on the ‘line-spring’ concept, whereby a two-dimensional antiplane analysis of the upward progression provides the relation between lithospheric thickness-averaged stress and slip used as a boundary condition in the generalized Elsasser plate model. The formulation results in a nonlinear integral equation for the rupture progression as a function of time and distance along strike. A simpler approximate single degree of freedom analysis procedure is described and shown to lead to instability results that can be formulated in terms of the slip-softening slope at the boundary falling below the elastic unloading stiffness of the surroundings. The results also indicate a delay of ultimate (seismic) instability due to the stiffer short versus long time asthenospheric response and predict a final period of self-driven creep toward instability. The procedures for prediction of rupture progression and instability are illustrated in detail for an elastic-brittle crack model of slip zone advance, and parameters of the model are chosen consistently with great earthquake slips and stress drops. For example, an effective crack fracture energy of the order 4×106 J/m2 at the peak, 7 to 10 km below surface, of a Gaussian bell-shaped distribution of fracture energy with depth, with variance of the order 5 km, simulating strength build-up in a seismogenic layer, leads to prediction of nominal seismic stress drops of 30 to 60 bars and slips of 2 to 5 m in great strike slip earthquake ruptures breaking 100 to 400 km along strike. Precursory surface straining in the self-driven stage is predicted to proceed at a distinctly higher rate over time intervals beginning 3 to 10 months before such an earthquake, this interval being greater for longer distances along strike over which the preseismic upward rupture progression takes place.

On work hardening and springback in plane strain draw forming

Abstract: Based on a simple forcebalance method, an approximate analysis of the plane strain, deep drawing of sheet metal is carried out under the assumptions of Coulomb friction and the membrane theory of shells. Using this analysis with an estimate of the ratio of ultimate to yield load in plane strain enables us to predict the maximum work hardening of the part. This hardening is shown to depend upon the draw bead settings, the frictional conditions on the die and punch, the wall angle of the draw die, and certain uniaxial material properties, chiefly, the ratio of ultimate to yield stress, but also including the rate sensitivity and the anisotropy parameter. The limitations of this theory (mainly, the neglect of bending effects) are discussed. Comparison is made between predicted work hardening and measured yield strengths in a bumper facebar of dual phase steel. Springback of a gently curved part is calculated for preloads and postloads. It is shown that postloads are more effective, and that they can, in principle, reduce springback to low levels in any typical automotive sheet metal. In practice, it is difficult to transmit forces of sufficient magnitude to the region under the punch for certain materials.

Boundary Element Method in Geomechanics

Abstract: 1 Introduction.- 2 Material Behaviour and Numerical Techniques.- 2.1 Introduction.- 2.2 Linear Elastic Material Problems.- 2.3 Nonlinear Elastic Material Problems.- 2.4 Inelastic Material Problems.- 2.5 Time-Dependent Problems.- 3 Boundary Integral Equations.- 3.1 Introduction.- 3.2 Governing Equations and Fundamental Solutions.- 3.3 Integral Equations.- 3.4 Body Force Problem.- 3.5 Prestress Force Problem.- 3.6 Temperature Shrinkage and Swelling.- 4 Boundary Integral Equations for Complete Plane Strain Problems.- 4.1 Introduction.- 4.2 Governing Equations and Fundamental Solutions.- 4.3 Integral Equations for Interior Points.- 4.4 Boundary Integral Equation.- 5 Boundary Element Method.- 5.1 Introduction.- 5.2 Discretization of the Integral Equations.- 5.3 Subregions.- 5.4 Traction Discontinuities.- 5.5 Thin Subregions.- 5.6 Solution Technique.- 5.7 Practical Application of Boundary Element on Linear Problems.- 6 Notension Boundary Elements.- 6.1 Introduction.- 6.2 Rock Material Behaviour.- 6.3 Method of Solution.- 6.4 Application of No-Tension in Rock Mechanics.- 7 Discontinuity Problems.- 7.1 Introduction.- 7.2 Plane of Weakness.- 7.3 Analysis of Discontinuity Problems.- 7.4 Numerical Applications.- 8 Boundary Element Technique for Plasticity Problems.- 8.1 Introduction.- 8.2 Elastoplastic Problems in One Dimension.- 8.3 Theory of Plasticity for Continuum Problems.- 8.4 Numerical Approach for the Plastic Solution.- 8.5 Practical Applications in Geomechanics.- 9 Elasto/Viscoplastic Boundary Element Approach.- 9.1 Introduction.- 9.2 Time-Dependent Behaviour in One Dimension.- 9.3 Elasto/Viscoplastic Constitutive Relations for Continuum Problems.- 9.4 Outline of the Solution Technique.- 9.5 Time Interval Selection and Convergence.- 9.6 Elasto/Viscoplastic Applications.- 10 Applications of the Nonlinear Boundary Element Formulation.- 10.1 Introduction.- 10.2 Strip Footing Problem.- 10.3 Slope Stability Analysis.- 10.4 Tunnelling Stress Analysis.- 11 Conclusions.- References.- Appendices.

The stress intensity factor for a crack perpendicular to the welding bead

Abstract: An analysis of the stress intensity factor due to the residual stress is made for a crack perpendicular to the welding joint in a large plate. The residual stress distribution is represented by a simple function which is chosen to satisfy the physical requirements for the residual stress and to simulate the commonly observed distribution. The stress intensity factor is obtained using customary method based on the superposition principle. The function chosen for the residual stress distribution leads to an exact expression of the stress intensity factor in a simple closed form. The solution yields somewhat conservative values of the stress intensity factor for large cracks and it may be conveniently used for practical applications.

Ductile versus brittle behavior of amorphous metals

Abstract: Ductile behavior of amorphous metals, their ability to sustain localized flow at high nominal stresses, is attributed to a mechanism which alleviates the severe stress conditions prevailing near potential cleavage flaws. The model problem of the plane strain deformation of an infinite block of non-linear visco-elastic material containing an elliptical hole is studied numerically and analytically. A strong dependence of the viscosity on the hydrostatic tension, a result of the increase in the number of viscous flow defects with dilatation, is the principal source of non-linearity. The analysis reveals that, under a constant remote strainrate, the initial elastic stress distribution ahead of the hole gives way, with time, to a more uniform stress distribution. Altering the stress distribution permits the remote (nominal) stress to achieve higher values before critical stress conditions are reached locally at the concentrator. At ordinary temperatures, amorphous metals are not in thermodynamic equilibrium; this motivates a modification of the constitutive law that reflects the kinetic difficulty of maintaining thermodynamic equilibrium under conditions of varying hydrostatic tension. Re-solving the elliptical hole problem with the modified constitutive law reveals a delay in the stress redistribution in front of the concentrator which may explain brittle fracture.

Fracture Penetration Through an Interface

Abstract: This paper presents a combined theoretical and experimental investigation of fracture penetration through an interface. The results have application to hydraulicfracture containment in sandstone or limestone reservoir strata bounded by shale layers. Several simplifying assumptions and approximations are made. We assume that the interface separates dissimilar but adhering materials that are elastic to first order. We consider only differences in material properties and take stresses to be locally uniform. Approximations are made that reduce the fourth-order equation of plane strain elasticity to the second-order Laplace equation. Crack shape is taken to be sinusoidal and fluid leakoff is ignored. Additional simplifying assumptions are made about the shape of the crack tip as it passes through the interface. Using a virtual work analysis, we derive a relation for internal fluid pressure required to extend the crack through the interface. This pressure is related to the equilibrium pressure needed to hold the crack barely