Elastic-plastic Transition Research Articles

Solution methods for thermo-elastic-plastic FEM and its application to centrifugal casting mold.

Three solution methods for thermo-elastic plastic finite element analysis were studied with special emphases paid to the treatment of elastic-plastic transition. These are the iterative convergent method using “effective stiffness matrix” and two modified rmin methods I and II. The computational results obtained were compared with each other in terms of the “followability” of the equivalent stress and the flow stress of the material.A simple and efficient algorithm to automatically determine time increment was also presented. It was shown that when this algorithm is combined with the modified rmin method II highly accurate solution can be obtained with less thermal load increments.Computations were performed on the basis of the modified rmin method II on centrifugal casting molds to analyze the influences of casting parameters on the stress-strain state of the molds.The results revealed the importance of the adiabatic effect of refractory coating to enhance the thermal fatigue resistance of the bore surface of mold. The influences of the pouring temperature, the mold wall-thickness and the preheating temperature of mold were also clarified.Mold warping or bowing was also discussed in conjunction with the “plastic work over a mold”, which was found to be decreased considerably by increasing the thermal resistance of coating, by lowering the pouring temperature and by thickening the mold wall.

Acceleration of final cooling of forgings in isothermal annealing

1. The temperature differential in a charge in final cooling of forgings is determined by the difference in cross sections of the forgings, the point of their location in the furnace, and the allowable specified rates and method of cooling. In the case of location of forgings with thin sections at the charging door in acceleration of the process of final cooling by gradual opening of the door the temperature differential between forgings of different cross sections (1000 and 330 mm) reaches 190°C and between the sections of a stepped forging 100°C. 2. Acceleration of final cooling by supplying air through the burners and partial opening of the door provides more stable cooling of the forgings than gradual opening of the door. As the result of elimination of inflow of cold air the temperature differential between the sections of a stepped forging does not exceed 30–60°C. 3. In acceleration of cooling by gradual opening of the door the maximum temperature differential is observed in the 540–350°C range, that is, for the majority of constructional steels in the area of the elastic-plastic transition, while in supplying air through the burners it is in the area of the plastic or the elastic condition of the metal. 4. In supplying air through the burners the cooling rate of the surfaces of the forgings of the charge does not exceed the specified values if the furnace is cooled at rates allowed by the specification. 5. To accelerate the final cooling process of production charges of forgings the method of supplying air through the burners with partial opening of the furnace door, which provides stable cooling of forgings with a minimum temperature differential in the charge, may be recommended.

Two Examples of Pathological Elastic-Plastic Behavior

We present two examples of dynamic elastic-plastic behavior in which the material undergoes infinitely many transitions in a finite time period. In the first example, a dynamic oscillator is excited by a pathological force varying as $t\sin ( t^{ - 1} )$. In the second, a pathological force is constructed in such a manner that the oscillator undergoes elastic-plastic transition at times determined by the Cantor set.

Constitutive Equations of Elastoplastic Materials With Anisotropic Hardening and Elastic-Plastic Transition

Constitutive equations of elastoplastic materials with anisotropic hardening and elastic-plastic transition are presented by introducing three similar surfaces, i.e., a loading surface on which a current stress exists, a subyield surface limiting a size of the loading surface and a distinct-yield surface representing a fully plastic state. The assumption of similarity of these surfaces leads the derived equations to remarkably simple forms. Also a more general rule of the kinematic hardening for the distinct-yield surface is incorporated into the constitutive equations. While they seem to be applicable to various materials, special constitutive equations of metals, for example, are derived from them and are compared with experimental data on a cyclic uniaxial loading of aluminum. A close correlation between theory and experiment is observed in this comparison.

Constitutive equations of elastoplastic materials with elastic-plastic transition

Constitutive equations of elastoplastic materials with elastic-plastic transition

Discussion: “Constitutive Equations of Elastoplastic Materials With Elastic-Plastic Transition” (Hashiguchi, K., 1980, ASME J. Appl. Mech., 47, pp. 266–272)

Discussion: “Constitutive Equations of Elastoplastic Materials With Elastic-Plastic Transition” (Hashiguchi, K., 1980, ASME J. Appl. Mech., 47, pp. 266–272)

Experimental estimation of the interaction parameter eO for ZnN

The use of dislocation measurements in Zinc single crystals reveals the presence of a yield point or flow stress corresponding to a Stages I–II (elastic to elastic-plastic) transition. This is commonly attributed to the presence of interstitials such as N. The mathematical models of Seeger and Dorn and Rajnak are used to evaluate certain microstructural parameters. These correlate favourably with experimental values. The excess work done over that to overcome the Peierls and frictional forces is then calculated per pair of pinning interstitials and equated to the energy of formation of a molecule of Zn 3N 2 which is believed to form. The results show that this assumption does lead to the stochiometry Zn 3N 2, thus verifying mathematical parameters from different equations using work-energy concepts. Use of the mechanical equivalent J also seems to indicate a favourable correlation for the stochiometry of Zn 3N 2, and for e O as calculated calorimetrically.

Abnormalities in the Membrane Material Properties of Hereditary Spherocytes

Mechanical measurements of intrinsic membrane material properties are used to characterize the defect in hereditary spherocyte membrane at a continuum level. The value of the surface elastic shear modulus is two-thirds as large as normal values, and the value of the yield shear resultant is one-third as large as normal values. The viscosity of the surface above the elastic-plastic transition appears normal. Under similar geometric conditions, the force required to fragment a hereditary spherocyte is about one-third as large as the force required to fragment a normal cell.

Constitutive Equations of Elastoplastic Materials With Elastic-Plastic Transition

Constitutive equations of elastoplastic materials with an elastic-plastic transition observed in the loading state after a first yield are presented by introducing a new parameter denoting the ratio of the size of a loading surface in the transitional state to that of a yield surface in the classical idealization which ignores the transitional state. These equations involve a reasonably simplified rule for the kinematic hardening. They would describe reasonably not only the hardening behavior but especially the softening behavior which requires our careful consideration about the elastic-plastic transition. From these equations, moreover, we derive plastic constitutive equations specifically of metals and granular media which exhibit very different plastic behaviors. Besides, brief discussions are provided concerning the existing constitutive equations describing the elastic-plastic transition.

Influence of texture on the analysis of thermoelastic/plastic anisotropy and the initial yielding of zircaloy tubes

Averaging procedures are developed to obtain texture factors relating the thermoelastic properties of single crystals of zirconium to polycrystalline properties. The effect of various approximations to the elastic properties of poly crystalline material and various assumed yield criteria on the transition from elastic to plastic behavior for a tube subjected to various boundary conditions is examined. It is found that thermal anisotropy has a greater effect on the elastic-plastic transition than does elastic anisotropy. The choice of yield condition can also affect appreciably the predicted yielding behavior of a tube.

Elastic-Plastic Deformation Behaviour of Inhomogeneous Materials Including Polycrystals

Elastic-plastic deformation behaviour of inhomogeneous materials such as polycrystals is investigated analytically using a plane geometrical model of inhomogeneous materials and finite element calculation. The deformation behaviour is expressed quantitatively making use of constraint ratio calculated from strain distribution in the model. The change of mean constraint ratio for polycrystals during elastic-plastic deformation is obtained. It is found that deformation proceeds between the conditions under constant strain and under constant stress for the most of the calculated range of plastic strains. When elastic inhomogeneity or anisotropy is present, however, the deformation mode becomes close to the constant strain deformation in the elastic-plastic transition stage. The difference between macroscopic yield of polycrystals and microscopic yield in grains due to the presence of elastic inhomogeneity is also discussed.

Elastic-Plastic Deformation Behaviour of Inhomogeneous Materials Including Polycrystals

Elastic-plastic deformation behaviour of inhomogeneous materials such as polycrystals is investigated analytically using a plane geometrical model of inhomogeneous materials and finite element calculation. The deformation behaviour is expressed quantitatively making use of constraint ratio calculated from strain distribution in the model. The change of mean constraint ratio for polycrystals during elastic-plastic deformation is obtained. It is found that deformation proceeds between the conditions under constant strain and under constant stress for the most of the calculated range of plastic strains. When elastic inhomogeneity or anisotropy is present, however, the deformation mode becomes close to the constant strain deformation in the elastic-plastic transition stage. The difference between macroscopic yield of polycrystals and microscopic yield in grains due to the presence of elastic inhomogeneity is also discussed.

Generalized procedures for the use of plane strain and elastic-plastic fracture mechanics options in the modernization of nuclear standards

In the design of nuclear power plants and the selection of required structural materials, the assurance of reliability in operation is an essential consideration. The need for analytical criteria for defining the adequacy of fracture toughness is particularly acute for pressure vessel materials. The 1972 revision to the American Society of Mechanical Engineers (ASME) Boiler and Pressure Vessel Code (Section III) has adopted linear elastic fracture mechanics (LEFM) methods as a means of assuring fracture-safe operation for nuclear vessels and components. It is noted that LEFM can be used to analyze only the behavior of metals subject to plane strain constraint (i.e. brittle behavior), while many steels when used in structural applications will behave in a ductile fashion. Thus, in the modernization of nuclear codes, there is the additional need to include the full range of fracture mechanics options for system design, that is, elastic-plastic and fully plastic fracture mechanics as well as the linear elastic procedures. The choice of a particular toughness regime for application of the metal (e.g., plane strain or elastic-plastic) can then be made by the designer or regulatory body. It follows that this decision will have a major implication on the selection of nuclear structural materials. This paper describes recent developments in the means for defining the full range of plane strain, elastic-plastic, and plastic fracture mechanics options available to the designer. Comparisons are made between these options and the fracture toughness requirements of the ASME Nuclear Code (Section III). Exisiting dynamic plane strain, K Id , data for structural metals are analyzed in concert with dynamic tear (DT) test trends. The limited temperature region of K Id applicability for these materials is shown to presage the elastic-plastic regime through which sharply increasing stress is required for fracture propagation whereby a leak-before-fail condition is ultimately attained. This phenomenon highlights the need to extend the analytical capabilities for fracture assurance into the non-brittle regime. The DT test is an effective engineering tool which, like the crack opening displacement (COD) concept, can be used to define the elastic-plastic and plastic-constraint transitions. The DT test procedure is fully rationalizable in terms of section size parameters and can be used independently or together with the K Id -temperature trend to predict the onset of the elastic-plastic and plastic regimes as a function of temperature and section thickness.

Analysis of Elastic-Plastic Grid Systems

An incremental load method for analyzing the piecewise linear behavior of elastic-plastic grid systems with various support conditions is presented. The assumption is made that the torsional rigidity of the members is negligible. The equilibrium and compatibility equations are written in terms of bending moments, plastic hinge rotations, load parameters and an elastic-plastic transition function. The use of this transition function allows the formulation to be applicable to every regime as well as facilitates the treatment of the reversal of plastic hinge rotations. A technique of linear scaling is employed to determine the points of transition including the state of incipient plastic collapse. The collapse mechanism is verified by means of the solution of a system of hinge rotation equations. The analysis provides the complete load-deflection history of the grid systems from the elastic regime, through all the intermediate regimes, up to and including the collapse regime. Numerical examples are given to illustrate the method of analysis.

Dynamic Analysis of Inelastic Multi-Degree Systems

A method is developed whereby the dynamic response of statically determinate and indeterminate elastic-plastic beams may be computed with a maximum degree of accuracy and with a minimum of computational difficulty. The development is applied to a mathematical model of the actual beam consisting of a system of concentrated masses connected by massless segments whose stiffnesses are identical to the stiffnesses of the corresponding parts of the original beam. The equations of motion are written in the usual manner, in which the structural resistance to deformation is expressed explicitly in terms of the deflections for any elastic-plastic phase of deformation. These equations of motion are solved by means of a single step forward numerical integration procedure. A method of determining the points of elastic-plastic and plastic-elastic transitions is described that, in conjunction with the numerical integration procedure, is suitable for programming on a high-speed digital computer. Several examples are given to show the versatility of the method, including the response of a continuous beam subjected to a moving load.

Elastic-Plastic Transition in Copper Crystals as Determined by an Etch-Pit Technique

Copper (99.999%) crystals with a dislocation density of 50/mm2 have been prepared. These crystals were stressed by applying a pure bending moment, and they were etched with a dislocation etch either before and after or while the stress was applied. The motion of dislocations was determined by observing the size and nature of the dislocation etch pits. The resolved stress necessary to move grown-in dislocations was about 4 g/mm2. Examples of dislocation motion under stress, then return motion when the stress was removed, and of multiple motion under stress were observed. Multiplication of dislocations occurred at a resolved stress of about 18 g/mm2. The observed phenomena are discussed in terms of simple dislocation theory.