Non-hardening Material Research Articles

COMBINED EXTRUSION OF GLASSES WITH A CONICAL BOTTOM. EXPERIMENTAL VERIFICATION OF THEORETICAL FORMULAS IN TRADITIONAL EXTRUSION WITHOUT HARDENING

The necessity of experimental verification of the accuracy of the theoretical calculation of the force of traditional extrusion of glasses with a conical bottom is substantiated. The extrusion of a non-hardening material simulating hot deformation is considered. The methods for performing specific calculations of the deforming force for various extrusion geometric parameters, both for free and constrained extrusion, are demonstrated in detail. Comparison with the obtained experimental data shows the high accuracy of the derived calculation formulas.

RADIAL EXTRUSION OF A PIPE BILLET WITH INTERNAL LEDGE. EXTENDED EXPERIMENTAL VERIFICATION OF THEORETICAL FORMULAS FOR EXTRUSION WITHOUT HARDENING

A continuation of the experimental verification of theoretical results, begun in the previous article, is outlined. The design and characteristics of the new experimental stamp are described in detail. Studies of extrusion of a non-hardening material – lead C1, simulating hot deformation, were carried out. The high accuracy of the obtained calculation formulas is confirmed.

RADIAL EXTRUSION OF A PIPE BILLET WITH INTERNAL LEDGE. EXPERIMENTAL VERIFICATION OF THEORETICAL FORMULAS ON AN UNHARDENED LEAD BILLET

The goals of experimental verification of theoretical results are formulated. The characteristics of the tools used, the geometric parameters of extrusion experiments, the strength characteristics of deformable materials, and their lubrication are described in detail. Studies of extrusion of a non-hardening material – lead C00, simulating hot deformation, were carried out. The high accuracy of the obtained calculation formulas is confirmed.

Plane strain wedge indentation revisited

For a plane strain wedge indentation problem, detailed comparisons between slip-line field theory (SLFT) solutions, experimental observations and finite-element analysis results are carried out. Several past remaining questions on this problem are explored. The finite-element computations appear to account for the nature of a modified slip-line field solution (Petryk H. 1980 Journal de Mécanique Appliquée 4 , 255–282) that assumes the existence of an additional fan-shaped slip-line field just above the boundary between yielding and unyielding regions. For obtuse-angled wedges, finite-element computations show that the indentation load could be smaller for a rough wedge than for a smooth wedge. The question that was raised in the literature with respect to the reversed order of required loads for smooth and rough wedges is not necessarily a paradox. It is revealed that with increasing friction coefficient, a distinct deformation mode resembling the dead metal cap solution in the SLFT is established immediately after passing the critical friction coefficient value at which the contact mode theoretically changes from frictional sliding to perfect sticking. The profiles of the bulged-out lip material remain straight in all the finite-element computations for non-hardening materials. Nonlinearity of the lip profiles frequently observed in experiments appears in the presence of strain hardening or strain-rate sensitivity.

PUNCHING CUPS WITH BOTTOM FLANGE BY DIRECT EXTRUSION WITH COUNTER-PUNCH. INITIAL EXPERIMENTAL VERIFICATION OF THEORETICAL RESULTS

The goals of experimental verification of theoretical results are formulated. The results of an experimental verification of the obtained theoretical formulas are presented, which make it possible to determine the most important parameters for extruding glasses with a counter-punch. The characteristics of the tools used, the geometric parameters of the extrusion experiments, the strength characteristics of deformable materials, and their lubrication are described in detail. Studies of extrusion of non-hardening material simulating hot deformation are carried out. The high accuracy of the obtained calculation formulas is confirmed.

Strain fields in cracked bodies under antiplane shear for a generalised non-hardening material law

An exact, closed form, solution is derived for the non-linear stress distribution in a cracked body under antiplane shear deformation. A generalised, non work-hardening, law is introduced to describe the material behaviour, and the stress and strain fields are derived in closed form. Such a new generalised material law includes the effect of a new parameter, a, which allows the transition from the ideally elastic behaviour (low strain regime) to the pure non-linear behaviour (large strain regime) to be modulated. A discussion is carried out on the features of the new solution and on the behaviour of stresses and strains close to and far away from the crack tip.

Effect of Surface Hardening on Dynamic Frictional Rolling Contact Behavior and Degradation of Corrugated Rail

The present study was undertaken to evaluate the effect of surface hardening technology on dynamic frictional rolling contact behavior and degradation of corrugated rail in Shenzhen. Characteristic parameters such as length and depth of corrugation were analyzed by means of a continuous measurement method based on the corrugation analysis trolley. The explicit finite element method for material hardening characteristics and real contact geometry was adopted to set up the 3D transient FE model of wheel and rail, after which the value and distribution of stress/strain as well as contact solutions could be obtained during frictional contact, and then the Archard wear model and simplified wear superposition method are integrated as a numerical simulation tool for rail wear after hardening. The simulation results show that laminar plasma surface hardening technology can increase residual stress and shear stress in quenched zones, leading to local stress concentration at their boundaries; the plastic strain in the matrix material is higher than that in the quenched zones, while the strain concentration is mainly focused on the matrix material. The hardening can remarkably reduce the rail wear along the corrugation wave, and the wear depth of material with hardening technology is about 36% of that of nonhardening material. Laminar plasma surface hardening technology can therefore restrain the development of rail corrugation.

The internal force relationship of rectangular and I-section for bi-linear hardening material with limit strain

Based on bilinear stress-strain constitutive law σ = f (e), the elastic to fully plastic analysis of bending of rectangular-section and bi-symmetrical I-section beams with the influence of axial load is presented for hardening material with limit strain. The variation of the applied bending moment with the axial force for the fully elastic, elastic-plastic, and fully plastic conditions is given in analytical form. The Internal force relationship of the elastic limit is the same for both hardening and non-hardening material and independent of the geometry of the beam section. However, for the elastic-plastic and plastic limits, the relationships are dependent of the hardening parameter β q, limit strain e lim and the geometry of the beam section for neutral axis (N.A) inside the cross section. When N.A outside the cross section, the relationships are dependent of hardening parameter β q and limit strain e lim but independent of the geometry of the beam section. The results given by the analytical expressions reduce to the ones for non-hardening material are in good agreement with the existing results.

Evaluation of the fracture toughness of brittle hardening materials by Vickers indentation

We propose a Vickers indentation cracking-based method for evaluating the fracture toughness of brittle materials exhibiting strain hardening. The approach is an extension of the recent method by Hyun et al. (2015) for non-hardening materials to hardening materials. The hardening material is simplified by an equivalent non-hardening material with an elevated modified yield strain so that the fracture toughness of a hardening material can be evaluated with the formula proposed by Hyun et al. The proposed extension is verified by comparison with experimental results from nanoindentation tests on Ge(100) and Si(100).

Modelling of microstructural evolution on complex paths of large plastic deformation

Abstract A procedure for the modelling of microstructural changes induced by non-uniform large plastic deformation of metals is developed. For a given plastic working process, a set of deformation paths for different initial locations of a material element is generated first for a non-hardening material by using the finite element method. Next, changes of a cellular microstructure and related hardening effects along each path are calculated by using a recently proposed model. The procedure is applied to a non-conventional process of cold extrusion assisted by cyclic rotation of the die. The evolution of microstructural parameters, their effect on strain hardening and a distribution map over the specimen cross-section are calculated.

Accurate and approximate integrations of Drucker–Prager plasticity with linear isotropic and kinematic hardening

Abstract Up to now, some explicit approximate integration schemes based on exponential maps, for non-hardening material obeying Drucker–Prager’s criterion, have been presented. Two new exponential-based approximate formulations, for associative Drucker–Prager plasticity are developed in this article. Both are consistent and explicit algorithms. The linear isotropic and Prager’s kinematic hardening behavior are assumed. Furthermore, an accurate solution for the constitutive equations is derived. The accuracies of the suggested approximate algorithms are assessed by creating related iso-error maps. In addition, by using piecewise strain load histories, and calculating computation times, the robustness and efficiency of the formulations are demonstrated.

Design of thick-walled cylindrical vessels under internal pressure based on elasto-plastic approach

By considering the Bauschinger effect and the yield criterion of Tresca, an exact elasto-plastic analytical solution for a thick-walled cylindrical vessel made of elastic linear-hardening material is derived. Having the working pressure and geometric dimensions of the vessel, the distribution of the hoop and equivalent stresses are optimized in the way that the distribution of stresses becomes smooth in the vessel wall. Based on two optimizing methods of the hoop and equivalent stresses, the best autofrettage pressure is determined. It is shown that this pressure is more than the working pressure and depends on the three following variables: Bauschinger effect, working pressure and geometric dimensions. In the next stage, the main task is to determine the wall thickness having the working pressure. To do this, two different design criteria namely; (i) optimizing the hoop stress distribution and (ii) assuming a suitable percent of yielding in the wall thickness are used. In the last step, for different types of structural materials under different working pressures, a number of different plots are given for the ratio of outer to inner radii and the best autofrettage pressure. It is shown that the design of vessels based on the elasto-plastic methods is much more economic than elastic methods. Also, it is seen that for a non-hardening material, the design of vessel is only done for the working pressure less than unit value.

A new experimental approach to evaluate friction in ring test

Abstract A real-time measuring method is proposed to evaluate the friction by using a ring test. In this approach, two experimental procedures are developed: the image processing that used to measure the radius of contraction inner surface (convex forming) at the high friction condition and the measurement of strain by using strain gage that is used to find the expansion inner radius (concave forming) at the low friction condition. An hardening material, aluminum, annealed at 1100 °F, is used for theoretical comparison and non-hardening materials that consist of tin (63%) and lead (37%) is adopted to calibrate various friction conditions. The size of specimen is machined to the standard size so the ratio of outer diameter to inner diameter to thickness is 6:3:2. A theoretical model of non-constant shear friction approach is used to construct the calibration curves. It is found that these methods present an excellent agreement with the experimental data and the results of prior literatures, specifically in evaluating the friction situation at concave forming.

Thermomechanical characterization of pure polycrystalline tantalum

The thermomechanical behavior of pure polycrystalline tantalum has been characterized over a wide range of strain rates, using the recently developed shear compression specimen [D. Rittel, S. Lee, G. Ravichandran, Experimental Mechanics 42 (2002) 58–64]. Dynamic experiments were carried out using a split Hopkinson pressure bar, and the specimen's temperature was monitored throughout the tests using an infrared radiometer. The results of the mechanical tests confirm previous results on pure Ta. Specifically, in addition to its significant strain rate sensitivity, it was observed that pure Ta exhibits very little strain hardening at high strain rates. The measured temperature rise in the specimen's gauge was compared to theoretical predictions which assume a total conversion of the mechanical energy into heat ( β = 1) [G.I. Taylor, H. Quinney, Proceedings of the Royal Society of London, vol. A, 1934, pp. 307–326], and an excellent agreement was obtained. This result confirms the previous result of Kapoor and Nemat-Nasser [R. Kapoor, S. Nemat-Nasser, Mech. Mater. 27 (1998) 1–12], while a different experimental approach was adopted here. The assumption that β = 1 is found to be justified in this specific case by the lack of dynamic strain hardening of pure Ta. However, this assumption should be limited to non-hardening materials, to reflect the fact that strain hardening implies that part of the mechanical energy is stored into the material's microstructure.

Limit load solution for mismatched welded plate and pressure vessel with a surface crack

Fracture behavior of a structure having a crack in the middle of a weld is influenced by the mechanical properties of the welded joint constituents (weld metal, base metal, heat-affected-zone) and their geometry. Therefore, defect assessment procedures have been developed in order to take into account all those affecting parameters. One of the key point for all those defect assessment procedures is to have limit load solutions of the mismatch structure. Such limit load solutions have been obtained by using 3D finite element solutions for different configurations such as plate with surface crack in tension or pressure vessel with axial external surface crack exposed to internal pressure, since those limit load have been obtained for non-hardening material exponent.

Micromechanics of coalescence in ductile fracture

Significant progress has been recently made in modelling the onset of void coalescence by internal necking in ductile materials. The aim of this paper is to develop a micro-mechanical framework for the whole coalescence regime, suitable for finite-element implementation. The model is defined by a set of constitutive equations including a closed form of the yield surface along with appropriate evolution laws for void shape and ligament size. Normality is still obeyed during coalescence. The derivation of the evolution laws is carefully guided by coalescence phenomenology inferred from micromechanical unit-cell calculations. The major implication of the model is that the stress carrying capacity of the elementary volume vanishes as a natural outcome of ligament size reduction. Moreover, the drop in the macroscopic stress accompanying coalescence can be quantified for many initial microstructures provided that the microstructure state is known at incipient coalescence. The second part of the paper addresses a more practical issue, that is the prediction of the acceleration rate δ in the Tvergaard–Needleman phenomenological approach to coalescence. For that purpose, a Gurson-like model including void shape effects is used. Results are presented and discussed in the limiting case of a non-hardening material for different initial microstructures and various stress states. Predicted values of δ are extremely sensitive to stress triaxiality and initial spacing ratio. The effect of initial porosity is significant at low triaxiality whereas the effect of initial void shape is emphasized at high triaxiality.

Large deformation plasticity and the Poynting effect

Abstract The aims of this paper are fourfold: (1) To develop a set of constitutive equations that are applicable to isotropic inelastic materials with large elastic and plastic strains using the multiconfigurational framework (Rajagopal, K.R., Srinivasa, A.R. Int. J. Plasticity 14 (1998) 945; Rajagopal, K.R., Srinivasa, A.R. Int. J. Plasticity 14 (1998), 948), in such a way as to generalize the central ideas (such as isotropy, constant elastic modulii, quadratic yield surfaces and non-hardening behavior) of the Prandtl–Reuss theory to finite deformations, (2) to examine the consequences of using a physically plausible criterion of maximum rate of mechanical dissipation, (3) to examine the relationship of the resulting models to the classical Prandtl–Reuss theory as well as other possible formulations (specifically those that rely on the use of a maximum plastic work postulate), and (4) to consider the effect of finite elastic strains on the response of the material subject to some simple homogenous deformations. By considering the response under simple shear, it is shown that the elastic-plastic counterpart of the well known Poynting effect in finite elasticity has a profound influence on the post-yield behavior of such materials. In particular, it is shown that this gives rise to a strain softening effect even though the overall response is that of a non-hardening material.

Constraint effects on plastic crack-tip fields for plane strain mode-I, II and III cracks in non-hardening materials

To explore constraint effects on fully plastic crakc-tip fields, analytical solutions are examined for mode-I, II and III loading in non-hardening materials under plane strain conditions. The results reveal that under mode-II and III loading the crack-tip stress fields are unique, and thus can be characterized by a `single parameter'. Under mode-I loading, however, the crack-tip stress field is non-unique but can be characterized by two sets of solutions or `two parameters'. One set of the solutions is the well-known Prandtl field and the other is a plastic T-stress field. This conclusion corroborates the observation of McClintock (1971) that the slip-line field is non-unique for plane strain tensile cracks. A two-term plastic solution which combines the Prandtl field and the plastic T-stress field with two parameters B1 and B2 can then characterize the crack-tip stress field of plane strain mode-I crack over the plastic region and quantify the magnitude of crack-tip constraints. These characters are similar to those for hardening materials. Analyses and examples show that the two-term plastic solution can match well with the slip-line field or finite element results over plastic region. Thus the parameters B1 and B2 can be used to characterize the constraint level for mode-I finite-sized crack specimens in non-hardening materials under plane strain conditions.

Fully plastic crack-tip fields for CCP and DECP specimens under tension in non-hardening materials

Detailed finite element analyses are performed for center cracked plate (CCP) and double edge cracked plate (DECP) in non-hardening materials under plane strain conditions. The objective is to systematically investigate the effects of deformation level, loading type, crack depth and specimen dimension on crack-tip fields and constraints of these two specimens. Special attention is placed on (a) under what conditions the slip-line fields can be present near the crack tip, and (b) determining what deformation mechanism makes the crack-tip fields significantly different in the two specimens at fully plastic state. The results reveal that (a) at load levels much smaller than the limit load (i.e., small-scale yielding) the crack-tip fields are close to the Prandtl field for both specimens, (b) the effects of crack depth a/W on the crack-tip field is not remarkable for CCP, but significant for DECP at the limit load, (c) as L/ W≥2.4 for CCP and L/ W≥2 for DECP, the crack-tip fields are independent of the specimen length L/W, (d) at the limit load, the crack face is under compression for all CCP, and (e) a compression (tensile) zone exists at the crack face of shallow (deep) cracked DECP. Moreover, it is found that there exist tensile and compressive stresses along the vertical centerline of specimen for both CCP and DECP which result in a bending moment M VL . The difference between M VL and the moment generated by the applied far-field loads makes the crack opening stress non-uniform along the remaining ligament. Recall that the slip-line fields for both the CCP and DECP have uniform opening stress along the ligament. At the limit load, therefore, the numerical crack-tip stress fields can only approach to, but cannot attain to, the slip-line fields for both CCP and DECP specimens. In addition, through comparison of the different limit loads given for DECP specimens, the present results indicate that the limit load formula given by Kumar et al. (EPRI, 1981) is valid only for 0.4≤ a/ W≤0.7, whereas the formula of Ewing and Hill (1967) can be used for any crack depth.

Characterization of constraint of fully plastic crack-tip fields in non-hardening materials by the three-term solution

The J- A 2 three-term asymptotic solution for a crack in a power-law hardening material has been successfully used to interpret the constraint effects due to finite specimen geometry and loading configurations. In the current paper, we study the mechanics behavior of the J- A 2 solution for a plane strain mode-I crack in very low hardening materials. The objective is to investigate the validity of the J- A 2 characterization, in an approximate sense, for the fully plastic crack-tip fields in non-hardening materials. In particular, the constraints at the crack tip for several conventional specimen geometries and loading configurations in non-hardening materials are studied under the framework of the J- A 2 description. The results indicate that within the plastic zone ahead of the crack tip the three-term solution can capture the essential features of fully plastic fields in various finite size specimens in non-hardening materials. Consequently A 2 can be effectively used as a constraint parameter in characterizing the crack-tip field in non-hardening materials.