The initiation of necking in rectangular elastic/plastic specimens under uniaxial and biaxial tension

Necking in a rectangular parallelepiped of incompressible elastic/plastic material under uniaxia tension is studied as a bifurcation problem. Approximate upper-bound bifurcation stresses are found which show that bifurcation can occur immediately after the load on the specimen reaches a maximum if the length of the specimen is sufficiently great compared to the width and thickness. A simple formula applicable to sufficiently thin specimens is obtained for the approximate bifurcation stress. Sufficient conditions for uniqueness are found for elastic/plastic solids subjected to a general homogeneous stress-field. The particular case of the rectangular specimen under equal biaxial tension is investigated further, and the magnitude of the bifurcation stress is found to be very sensitive to the particular boundary conditions imposed.     

Necking of biaxially stretched elastic-plastic circular plates

The necking of an elastic-plastic circular plate under uniform radial tensile loading is investigated both within the framework of the three-dimensional theory and within the context of the plane-stress approximation. Attention is restricted to axisymmetric deformations of the plate. The material behavior is described by two different constitutive laws. One is a finite-strain version of the simplest flow-theory of plasticity and the other is a finite-strain generalization of the simplest deformationtheory, which is employed as a simple model of a solid with a vertex on its yield surface. For an initially uniform plate made of an incompressible material, bifurcation from the uniformly stretched state is studied analytically. The regimes of stress and moduli where the governing axisymmetric three-dimensional equations are elliptic, parabolic or hyperbolic are identified. The plane-stress local-necking mode emerges as the appropriate limiting mode from the bifurcation modes available in the elliptic regime. In the elliptic regime, the main qualitative features of the bifurcation behavior are revealed by the plane-stress analysis, although three-dimensional effects delay the onset of necking somewhat. For the deformation theory employed here, the first bifurcation modes are encountered in the parabolic regime if the hardening-rate is sufficiently high. These bifurcations are not revealed by a plane-stress analysis. For a plate with an initial inhomogeneity, the growth of an imperfection is studied by a perturbation method, by a plane-stress analysis of localized necking, and by numerical computations within the framework of the three-dimensional theory. When bifurcation of the corresponding perfect plate takes place in the elliptic regime, the finite element results show that the plane-stress analysis gives reasonably good agreement with the numerical results. When bifurcation of the corresponding perfect plate first occurs in the parabolic regime, then a bifurcation of the imperfect plate is encountered, that is, the finite element stiffness matrix ceases to be positive definite.     

Asymmetric bifurcations of thick-walled circular cylindrical elastic tubes under axial loading and external pressure

In this paper, we consider bifurcation from a circular cylindrical deformed configuration of a thick-walled circular cylindrical tube of incompressible isotropic elastic material subject to combined axial loading and external pressure. In particular, we examine both axisymmetric and asymmetric modes of bifurcation. The analysis is based on the three-dimensional incremental equilibrium equations, which are derived and then solved numerically for a specific material model using the Adams–Moulton method. We assess the effects of wall thickness and the ratio of length to (external) radius on the bifurcation behaviour.     

Bifurcation of inflated circular cylinders of elastic material under axial loading—I. Membrane theory for thin-walled tubes

Bifurcations of circular cylindrical elastic tubes subjected to inflation combined with axial loading are analysed. Membrane tubes are considered in detail as a background to the more difficult analysis of thickwalled tubes described in the companion paper (Part II). Our results for membranes reinforce and extend those given by R.T. Shield and his co-workers.Two modes of bifurcation are investigated: firstly, a bulging (axisyrmmetric) mode; secondly, a prismatic mode in which the cross-section of the tube becomes non-circular. Necessary and sufficient conditions for the existence of modes of either type are given in respect of an arbitrary (incompressible isotropic) form of elastic strain-energy function. For a closed tube with a fixed axial loading many features of the results have close parallels with recent findings by D.M. Haughton and R.W. Ogden for spherical membranes. On the other hand, some results for tubes with fixed ends have no such parallel. In particular, bifurcation may, under certain conditions, occur before the inflating pressure reaches a maximum. A combination of the two modes is interpreted in terms of bending for a tube under axial compression, and the relative importance of the bending and bulging modes is discussed in relation to the length to radius ratio of the tube. The analytical results are illustrated for specific forms of strain-energy function. Corresponding analysis is given for thick-walled tubes in Part II.     

Necking of pressurized spherical membranes

The necking of spherical membranes subject to a prescribed increase in enclosed volume is investigated. Attention is restricted to axisymmetric deformations. The materials considered are incompressible, isotropic, time-independent and incrementally linear. A complete set of axisymmetric bifurcation modes is considered and a simple relation is found to govern the critical stress for bifurcation into a given mode. The limiting critical stress and the corresponding mode for short wavelengths are investigated and related to the results obtained from an independent local-necking analysis. Two perturbation methods are employed to study the growth of initial imperfections: one is valid for arbitrary modes, but restricted to small deviations from sphericity, and the other is valid only for the local-necking mode, but is not restricted to small deviations. The effect of path-dependent material behavior on the onset of local necking is explored. Path-dependent material behavior is found to encourage the preferential growth of short wavelength imperfections. Path-independent materials are shown to exhibit significant sensitivity to initial imperfections in the localized-necking mode, although this sensitivity is far less than for a path-dependent material. When account is taken of initial material-property inhomogeneities as well as initial thickness imperfections, it seems that no definite conclusion can be drawn concerning the appropriateness or inappropriateness of an explanation of the onset of localized necking based on a smooth yield-surface plasticity theory and assuming the presence of such initial inhomogeneities.     

A Finite Strain Analysis of Cavity Formation at a Rigid Inhomogeneity

The formation of a cavity by inclusion-matrix interfacial separation is examined by analyzing the response of a plane rigid inclusion embedded in an unbounded incompressible matrix subject to remote equibiaxial dead load traction. A vanishingly thin interfacial cohesive zone, characterized by normal and tangential interface force-separation constitutive relations, is assumed to govern separation behavior. Rotationally symmetric cavity shapes (circles) are shown to be solutions of an interfacial integral equation depending on the strain energy density of the matrix, the interface force constitutive relation and the remote loading. Nonsymmetrical cavity formation, under rotationally symmetric conditions of geometry and loading, is treated within the theory of infinitesimal strain superimposed on a given finite strain state. Rotationally symmetric and nonsymmetric bifurcations are analyzed and detailed results, for the Mooney–Rivlin strain energy density and for an exponential interface force-separation law, are presented. For the nonsymmetric rigid body displacement mode, a simple formula for the critical load is presented. The effect on bifurcation behavior of interfacial shear stiffness and other interface parameters is treated as well. In particular we demonstrate that (i) for the smooth interface nonsymmetric bifurcation always precedes rotationally symmetric bifurcation, (ii) unlike rotationally symmetric bifurcation, there is no threshold value of interface parameter for which nonsymmetric bifurcation will not occur and (iii) interfacial shear may significantly delay the onset of nonsymmetric bifurcation. Also discussed is the range of validity of a nonlinear infinitesimal strain theory previously presented by the author (Levy [1]). This revised version was published online in July 2006 with corrections to the Cover Date.     

Effects of material anisotropy and inhomogeneity on cavitation for composite incompressible anisotropic nonlinearly elastic spheres

The effects of material anisotropy and inhomogeneity on void nucleation and growth in incompressible anisotropic nonlinearly elastic solids are examined. A bifurcation problem is considered for a composite sphere composed of two arbitrary homogeneous incompressible nonlinearly elastic materials which are transversely isotropic about the radial direction, and perfectly bonded across a spherical interface. Under a uniform radial tensile dead-load, a branch of radially symmetric configurations involving a traction-free internal cavity bifurcates from the undeformed configuration at sufficiently large loads. Several types of bifurcation are found to occur. Explicit conditions determining the type of bifurcation are established for the general transversely isotropic composite sphere. In particular, if each phase is described by an explicit material model which may be viewed as a generalization of the classic neo-Hookean model to anisotropic materials, phenomena which were not observed for the homogeneous anisotropic sphere nor for the composite neo-Hookean sphere may occur. The stress distribution as well as the possible role of cavitation in preventing interface debonding are also examined for the general composite sphere.     

Analysis of necking based on a one-dimensional model

Dimensional reduction is applied to derive a one-dimensional energy functional governing tensile necking localization in a family of initially uniform prismatic solids, including as particular cases rectilinear blocks in plane strain and cylindrical bars undergoing axisymmetric deformations. The energy functional depends on both the axial stretch and its gradient. The coefficient of the gradient term is derived in an exact and general form. The one-dimensional model is used to analyze necking localization for nonlinear elastic materials that experience a maximum load under tensile loading, and for a class of nonlinear materials that mimic elastic-plastic materials by displaying a linear incremental response when stretch switches from increasing to decreasing. Bifurcation predictions for the onset of necking from the simplified theory compared with exact results suggest the approach is highly accurate at least when the departures from uniformity are not too large. Post-bifurcation behavior is analyzed to the point where the neck is fully developed and localized to a region on the order of the thickness of the block or bar. Applications to the nonlinear elastic and elastic-plastic materials reveal the highly unstable nature of necking for the former and the stable behavior for the latter, except for geometries where the length of the block or bar is very large compared to its thickness. A formula for the effective stress reduction at the center of a neck is established based on the one-dimensional model, which is similar to that suggested by Bridgman (1952).     

Computational analysis of stress-based forming limit curves

This article, through computational analyses, examines the validity of using the stress-based and extended stress-based forming limit curves to predict the onset of necking during proportional loading of sheet metal. To this end, a model material consisting of a homogeneous zone and a zone that has voids (material inhomogeneity) is proposed and used to simulate necking under plane strain and uni-axial stress load paths. Results of the in-plane loading computations are used to construct a strain-based formability limit curve for the model material. This limit curve is transformed into principal stress space using the procedure due to Stoughton [Stoughton, T.B., 2000. A general forming limit criterion for sheet metal forming. International Journal of Mechanical Sciences 42, 1–27]. The stress-based limit curve is then transformed into equivalent stress and mean stress space to obtain an Extended Stress-Based Limit Curve (XSFLC). When subjected to three-dimensional loading, the model material is observed to display a variety of responses. From these responses, a criterion for the applicability of the XSFLC to predict the onset of necking in the model material when it is subjected to three-dimensional loading is obtained. In the context of straight tube hydroforming, to provide support for the use of the XSFLC, it is demonstrated that the criterion is satisfied.     

SECOND ORDER EFFECTS IN AN ELASTIC HALF－SPACE ACTED UPON BY A NON－UNIFORM NORMAL LOAD

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Fluid-pressure eigenstates and bifurcation in tension specimens under lateral pressure

An analysis is presented of an eigenstate that may be significant in deformation processes where part of the surface of a body is subjected to loading by uniform fluid pressure. The ‘fluid-pressure eigenstate’ is a configuration in which quasi-static incremental deformation is possible under surface traction-rates that are related to the instantaneous velocity field in a certain way, the fluid pressure being momentarily stationary. Deformation processes exist such that, given certain rate boundary-conditions, uniqueness of the incremental deformation is guaranteed at every instant up to a fluid-pressure eigenstate. For a cylindrical specimen, of arbitrary cross-section, of elastic/plastic or incompressible, finite elastic material it is shown that the first fluid-pressure eigenstate to be reached on a path of uniform stretching corresponds to the instant at which the ‘effective load’ reaches a maximum. No fluid-pressure eigenstates are reached in isotropic Cauchy-elastic solids under all-round fluid pressure loading provided the physically reasonable conditions that the instantaneous bulk and shear moduli remain positive are satisfied.     

Bifurcation conditions for a thick elastic plate under thrust

A thick rectangular plate of incompressible isotropic elastic material is subjected to a pure homogeneous deformation by tensile forces or thrusts applied to a pair of opposite faces. The theory of small deformations superposed on finite deformations is applied to determine the critical conditions under which bifurcation solutions (i.e. adjacent equilibrium positions) can exist. The adjacent equilibrium positions considered are those for which the superposed deformation is two-dimensional and is coplanar with the loading force and the thickness direction of the plate, the faces of the plate normal to its thickness being force-free. A number of theorems relating to the critical conditions for superposed deformations of the flexural and barreling types are derived under conditions on the strain-energy function more general than those employed in earlier work. It is also shown how these results can be applied to the determination of the bifurcation conditions corresponding to any specified strain-energy function.     

Creep transition in torsion

Stresses for a circular cylinder of compressible material subjected to torsion are derived in closed form for steady state creep. It is shown that the asymptotic solution through stress leads from elastic state to plastic and then to creep and through stress difference leads to the creep state. The effect of compressibility is presented graphically. The results indicate that the value of maximum shear stress for a cylinder of compressible material is greater than that for an incompressible material and increases with the increase in a measure index n. For an incompressible material, as a particular case, the results obtained are the same as given by Marin [9].     

Bifurcation phenomena in the plane compression test

A basic, compression, bifurcation problem is studied by methods similar to those used by R. Hill and J. W. Hutchinson (1975) for the corresponding tension problem. Bifurcations from a state of homogeneous in-plane compression loading are investigated for a rectangular block of incompressible material constrained to undergo plane deformations. The sides of the block are tractionfree, and it is loaded compressively by a uniform, shear-free, relative displacement of its ends. For a wide class of incrementally-linear time-independent materials only two instantaneous moduli enter into the analysis. Diffuse modes of both symmetric and antisymmetric bifurcation are examined in the elliptic regime of the governing equations, and the possibility of localized modes is considered both inside and outside this regime. Lowest bifurcation stresses are computed for essentially the entire range of possible combinations of material properties and geometry, and these are compared with results obtained by Hill and Hutchinson for the tension problem. The limiting case of large thickness (the semi-infinite block) is considered, confirming the results of M. A. Biot (1965).     

Crack tip field in a linear elastic–plastic strain-hardening material

The strip necking model for strain-hardening materials is studied in this paper, in which the stress distributed over the strip necking zone is assumed to be ultimate stress. The bi-linear stress–strain relation which can model certain features of plastic flow is adopted in this model. The stress and strain fields are calculated based on this model in this paper. The size of the strip necking region is determined by balancing the stress intensity factor due to remote loading with that due to assumed closing forces equal to the ultimate tensile strength of the material distributed over the strip necking zone. It is interesting that the strip necking region size and the crack tip opening displacement depend not only on the remote load, but also the material hardening parameters, which is different from the results of strip yield model. The results agree with experiments well, and the model has wider application.     

Thermistor analog study of dynamic shear in model viscoelastic materials

An electric transmission line analog for studies of the dynamic plane shear of an incompressible viscoelastic material which has a temperature dependent viscosity is described. Thermistors are used to simulate the temperature dependent viscosity. Criteria are established for the similarity of the line and the material. Experiments analogous to constant rate of deformation studies of an elastic material and a material with a single viscoelastic relaxation time are described. More detailed experiments analogous to the deformation of a viscous material at a constant rate of deformation and at constant stress are also described. These show phenomena analogous to necking and fracture.     

Stability,bifurcation and post-critical behavior of a homogeneously deformed incompressible isotropic elastic parallelepiped subject to dead-load surface tractions

We study the equilibrium homogeneous deformations of a homogeneous parallelepiped made of an arbitrary incompressible, isotropic elastic material and subject to a distribution of dead-load surface tractions corresponding to an equibiaxial tensile stress state accompanied by an orthogonal uniaxial compression of the same amount. We show that only two classes of homogeneous equilibrium solutions are possible, namely symmetric deformations, characterized by two equal principal stretches, and asymmetric deformations, with all different principal stretches. Following the classical energy-stability criterion, we then find necessary and sufficient conditions for both symmetric and asymmetric equilibrium deformations to be weak relative minimizers of the total potential energy. Finally, we analyze the mechanical response of a parallelepiped made of an incompressible Mooney–Rivlin material in a monotonic dead loading process starting from the unloaded state. As a major result, we model the actual occurrence of a bifurcation from a primary branch of locally stable symmetric deformations to a secondary, post-critical branch of locally stable asymmetric solutions.     

Axisymmetric bifurcation analysis in soils by the tangential-subloading surface model

This article discusses localized bifurcation modes corresponding to shear band formation and diffuse bifurcation modes corresponding to bulge formation for cylindrical soil specimen subjected to an axisymmetric load under undrained conditions. We employ the tangential-subloading surface model, which exhibits the characteristic regimes of the governing equations: elliptic, hyperbolic and parabolic. Also, conditions for shear band formation, shear band inclination, diffuse bulging formation, and the long and short wavelength limits of diffuse bulging modes are discussed in relation to material properties and their state of stress, i.e. the stress ratio and the normal-yield ratio. Tangential-plastic strain rate term is required for the analyses of shear band and diffuse bulging. The shear band and the diffuse bulging are generated in not only normal-yield but also subyield states and they are severely affected by the normal-yield ratio describing the degree of approach to the normal-yield state.     

Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.     

Effect of pure mode I,II or III loading or mode mixity on crack growth in a homogeneous solid

Crack growth is analyzed numerically under combined mode I, II and III loading, or under loading in one of these modes alone. The solid is a ductile metal modelled as elastic–plastic, and the fracture process is represented in terms of a cohesive zone model. The analyses are carried out for conditions of small-scale yielding, with the elastic mixed mode solution applied as boundary conditions on the outer edge of the region analyzed. For pure mode I loading crack growth continued far beyond the maximum fracture toughness shows that the predicted subsequent steady-state toughness is well below the maximum. The reason for this is discussed in terms of the local stress and strain fields around the tip. For pure mode II or mode III loading it is shown that there is no maximum before the steady-state. Also results for different mixed mode conditions are presented and discussed in relation to the results for loading in only one mode. Most of the results are based on assuming that the peak tractions for tangential separation are equal to that for normal separation, but it is shown that a relatively smaller peak traction for tangential separation may significantly affect the predictions.