Uniaxial Constitutive Equation Research Articles

Improved concept of representative directions: cluster approach

The concept of representative directions is a method for constitutive modelling that generalises uniaxial constitutive equations to the general multiaxial case. The simplicity of the concept allows both novice and experienced users to develop advanced material models, covering a wide range of nonlinear phenomena. This paper introduces the cluster approach, a new version of the concept that operates with clusters of fibres. Similar to the original concept, the cluster approach ensures objectivity and inherits the thermodynamic consistency of uniaxial models. The paper details the computational algorithms and presents numerical tests, highlighting the advantages of the new approach. Due to the smearing of fibres in orientation space, the cluster approach efficiently represents initially isotropic material behaviour with fewer clusters, making it computationally more efficient than the classical concept of representative directions. As a demonstration, the paper shows the adequacy of the cluster approach in capturing the actual inelastic behaviour of certain polymers and metals.

A damage-coupled unified viscoplastic constitutive model for prediction of forming limits of 22MnB5 at high temperatures

Hot stamping of the boron steel 22MnB5 has been widely used to produce high strength parts in automobile industry. The forming limits of 22MnB5 sheets are governed by the damage evolution in hot stamping. The aim of this work is to formulate a constitutive model to predict the forming limits of 22MnB5 sheets under hot stamping condition according to continuum damage mechanics (CDM). The authors developed a set of damage-coupled viscoplastic constitutive equations to describe the thermal deformation behavior of 22MnB5 steel under uniaxial tensile state. The material constants of uniaxial constitutive equations were determined from the tensile testing data at a temperature range of 600–900 °C and a strain rate range of 0.01–10 s−1. The uniaxial damage-coupled constitutive equations were generalized to the plane stress state by introducing a multiaxial damage factor considering the effect of stress state on damage evolution. Hot Nakajima testing was conducted to get the forming limits of 22MnB5 sheets at different deformation temperatures and punch speeds. The material constants of multiaxial damage factor were calibrated and confirmed from the experimental data of Nakajima testing. The Nakajima test was numerically simulated by implementing the established constitutive equations via the user material subroutine VUMAT. The prediction by the numerical simulation agrees with the experimental results. Using the determined plane stress damage-coupled constitutive model, the forming limits of 22MnB5 sheets can be predicted under the hot stamping condition.

An Industrial Application of the Continuum Damage Mechanics (CDM) Model for Predicting Failure of AA6082 under HFQ® Process

The failure prediction for forming of high strength complex shaped panel made of AA6082 aluminium alloy during the solution heat treatment cold die forming and quenching process (HFQ®), is presented. Experimental trails have been carried out to optimise the blank shape on the success of forming the complex-shaped aerospace panel component (wing stiffener part). Using the traditional set of uniaxial constitutive equations is difficult to predict the forming features in real hot stamping, especially taken the varied stress states during the forming process. A multi-axial viscoplastic constitutive model based on continuum damage mechanics (CDM) has been developed to describe the deformation behaviour of aluminium alloys under hot/warm stamping conditions. By linking such CDM-based constitutive equations into complete finite element solver, PAM-STAMP in this case, via external Subroutine, as a User Defined Material (UDM), the formability for any process conditions, cold or hot and low or high speed, could be predicted. In this paper the CDM model is verified using an experimental forming trail of complex aluminium component. The results show that the CDM model can be used to provide accurate formability and failure predictions.

On the Creep Behavior of Auto Glasses at High Temperatures

The experiments of a creep for auto glasses at high temperatures and different stress levels are conducted. It reveals nonlinear deformation behaviors of auto glass at those temperatures. Univer different temperatures and loadings, the creep curves occur in similar shapes. The primary creep is not obvious, but the steady and accelerating creep processes are dominant. And all fracture data follow the Monkman-Grant relationship, not influenced significantly by temperatures and stresses. Based on the experiments, this paper presents a nonlinear constitutive equation to describe the creep of glasses. It is found in comparison the theoretical prediction with experimental data that the nonlinear uniaxial constitutive equation proposed here is able to better describe the creep process of auto glasses at high temperatures.

Superplastic Flow: Phenomenology and Mechanics. Engineering Materials Series

9R30. Superplastic Flow: Phenomenology and Mechanics. Engineering Materials Series. - KA Padmanabhan (Indian Inst of Tech, Kanpur, 208016, India), RA Vasin (Moscow State Univ, Moscow, 119899, Russia), FU Enikeev (Inst of Metals Superplasticity Prob, Ufa, 450081, Russia). Springer-Verlag, Berlin. 2001. 363 pp. ISBN 3-540-67842-5. $99.00. Reviewed by MJ Zyczkowski (Inst of Mech and Machine Des, Politechnika Krakowska, Cracow Univ of Tech, ul Warszawska 24, 31-155 Krakow, Poland).Superplasticity is defined as the ability to exhibit very large tensile elongation prior to failure. From the viewpoint of solid mechanics, it corresponds to finite-strain plasticity and viscoplasticity including metal forming processes. On the other hand, from the viewpoint of materials science, the most important is the microstructure that determines necessary properties, finding the composition of alloys, the ranges of temperature and strain rate making it possible to obtain large elongations. The classical example of a superplastic material is that of the eutectoid system of approximately 80% aluminum and 20% zinc. The book under review aims at a presentation of superplasticity both from the mechanical and from the metallurgical point of view. It is divided into six chapters and four appendices. The first introductory chapter gives historical background, basic concepts, and typical mathematical descriptions of uniaxial state of stress. Particular attention is paid to Norton’s power creep law and its direct generalization to viscoplasticity—they are the most frequently used in superplasticity. The second chapter, Mechanics of solids, presents fundamental concepts of continuum mechanics, in particular of plasticity and creep, and discusses various aspects of experimental investigations in solid mechanics. Chapter 3, the longest, Constitutive equations for superplastics (80 pp) is in its majority devoted to uniaxial states and just the last 15 pages deal with the general multiaxial states and use tensorial notation. The uniaxial constitutive equations are divided into phenomenological and physical equations. The first group discusses many generalizations of Maxwell’s and Bingham’s bodies and the relevant mechanical models including nonlinear viscous elements. Examples of the materials conforming to particular models are supplied in most cases. Physical equations include microstructural parameters like the grain size, but the authors state that the problem of validity and meaning of physical parameters is beyond the scope of their book. Multiaxial states are restricted to the strain-hardening creep and Ilyushin’s theory of elastic-plastic processes. Chapters 4 and 5 consider applications to the problems of metalworking: bulging, sheet forming, bulk forming, extrusion, drawing, and deformation processing. Numerical methods are discussed in detail, but several analytical solutions are given as well. Short Chapter 6 is devoted to theoretical and experimental perspectives of superplasticity. Two appendices deal with finite-strain kinematics of solids and its applications. The remaining two discuss dimensional analysis and group properties of thermoviscoplasticity. The book is written clearly, from the viewpoint of mechanics, physics, and materials science, with mathematics kept at an intermediate level. One can agree with the authors that it is intended for a broad variety of readers: researchers working in superplasticity and related topics, engineers in industry, and students. It contains many original results of the authors shown at the background of the present state-of-the-art. The list of references quotes 621 entries plus 15 ascribed separately to Chapter 2; they undoubtedly cover the majority of related books and papers. The subject index is well prepared, whereas the author index is lacking. The theoreticians may be disappointed by a scarce treatment of constitutive equations for multiaxial states, in particular valid for finite strains (though such equations may be found in many papers on plasticity or viscoplasticity). On the other hand, the students will be happy not to see very complicated formulas based on difficult concepts introduced in such cases. The book contains some inconsistencies. For example, on page 35, the authors state that the analysis in superplasticity should follow the theory of finite strain, but on the same page they note that all the problems in their book are discussed for small deformation. On page 37, the authors write: “Only logarithmic strain tensor is popular and clear among specialists in superplasticity,” but the general definition of such a tensor is not given (like, for example, by J Betten, Kontinuumsmechanik, Springer 1993, p 44). Nevertheless, in conclusion Superplastic Flow: Phenomenology and Mechanics may be recommended to the researchers, engineers, and students, in view of its originality, comprehensive treatment, valuable references, and formulation of many directions of future research.

Superplastic Injection Forming of Magnesium Alloys

Intrinsic difficulty of metal forming in magnesium alloys must be overcome by new processing. Superplatic injection forming is one of the most promising methodologies. Stroke velocity controlled forming system was developed to evaluate the formability of AZ91 at 573 K both in upsetting and backward-extrusion modes. Uniaxial constitutive equation was used together with consideration of grain growth to predict the stress-strain rate and the stress-stroke relations in upsetting experiments. Double flange, thin-walled cup can be superplastically formed from a cylindrical billet.

Large Deformation Elastic-Plastic Analyses of Steel Damper under Cyclic Loading.

A damper made of coiled steel is often used for base isolation systems of buildings. Since large relative displacements occur at the isolation system, large deformation elastic-plastic analysis is necessary in the designing of the damper. In this paper a method of the analysis using the degenerated beam element is presented. A simple uniaxial constitutive equation that can be used in cyclic loadings is proposed. The equation uses a bilinear function for a skeleton curve and the Ramberg-Osgood functions for hysteresis curves. A simple method for the modeling of slips at boundaries of the damper is also proposed. A method for evaluating parameters of the constitutive equation and the model of the slip is shown. With the use of the identified parameters, the results obtained by the analyses agreed well with those of experiments.

Biaxial deformation of Ti-6Al-4V and Ti-6Al-4V/TiC composites by transformation-mismatch superplasticity

Gas-pressure bulge forming of unreinforced Ti-6Al-4V and TiC-reinforced Ti-6Al-4V was performed while cycling the temperature around the allotropic transformation range of the alloy (880–1020 °C). The resulting domes exhibited very large strains to fracture without cavitation, demonstrating for the first time the use of transformation-mismatch superplasticity under a biaxial state of stress for both an alloy and a composite. Furthermore, much faster deformation rates were observed upon thermal cycling than for control experiments performed under the same gas pressure at a constant temperature of 1000°C, indicating that efficient superplastic forming of complex shapes can be achieved by transformation-mismatch superplasticity, especially for composites which are difficult to shape with other techniques. However, the deformation rate of the cycled composite was lower than for the alloy, most probably because the composite exhibits lower primary and secondary isothermal creep rates. For both cycled materials, the spatial distribution of principal strains is similar to that observed in domes deformed by isothermal microstructural superplasticity and the forming times can be predicted with existing models for materials with uniaxial strain rate sensitivity of unity. Thus, biaxial transformation-mismatch superplasticity can be modeled within the well-known frame of biaxial microstructural superplasticity, which allows accurate predictions of forming time and strain spatial distribution once the uniaxial constitutive equation of the material is known.

Time dependent damage in half-plane part 2: Traction loading

Transient time dependent stress and strain distribution are obtained for a half-plane made of 4340 steel. A traction load is applied dynamically at a global strain rate of 5·10 2 s −1. Unlike the classical theory of incremental plasticity, that assumes the same constitutive relation for all elements, the surface/volume energy density theory derives the uniaxial constitutive equation for each element individually according to the local strain rate and strain rate history. Meterial elements are shown to undergo cooling and heating as a result of thermal/mechanical interaction. The equivalent uniaxial response for the traction boundary condition therefore differed from that of the displacement boundary condition considered in Part 1 [1] of this communication. Effective stress and effective strain variations for the local elements next to the applied load exhibited considerable softening after hardening while the plasticity solution increased monotonically. These differences are attributed to the neglect of dilatational and change in local strain rate effects in plasticity in addition to assuming that unloading is parallel to the load path. Damage of the material elements is also discussed in connection with the surface and volume energy density when they reach their respective critical values.

Uniaxial Constitutive Equation of Ice From Beam Tests

A method is proposed to obtain ice uniaxial stress, strain, strain-rate relations from beam tests. The basic advantage of the proposed analytical technique is that it is a direct method of reducing beam test data. So, no assumption is made with regard to the ice constitutive behavior. The proposed method is an extension of Gillis and Kelly’s procedure to account for different ice response in tension and compression. It is also an extension of the procedure reported by Mayville and Finnie to account for ice response dependence on strain rate. Furthermore, it is shown that the expressions presented by Mayville and Finnie are only valid when the bending moment, with respect to the zero strain axis, is assumed independent of the centroidal extensional strain. A simple example of a linear elastic beam with a Young’s modulus that varies linearly with the beam depth is worked out to show that these earlier given expressions are not applicable in that case.

Singular History Integral for Creep Rate of Concrete

A uniaxial constitutive equation describing the deviations from the linear principle of superposition of aging concrete at constant moisture content and temperature is presented. Both the creep increase (flow) at high stress and the stiffening (adaptation) due to low sustained compressive stress are modeled, the latter being of principal interest. The constitutive equation expresses the creep rate as a history integral with a singular kernel involving the time lag of creep strain. The integral has the property that the strain response is proportional to the stress history but depends nonlinearly on the stress history when nonproportional stress histories are superposed. The double power law for aging creep is a special limiting case. The constitutive relation is also explained in terms of the rate-process (activation energy) theory for the rate of bond ruptures causing creep. For structural creep problems, a corresponding step-by-step integration algorithm which correctly captures the asymptotic properties of the integral is also developed. A good agreement with test data from the literature is achieved.