Predictions of bifurcation and instabilities during dynamic extension

Dynamic bifurcation and flow instabilities of cylindrical bars, made of an incompressible strain hardening plastic material, are investigated. A Lagrangian linear perturbation analysis is performed to obtain a fourth order partial differential equation which governs the evolution of the perturbation. The analysis shows that inertia slows down the growth of long wavelengths while bidimensional effects conjugated to strain hardening extinct short wavelengths. The present approach is applied successfully to the analysis of bifurcation and instabilities in (i) a rectangular block during plane strain extension, (ii) a circular bar during uniaxial extension. New results are obtained in the case of rate independent materials and a synthetical point of view is obtained for rate dependent behaviors.     

Evolution of fatigue crack corrosion from surface irregularities

A moving boundary model is presented for crack nucleation and growth from surface flaws. It concerns with chemical attack that results in material dissolution. A controlling mechanism for evolution is the rupture of a brittle corrosion-protective film that is built up along the corroding surface. The evolution rate is a function of the degree of protective film damage caused by the surface straining. The problem is formulated for an elastic body containing a single and double pits. Low-frequency cyclic loading is considered. Numerical solution is proposed. The behaviours of a growing crack and of two competing cracks are described. Stages of incubation, blunting and steady-state growth characterise a single crack evolution. The steady-state growth rate is found independent of the initial geometry. Stages of independent growth, interactive growth and arrest of one crack characterise the evolution of two competing cracks. The lengths of the arrested cracks are presented as functions of the ratio between the pit depth for a series of different distances between the pits. It is emphasized that the solutions correspond to a homogeneous material. Further work is required to account for the material microstructure.     

Second-order bifurcation in elastic-plastic solidS

Second-order rate constitutive equations are formulated for a time-independent elastic-plastic material, obeying the normality flow rule with a smooth yield surface. Under specified regularity restrictions imposed on the involved fields, the regular second-order rate boundary value problem with quasistatic accelerations as unknowns is posed. It is shown that every solution of this generally non-linear rate problem is governed by a variational principle and that the corresponding functional reaches a strict absolute minimum, provided the solution satisfies a sufficient uniqueness condition. With the same incrementally linear comparison solid, Hill's exclusion condition rules out not only a first- but also a second-order bifurcation. The criticality of the exclusion condition is discussed and conditions are indicated under which a second-order bifurcation becomes possible, while the first-order rate problem is still uniquely solvable.     

Bifurcation and stability of homogeneous deformations of an elastic body under dead load tractions with Z 2 symmetry

An analysis is given of bifurcation and stability of homogeneous deformations of a homogeneous, isotropic, incompressible elastic body subject to three perpendicular sets of dead-load surface tractions of which two have equal magnitude. A minimization problem is formulated within the framework of non-linear elasticity, which leads to a bifurcation problem with Z ₂ symmetry. Various bifurcation diagrams are deduced by using singularity theory, and stabilities of solution branches are examined.     

Surface waves in a deformed isotropic hyperelastic material subject to an isotropic internal constraint

An isotropic elastic half-space is prestrained so that two of the principal axes of strain lie in the bounding plane, which itself remains free of traction. The material is subject to an isotropic constraint of arbitrary nature. A surface wave is propagated sinusoidally along the bounding surface in the direction of a principal axis of strain and decays away from the surface. The exact secular equation is derived by a direct method for such a principal surface wave; it is cubic in a quantity whose square is linearly related to the squared wave speed. For the prestrained material, replacing the squared wave speed by zero gives an explicit bifurcation, or stability, criterion. Conditions on the existence and uniqueness of surface waves are given. The bifurcation criterion is derived for specific strain energies in the case of four isotropic constraints: those of incompressibility, Bell, constant area, and Ericksen. In each case investigated, the bifurcation criterion is found to be of a universal nature in that it depends only on the principal stretches, not on the material constants. Some results related to the surface stability of arterial wall mechanics are also presented.     

Uniqueness and stability of rigid-plastic spherical shells subjected to finite deformation under hydrostatic pressure

The rate problem for rigid-plastic strain-hardening deformations in structures subjected to prescribed hydrostatic pressure surface load is stated rigorously with due account of finite deformations. From the basic theory, a complete solution for admissible stress and velocity fields occurring at bifurcation is obtained for the problem of a spherical shell under arbitrary combinations of internal and external pressures. An earlier proven, sufficient condition for the uniqueness of continuing quasi-static deformation of a spherical shell is shown to be one of necessity. In the case of solely external pressure, it is shown that buckling modes are excluded by attention to an isotropically strain-hardening material with a non-singular yield surface. For preponderant internal pressure, however, it is possible for the predicted bifurcation mode to occur under increasing pressure.     

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.     

Prediction of ductile fracture in tension by bifurcation,localization, and imperfection analyses

In this investigation, it is shown that the onset of ductile fracture in tension can be interpreted as the result of a supercritical bifurcation of homogeneous deformation and that this fact can be applied to predict ductile fracture initiation of materials with general imperfections or flaws. We focus on one dimensional quasi-static simple tension for rate-independent isotropic plastic materials. For deformation beyond the bifurcation point, multiple equilibrium paths appear. The homogeneous deformation, as one of the equilibrium paths, loses stability while the inhomogeneous paths are stable, thus indicating the occurrence of strain localization. This investigation also provides a physical example for the application of the Lambert W function in material localization analyses. Material instability is treated as the instability of a static system with dynamic perturbation. We also address the presence of microvoids in a power law plastic material as an unfolding of the supercritical pitchfork bifurcation. The imperfect system, idealized as spherical voids within the plastic matrix, is analyzed using the familiar Gurson model which is based on the presumption of a randomly voided material and characterized by the volume fraction of voids. If, in addition, the sizes of the microvoids are known, this then provides a length scale for the imperfection zone. In this manner, relevance to the sample size effects of strain-to-failure for ductile fracture initiation is addressed by considering separate zones with variations in void volume fractions. Fracture initiation predictions are presented and compare very well to existing experimental results.     

Non-linear interactions in the flexible multi-body dynamics of cable-supported bridge cross-sections

An elastic section model is proposed to analyze some characteristic issues of the cable-supported bridge dynamics through an equivalent planar multi-body system. The quadratic non-linearities of the four-degree-of-freedom model essentially describe the geometric coupling which may strongly characterize the dynamic interactions of the bridge deck and a pair of identical suspension cables (hangers or stays). The linear modal solution shows that the flexural and torsional modes of the deck (global modes) typically co-exist with symmetric or anti-symmetric modes of the cables (local modes). The combinations of parameters which realize remarkable 2:1:1 internal resonance conditions among one of the global modes (with higher natural frequency) and two local modes (with lower and close natural frequencies) are obtained by virtue of a multiparameter perturbation method. The non-linear response of the resonant systems shows that the global deck motion – directly forced at primary resonance by an external harmonic load – can parametrically excite the local cable motion, when the deck vibration amplitude overcomes the critical value at which a period-doubling bifurcation occurs. The relevant effects of both viscous damping and internal detuning on the instability boundaries are parametrically investigated. All the internal resonance conditions as well as the critical vibration amplitudes are expressed as an explicit, though asymptotically approximate, function of the structural parameters.     

Localization in elasto-plastic materials: Influence of the plasticity yield surface in biaxial loading conditions

The influence of the plasticity yield surface on the development of instabilities in plane plates in biaxial loading is analyzed in order to understand and simulate the localization pattern observed in an expanding hemisphere experiment. First, a criterion for the activation of slip bands is formulated in the form of a critical hardening coefficient: it is particularized to the Von Mises and Tresca surfaces. In the Von Mises case, the criterion gives a strongly negative hardening coefficient in biaxial loading conditions different from the ones of plane strain. In the Tresca case, the criterion is fulfilled for a perfectly plastic material in uniaxial and biaxial loading; besides, in equi-biaxial loading, two possible orientations for slip bands are exhibited; this can be understood, with a few approximations, by the existence of a vertex point on the Tresca yield surface which give additive degrees of freedom for the direction of the plastic strain rate. Second, the development of localization in the loading conditions met in an expanding hemisphere experiment is simulated using both plasticity yield surfaces; whereas the Von Mises simulation does not localize, the Tresca simulation exhibits a pattern composed of a network of shear bands of different orientations; this pattern is not far from the pattern observed experimentally.     

Microstructural effects on shear instability and shear band spacing

A constitutive relation that accounts for the thermally activated dislocation motion and microstructure interaction is used to study the stability of a homogeneous solution of equations governing the simple shearing deformations of a thermoviscoplastic body. An instability criterion and an upper bound for the growth rate of the infinitesimal deformations superimposed on the homogeneous solution are derived. By adopting Wright and Ockendon's postulate, i.e., the wavelength of the dominant instability mode with the maximum growth rate determines the minimum spacing between shear bands, the shear band spacing is computed. The effect of the initial dislocation density, the nominal strain-rate, and parameters describing the initial thermal activation and the initial microstructure interaction on the shear band spacing are delineated.     

Anisotropic parameter identification using inhomogeneous tensile test

In this contribution, an inverse identification strategy of constitutive laws for elastoplastic behaviour is presented. The proposed inverse algorithm is composed on an appropriate finite element calculation combined with an optimisation procedure. It is applied to identify material anisotropic coefficients using a set up of easy performed laboratory tests. The used experimental data are the plane tensile test and the off axes tensile tests. The identified behaviour models are mainly based on Hill's quadratic yield criterion. Two cases of this yield criterion have been considered: the transverse isotropic and the orthotropic one under an associated and non-associated flow rule assumptions for each case. The yield surface has been assumed to expand isotropically (isotropic strain hardening law) as a function of the plastic work.In order to better describe anisotropic plastic properties of the studied materials, a recently planar anisotropic yield function is used. It is a non-quadratic yield criterion which takes account of anisotropic yield stresses as well as anisotropic strain ratios. It is subsequently shown that the agreement between inverse identification results and experimental measurements were improved.We prove also that the presented strategy is a good alternative to the simplified homogeneous tests assumption, especially for the plane tensile test.     

Richtmyer-Meshkov instability of an interface between two media due to passage of two successive shocks

The instability of a free surface of aluminum after passage of two shocks that follow one after the other at a certain time interval is studied numerically. The first shock is rather strong (the postshock pressure is about 75 GPa). It is shown that if at the moment when the second shock arrives at the free surface, the perturbation evolution is nonlinear, then, in contrast to the linear stage, the change in the growth rate of the amplitude depends weakly on the wavelength of the initial perturbation. A formula is proposed which describes the effect of the second shock on the amplitude growth rate and in which the main structure of Richtmyer's formula is preserved. It is demonstrated that the parameters of the second shock that ensure freezing of the instability can be determined using only the growth rate of the amplitude. Computing Center, Russian Academy of Sciences, Moscow 117967. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 41, No. 1, pp. 28–37, January–February, 2000.     

Void growth and coalescence in single crystals

Void growth and coalescence in single crystals are investigated using crystal plasticity based 3D finite element calculations. A unit cell involving a single spherical void and fully periodic boundary conditions is deformed under constant macroscopic stress triaxiality. Simulations are performed for different values of the stress triaxiality, for different crystal orientations, and for low and high work-hardening capacity. Under low stress triaxiality, the void shape evolution, void growth, and strain at the onset of coalescence are strongly dependent on the crystal orientation, while under high stress triaxiality, only the void growth rate is affected by the crystal orientation. These effects lead to significant variations in the ductility defined as the strain at the onset of coalescence. An attempt is made to predict the onset of coalescence using two different versions of the Thomason void coalescence criterion, initially developed in the framework of isotropic perfect plasticity. The first version is based on a mean effective yield stress of the matrix and involves a fitting parameter to properly take into account material strain hardening. The second version of the Thomason criterion is based on a local value of the effective yield stress in the ligament between the voids, with no fitting parameter. The first version is accurate to within 20% relative error for most cases, and often more accurate. The second version provides the same level of accuracy except for one crystal orientation. Such a predictive coalescence criterion constitutes an important ingredient towards the development of a full constitutive model for porous single crystals.     

Postbifurcation behavior of wrinkles in square metal sheets under Yoshida Test

The stretching of a square sheet along one of its diagonals, called “Yoshida Test”, has been developed to simulate the wrinkling behavior in press forming of steel sheets into autobody panels. The finite deformation, onset of wrinkling, and growth of wrinkles in such a specimen are investigated. Hill's yield criterion for sheet materials having the normal anisotropy and Hill's quasistatic bifurcation criterion are employed. The growth of wrinkles in the finite deformation process is incrementally and numerically determined by a thin shell finite element in a convected coordinate system. The Lagrangian formulation of the thin shell finite element is based on Hill's variational principle for elastic-plastic solids, a modification of Love-Kirchhoff postulates and a quasiconforming element technique. The shell element fulfills the interelement C¹ continuity condition in a variational sense. Reasonable agreements between the present numerical results and available analytical and experimental results are shown.     

Perturbation solution to heat conduction in melting or solidification with heat generation

The Stefan problem involving a source term is considered in this technical note. As an example, planar solidification with time-dependent heat generation in a semi-infinite plane is solved by use of a perturbation technique. The perturbation solution is validated by reducing the problem to the case without heat generation whose exact solution is available. An application to the case with constant heat generation is presented, for which a closed-form solution is obtained. The effects of heat generation and Stefan number on the evolution of solidification are examined using the perturbation solution.     

On the adjacent equilibrium method in the stability theory of conservative elastic continua

The relationship of the adjacent equilibrium method, the regular perturbation method and the energy method for neutral equilibrium is studied. It is shown that unlike the adjacent equilibrium method, the regular perturbation method yields, for the problems under consideration, non-homogeneous perturbation equations and that adjacent states of equilibrium do not exist at the bifurcation point. These results are then compared with the result of the energy criterion for neutral equilibrium V₂[u] = 0. It is found that although the physical arguments are different in the three methods, the resulting stability equations are identical; thus showing why the adjacent equilibrium argument, even for cases when it is incorrect, yields correct critical loads. This is followed by a discussion of an incorrect derivation of a stability condition and a notion about a load type introduced in the stability literature, which are consequences of the assumption of the general existence of adjacent equilibrium states at bifurcation points.     

Bounds to bifurcation stresses in solids with non-associated plastic flow law at finite strain

In the present paper, Hill's theory of bifurcation and stability in solids obeying normality is generalized to include a non-associated flow law. A one-parameter family of linear comparison solids has been found that admits a potential and has the property that if uniqueness is certain for the comparison solid then bifurcation and instability are precluded for the underlying elastic-plastic solid. The uniqueness criterion derived may be used as a device to determine lower bounds to the magnitudes of primary bifurcation and instability stresses which are ordinarily unknown. A second linear solid is introduced whose constitutive relations have the same form as the elastic-plastic solid “in loading”. The first eigenstate of this solid gives an upper bound to the primary bifurcation state of the underlying elastic-plastic solid. The search for the genuine primary bifurcation state is therefore replaced by a search for upper and lower bounds in the situation when normality fails to hold. The theory is applied to problems of homogeneous stress states.