Plane Strain Loading Conditions Research Articles

Biaxial Tensile Testing of Cruciform Samples to Determine the Impact of Pre-Strain on the Forming Limit Diagram of Aluminum Alloy 5052 through Finite Element Simulation

The present work focuses on the effect of pre-strain on the forming limit curves (FLCs) of aluminium alloy 5052 sheet of 2.5 mm thickness using cruciform samples. To identify the effect of pre-strain on the FLC, the finite element simulations are performed on cruciform samples. The cruciform samples are deformed to a small level of pre-strain in different strain paths. Thereafter, the further deformation under the various strain paths to measure the effect of pre-strain on the forming limits are carried out. Pre-strain in uniaxial, plane strain and biaxial loading conditions are considered in this work. For each case, i.e., uniaxial, plane strain and biaxial conditions two pre-strain conditions are considered. The cruciform samples are carefully thinned in the central region to promote development of large plastic strains there and eventually to neck and fail. The output obtained through simulations are presented in terms of forming limit curves for various strain paths.

Couple-stress effects in a thin film bonded to a half-space

This study investigates the contact mechanics of a thin film laying on an elastic substrate within the context of couple-stress elasticity. It aims to introduce the effects of material internal length scale, which has proven an effective way of modeling structures at micro- to nano-scales, allowing to capture their size-dependent behavior. Specifically, stress analysis for a thin film bonded to a couple-stress elastic half-space is considered under plane strain loading conditions by assuming that both shear stress and couple tractions are exchanged between the thin film and the substrate. The problem is converted to a singular integral equation, which is solved by expanding the shear stress tractions as a Chebyshev series. The results show that the introduction of couple tractions decreases the shear stress tractions and the axial load in the thin film. When the characteristic length is sufficiently small, but still finite, the results for classical elastic behavior are approached.

Soft elastic composites: Microstructure evolution, instabilities and relaxed response by domain formation

This work provides a review of several complementary analytical methods for characterizing the macroscopic response of ‘soft’ elastic composites subjected to large deformations. For this purpose, we first recall the variational definitions of the homogenized response for hyperelastic composites with periodic and random microstructures. We distinguish between the principal solution for the ‘mesoscopic’ stored-energy function, which is based on one-cell-periodic solutions for the periodic microstructures and on the macroscopically uniform solution for the random microstructures, and the ‘relaxed’ or ‘macroscopic’ homogenized stored-energy function, which is obtained after the onset of ‘microscopic’ (periodic on multiple cells) or ‘macroscopic’ (long wavelength) instabilities. The various techniques include variational bounds of the Voigt, and generalized Reuss and Hashin–Shtrikman types, variational estimates based on the use of suitably optimized ‘linear comparison composites,’ and bounds based on the relaxation of the principal solution for the mesoscopic energy by means of a generalized Maxwell procedure consisting in the quasiconvexification of the energy and leading to domain formation. Although the methods are completely general, for simplicity, we review here illustrative results for neo-Hookean laminates and fiber-reinforced elastomers subjected to plane strain loading conditions. Among the main results, it was found that the predicted formation of twins or lamellar domains and associated soft modes of deformation for the laminates is consistent with experimental observations for thermoplastic elastomers with highly ordered lamellar microstructures. In addition, it was found that the predictions for the macroscopic instabilities by means of the relaxation of the mesoscopic energy are generally different (more conservative) and more consistent with the variational definitions of the macroscopic homogenized energy than those based on loss of ellipticity of the incremental elasticity homogenization problem.

Homogenization, macroscopic instabilities and domain formation in magnetoactive composites: Theory and applications

This work deals with the stability and macroscopic response of soft magnetoelastic composites under combined magnetic and mechanical loading. It is known that, as in the purely mechanical setting, soft composites with suitably designed microstructures can undergo macroscopic (long wavelength) instabilities under certain special loading conditions. The mechanism for such instabilities, and their effects on the macroscopic response of the composites, are less well understood. We investigate here both the onset the instabilities, as well as the macroscopic response after the instabilities, by means of a ‘twinning’ or domain formation mechanism. For this purpose, we propose a generalized Maxwell construction, which builds on the work of Ball and James (1987) for shape-memory alloys and properly accounts for mechanical and magnetic compatibility requirements between ‘twins’ or domains. Thus, generalizing the work of Avazmohammadi and Ponte Castañeda (2016) for the purely mechanical response of reinforced elastomers, this is accomplished by means of the relaxation or quasiconvexification of the ‘principal’ homogenized solution, i.e., the solution prior to the onset of an instability. The methodology is then applied to two-phase magnetoelastic laminates under plane-strain loading conditions with a magnetic field applied in the plane of deformation. We find that the principal solution generally loses (strict) global rank-one convexity before it loses strong ellipticity, as the laminate finds stability by forming ‘twinned’ lamellar domains. In addition, we use the estimates to characterize the ‘phase diagrams’ for the laminates and to describe in detail their macroscopic response when the magnetic field is applied either parallel or perpendicular to the layers. We find that the formation of the lamellar domains leads to a softer mechanical response – including perfectly soft modes of deformation – which can be either facilitated or impeded, depending on the direction of the applied magnetic field. This mechanism opens up novel strategies for the design of magnetoactive elastomer sensors and actuators.

Experimental Determination of Fracture Toughness of Woven/Chopped Glass Fiber Hybrid Reinforced Thermoplastic Composite Laminates

Polymer composites have a wide share among engineering materials. It is important that the material properties are known before being used in industrial applications. Damage behavior needs to be determined in order to safely forming of laminated composites. Propagation characteristics of existing cracks for determining damage are among the current research topics of the researchers. In this study, the fracture toughness of the composite structure was investigated by performing compact tensile and compact compression tests for hybrid fiber reinforced polypropylene composite laminates which have three types of composition having various thicknesses and fiber contents, woven and/or chopped glass fiber reinforcement. The critical energy release rates of fiber and matrix in both tensile and compressive fracture cases were determined in pre-cracked specimens under plane-strain loading conditions. The damage mechanisms of the composite materials used in the present study were described as fiber breakage/buckling of longitudinal and matrix crack/crushing of transverse. As a result of the longitudinal tension, the damage progressed gradually as translaminar fiber breaking in materials containing continuous fibers. In the transverse tension process, fiber-matrix separation caused intralaminar deformation in the materials. The highest fracture critical energy release rate was found in the material with the maximal fiber layer.

Investigation on springback behaviours of hexagonal close-packed sheet metals

In this study, a novel analytical method for predicting bending and springback behaviours of hexagonal close-packed (HCP) sheet metals is presented. The proposed analytical approach is developed by using the Cazacu-Barlat 2004 asymmetric yield function and isotropic plastic hardening rule. This model can be used to determine bending moment-curvature relationships and springback of HCP metals under uniaxial and plane strain loading conditions. Furthermore, to capture the nonlinearity in unloading and to improve springback prediction, the variable elastic modulus approach is implemented in the proposed model. The proposed new model reveals that reverse effects of the back force on springback behaviours cannot be found under the plane strain condition, which could not be found by using any existing models. Moreover, the analytical model is implemented into Abaqus via UMAT subroutine for its application in complex cases, and a numerical model is then developed as a showcase. The proposed methods are validated by using those experimental results available in literature. The results show considerable improvements by considering the plane strain condition and nonlinear unloading.

Effect of Material Strength on Ductile Failure of Steel in Pressure Vessel Design

Abstract Pressure vessels and their components are commonly designed with the ASME Boiler and Pressure Vessel Codes. One of the requirements when pursuing the design by analysis route is to design these equipment against ductile local failure criterion provided in the codes. However, the ductile local failure criterion in the ASME codes only accounts for the stress triaxiality (T) as a stress state measure. Recent work has shown that ductile failure highly depends on the stress state characterized by both T and the Lode parameter L, which is related to the third deviatoric stress invariant. In this study, the effect of stress state characterized by both T and L is investigated for six different steel grades with different material strength levels. To establish the ductile failure loci for the six steel grades with respect to T and L, experiments were conducted on two different specimen geometries. The L parameter is controlled by the specimen configuration, where the round notched bar specimen corresponds to axisymmetric tensile conditions (L = −1) and the flat notched specimen corresponds to plane strain loading conditions (L = 0), whereas T is controlled by introducing a notch at the center of the specimens. A Lode sensitivity parameter (LS) is defined based on the experimental results and revealed that the steel grades with ultimate strength higher than a certain threshold value (450 MPa) exhibit sensitivity to the Lode parameter. The Lode sensitivity was quantified, and the results showed that the LS increases with increase in the ultimate strength of the steel grade. The results were incorporated to enhance the original ASME local failure criterion by accounting for T, L, and LS to accurately assess ductile failure in high-strength steels. The application of the enhanced failure locus in a design analysis of a pressure vessel made of a high-strength steel grade is demonstrated, which showed that the original ASME criterion, as compared to the enhanced criterion in this study, is not capable of predicting ductile failure and hence rendering a rather nonconservative design. It is concluded that the enhanced local failure criterion is recommended to be used for the design of pressure vessels and their components made of steel grades with an ultimate strength higher than the threshold value.

On the critical state characteristics of methane hydrate-bearing sediments

Abstract Reasonable prediction of marine ground stability is vital for the safe exploitation of methane gas from the seabed and development of relevant technologies for gas production. Constitutive models based on a critical state soil mechanics framework for methane hydrate-bearing sediments have been proposed in recent decades. They required careful calibration of laboratory results to obtain accurate information on model parameters. This study presents a comprehensive analysis of triaxial shear test data to examine the effects of hydrate saturation on the geomechanical characteristics of methane hydrate-bearing sediments, particularly at the critical state. Multiple critical state lines are identified for methane hydrate-bearing sediments with different levels of hydrate saturation on the specific volume and logarithm of mean stress plane. A rise in hydrate saturation shifts the critical state line of methane hydrate-bearing sediments upward and slightly increases the gradient of critical state line in the v − ln p ′ plane. Non-unique critical state lines are observed for methane hydrate-bearing sediments under plane-strain loading conditions. Considering methane hydrate saturation as an extra dimension of critical states, the critical state line in the plane of specific volume and logarithm of mean stress, becomes a three-dimensional critical state surface.

On the optimality of the variational linear comparison bounds for porous viscoplastic materials

This paper is concerned with the optimality of the variational bounds of the Hashin-Shtrikman type (VHS) for nonlinear composites, first obtained by Ponte Castañeda (1991a) by means of the corresponding HS bounds for suitably optimized linear comparison composites (LCCs). For simplicity, this problem is addressed in the context of porous viscoplastic materials with incompressible, isotropic matrix phase and two-dimensional microstructures, subjected to plane-strain loading conditions. Although special, this case—exhibiting infinite heterogeneity contrast and compressible macroscopic response—is expected to be fully representative of more general three-dimensional porous materials, as well as more general two-phase, well-ordered composites. Thus, it is shown that the VHS bounds, which were originally derived for the class of nonlinear composites with statistically isotropic microstructures, are in fact attained over the larger class of microstructures with anisotropic nonlinear response but isotropic linear response. By appealing to an exact variational representation for the effective potential of finite-rank nonlinear laminates, it is shown that there exist certain values of the applied macroscopic stress for which the finite-rank laminate (closed-cell porous) microstructures attaining the linear HS bounds also attain the nonlinear VHS bound. Explicit results are obtained in the ideally plastic limit for the yield surface of the finite-rank laminates attaining the VHS bound. In particular, the results of the paper highlight the fact that bounds for nonlinear composites are much more sensitive to microstructural details than bounds for linear composites.

On anisotropic plasticity models using linear transformations on the deviatoric stress: Physical constraints on plastic flow in generalized plane strain

The generation of anisotropic yield functions from isotropic yield criteria by applying linear transformations on the deviatoric stress tensor has many advantages for simulations of sheet metal forming. Linear transformations maintain convexity and can enforce incompressibility while multiple transformations can be employed depending upon the severity of the anisotropy to be described. The present study reveals that the use of linear transformations on associated yield functions or plastic potential functions requires a series of constraints to be imposed for generalized plane strain loading conditions to be physically-consistent with the assumptions of pressure-independent plasticity. A recently proposed plastic constraint for shear loading based upon experimental observations is shown to be part of a more general constraint upon the plastic potential that applies whenever the third deviatoric stress invariant is zero. Due to these constraints, models such as an associated Yld2000 may become over-constrained and either a non-associated or higher-order associated yield criterion may be required. A review of the experimental data for plane strain tension and shear in the literature provides strong support for the enforcement of the plastic constraints for FCC and BCC materials but not for HCP materials since plastic flow will be slip or twinning dominant depending upon the stress state. Several case studies are considered to demonstrate that the plastic flow constraints can be satisfied or significantly violated during the calibration procedure. In particular, improvements in the prediction of the thinning distribution in hole expansion tests reported in the literature can be attributed in part to the enforcement of the plane strain constraints on an associated Yld2000 model for a mild steel. Non-associated plasticity allows for a simpler yield function to be readily calibrated using experimental data while a more complex plastic potential can be selected to satisfy plastic anisotropy and the plane strain constraints. Enforcement of the plane strain constraints is not required for a non-associated yield function, but if enforced, will ensure that the highest yield stress will occur in plane strain and should maximize plastic dissipation during plane strain localization.

Limit Strain Characterization in an Aluminum Die-Quenching Process

This work examines the nature of the strain distributions and limit stains during die quenching of a 7000-series aluminum alloy sheet. Forming limit experiments using limiting dome height (LDH) specimens were performed under plane-strain loading conditions using a 100 mm hemispherical Nakazima punch. Strains were measured using in situ stereoscopic digital image correlation (DIC). Two forming processing routes were examined: (i) an intermediate quench and form (IQF) processing route in which the LDH coupons were solutionized, quenched to a preset temperature, and isothermally formed and (ii) a die-quench (DQ) process where the LDH coupons were solutionized, and quenched and formed simultaneously with room temperature (RT) tooling under non-isothermal conditions. The DQ processing route was devised to understand the formability of the alloy under practical die-quenching conditions, while the IQF route was meant to understand the influence of temperature on the formability of the material. The DQ processing route exhibited the best formability from a localization standpoint; however, it was found that at deformations in excess of 0.5 major true strain, an orange-peel defect was present. The IQF process with room temperature tooling may be comparable to a W-temper forming operation. For these conditions, significant Portevin-Le-Chatelier (PLC) bands were present and the formability was approximately 75% less than for the DQ process. All formability results are summarized and a discussion of the interpretation of forming limits under the diffuse necking conditions associated with elevated temperature forming is presented.

On the strength of transversely isotropic rocks

SummaryAccurate prediction of strength in rocks with distinct bedding planes requires knowledge of the bedding plane orientation relative to the load direction. Thermal softening adds complexity to the problem since it is known to have significant influence on the strength and strain localization properties of rocks. In this paper, we use a recently proposed thermoplastic constitutive model appropriate for rocks exhibiting transverse isotropy in both the elastic and plastic responses to predict their strength and strain localization properties. Recognizing that laboratory‐derived strengths can be influenced by material and geometric inhomogeneities of the rock samples, we consider both stress‐point and boundary‐value problem simulations of rock strength behavior. Both plane strain and 3D loading conditions are considered. Results of the simulations of the strength of a natural Tournemire shale and a synthetic transversely isotropic rock suggest that the mechanical model can reproduce the general U‐shaped variation of rock strength with bedding plane orientation quite well. We show that this variation could depend on many factors, including the stress loading condition (plane strain versus 3D), degree of anisotropy, temperature, shear‐induced dilation versus shear‐induced compaction, specimen imperfections, and boundary restraints.

The Formation and Evolution of Shear Bands in Plane Strain Compressed Nickel-Base Superalloy

The formation and evolution of shear bands in Inconel 718 nickel-base superalloy under plane strain compression was investigated in the present work. It is found that the propagation of shear bands under plane strain compression is more intense in comparison with conventional uniaxial compression. The morphology of shear bands was identified to generally fall into two categories: in “S” shape at severe conditions (low temperatures and high strain rates) and “X” shape at mild conditions (high temperatures and low strain rates). However, uniform deformation at the mesoscale without shear bands was also obtained by compressing at 1050 °C/0.001 s−1. By using the finite element method (FEM), the formation mechanism of the shear bands in the present study was explored for the special deformation mode of plane strain compression. Furthermore, the effect of processing parameters, i.e., strain rate and temperature, on the morphology and evolution of shear bands was discussed following a phenomenological approach. The plane strain compression attempt in the present work yields important information for processing parameters optimization and failure prediction under plane strain loading conditions of the Inconel 718 superalloy.

A Note on the Effect of Intermediate Principal Stress on the Onset of Strain Localization in Granular Soils

Shear band formation, as an instability problem, is often analyzed as a bifurcation problem under plane strain loading condition where the effect of intermediate principal stress is ignored. However, the effect of intermediate principal stress has a great impact on the strain localization of sands and it has already been investigated experimentally. In this study, “Single Hardening” constitutive model is employed to capture the stress–strain relationships and to predict the onset of strain localization under the true triaxial condition. The failure in dense sands is found to be caused by shear band initiation near plane strain loading condition and not by the theoretical failure employed in the Single Hardening model.

A constitutive model based on the modified generalized three-dimensional Hoek–Brown strength criterion

A new constitutive model is proposed based on the generalized three-dimensional (3D) Hoek-Brown strength criterion which was proposed by the authors in 2007 and 2008 and later modified in 2013. The model involves a 3D continuous multi-segment plastic flow rule that can not only consider the effect of confining stress on the plastic flow and volumetric deformation, but also avoid the use of uncertain parameters such as dilatancy angle. The constant volume and maximum tensile stress flow rules are adopted at the high compressive confining stress condition, and the tensile stress condition, respectively. A new interpolation between the high compressive confining stress condition and the tensile stress condition is proposed to ensure the continuity of the plastic potential function over the entire stress space. The proposed constitutive model is based on the assumption that the rock mass is elastic-perfectly plastic but can be extended to the case when a rock mass shows strain-hardening behavior. The new constitutive model has been successfully implemented in a 3D finite element code, and validated by analyzing a circular ring under plane-strain loading condition and a model test tunnel with detailed instrumentation and measurements and then comparing the numerical results with the corresponding analytical solutions and experimental results. Finally, the new constitutive model is successfully applied to analyze a real highway tunnel excavated in rock mass with detailed instrumentation and measurements. The obtained tunnel roof and horizontal convergence displacements using the new constitutive model based on the modified GZZ criterion are in good agreement with those from the field measurements, further proving the importance in considering the contribution of intermediate principal stress to rock strength.

Effect of loading conditions on nucleation of nano void and failure of nanocrystalline aluminum: An atomistic investigation

Deformation mechanism, nano void nucleation and failure of nano crystalline aluminum are investigated for uniaxial, plane strain and equi biaxial loading conditions using atomistic simulation technique. Nano deformation twin formation is noticed for all three loading conditions. Grain boundary triple point is the site of nano void nucleation and the fracture mode is intergranular type. Early nano void nucleation and failure are observed in plane strain loading condition in comparison with uniaxial and equi biaxial loading conditions.

Determination of forming limit diagrams of AA6013-T6 aluminum alloy sheet using a time and position dependent localized necking criterion

The forming limit behaviour of AA6013-T6 aluminium alloy sheet was characterized under isothermal conditions at room temperature (RT) and 250°C using limiting dome height (LDH) tests. Full field strain measurements were acquired throughout testing using in situ stereoscopic digital image correlation (DIC) techniques. Limit strain data was generated from the resulting full field strain measurements using two localized necking criteria: ISO12004- 2:2008 and a time and position dependent criterion, termed the “Necking Zone” (NZ) approach in this paper, introduced by Martinez-Donaire et al. (2014). The limit strains resulting from the two localization detection schemes were compared. It was found that the ISO and NZ limit strains at RT are similar on the draw-side of the FLD, while the NZ approach yields a biaxial major limit strain 14.8% greater than the ISO generated major limit strain. At 250°C, the NZ generated major limit strains are 31-34% greater than the ISO generated major limit strains for near uniaxial, plane strain and biaxial loading conditions, respectively. The significant variance in limit strains between the two methodologies at 250°C highlights the need for a validation study regarding warm FLC determination.

Macroscopic constitutive relations for elastomers reinforced with short aligned fibers: Instabilities and post-bifurcation response

This paper is concerned with the characterization of the macroscopic response and possible development of instabilities in a certain class of anisotropic composite materials consisting of random distributions of aligned rigid fibers of elliptical cross section in a soft elastomeric matrix, which are subjected to general plane strain loading conditions. For this purpose, use is made of an estimate for the stored-energy function that was derived by Lopez-Pamies and Ponte Castañeda (2006b) for this class of reinforced elastomers by means of the second-order linear comparison homogenization method. This homogenization estimate has been shown to lose strong ellipticity by the development of shear localization bands, when the composite is loaded in compression along the (in-plane) long axes of the fibers. The instability is produced by the sudden, collective rotation of a band of fibers to partially release the high stresses that develop in the elastomer matrix when the composite is compressed along the stiff, long-fiber direction. Consistent with the mode of the impending instability, a lower-energy, post-bifurcation solution is constructed where “striped domain” microstructures consisting of layers with alternating fiber orientations develop in the composite. The volume fractions of the layers and the fiber orientations within the layers adjust themselves to satisfy equilibrium and compatibility across the layers, while remaining compatible with the imposed overall deformation. Mathematically, this construction is shown to correspond to the rank-one convex envelope of the original estimate for the energy, and is further shown to be polyconvex and therefore quasiconvex. Thus, it corresponds to the “relaxation” of the stored-energy function of the composite, and can in turn be viewed as a stress-driven “phase transition,” where the symmetry of the fiber microstructures changes from nematic to smectic.

A model for porous single crystals with cylindrical voids of elliptical cross-section

This work presents a rate-dependent constitutive model for porous single crystals with arbitrary number of slip systems and orientations. The single crystal comprises cylindrical voids with elliptical cross-section at arbitrary orientations and is subjected to general plane-strain loadings. The proposed model, called modified variational model (MVAR), is based on the nonlinear variational homogenization method, which makes use of a linear comparison porous single crystal material to estimate the response of the nonlinear porous single crystal. The MVAR model is validated by periodic finite element simulations for a large number of parameters including general in-plane crystal anisotropy, general in-plane void shapes and orientations, various creep exponents (i.e., nonlinearity) and general plane strain loading conditions. The MVAR model, which at the present state involves no calibration parameters, is found to be in good agreement with the finite element results for all cases considered in this work. The model is then used in a predictive manner to investigate the complex response of porous single crystals in several cases with strong coupling between the anisotropy of the crystal and the (morphological) anisotropy induced by the shape and orientation of the voids.

Approximate solution for crack interacting with a gas-filled void

Approximate analytical solution for crack interacting with a gas-filled void is developed on the basis of Eshelby equivalent inclusion theory and transformation toughening theory. As validated by detailed finite element analysis the approximate solution has good accuracy for prediction of mode I, mode II and I/II mixed mode crack-tip stress intensity factors under plane strain loading conditions.