Foam rheology: Relation between extensional and shear deformations in high gas fraction foams

A two-dimensional hexagonal foam cell model is used to derive analytic expressions for the bulk stress tensor and foam microstructure for any small homogeneous deformation. We show that calculations done for deformations where the principal axes of stress and strain coincide, such as in extension, are sufficient to provide all information about shear deformation. The stresses and foam structure for any given strain and initial cell orientation in shear bears a unique relation to a different strain and orientation in extension. Such a mapping is obtained using the assumption that the principal axes of strain and stress corotate with each other. This in turn implies that high gas fraction foams follow the Lodge-Meissner relation, i.e. the ratio of the normal-stress difference to the shear stress equals the shear strain. The spatially periodic structure of foam along with the fact that the cell centers move affinely with the bulk, makes the above assumption a justifiable one.     

Strain measurements on hydrostatically stressed materials using differential-transformer transducers

An apparatus for directly measuring orthogonal strains of hydrostatically stressed polymeric materials is described. The compliance of the principal dimensions of regular-shaped specimens permits calculation of bulk compliance, bulk modulus, and anisotropy that may exist in the material. The application of differential transformers to hydrostatic strain measurements is discussed.     

ZrCuNiAlAg块体非晶合金JH-2模型研究

为了更好的进行ZrCuNiAlAg块体非晶合金药型罩的爆炸成形及侵彻仿真研究,首要就是建立其材料模型。本文结合ZrCuNiAlAg块体非晶合金力学性能试验结果计算得到了材料的JH-2模型参数,研究确定了ZrCuNiAlAg块体非晶合金JH-2模型。为了验证ZrCuNiAlAg块体非晶合金JH-2模型的准确性,采用Autodyn建立了平板撞击试验有限元模型,模拟了ZrCuNiAlAg块体非晶合金材料在高压、高应变率等环境条件下的变形过程,仿真计算得到的靶板背面自由面粒子速度与试验结果相比,速度平均偏差均在3%以内,表明ZrCuNiAlAg块体非晶合金JH-2模型能很好的描述该材料在大变形、高应变率、高压等环境条件下的力学行为,验证了ZrCuNiAlAg块体非晶合金JH-2模型准确性。     

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.     

Non-steady plane-strain ideal plastic flow

Ever since the ideal forming theory has been developed for process design purposes, application has been limited to sheet forming and, for bulk forming, to two-dimensional steady flow. Here, application for the non-steady case was performed under the plane-strain condition based on the theory previously developed. In the ideal flow, material elements deform following the minimum plastic work path (or mostly proportional true strain path) so that the ideal plane-strain flow can be effectively described using the two-dimensional orthogonal convective coordinate system. Besides kinematics, for a prescribed final part shape, schemes to optimize a preform shape out of a class of initial configurations and also to define the evolution of shapes and boundary tractions were developed. Discussions include the two problematic issues on internal tractions and the non-monotonous straining. For demonstration purposes, numerical calculations were made for a bulk part under forging.     

Prediction of forming limit curves using an anisotropic yield function with prestrain induced backstress

The Forming Limit Diagram (FLD), a plot of the maximum major principal strains that can be sustained by sheet materials prior to the onset of localized necking, is a useful concept for characterizing the formability of sheet metal. Both experimental and numerical results in the literature have shown that the level of the FLD is strongly strain path dependent and the prediction of FLD depends on the shape of the initial yield function and its evolution. In this work, to improve the accuracy of FLD prediction under nonlinear strain paths for a given material, the evolution of the yield function is proposed in terms of the changes of its center and its curvature. The center of the subsequent yield surface after preloading and unloading will be determined via a backstress tensor, and the curvature change will be reflected by changing the exponent in the yield function. Both parameters are functions of the effective plastic strain and will be determined using the forming limit strains obtained from two nonlinear tests. Using this approach, a combination of Marciniak–Kuczynski (M–K) analysis (Marciniak, Z., Kuczynski, K. 1967. Limit strains in the processes of stretch-forming sheet metal. Int. J. Mech. Sci. 9, 609.) and a general anisotropic yield criterion developed by Karafillis and Boyce (Karafillis, A.P., Boyce, M.C. 1993. A general anisotropic yield criterion using bounds and transformation weighting tensor, J. Mech. Phys. Solids, 41, 1859) is used to predict nonlinear FLDs of both Al2008-T4 and Al6111-T4. Excellent agreements were obtained between computed FLDs with the experimental data of Graf and Hosford (Graf, A., Hosford, W.F. 1993a. Calculations of forming limit diagrams for changing strain paths. Metall. Trans. A. 24, 2497; Graf, A., Hosford, W.F. 1993b. Effect of changing strain paths on forming limit diagrams of Al 2008-T4. Metall. Trans. A. 24, 2503; Graf, A., Hosford, W.F. 1994. The influence of strain path changes on forming limit diagrams of Al 6111-T4. Int. J. Mech. Sci. 36, 897). This prediction capability provides a powerful tool in the design and optimization process of 3D sheet metal forming where strain path changes are inevitable.     

Principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string

To investigate the principal resonance in transverse nonlinear parametric vibration of an axially accelerating viscoelastic string, the method of multiple scales is applied directly to the nonlinear partial differential equation that governs the transverse vibration of the string. To derive the governing equation, Newton‘s second law, Lagrangean strain, and Kelvin‘s model are respectively used to account the dynamical relation, geometric nonlinearity and the viscoelasticity of the string material. Based on the solvability condition of eliminating the secular terms, closed form solutions are obtained for the amplitude and the existence conditions of nontrivial steady-state response of the principal parametric resonance. The Lyapunov linearized stability theory is employed to analyze the stability of the trivial and nontrivial solutions in the principal parametric resonance. Some numerical examples are presented to show the effects of the mean transport speed, the amplitude and the frequency of speed variation.     

Numerical simulation of roll forming for channel section with outer edge

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Analytical Solutions to Expansion of Cylindrical Cavity in Linear Softening Soil

Based on the results of conventional triaxial compression tests for a soil, a trilinear elasto-plastic model is proposed to simulate the stress-strain softening curve. According to this curve, the constitutive relation between the bulk strain and two pr…     

Failure prediction in anisotropic sheet metals under forming operations with consideration of rotating principal stretch directions

An approximate macroscopic yield criterion for anisotropic porous sheet metals is adopted to develop a failure prediction methodology that can be used to investigate the failure of sheet metals under forming operations. Hill's quadratic anisotropic yield criterion is used to describe the matrix normal anisotropy and planar isotropy. The approximate macroscopic anisotropic yield criterion is a function of the anisotropy parameter R, defined as the ratio of the transverse plastic strain rate to the through-thickness plastic strain rate under in-plane uniaxial loading conditions. The Marciniak–Kuczynski approach is employed here to predict failure/plastic localization by assuming a slightly higher void volume fraction inside randomly oriented imperfection bands in a material element of interest. The effects of the anisotropy parameter R, the material/geometric inhomogeneities, and the potential surface curvature on failure/plastic localization are first investigated. Then, a non-proportional deformation history including relative rotation of principal stretch directions is identified in a critical element of a mild steel sheet under a fender forming operation given as a benchmark problem in the 1993 NUMISHEET conference. Based on the failure prediction methodology, the failure of the critical sheet element is investigated under the non-proportional deformation history. The results show that the gradual rotation of principal stretch directions lowers the failure strains of the critical element under the given non-proportional deformation history.     

An elasto-plastic constitutive model with plastic strain rate potentials for anisotropic cubic metals

An elasto-plastic constitutive model with the plastic strain rate potential was developed for finite element analysis. In the model, isotropic-kinematic hardening was incorporated under the plane stress condition for anisotropic sheet cubic metal forming analysis. The formulation is general enough for any homogeneous plastic strain rate potential (with the first-order homogeneous effective strain rate) but the plastic strain rate potential Srp2004-18p was considered here. Attention was focused on the development of the elasto-plastic transition criterion and the effective stress update algorithm. Also, to assure the quadratic convergence rate in Newton’s method, the elasto-plastic tangent modulus was analytically derived. Accuracy and convergence of the stress update algorithm were assessed by the iso-error maps, whereas stability of the algorithm was confirmed by analytical procedure. Validations were performed for the examples of the circular cup drawing, 2D draw-bending and unconstrained cylindrical bending tests, utilizing aluminum sheet alloys.     

Forming limit diagram prediction of tailor welded blank by modified M–K model

The Marciniak and Kuczynski (M–K) model for necking prediction in sheet metal forming was based on the in-plane forming. Bending which was resulted from out-of-plane forming was not considered in the M–K model. Whereas most of the sheet metal forming processes and also standard test of hemispherical punch for forming limit diagram are out-of-plane forming, it is important to consider bending effect in the M–K model. Therefore, in this study bending strain is added to stretching strain of M–K model and a new model is presented for forming limit diagram (FLD) prediction. This modified M–K (MM–K) model is written in the python programming language and it is used as a post-processing criterion for FLD prediction in the commercial software Abaqus. The MM–K model was used to predict FLD and weld line movement in the tailor welded blank forming. It was found that the predicted results by MM–K model are in a good agreement with experimental data.     

Stability of bubbles in wax-based oleofoams: decoupling the effects of bulk oleogel rheology and interfacial rheology

Oleofoams are dispersions of gas bubbles in a continuous oil phase and can be stabilized by crystals of fatty acids or waxes adsorbing at the oil-air interface. Because excess crystals in the continuous phase form an oleogel, an effect of the bulk rheology of the continuous phase is also expected. Here, we evaluate the contributions of bulk and interfacial rheology below and above the melting point of a wax forming an oleogel in sunflower oil. We study the dissolution behaviour of single bubbles using microscopy on a temperature-controlled stage. We compare the behaviour of a bubble embedded in an oleofoam, which owes its stability to both bulk and interfacial rheology, to that of a bubble extracted from the oleofoam and resuspended in oil, for which the interfacial dilatational rheology alone provides stability. We find that below the melting point of the wax, bubbles in the oleofoam are stable whereas bubbles that are only coated with wax crystals dissolve. Both systems dissolve when heated above the melting point of the wax. These findings are rationalized through independent bulk rheological measurements of the oleogel at different temperatures, as well as measurements of the dilatational rheological properties of a wax-coated oil-air interface.

     

On Computational Issues in Large Deformation Analysis of Rubber Bushings*

ABSTRACT

Accurate bushing analysis requires a locking free finite element formulation, an appropriate selection of the strain energy density function, and an adequate use of bulk modulus to assure numerical stability and accuracy. In this paper, the pressure projection finite element method is employed. The method projects displacement-calculated pressure onto a lower order pressure field, based on the Babuska-Brezzi condition, to avoid volumetric locking and pressure oscillation. Mooney-Rivlin and Cubic strain energy density functions are used to study the material effect on the predicted rubber behavior in tension-compression and shear deformation modes, and the need to use a higher order strain energy density function for bushing analysis is identified. The effect of bulk modulus on bonded rubber behavior in bushings with respect to bushing shape factor is studied, and the minimum allowable bulk modulus to impose incompressibility in bushing analysis is characterized. The load-deflection response of annular bushings subjected to axial, torsional, and radial deformations are analyzed and results are compared to linear approximations. An effort is made to demonstrate how a Mooney-Rivlin model cannot capture load-displacement nonlinearities in bushing axial and torsional deformations. Two- and three-dimensional results are compared and the applicability of two-dimensional analysis is discussed.     

The multi-axial deformation behavior of bulk metallic glasses at high homologous temperatures

In this work, we study the behavior of a recently-developed Lanthanum-based bulk metallic glass under uniaxial and multi-axial stress-states using the constitutive model developed by Thamburaja and Ekambaram (2007). The material parameters in the constitutive model are fitted to match the stress–strain responses obtained from a set of simple compression experiments conducted at temperatures within the supercooled liquid region under a variety of strain rates spanning approximately three decades. With the material parameters calibrated, we show that the aforementioned constitutive model is able to accurately predict the force vs. displacement responses of representative experiments conducted under multi-axial stress-states at temperatures within the supercooled liquid region, namely three-point bending and the superplastic forming of a miniature gear component. In particular, the evolution of the specimen geometry during the deformation under multi-axial loading conditions are also well-predicted by the constitutive model.     

Computational analysis of stress-based forming limit curves

This article, through computational analyses, examines the validity of using the stress-based and extended stress-based forming limit curves to predict the onset of necking during proportional loading of sheet metal. To this end, a model material consisting of a homogeneous zone and a zone that has voids (material inhomogeneity) is proposed and used to simulate necking under plane strain and uni-axial stress load paths. Results of the in-plane loading computations are used to construct a strain-based formability limit curve for the model material. This limit curve is transformed into principal stress space using the procedure due to Stoughton [Stoughton, T.B., 2000. A general forming limit criterion for sheet metal forming. International Journal of Mechanical Sciences 42, 1–27]. The stress-based limit curve is then transformed into equivalent stress and mean stress space to obtain an Extended Stress-Based Limit Curve (XSFLC). When subjected to three-dimensional loading, the model material is observed to display a variety of responses. From these responses, a criterion for the applicability of the XSFLC to predict the onset of necking in the model material when it is subjected to three-dimensional loading is obtained. In the context of straight tube hydroforming, to provide support for the use of the XSFLC, it is demonstrated that the criterion is satisfied.     

Evolution of strain localization in glassy polymers: A numerical study

An explicit numerical implementation is described, for a constitutive model of glassy polymers, previously proposed and validated. Then it is exploited within a Finite Element continuum model, to simulate spontaneous strain localization (necking) occurring during extension of a prismatic bar of a typical glassy polymer. Material parameters for atactic polystyrene are employed. The material model is physically based and highly non-linearly viscoelastic. Three of its principal features are critical in simulations of strain localization: rate-dependence of plastic flow stress; strain-induced structural rejuvenation, represented by increase of Tool’s fictive temperature and leading to pronounced post-yield strain softening; and molecular alignment during extension, giving rise to strain-hardening. In all simulations there is a peak in nominal stress, satisfying the condition for localization to occur. Nevertheless, the simulations show that the process of strain localization varies considerably, depending on details of the extension sequence and on assumed values for certain material parameters. A characteristic feature observed is that strain localization in such a material occurs in two stages. There is an initial spurt associated with strain-softening, followed by a slower growth of localization that eventually subsides, ultimately giving way to uniform extension of the neck. But the details of evolution of the strain distribution vary greatly. The rapidity and severity of localization are increased by decreased temperature, increased strain-rate or greater structural rejuvenation. A simple one-dimensional stability analysis is successful in explaining the results.     

Void coalescence within periodic clusters of particles

The effect of particle clustering on void damage rates in a ductile material under triaxial loading conditions is examined using three-dimensional finite element analysis. An infinite material containing a regular distribution of clustered particles is modelled using a unit cell approach. Three discrete particles are introduced into each unit cell while a secondary population of small particles within the surrounding matrix is represented using the Gurson-Tvergaard-Needleman (GTN) constitutive equations. Deformation strain states characteristic of sheet metal forming are considered; that is, deep drawing, plane strain and biaxial stretching. Uniaxial tensile stress states with varying levels of superimposed hydrostatic tension are also examined.The orientation of a particle cluster with respect to the direction of major principal loading is shown to significantly influence failure strains. Coalescence of voids within a first-order particle cluster (consisting of three particles) is a stable event while collapse of inter-cluster ligaments leads to imminent material collapse through void-sheeting.