Large Deformation Effects Research Articles

In-Plane Compression: A New Laboratory Forming Technique to Study the Effect of Large Drawing Deformations in Sheets on Their Subsequent Mechanical Behavior

Abstract The deformation mode or strain path imposed during sheet forming can have a strong influence on the mechanical behavior of materials, both during and after forming. To study strain path effects, laboratory forming techniques capable of producing deformed samples that are flat, of sufficient size, and relatively free from strain gradients are desirable. A new laboratory forming technique, called in-plane compression, has been developed in this study to produce samples in which the strain state approximates that observed in the flange of a partially drawn cylindrical deep-drawn cup, that is, ϵ2 < 0, ϵ1 and ϵ3 > 0, where ϵ1, ϵ2, and ϵ3 are the principal logarithmic strains. These samples exhibit good strain uniformity and are of adequate size (31.8 by 76.2 mm by sheet thickness) to permit subsequent measurement of the postformed mechanical properties. For a 552-MPa (80-ksi) yield-strength high-strength low-alloy (HSLA) steel, it is shown that the postformed tensile properties after in-plane compression agree well with those observed after cup drawing, thus validating the use of in-plane compression to simulate drawing deformations.

塑性大変形を伴う接触問題の解析

In this paper we propose the rational computational strategy for the contact problems included the effect of plastic large deformations in order to analyse an axisymmetric progressive plastic buckling of cylindrical shell under an axial compression. The major contents of this study are summarized as follows; 1) the description of the procedure to introduce the constrained conditions forced on the contact surfaces to the incremental virtual work principle through LAGRANGE's multipliers. 2) the explanation of the logic that determine the contact state at each deformation stage and supply the informations for following step. 3) the introduction of the triple iterative scheme of computation which would combine two nonlinear terms, namely the effective plastic loads and the frictional slide loads, and the factor controlling the amplitude of displacement increments at the stage when contact condition change. In connection with this formulation we use the standard FE technique. The result of computation in the case of Al-Alloy tube confirms us that the present strategy is feasible to examine the collapse process and is applicable to certain class of contact problems.

Viscoelastic properties of linear polymers in the fluid state and their transition to the high‐elastic state

AbstractThe experimental results of the Viscoelastic properties of linear polymers of narrow molecular weight distribution (MWD) and of their mixtures have been analyzed and generalized. Based on the study of the properties of polymers of narrow MWD, we propose a classification of high molecular weight compounds. It specifies a distinct boundary between oligomers and polymers, assuming that the most important feature of polymers is the manifestation of large high‐elastic recoverable deformations of entropy character. For polymers to be characterized, not the absolute molecular weight is essential, but the molecular weight referred to the boundary values. The corresponding state for polymers is attained at temperatures 100°C away from the glass temperature. The transition from the fluid to the high‐elastic state with increasing deformation rate (or frequency for cyclic deformation) has been studied. Transition to the high‐elastic state takes place over a narrow stress range (0.1‐1.0 dynes/cm2), independent of molecular weight, whereas the critical deformation rates (frequencies), like viscosity, depend greatly on molecular weight. An increase in the amount of deformation shifts, to u certain extent, this transition to lower Kites of deformation (frequencies). In the region of deformation rates (frequencies) corresponding to the high‐elastic state, the effect of large deformations during shear manifests itself largely in the tear‐off of polymers Iron, the confining surfaces and in specimen rupture. Polydispersity has a strong effect on the properties of polymeric systems. As the rate of deformation is increased, the transition proceeds successively from the higher molecular weight components. This relaxational transition is tantamount to a change of the structure for polymeric systems. It is responsible for non‐linear, particularly, non‐Newtonian behavior of such systems. The transition to the high‐elastic state and all the related phenomena are observed also in concentrated solutions of high molecular weight polymers. The long‐term durability of un‐cured rubbers in the high‐elastic state is described by the same relationships.

Nonlinear finite element analysis of shell structures using the semi-loof element

The Semi-Loof Shell element originally developed by Irons [2] for linear elastic analysis of thin shell structures is formulated to include large deflection and plastic deformation effects. In this paper the details of the finite element formulation of the problem using total Lagrangian coordinate systems are presented and different element matrices are given. For plastic materials following the Prandtl-Reuss flow rule with isotropic strain hardening a multi-layer approach using a subincremental technique is employed. Numerical results on the performance of the element for a variety of applications are presented. These computer studies include complete load-deflection curves into the post-buckling range and comparisons are made with other existing results. Current experience with the element indicates that it is a reliable and competitive element for nonlinear analysis of shells of general geometry.

A finite element method for studying the transient non‐linear thermal creep of geological structures

AbstractA finite element theory, suitable for describing the long‐term transient thermal creep of geomechanical structures where the material obeys an arbitrary type of creep law, is presented. The method takes into account large deformation effects, is stable for the large time steps required to model geophysical phenomena and accurately simulates changing, incompressible, plastic flow fields. Applications of the theory to the prediction of long‐term creep and creep rupture of simple engineering structures, are given. The theory is also applied to predict the thermal creep of layered media to study the mechanics of folding and rift formation.

Large Amplitude Cyclic Deformation of a Crosslinked Epoxy Resin in the Transition Zone

Abstract Absolute values of complex dynamic shear modulus at different frequencies and deformation amplitudes in the transition zone from glassy state to rubbery state were measured for an epoxy resin cured with diethylenetriamine. A sharp decrease of the modulus, truncation of the glassy state region and shifting of the transition zone to lower temperatures have been observed as an effect of large amplitude deformations above certain critical values. The maximum in the modulus values decrease and the minimum of the critical amplitude both take place at glass transition temperature. The values of the maximum and the minimum, as well as the temperature when they occur, depend upon frequency. The transition temperature, determined by the point of intersection of straight lines which express the modulus temperature relation in the glassy state and in the transition zone, can be frequency independent at large deformation amplitudes, contrary to the known behavior of the polymers at small deformation amplitude...

円筒殻の塑性大変形の擬平衡形要素による解析

To analyse the plastic large deformation problems of cylindrical shell by the incremental method, we develope the pseudo-equilibrium type element based on the principle of modified complementary virtual work. Then, we examine the developing process of plastic deformation and stress distributions when a cylinder is subjected to an axial compression force (one of the complicated structural problems involving plastic large deformation, variable boundaries and dynamic effects). Here we treat this problem as the static one and calculate the two cases of boundaries (1) simple supports (2) clamped supports. In the sample, the finite element solutions are performed for L/R=4, R/T=20, T=1mm and 2024-T 4 Al-Alloy material. But it is easy to employ any type of strain hardening rules and to extend the above method to the dynamic problems. As a result, we find that, unlike the elastic solution, one half-wave in the case of simple supports is formed at the end and outwards from the original generating line of the shell. It coincides with the experimental facts.

Nonlinear response of drained clay to footings

An analytical study of the static response of a homogeneous drained clay stratum to footing loads is presented. Clay is modelled as a linear elastic-perfectly plastic material with the Drucker-Prager yield condition and associated flow rule. The effect of large deformations on the response of the clay is included in the analysis. Numerical results are obtained within the framework of the finite element method and a step-by-step integration procedure. In particular a single, strip surface footing bearing on a finite stratum of overconsolidated clay is considered. The footing is assumed to be rigid and the interface between the footing and soil may be either smooth or rough. The base of the soil stratum is rigid and perfectly rough. A plane strain condition is assumed. Load-displacement curves, stress distributions, zones of yielding and velocity fields are presented. It was found that for a rough footing bearing on a weightless clay, the mode of failure corresponds to the so-called Prandtl velocity field. For a smooth footing, however, the failure mechanism contains elements of both the Hill and Prandtl type velocity fields.

Nonlinear response of undrained clay to footings

An analytical study of the static response of a homogeneous undrained clay stratum to footing loads is presented. Clay is modeled as a linear elastic-perfectly plastic material with the von Mises yield condition and associated flow rule. The effect of large deformations on the response of the clay is included in the analysis. Numerical results are obtained within the framework of the finite element method and a step-by-step integration procedure. In particular a single, strip surface footing bearing on a finite stratum of undrained clay is considered. The footing is assumed to be rigid and the interface between the footing and soil may be either smooth or rough. The base of the soil stratum is rigid and perfectly rough. A plane strain condition is assumed. Load-displacement curves, stress distributions, zones of yielding and velocity fields are presented. It was found that for a rough footing bearing on a undrained clay, the mode of failure corresponds to the so-called Prandtl velocity field. For a smooth footing, however, the failure mechanism appears to be a combination of the Hill and Prandtl type velocity fields.

Dynamic transient behaviour of two‐ and three‐dimensional structures including plasticity, large deformation effects and fluid interaction

AbstractThe finite element method is employed in the prediction of the dynamic transient response of two‐ and three‐dimensional solids exhibiting geometric (large deformations) and material (elasto‐plastic) non‐linearities. Explicit time marching schemes are adopted for integration of the dynamic equilibrium equation and a diagonal ‘lumped’ mass matrix is employed with a special procedure applicable to parabolic isoparametric elements. A variety of problems are presented including a solid/fluid interaction situation, and the method is shown to be able to solve economically many problems of dynamic or catastrophic nature which can occur in such structures as nuclear reactors, containment vessels, etc.

Dynamic Response of Membranes with Finite Elements

A general set of finite-element equations is developed for treating the nonlinear dynamic response of membranes consisting of complex nonlinear materials. The approach proposed by Key (1974) for a most complete treatment of the finite-element method to the nonlinear dynamic analysis of solids is used in developing the nonlinear equations of motion for membranes with cubic quadrilateral finite elements. The computations are kept general with respect to material modeling by computing the deformation gradient and requiring a constitutive subroutine to supply the Cauchy stress. An example calculation of the vibration of a square membrane demonstrates the accuracy of the formulation for small deformation theory and shows the nonlinear effect of large deformations.

Effect of initial deformations on wave front propagation in anisotropic plates

The effects of large amplitudes and initial deformations on shock waves and acceleration waves propagating in fiber-reinforced laminated plates are investigated. Three cases are discussed, namely the large amplitude shock under initial in-plane deformations, small amplitude waves under in-plane deformations, and small amplitude waves propagating in a plate with large deflection. It is found that the in-plane force has a substantial effect on the transverse shear mode but little effects on other modes. The large initial deflection, however, is found to have considerable effects on all modes. A general procedure for constructing the wave surfaces is also presented.

Non-linear bending of beams of variable cross-section

For the non-linear bending of cantilever beams of variable cross-section, the effect of large deformations, but with linear elasticity, is considered. The governing integral equation is solved by a numerical iterative procedure. Results for some typical cases are obtained and compared with some of those available in the literature.

Some Effects of Kinetic Heating on the Stiffness of Thin Wings

This paper presents a preliminary analysis of the effects of kinetic healing at supersonic speeds on the torsional and flexural stiffnesses of thin solid wings. The main investigation is based on the small deflexion theory, but the scope of the analysis for torsion is extended to cover the effects of large deformations.