韧性断裂准则

Development of a combined tension–torsion experiment for calibration of ductile fracture models under conditions of low triaxiality $Stephen M.Graham a ,n ,Tingting Zhang b ,Xiaosheng Gao b ,Matthew Hayden caDepartment of Mechanical Engineering,United States Naval Academy,590Holloway Rd,Annapolis,MD 21402,USA bDepartment of Mechanical Engineering,The University of Akron,Akron,OH 44325,USA cAlloy Development and Mechanics Branch,Naval Surface Warfare Center,West Bethesda,MD 20817,USAa r t i c l e i n f oArticle history:Received 11November 2010Received in revised form 8September 2011Accepted 17October 2011Available online 25October 2011Keywords:Ductile fractureExperimental technique Tension TorsionStress triaxiality Lode parametera b s t r a c tDevelopments in computational mechanics have given engineers tools to predict the evolution of damage in complex structures.Damage models have been developed that relate failure strain to stress triaxiality and Lode angle.Calibration of these models has traditionally relied on specimens that exhibit high triaxiality and limited Lode angle.This paper presents a specimen that can be tested in combined tension and torsion to achieve low triaxiality over a range of Lode angle.Numerical analysis of the specimen shows that it exhibits uniformity of stress–strain and stable values of triaxiality and Lode angle as plastic strain develops,both of which are desirable characteristics for calibration of ductile failure models.The design of a new displacement and rotation gage is presented that allows non-contact measurement at the gage section.Experimental results are used to develop the failure surface for 5083aluminum.Published by Elsevier Ltd.1.IntroductionDevelopments in computational mechanics have enabled engi-neers to conduct stress analysis and perform failure predictions on complex structures.When stresses exceed the yield point of the material in a structure,predictions of stress,strain,and damage will be highly dependent on the constitutive model used to characterize plasticity and damage.Therefore,accurate predictions of structural integrity depend on accurate material models.Development of these models requires constitutive theories that properly account for the influence of stress on yielding and plastic flow,and on experimental methods to calibrate the models.Calibration for parts of a structure that exhibit high constraint is readily done using simple tensile type specimen geometries.However,it is a particular challenge for low constraint areas such as thin sections,regions under predominant shear,and at surfaces because the specimen geometry and loading required to achieve low triaxiality are more complex.Since failure in structures often initiates in thin sections or at surfaces,calibration could significantly influence the accuracy of failure predictions.The most widely used approach for characterizing plasticity of ductile metals has been the classical J 2plasticity theory.In this theory the yield surface depends only on the second invariant of the stress deviator,J 2.However,it has been known for some time that yielding in geomaterials is sensitive to not only the hydro-static stress,but also to the Lode angle,which is related to the third invariant of the deviator stress tensor [1–5].Recently the influence of hydrostatic stress and Lode angle on yielding in ductile metals has been investigated [6–15].Damage models have been developed [6–8]that relate failure strain to the stress triaxiality,which is defined as the ratio of the hydrostatic stress to equivalent stress.These models have recently been extended to include the influence of Lode angle [9,11–14].Before continuing this review of past work,it is useful to define the various parameters used in ductile fracture models and to show how they are related.Triaxiality,T ,is universally recognized as the ratio of the mean stress over the von Mises equivalent stress,although the symbol used to represent it varies.T ¼Z ¼s m s eð1ÞThe mean stress is defined ass m ¼13ðs 1þs 2þs 3Þ¼I 13ð2Þs 1,s 2,and s 3are the principal stresses and I 1is the first invariant of the stress tensor.Contents lists available at SciVerse ScienceDirectjournal homepage:/locate/ijmecsciInternational Journal of Mechanical Sciences0020-7403/$-see front matter Published by Elsevier Ltd.doi:10.1016/j.ijmecsci.2011.10.007$Role of the funding source:this work was funded by the Advanced Combatant Materials Program through the Naval Sea Systems Command (NAVSEA)and the Metals Division of the Survivability,Structures,and Materials Department at the Naval Surface Warfare Center,Carderock Division (NSWCCD).NAVSEA had no direct involvement in the collection,analysis and interpretation of data,in writing the report,or in the decision to submit the paper for publication.nCorresponding author.Tel.:þ14102936519;fax:þ14102933041.E-mail address:smgraham@ (S.M.Graham).International Journal of Mechanical Sciences 54(2012)172–181The Von Mises equivalent stress iss e ¼1ffiffiffi2p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðs 1Às 2Þ2þðs 2Às 3Þ2þðs 3Às 1Þ2q ¼ffiffiffiffiffiffiffi3J 2p ð3ÞThere are many different variations on the definition of theLode angle,y ,and related parameters in the literature.In the early work of Lode [16],the Lode angle is defined as the angle measured counter-clockwise from the projection of the s 1axis to a point on a deviatoric plane (constant hydrostatic stress),as shown in Fig.1.s 10,s 20,and s 30are the projections of the principal stress axes on a deviatoric plane.P (x ,r ,y )are the Haigh-Westergaard cylindrical coordinates.x ¼I1ffiffiffi3p ¼ffiffiffi3p s mð4Þr ¼ffiffiffiffiffiffiffi2J 2p ¼ffiffiffi2r s e ð5Þcos ð3y Þ¼ffiffiffiffiffiffi27p J 32J 3=22¼27J 32s 3eð6ÞFor an isotropic material the ordering of the principal stresses does not influence the material behavior,therefore the stress space has three symmetry axes,as shown in Fig.1,and the Lode angle is limited to 0r y r ðp =3Þ.Xue [17]defined an alternative Lode angle,y L ,relative to the shear meridian axis [18],as shown in Fig.2.The shear meridian axis (y ¼301or p /6)represents all states of stress that can be formed by combining a state of pure shear with a hydrostatic stress.There is also a tensile meridian axis (y ¼01)representing states of stress formed by combining uniaxial tension with hydrostatic stress,and a compression meridian axis (y ¼601or p /3)formed by combining uniaxial compression with a hydrostatic stress.The two angles are related by y L ¼y Àðp =6Þand Àðp =6Þr y L r ðp =6Þ.This angle is related to the principal stresses by the following equation,where s 1,s 2,s 3are the principal values of the deviatoric stress tensor (s i ¼s i Às m ).y L ¼tan À11ffiffiffi3p 2s 2Às 1Às 313ð7ÞXue also defines a parameter that is the relative ratio of theprincipal deviatoric stresses.w ¼s 2Às 3s 1Às 3ð8ÞThe values of these parameters for common specimen types are compared in Table 1.Barsoum and Faleskog [13]use the parameter m to describe Lode angle dependence.m ¼2s 2Às 1Às 3s 1Às 3ð9ÞThis parameter is equivalent to the quantity in parentheses in Eq.(7),thereforey L ¼tan À1m ffiffiffi3p ð10ÞBai and Wierzbicki [14]define a normalized third deviatoric stress invariant,x ,which is not the same as the Haigh–Westergaard coordinate.x ¼27J 3s e¼cos ð3y Þð11Þwhere the range of the Lode angle is 0r y r ðp =3Þcorresponding to 1Z x Z À1.They also introduce a normalized Lode angle.y ¼1À6yp¼1À2pcos À1xð12ÞThis Lode angle parameter has the same range as x .Xue and Wierzbicki [19]introduce another variation on the Lode angle that is anti-symmetric about the tensile meridian axis.tan ðy ’L Þ¼Àm ffiffiffi3p ð13Þσ1’3’Fig.1.Deviatoric plane showing definition of Lode angle,y .σ1’3’P ( , Fig.2.Deviatoric plane showing definition of Lode parameter y L and its relation to the Lode angle,y .Table 1Comparison of Lode angle parameters for three stress states.Stress stateMeridian axisyy Lwmxyy 0Ly gx gGeneralized tension (axisymmetric tension)Tensile ðs 14s 2¼s 3Þ0Àðp =6Þ0À1þ1þ1ðp =6ÞÀðp =6Þþ1Pure torsion and plane strainShear ðs 2¼ð1=2Þðs 1þs 3ÞÞðp =6Þ00.5000000Generalized compression (axisymmetric compression)Compression ðs 1¼s 24s 3Þðp =3Þðp =6Þþ1þ1À1À1Àðp =6Þðp =6ÞÀ1S.M.Graham et al./International Journal of Mechanical Sciences 54(2012)172–181173The prime has been added to differentiate it from the Barsoum and Faleskog y L .The parameters presented thus far are based on the ordering of principal stresses s 1Z s 2Z s 3.Gao et al.[10]modify the defini-tion of x based on the ordering s 2Z s 3Z s 1as presented in [20].x g ¼cos 3y g þp ¼cos3y g þpð14Þwhere y g is measured from the shear meridian axis,which is now 301(p /6)counter-clockwise from the s 02axis,in a manner similar to y L .Adding p /6makes the angle anti-symmetric about the shear meridian axis,as can be seen in Table 1.In a study by Gao et al.[11],notched round and flat tensile specimens were used to investigate the influence of hydrostatic stress and Lode angle on ductile failure in DH36steel.Varying the notch radius allowed them to obtain a range of triaxiality.For the round tensiles,notch radii ranging from 1.12to 8.94mm yielded triaxiality of 1.3to 0.9,respectively,for DH36steel.The triaxiality depends not only on the specimen geometry,but also on the plastic behavior of the material.In a different study on 5083aluminum [10],the same specimen geometries yielded triaxiality ranging from about 1.6to 0.8.However,the axisymmetry of a notched tensile specimen causes it to have the same Lode angle (y ¼01,x ¼þ1),regardless of the notch geometry and material.Consequently,it does not allow investigation of Lode angle effects.Notched flat tensile specimens,referred to as plane strain specimens,were used to achieve triaxiality of 0.8–1.1(depending on the notch radius)for DH36steel at a Lode angle of y ¼301(x ¼0).They point out that alternative specimen designs are required to more accurately calibrate the failure surface at Lode angles between 0and 1because curvature of the failure surface is greater at low triaxiality,so failure strain becomes more sensitive to x [11].Wierzbicki et al.[21]developed a unique butterfly shaped specimen that is tested in combined tension and shear in order to investigate failure at low triaxiality.This specimen achieves triaxiality over the range of À0.191to 1.01and À0.503r x r 0.858;however,it requires some complex machining and a unique apparatus to perform the tests.The variation in triaxiality and Lode angle during evolution of plastic deformation required that average values of triaxiality and x be calculated by integrating over strain.It is also possible to achieve low triaxiality and a range of Lode angle using a simple notched tubular specimen tested in com-bined tension and torsion [13].Faleskog et al.were able to achieve triaxiality from about 0.3–1.3and Lode parameter À1r m r 0,which from Table 1corresponds to 0r x r 1,for a high yield strength and low strain hardening steel.Gao et al.[22]developed an alternative specimen design that is similar to the notched tubular specimen used by Faleskog et al.Their specimen is a modified version of a specimen used by Lindholm et al.[23]for conducting high-rate torsion tests.The objective of this paperis to investigate the distribution of stresses,triaxiality and Load anglein the gage section of the modified Lindholm specimen for use in calibrating ductile fracture models under conditions of low triaxiality and varying Lode angle.2.Numerical analysisThe ideal specimen for calibration of ductile failure models would have three characteristics:(1)uniform stress and strain in the gage section,(2)ability to achieve a range of triaxiality and Lode angles that fall between the tension and shear cases,and (3)stationary triaxiality and Lode angle as plasticity evolves up to failure.The rationale behind each of these characteristics,along with numerical analysis to characterize the modified Lindholm specimen,will be presented in the following.It is not possible to directly measure internal stress in a combined tension þtorsion calibration specimen,therefore it is necessary to use numerical stress analysis to determine the triaxiality,Lode angle,and equivalent plastic strain in the specimen at the time of failure.The quantities that can be readily measured in the test are the applied force,deflection,applied torque,and rotation.The model must accurately reproduce the force–displacement and torque–rotation behavior of the specimen to validate the accuracy of the computed stresses.It is necessary to determine the key stress parameters at the location and time of failure.If the stresses in the gage section of the specimen are uniform,then the precise location of failure does not have to be determined as long as it occurs within the uniform stress region.Therefore,a specimen design that promotes uniformity of stresses obviates the need to determine failure location.In an effort to provide a relative comparison with established approaches from the literature,numerical stress analysis of the modified Lindholm specimen will be compared with the specimen used by Barsoum and Faleskog [13].The dimensions of the specimen they used,which will be referred to as the Notched Tube (NT)specimen,are shown in Fig.3.The dimensions of the modified Lindholm specimen are shown in Fig.4.The latter differs from the original Lindholm specimen in that it is longer and the ends are cylindrical rather than hexagonal.A cylindrical surface was required to grip the specimen in hydraulic collet grips in order to apply tension.The original Lindholm specimen was designed for pure torsional loading.The gage sections of the NT and Lindholm speci-mens are compared in Fig.5.Even though they are close to the same overall size,the gage sections are significantly different.The NT specimen has both internal and external circular notches,while the modified Lindholm specimen is notched only on the outside with a trapezoidal notch profile.The gage section in the modified Lindholm is uniform over 2.54mm (0.1in),while the gage section in theNTFig.3.Dimensions of NT specimen [13].All dimensions in mm.S.M.Graham et al./International Journal of Mechanical Sciences 54(2012)172–181174specimen is curved throughout.The gage section width (wall thickness)for the NT specimen is 1.2mm at the narrowest point,which is about twice the thickness of the wall throughout the gage section of the Lindholm specimen (0.74mm).The thin wall in the gage section acts to minimize the variation in stress through the wall thickness.The very tight tolerance on concentricity shown in Fig.4is intended to ensure axisymmetric deformation in the gage section.A series of finite element stress analyses were conducted to compare the distributions of stress and effective plastic strain in the gage section of the NT specimen and the Lindholm specimen under different combinations of tension and torsion.The stress state for both specimens will be characterized using the equivalent plastic strain,e p ,triaxiality parameter,T ,and the Lode parameter,m .A sense for the physical significance of the Lode parameter m can be gained by re-arranging the equation as follows [13]:m ¼s 2Àðs 1þs 3Þ=2ðs 1Às 3Þ=2ð15ÞLooking at this in terms of Mohr’s circle,the second term in the numerator,ðs 1þs 3Þ=2,is the center of the largest circle,and the term in the denominator is the corresponding radius.Therefore,m can be interpreted as the relative offset of the second principal stress from the center of the largest Mohr’s circle.For the purposes of this comparison,the material is assumed to follow the J 2flow plasticity theory and the same material is used to model both specimens.All the analysis here uses material properties of Weldox 960,which is the same material used by Barsoum and Faleskog [13].s ¼E eif s r s 0s s 0¼e e 0Nifs 4s 0ð16Þwhere E ¼208GPa,v ¼0.3,s 0¼956MPa,e 0¼0.0046and N ¼0.059.Analysis was conducted using ABAQUS and both specimens were modeled using 4-node axisymmetric elements with twist (CGAX4R).The element size in the gage section of both specimens was approximately 0.05mm Â0.05mm.The finite element meshes for the two specimens are shown in Fig.6.An axial displacement,u ,and a twist angle,y ,are imposed on the top end of each specimen while the bottom end is constrained against axial or rotational motion.Because the geometries of the Lindholm specimen and the NT specimen are different,they can not achieve the same values of both T and m concurrently.In the following,the tension–torsion ratio based on displacement is defined as:c ¼u R yð17ÞIn this equation R is the mean radius in the center of the gage section,which is 12mm for the NT specimen and 6.91mm for the Lindholm specimen.During the loading process the tension–torsion ratio,c ,is kept constant.The stress distributions presented in the following are all along a radial line in the center of the gage section,which is on the symmetry plane of the specimen.3.Numerical analysis results 3.1.Pure torsionFig.7compares the through-thickness distribution of T ,m and e p for pure torsion loading (c ¼0)at the time when the equivalent plastic strain,e p ,reaches a level of about 0.37.For pure torsion,T and m should both be zero.For the NT specimen,r m represents the mid-radius in the gage section (12mm)and t n is the half net thickness in the narrowest part of the notch (0.6mm).For the Lindholm specimen,r inner represents the inner radius (6.54mm)and t n is the wall thickness in the gage section (0.735mm).It is apparent that T and m are very close to zero through the whole thickness for both specimens.For the Lindholm specimen,the effective plastic strain remains constant through the thickness.3.2.Torsion–tensionIn an effort to provide a comparison under similar conditions,the tension–torsion ratios applied to the different specimens were chosen to produce similar m values.For the NT specimen,c ¼0.194,and for the Lindholm specimen,c ¼0.068.These values are chosen such that the two specimens produce m values of approximately À0.2.Fig.8compares the through-thickness distribution of T ,m and e p at the time when e p in the element at the inner surface of the Lindholm specimen,and the center if the NT specimen,reaches 0.02.Fig.9compares the through-thickness distribution of T ,m and e p at a higher strain level of 0.20.The plastic strain at yieldisFig.4.Dimensions of Lindholm specimen [23].All dimensions inmm.parison of gage sections in NT and Lindholm specimens (to scale).S.M.Graham et al./International Journal of Mechanical Sciences 54(2012)172–1811750.002,so these strain levels are well beyond yield.Fig.10compares the through-thickness distribution of T ,m and e p at the time when e p reaches 0.44.The thickness variations in these parameters are summarized in Table 2.From the above figures and the table we can see that e p remains almost constant through thickness for the Lindholm specimen.Triaxiality in the NT specimen is the lowest at the surfaces and reaches a maximum near the center,while T in the Lindholm specimen is the highest at the inner surface and exhibits a small gradual decrease through the thickness.Lode parameter shows a gradual increase from inner to outer surfaces in both specimens.In general,the stress and strain distributions in the modified Lindholm specimen exhibit good uniformity,thereby obviating the need to determine exact failurelocation.Fig.6.Finite element meshes for the:(a)NT specimen and (b)modified Lindholm specimen.0(r-r inner )/t n-1-0.500.51-1(r-r m )/t n-1-0.500.51μTεpεpTμ-0.50.5010.250.50.751Fig.7.Through-thickness distribution of T ,m and e p at the mid-section of the specimen for e p ¼0:37:(a)NT specimen and (b)Lindholm specimen.0(r-r inner )/t n-1-0.500.51-1(r-r m )/t n-1-0.500.51μTεpεpTμ-0.500.510.250.50.751Fig.8.Through-thickness distribution of T ,m and e p at the mid-section of the specimen for effective plastic strain of 0.02:(a)NT specimen and (b)Lindholm specimen.S.M.Graham et al./International Journal of Mechanical Sciences 54(2012)172–181176It is interesting to compare the two specimens at similar levels of triaxiality.In this case the Lode parameter will be different for the two.In an effort to achieve the same value for T ,an analysis was run where c ¼0:194for Faleskog specimen and 0.368for the Lindholm specimen.These values were chosen such that the two specimens produced similar T values,which were around 0.36.Fig.11compares the through-thickness distribution of T ,m and e p at the time when e p in the element at the inner surface for0(r-r inner )/t n-1-0.500.51-1(r-r m )/t n-1-0.500.51μTεpεpTμ-0.500.510.250.50.751Fig.9.Through-thickness distribution of T ,m and e p at the mid-section of the specimen for effective plastic strain of 0.20:(a)NT specimen and (b)Lindholm specimen.0(r-r inner )/t n-1-0.500.51-1(r-r m )/t n-1-0.500.51μTεpεpTμ-0.500.510.250.50.751Fig.10.Through-thickness distribution of T ,m and e p at the mid-section of the specimen for effective plastic strain of 0.44:(a)NT specimen and (b)Lindholm specimen.Table 2Summary of thickness variations of ductile fracture parameters.NT specimenLindholm specimene pTme pTmFig.80.02–0.0350.131–0.258À0.120to À0.2380.020–0.0230.046–0.063À0.120to À0.165Fig.90.20–0.3150.166–0.356À0.168to À0.2150.20–0.220.064–0.094À0.167to À0.247Fig.100.44–0.740.250–0.450À0.175to À0.2450.44–0.480.062–0.086À0.160to À0.230Fig.110.14–0.220.148–0.333À0.162toÀ0.2060.14–0.140.290–0.380À0.830toÀ0.970(r-r inner )/t n-1-0.50.51-1(r-r m )/t n-1-0.500.51μTεpεpTμ-0.500.510.250.50.751Fig.11.Through-thickness distribution of T ,m and e p at the mid-section of the specimen when e p reaches 0.14:(a)NT specimen and (b)Lindholm specimen.S.M.Graham et al./International Journal of Mechanical Sciences 54(2012)172–181177Lindholm specimen,and the center of the NT specimen,reaches 0.14.We can see that once again e p remains constant through thickness for the Lindholm specimen (see summary in Table 2).The Lindholm specimen does show a small decrease in T going from the inner to outer radius.For this analysis,where m is approaching À1,the Lindholm specimen exhibits more variation in Lode parameter than the NT specimen.As mentioned previously,uniformity of stress and strain in the gage section is one desirable characteristic for a calibration specimen.Although it is not apparent from these graphs,the stress and strain distributions in the gage section of the Lindholm specimen are also uniform in the axial direction.The large ratio of gage section length to wall thickness (2.54/0.735¼3.5)ensures that the material over a large percentage of the gage length is not influenced by the transition at the ends.This is a desirable characteristic because it obviates the need to determine the exact axial location of the failure initiation.The second desirable characteristic is to achieve a range in triaxiality and Lode angle through control of the loading.For the NT and modified Lindholm specimens,this is achieved through choice of the tension–torsion ratio.For the Butterfly specimen,it is achieved by rotating the specimen axis relative to the loading axis to place the specimen under combined tension þshear.The different values of triaxiality and Lode parameter that can be achieved for each specimen are compared in Fig.12.The T and x values for the Butterfly specimen are average values up to failure.Lode angle for the NT specimen is presented in terms of Àm .It is important to note that although m and x are equal and opposite in sign at the meridian axes,in general m a Àx between these axes.The Lindholm results shown are for 5083aluminum,the NT results are for Weldox 960[13]and the Butterfly results are for A710steel [21].The values of triaxiality and Lode parameter achieved are partly dependent on the material properties,so this comparison only provides a general idea of the values that can be obtained and is not intended to isolate the influence of geometry.The Lindholm specimen behavior evolves from T ¼0and x ¼0under pure torsion to T ¼0.35and x ¼1at c ¼0.1.This is nearly the stress state of a smooth tensile specimen (T ¼0.33,x ¼1).As c is increased further,the stress state approaches plane stress bi-axial tension (s 1¼2s 2and s 3¼0)where T ¼0.58and x ¼0.This evolution results from the constraint imposed by the ends of the specimen outside the gage section as they inhibit circumferentialstrain,thereby inducing a circumferential stress.The stress state starts off on the shear meridian axis,gradually moves over to the tensile meridian axis at c ¼0.1and then moves back over to the shear meridian axis.Similar behavior is observed for the other two specimens,although the maximum Lode parameter occurs at different values of Triaxiality.The third desirable characteristic of a calibration specimen is to have minimal variation in triaxiality,Lode angle and effective plastic strain as plastic deformation evolves.Ideally,these para-meters would be stationary during plastic deformation all the way to failure.This would eliminate the need to calculate average values by integrating over strain.The evolution of triaxiality and Lode parameters m and x with increasing effective plastic strain is shown in Fig.13at tension–torsion ratios of c ¼0.194for the NT specimen and c ¼0.068for the Lindholm specimen,which were chosen to obtain similar m values of about À0.20.It is apparent that the modified Lindholm specimen quickly approaches nearly stationary values as plastic strain progresses.4.Material and experimental methodsExperiments were conducted using the modified Lindholm specimen over a range of tension–torsion ratios to calibrate a constitutive model for plasticity,and thereby validate the finite element model for future prediction of stresses and strains at ductile fracture.The tests were conducted in a two-axis servo-hydraulic load frame with axial and rotational actuators and hydraulic collet grips.The grips were carefully aligned to prevent buckling and development of asymmetric stresses in the relatively thin-wall gage section of the specimens.All tests were conducted in computer control,where axial extension and rotation were ramped with time to obtain a specific tension–torsion ratio and strain rate.Careful tuning of the servo-valve controllers was required to properly synchronize the axial and rotational actua-tors.In early tests it was observed that improper tuning led to a small time-lag in axial deflection that had a significant effect on the resulting development of plasticity in the specimen.In order to more accurately monitor and control the deforma-tion in the gage section of the specimen,a local extension plus rotation transducer that mounted just outside the gage section was developed.The transducer,shown in Fig.14,usesTriaxiality (T)L o d e P a r a m e t e r (ξ)00.20.40.60.81L o d e P a r a m e t e r (-μ)parison of Lode parameter and triaxiality over a range of applied loading conditions for the modified Lindholm,Butterfly and NT specimens.S.M.Graham et al./International Journal of Mechanical Sciences 54(2012)172–181178。