Modeling and Simulation for Material Selection and Mechanical Design Part 8 pps

A Flow Curves In order to characterize the strain hardening behavior of metallic materials during plastic deformation, one has to determine experimentally the relation _ sY ¼ sY ðeeq ; eeq ; TÞ that defines the dependence of the flow stress sY on the _ plastic parts of the equivalent strain eeq , the equivalent strain rate eeq and on the temperature T Flow curves are defined as the relation sY ¼ sY ðeeq Þ _ determined for eeq ¼ const: at a constant temperature They are often determined in compression test, taking into consideration the influence of friction They are also to be determined in tension test up to the ultimate force assuming uniform deformation 1 Empirical Relations The flow curves are almost described by power laws The oldest of these relations, introduced in 1909 by Ludwik [1], is given by sY ¼ K0 þ Ken ð5Þ This relation allows a good description of the flow curves of materials having a finite elastic limit For a plastic strain ðe ¼ 0Þ, the flow stress equals K0 It leads, however, to an infinite value for the slope of the curve @ sY =@ e at the yield point A simplified form of this equation sY ¼ Ken ð6Þ was suggested by Hollomon [2] Because of its simplicity, it is till now the most common relation applied for the description of the flow curve However, no yield point is considered by this relation as sY ¼ 0 for e ¼ 0 Especially for materials with a high yield point or materials previously deformed, the flow stress cannot be described well by this relation in the region of small strains A more adequate description is achieved by the Swift relation [3] sY ¼ KðB þ eÞn ð7Þ For e ¼ 0, a yield point is considered with a value of sY ¼ KBn An alternative description sY ¼ a þ b½1 À expðÀceފ ð8Þ was introduced by Voce [4] and is well applicable for the range of small strains Figure 1 shows the optimum fit achieved by the four equations (5–8) for the flow curves of an austenitic steel at different temperatures in the range of relatively small strain up to 0.2 The figure shows that the Swift relation and the Voce-relation describe well the flow curves in the relative Copyright 2004 by Marcel Dekker, Inc All Rights Reserved If the parameters k1 and k2 are considered to be constants, the flow curves follow by: s ¼ s0 þ ðs1 À s0 Þ½1 À expðÀe=eà ފ ð12Þ This equation is identical with the empirical Voce relation In the range of relatively small strains, it fits the experimental data very well However, it fails to describe the flow curves in the range of high strains because the experimental results for the flow stress do not asymptotically approach a definite value [6] The following modification can be suggested, to yield an evolution equation that describes well the strain hardening in the range of high pffiffiffi strains The parameter k1 ¼ 1=ðl rÞ, where l is the dislocation free path This parameter can be considered as a function of strain and may be expressed as k1 ¼ kð1 þ ceÞ The evolution equation of the flow stress becomes ! ds k kc K2 s ð13Þ ¼ þ eÀ 2 de 2aGb 2aGb Figure 2 Flow curve of the austenitic steel X8CrNiMoNb16-16, described by Eqs (13) and (14) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved The solution of this differential equation is s ¼ C1 þ C2 e þ C3 ½1 À expðÀC4 eފ ð14Þ where C1 is the yield stress, C2 ¼ kc=ðk2 aGbÞ, C3 ¼ kð1 þ 2c=k2 Þ=ðk2 aGbÞ; and C4 ¼ k2 =2 This equation is identical with the empirical relation introduced in Ref [7] It is found to give the optimum fit for the experimental results of several materials (Fig 2) However the determination of its parameter needs some more effort It should be mentioned that also the empirical Swift relation given by Eq (5) fits well the experimental data in this strain range B Influence of Strain Rate and Temperature Figure 3a shows an example for the influence of increasing temperature on the flow stress for given values of strain and strain rate [8] Considering the slope ds=dT, three different temperature ranges can be defined: (A) range of low temperatures, between absolute zero and about 0.2 of the absolute melting point, where the influence of the temperature on the flow stress is great The material behavior is governed by thermally activated glide, (B) range of intermediate temperatures between 0.2 and 0.5 of the absolute melting temperature Only a slight influence of strain rate and temperature on the flow stress is usually observed in this range, and (C) range of temperatures higher than 0.5Tm in which the flow stress depends highly on the temperatures because of the dominance of diffusion-controlled deformation processes The influence of the strain rate variation [9] is represented in Fig 3b Three different strain rate ranges can also be recognized according to the Figure 3 (a) Temperature influence on the yield stress of NiCr22Co12Mo9 at _ e ¼ 3  10À4 secÀ1 [8] (b) Influence of stain rate on shear yield stress of mild steel [9] Copyright 2004 by Marcel Dekker, Inc All Rights Reserved _ variation of @s=@ ln e: (I) range of low strain rates with only a slight influence of the strain rate due to athermal glide processes, (II) range of intermediate and high strain rates with relatively high strain rate sensitivity due to thermal activated glide mechanisms, and (III) range of very high strain rates where internal damping processes dominate and a very high strain rate sensitivity is observed The boundary between the ranges (I) and (II) depends on the temperature Overviews concerning the mechanical behavior under high strain rates are represented, e.g in Refs [10,11] To estimate the mechanical behavior over wide ranges of strain rate and temperature, constitutive equations must be established taking the time dependent material behavior into consideration A visco-plastic behavior is often assumed by using, for example, the Perzyna equation [13] _ eij ¼ _ Sij 1 À 2v @f _ skk dij þ 2ghFðFÞi þ 2m 2E @sij ð15Þ where m is the shear modulus, f is square root of the second invariant of the stress deviator Sij and F ¼ (f=k) À1 is the relative difference between f and pffiffiffi the shear flow stress k ¼ sF = 3 The function FðF Þ is often estimated using simple rheological models assuming FðF Þ ¼ F and leading to linear relation of the type s ¼ sF ðeÞ þ Z_ which is acceptable for metals only at strain rates e >103 secÀ1 1 Empirical Relations Different empirical relations could be implemented in Eq (15) With FðF Þ ¼ expðF =aÞ À 1 or FðF Þ ¼ F 1=m , the corresponding relations between stress and stress rate in the uniaxial case are identical with the empirical relations introduced 1909 by Ludwik [14] _ s ¼ sF ðe; TÞ½1 þ a lnð1 þ e=aފ ð16Þ s ¼ sF ðe; TÞ½1 þ ð_ =aà Þm Š e ð17Þ The influence of temperature on the flow stress is also described by dif_ ferent relations of the type s ¼ sðe; eÞf ðT=Tm Þ where Tm is the absolute melting point of the material, such as _ s ¼ s0 ðe; eÞ exp½ÀbT=Tm Š ð18Þ or according to Ref [15] _ s ¼ s0 ðe; eÞ½1 À ðT=Tm Þv Š Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð19Þ On applying such empirical relations, the flow stress is usually e represented by s ¼ ft ðeÞf2 ð_ Þf3 ðTÞ as a product of three separate functions of strain, strain rate and temperature This is a rough approxima_ tion especially in the case of moderate strain rates of e < 103 secÀ1 However, the basic problem is that nearly all the parameters of these empirical equations can only be regarded as constants only within rela_ tively small ranges of e, e, and T The determination of the functional behavior of the parameters requires a great number of experiments Therefore, constitutive equations based on structure-mechanical models are gaining increasing interest as they can improve the description of the mechanical behavior in wider ranges of strain rates and temperature and may, if carefully used, allow for the extrapolation of the determined relations 2 Structure-Mechanical Models The macroscopic plastic strain rate of a metal that results from the accumulation of sub-microscopic slip events caused by the dislocation motion is given by _ e ¼ brm v=MT ð20Þ In this equation, the Burger vector b and the Taylor factor MT are constants for a given material whereas the mobile dislocation density rm is mainly a function of strain The relation between the dislocation velocity v and the stress was experimentally determined for several materials [16] It can be represented in the range of low stresses by a power law v ¼ v0 ðs=s0 ÞN At very high stresses, the dislocation velocity approaches qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi asymptotically the shear wave velocity cT and s ¼ avn = 1 À ðv=cT Þ2 3 Athermal Deformation Processes In the range of intermediate temperatures and low strain rates (combined ranges B and I), and at relatively low temperatures, i.e., less than 0.3 of the absolute melting point Tm, the influence of strain rate and temperature _ depends on the e-range of the deformation process Below a specific value of the strain rate, that depends on temperature, only a slight influence of strain rate and temperature on the flow stress is observed In this region I, athermal deformation processes are dominant, in which the dislocation motion is influenced by internal long range stress fields induced by such barriers as grain boundaries, precipitations, and second phases The flow stress varies with temperature in the Copyright 2004 by Marcel Dekker, Inc All Rights Reserved same way as the modulus of elasticity The influence of strain rate can be described by s¼C EðTÞ m _ e EðT0 Þ ð21Þ where E is the modulus of elasticity and m is of the order of magnitude of 0.01 4 Thermally Activated Deformation In the ranges of low temperatures (A) and intermediate to high strain rates (II), the dislocation motion is increasingly influenced by the short range stress fields induced by barriers like forest dislocations and solute atom groups in fcc-materials or by the periodic lattice potential (Peierls-stress) in bcc materials If the applied stress is high enough, these barriers can immediately be overcome At lower stresses, a waiting time Dtw is required until the thermal fluctuations can help to overcome the barrier A part of the dislocation line becomes free to run, in the average, a distance sà until it reaches the next barrier within an additional time interval Dtm The mean dislocation velocity is given by v ¼ sà =ðDtw þ Dtm Þ The waiting time Dtw equals the reciprocal value of the frequency n of the overcoming attempts If the strain rate is lower than ca 103 secÀ1, it can be assumed that Dtw 4 Dtm The relation between strain rate _ _ _ and stress is then given by e ¼ e0 ðeÞ exp½ÀDG=kT Š where e0 ¼ brm n 0 sà =MT The activated free enthalpy DG depends on the difference sà ¼ s À sa between the applied Rstress and the athermal stress according to kT lnð_ 0 =_ Þ ¼ DG ¼ DG0 À V à dsà where V à ¼ bl à sà =MT is the e e reduced activation volume For given stress and strain, the value of T lnð_ à =_ Þ is constant for all e e _ temperatures and also for all strain rate values between e0 exp½ÀDG0 = _ ðkTފ and e0 This means that the increase of stress at constant strain with decreasing temperature or with increasing strain rates is the same, as long as the values of DG ¼ kT lnð_ à =_ Þ are equal in both cases e e Depending upon the formulation of the function V à ðsà Þ; different _ _ relations for e ¼ e0 ðsÞ were proposed in Refs [17–21] The most common are the relation introduced by Vohringer [19,20] and by Kocks ¨ et al [21] _ _ e ¼ e0 ðeÞ exp À & ! 'q ! DG0 s À sa p 1À kT s0 À s a Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð22Þ and that by Zerilli and Armstrong [22,23] &  ' ! _ e0 DG0 k ln s À sa ¼ à exp À b0 þ T _ DG0 V0 e ð23Þ 5 Transition to Linear Viscous Behavior At strain rates higher than some 103 secÀ1, the stress is high enough to the extent that Dtw vanishes Only the motion time Dtm is to be considered The dislocation run with high velocity throughout the lattice and damping effects dominate The dislocation velocity v ¼ sà =Dtm can then be given by v ¼ bðt À th Þ=B according to Ref [24] The flow stress follows the relation s ¼ sh ðe; TÞ þ Z_ e ð24Þ with Z ¼ MT B=ðb2 Nm Þ This relation is validated experimentally in Ref [9] as well as by Sakino and Shiori [25], as shown in Fig 4a A continuous transition takes place, when the strain rate is increased from the thermal activation range (II) to the damping range (III) This can be described in two different ways: regarding the dislocation velocity to be equal to v ¼ sà =ðDtw þ Dtm Þ, the strain rate can be represented by  & ! 'q  !À1 DG0 s À sa p x _ ¼ e0 exp _ 1À ð25Þ e þ kT s0 À sa s À sh where x is a function of strain Alternatively, the continuous transition can be described by an additive approximation The stress is regarded to be the sum of the athermal, the thermal activated and the drag stress components According to this approximation, s % sa þ sth þ Z_ where sth is the thermal e activated component of stress determined from Eq (22) or (23) 6 Diffusion-Controlled Deformation In the range of high temperatures (C), the deformation is governed by strain hardening and diffusion-controlled recovery processes !  s n Q1 _ _ exp À e ¼ e0 ð26Þ RT G At very high temperatures and low stresses ! sO 1 pdDB _ Dv 1 þ ed;e ¼ 14 kT d2 dDv Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð27Þ C Material Laws for Wide Ranges of Temperatures and Strain Rates Material laws that describe the flow behavior over very wide ranges of temperatures and strain rates are needed for the simulation of several deformation processes, such as high-speed metal cutting In this case, different physical mechanisms have to be coupled by a transition function Fig 5 shows the dependence on the stress with the strain rate at different temperatures for a constant strain Three main mechanisms can be distinguished: (a) _ diffusion-controlled creep processes with ecr / sNðTÞ in the region (1) of low strain rates and high temperatures, (b) dislocation glide plasticity with _ mðTÞ in the region (2) of intermediate temperatures and strain rates, s / epl e _ and (c) viscous damping mechanism with s ¼ sG þ Zð_ À eG Þ in the region _ > 1000 secÀ1 in the region (3) of very high strain rates e 1 Visco-plastic Material Law For a continuous description over the different ranges, the strain rates have to be combined [27] to obtain À Á _ _ _ ð28Þ e ¼ ð1 À MÞ ekr þ epl þ M_ damping e with the transition function M ¼ 1 À exp½Àð_ =_ G Þm Š The complete strain e e rate range can be described by  NðTÞ  1=mðTÞ ! s s _ _ e ¼ ð 1 À MÞ eà þ s0 ðT; eÞ sH ðT; eÞ   s À sG ðT; eÞ _ þ eG ðT; eÞ þM ð29Þ Z _ with eà ¼ 1 secÀ1 The parameters and functions s0(T, e), sH(T, e), m(T), N(T), and Z have to be determined by curve fitting in the individual regions _ (1)–(3), whereas the parameters sG and eG are determined requiring that the _ follows a continuous function in the transition region: derivative @s=@ e _ e e sG ¼ sH ð_ G =_ à Þm and eG ¼ ðmsH =ZÞ1=ð1ÀmÞ The values of the parameter used are given in Ref [28] An exception of the rule of the reduction of flow stress with increasing temperature is the influence of dynamic strain hardening observed in ferritic steel at temperatures between 2008C and 4008C, where the flow stress increases towards a local maximum It is caused by the interaction between moving dislocations and diffusing interstitial atoms The additional stress can be described by Ds ¼ að_ Þ exp½ÀfðT À bð_ ÞÞ=cð_ Þg2 Š, With this addie e e tional term, the dependence of flow stress of steel Ck45 (AISI 1045) on temperature and strain rate is determined [28] and represented in Fig 6 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved 2 Adiabatic Softening Flow curves determined in the range of high strain rates are almost adiabatic, since the deformation time is too short to allow heat transfer The major part of the deformation energy is transformed to heat while the rest is consumed by the material to cover the increase to internal energy due to dislocation multiplication and metallurgical changes On strain increase by de, the temperature increases according to dT ¼ k 0:9 s de rc ð30aÞ where the factor 0.9 is the fraction of the deformation work transformed to heat, s is the current value of the flow stress which is already influenced by the previous temperature rise and k is the fraction of energy remaining in the deformation zone At low strain rate, there is enough time for heat transfers out of the deformation zone and the temperature increase is negligible In this case, k ¼ 0 On the other hand, the deformation process is almost adiabatic at high strain rate and k ¼ 1 A continuous transition from the isothermal deformation under quasi-static loading to the adiabatic behavior under dynamic loading can be achieved considering k as a function of strain rate in the form Figure 7 Quasi-static and adiabatic flow curves of unalloyed fine grained steel Copyright 2004 by Marcel Dekker, Inc All Rights Reserved   _ 1 4 e kð_ Þ ¼ þ arctan e À1 _ 3 3p ead ð30bÞ _ The transition strain rate ead depends on the thermal properties of the _ material If the temperature of the surroundings is the room temperature, ead is around 10þ1 secÀ1 As the flow stress usually decreases with increasing temperature, the flow curve shows a maximum (Fig 7) A thermally induced mechanical instability can take place leading to a concentration of deformation, a localization of heat and even to the formation of shear bands An overview of different criteria for the thermally induced mechanical instability is presented in Ref [29] The adiabatic flow curve can be deter_ mined numerically for an arbitrary function sðe; e; TÞ for the shear stress which has been determined in isothermal deformation tests In order to obtain a closed-form analytical solution demonstrating the adiabatic flow _ behavior, the simple stress–temperature relation s ¼ siso ðe; eÞCðDTÞ can be used [30,31] In this case, the change of temperature can simply be determined by separation of variables and integration For example, _ s ¼ siso ðe; eÞ 1 À m ! T À T0 ; Tm s ¼ siso 0:9km exp À rcTm ! Z siso de !À1 ! Z T À T0 0:9kb _ s ¼ siso ðe; eÞ exp Àb siso de ; s ¼ siso 1 þ Tm rcTm ð31Þ ð32Þ  c Tm is the absolute melting point of the material, r and are the mean values of density and specific heat in the temperature range considered Around room temperature, the product rc lies between 2 and 4 MPa=K for most of the materials For a rough approximation, it can be assumed that (rcTm =0.9) % 3Tm in MPa using Tm in K Many experimental investigations e.g Ref [32] were carried out in order to determine the temperature dependence of the flow stress Up to a homologous temperature of 0.6, the stress–temperature relation can be described better by Eq (35) than by Eq (34), showing values of b between 1 and 4 Therefore, only Eq (35) will be considered in the following discussion If the isothermal stress can be simply described by e siso % Ken Fð_ Þ ð33Þ _ the flow stress, determined in an adiabatic test with constant e, is then given by Copyright 2004 by Marcel Dekker, Inc All Rights Reserved sad ¼ Ken Fð_ Þ 1 þ e ka Ke1þn Fð_ Þ e ð1 þ nÞTm !À1 ð34Þ where a ¼ 0:9b=ðÞ The parameter a can be considered as approximately rc constant represented by its mean value over the deformation process which is of the order of magnitude of 1 K=MPa The flow curve shows a maximum smax at the critical strain ec , where ! ! nð1 þ nÞTm 1=ð1þnÞ KFð_ Þ nð1 þ nÞTm n=ð1þnÞ e ; smax ¼ ð35Þ ec ¼ 1 þ n k aKFð_ Þ k aKFð_ Þ e e and the parameters K and a can be estimated by K¼ 1 þ n smax ; Fð_ Þ en e c ka ¼ nTm smax ec ð36Þ the remaining unknown parameter n can be determined by fitting the curve the adiabatic flow curve [12] Similar to the process of neck formation in a tensile specimen, the existence of a stress maximum leads to mechanical instability Especially after reaching the stress maximum, a great part of the specimen is unloaded elastically causing further deformation localization In dynamic torsion tests, the deformation localization leads to a heat concentration and hence a higher local temperature rise and a high shear strain concentration Coffey and Armstrong [33] introduced a global temperature localization factor which is the ratio of the plastic zone volume to the total specimen volume The influence of inhomogeneity on the strain distribution has been demonstrated by using a simple model [34] which represents the torsion specimen by two slices, a reference one and another slice with slight deviations in strength or dimensions Furthermore, the deformation localization could be traced during the torsion test by observing the deformation of grid lines on the specimen surface by means of high-speed photography [35,36] The influence of adiabatic softening can be illustrated in the case of compression test at high strain rates Due to friction between the cylindrical specimen and the loading tools, a compression specimen becomes a barrel form during the test In an etched cross-section of a quasi-statically tested specimen, two conical zones of restricted deformations can often be recognized after quasi-static upsetting The deformed geometry is symmetrical about the midplane _ (Fig 8) An FE-simulation is carried out for a compression test with e¼ 0.001 secÀ1 considering stain hardening according to Eq (8) and friction at the upper and lower surfaces by a coefficient m ¼ 0.1 The computational results indicate that the maximum values of equivalent stress as well Copyright 2004 by Marcel Dekker, Inc All Rights Reserved _ Figure 9 Quasi-static compression test on cylindrical specimens with e ¼ 0:001 secÀ1 (a) Etched section of a SMnPb-steel (b) Distribution of Mises’ stress (c) Distribution of the equivalent plastic Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 10 Etched longitudinal section of a cylindrical compression specimen of Armco iron loaded dynamically (_ ¼ 5000 secÀ1) e (Fig 12a) The combination of experiment and finite-element simulation allows examining the possibility of extrapolation of materials laws to the range of very high strains and strain rates [28] In addition, valuable information can be obtained for the optimization of the width B in shear Figure 11 Distribution of Mises’ stress and equivalent plastic strain in a _ compression specimen after dynamic test with e ¼ 5000 secÀ1 , friction coefficient m ¼ 0.1 Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 13 Distribution of the equivalent stress and the equivalent stress in the shear zone of a hat-shaped specimen [28] (a) Equivalent strain (b) Equivalent stress in MPa according to von Mises Copyright 2004 by Marcel Dekker, Inc All Rights Reserved displacement–time function of the upper surface is applied to upper the nodes In order to reduce the total number of elements, the lower die is idealized using the so-called infinite elements The material law is determined in compression tests at different temperatures with strain rates up to 7500 secÀ1 As discussed above, one e can assume a linear viscous behavior according to s ¼ sh ðeÞ þ Z_ , when _ e > 2000 secÀ1 and the damping mechanism dominates The simulation should examine the accuracy of the reproduction of the force–displacement curves determined experimentally for this geometry The distributions of the von Mises equivalent stress and equivalent strain, represented in Fig 13, show a great non-uniformity High strain concentrations exist at the two diagonally opposite corners of the deformation zone In these regions, the strain rate is so high that the influence of adiabatic softening is more than compensated and high stress values are determined there Examples for the force–displacement curves determined experimentally and computed by the FEM are shown in Fig 14 for different values of the shear zone width B The deviation of the computed curves from the experimental ones is relatively small Therefore, it can be assumed that _ the material law determined in the range of high strain rates e > 2000 secÀ1 Figure 14 Force–displacement curves of the hat specimen Markers: experimental results, curves: FE simulation (From Ref 28.) Copyright 2004 by Marcel Dekker, Inc All Rights Reserved can be extrapolated to much higher strain rates assuming the dominance of the viscous damping mechanism according to Eq (24) This result is consistent with the experimental results of Sakino and Shiori (Fig 4a) Such material laws allow the simulation of different metal forming as well as metal cutting processes They can be validated by high-speed metal cutting tests [38] Structural damage during high rate tensile deformation can be accounted for by introducing a damage function [39] II CYCLIC DEFORMATION BEHAVIOR A Phenomenological Approach _ If a specimen is extended with a constant strain rate e0 , the stress increases first according to Hooke’s law of elasticity till the elastic limit is reached Then, a plastic deformation begins accompanied with a non-linear hardening After reaching an arbitrary total strain e1tot ¼ e1el þ e1pl , the strain rate changes to (À_ 0 ) At first, the material is unloaded elastically and is then e compressed until a total strain of (Àe1tot ), as represented in Fig 15 It can be clearly observed that the plastic compression begins at an = elastic limit Re , whose absolute value much smaller than the initial value Figure 15 Stress–strain diagram of an experiment with a change of the loading direction Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Re In general, it can be stated that a previous tensile deformation reduces the compression elastic limit Also, a compression deformation reduces the subsequent elastic limit under tension This phenomenon,known as Bauschinger effect, is characteristic for the behavior of the material under cyclic loading in the low cycle fatigue range With further cyclic loading, the stress range increases usually due to strain hardening (Fig 16) If the material is highly pre-deformed or hardened, a cyclic softening takes place and the stress range decreases with increasing number of cycles The rate of change of the stress range Ds decreases with the number of cycles and approaches a stationary value and the hysteresis loop remains unchanged The strain hardening phenomena under cyclic loading can be classified in two terms: (a) Isotropic deformation resistance sF , that includes the yield point as well as the isotropic change of the flow stress It increases (or decreases) monotonically with the number of Rcycles, depending upon the specific plastic P deformation work s de, or on the accumulated plastic strain jdej Its variation with strain can be considered as a result of the increase of the density of immobile Figure 16 Stress–strain diagram of an austenitic steel under cyclic loading with a constant range of the total strain of De ¼ 0:0066 at 6508C Copyright 2004 by Marcel Dekker, Inc All Rights Reserved dislocation with an additional influence of the changing microscopic residual stress state (b) Kinematic hardening or internal back stress si that depends on the direction of the deformation and the loading history and accounts for the Bauschinger effect It may result from the reversible interactions of mobile dislocations with obstacles, such as in the cases of pile ups or bowing between particles Figure 17 shows the influence of these stress parts in the biaxial case on the form of the Mises-ellipse The isotropic hardening leads to an equal increase of the ellipse in all directions, while the kinematic hardening shifts the ellipse in the loading direction Usually, both of these hardening types are to be expected during plastic deformation During uniaxial cyclic deformation, the influence of the strain rate can be considered in two ways: ðs À si Þ2 ¼ s2 Fð_ pl Þ e F ð s À si Þ 2 ¼  Ã2 sF þ Cð_ pl Þ e ð37aÞ ð37bÞ If the direction of the strain rate, and hence its sign, is suddenly changed, the sign of the isotropic material resistance sF changes at once In contrast, the value of internal back stress si changes gradually with increasing deformation approaching asymptotically a stationary value sis with the same sign as the strain rate In the first cycle, the stress equals sF0 þ si0 , at the beginning of the plastic deformation where si0 is approximately equal to 0 for annealed materials With increasing strain, both of sF and si increase approaching the stationary Figure 17 Mises’ ellipse after: (a) isotropic hardening, (b) kinematic hardening, and (c) mixed mode hardening Copyright 2004 by Marcel Dekker, Inc All Rights Reserved values sF1 and sis On reaching the maximum strain of Detot =2, the maximum stress is given by smax1 ¼ sF1 þ si1 If the loading direction is changed from tension to compression, the isotropic material resistance changes from (þsF1 ) to (ÀsF1 ) at once The material is first unloaded and the stress drops by the amount of sF1 With further reduction of length, plastic compression start when the stress is reduced by 2sF1 During this short time, the internal back stress si remains unchanged at the value si1 After the beginning of plastic compression, it starts to decrease gradually approaching a new stationary e value ðÀsis Þ that corresponds to the new strain rate of (À_ ) Each time when the strain rate changes from (þ_ ) to (À_ ) in an arbie e trary cycle, the stress drops during the elastic deformation by DsF ¼ 2sF and then gradually by Dsi during the plastic deformation of the half cycle On the next reverse (À_ ) to (þ_ ), a stress decreases first by DsF and then grae e dually by Dsi The stress ranges is Ds ¼ DsF þ Dsi The maximum and the minimum stress smax ¼ sF þ Dsi =2 and smin ¼ ÀsF À Dsi =2 The range DsF is defined as the difference between the maximum stress and the elastic limit in the subsequent compression phase Especially in cyclic deformation, it is rather difficult to exactly determine the elastic limit, i.e., the transition point between the elastic and plastic deformation ranges because this transition is almost gradual However, this point can be easily estimated if the hysteresis loop (Fig 18a) is differentiated and ds=detot is represented as a function of s (Fig 18b) Starting at minimum stress (point 1), the slope of the curve remains approximately constant and equals the modulus of elasticity till point 2 is reached Then the slope decreases to a Figure 18 (a) Hysteresis loop, and (b) the derivative with respect to the total strain as a function of the stress Copyright 2004 by Marcel Dekker, Inc All Rights Reserved relatively low value at point 3 of the maximum stress The stress–strain curve rotates to point 4 of maximum strain The slope decreases to À1, changes to þ1 and decreases again to the value of the modulus of elasticity This value should remain unchanged till point 5, where plastic compression begins and the slope decreases again till reaching point 6 Between point 6 (minimum stress) and point 1 (minimum strain), the value of slope changes to þ1 then to þ1 and decreases to the value of the modus of elasticity The relation between ds=detot and s can be linearized in the plastic ranges between the points 2 and 3 as well as between points 5 and 6 The intersection of these linear relations with the elastic relation ds=detot ¼ E defines the value of range DsF of the isotropic material resistance The range of the internal back stress is the defined by Dsi ¼ Ds þ DsF : Repeating the procedure represented in Fig 18 for the different cycles, one obtains Fig 19a For each half cycle, the value of DsF and the linear relation between ds=de and s can be determined The isotropic component sF as well varies monotonically and continuously with increasing number of cycles However, it can be assumed that its value is constant within arbitrary half cycles and changes only at the beginning of the next one, if the total number of cycles is great enough In this case dsi ds ¼ de de within each half cycle The linear relation ! dsi si À si0 ¼E 1À detot sis À si0 ð38Þ ð39Þ can be written for the internal back stress as well The stationary value sis varies with the number of cycles The isotropic material resistance sF as well as the stationary value sis of the internal back stress si are represented in Fig 19b as functions of the accumulated strain These relations may be described by sF ¼ sF0 þ ðsF1 À sF0 Þ½1 À expðÀCF eacc ފ ð40Þ sis ¼ sis0 þ ðsis1 À sis0 Þ½1 À expðÀCi eacc ފ ð41Þ yielding the evolution equations dsF ¼ CF ðsFs À sF Þ deacc ð42Þ dsis ¼ Ci ðsis1 À sis Þ deacc ð43Þ Copyright 2004 by Marcel Dekker, Inc All Rights Reserved Figure 20 Relation between stress derivative and internal back stress si (a) Derivative with respect to the total stress (b) Derivative with respect to plastic strain achieved by considering two internal variables for the internal back stress instead of only one as considered here [40] With depl ¼ detot À dsi =E, the derivative of the internal back stress with respect to the plastic strain can be obtained from Eq (39) as dsi sis À si0 ¼E À1 depl si À si0 ! ð44Þ A comparison of this relation with the experimental results is represented in Fig 20b B Constitutive Equation of Cyclic Behavior In contrast to these experimental facts, serious simplifications are usually made to reduce the number of the material parameters involved in computation The hyperbolic function of Eq (8) is simply linearized yielding dsi ¼ C À gsi depl Copyright 2004 by Marcel Dekker, Inc All Rights Reserved ð45Þ where C and g are material constants The stationary value sis is considered constant assuming that the material follows the Masing-rule and Eq (5) is reduced to sis ¼ C=g ð46Þ Lemaitre and Chaboche [41,42] introduced a non-linear isotropic= kinematic hardening model, which provides predictions that are near to the experimental evidence This model is applicable for isotropic incompressible materials The yield surface is defined by the function F ¼ fðsij À Xij Þ À s0 ¼ 0 ð47Þ where s0 is the yield stress that is equivalent to the isotropic material resistance sF and Xij is the tensor of the internal back stress denoted sis in the uniaxial case The function f ðsij À Xij Þ equals the equivalent Mises stress when the back stress X is taken into consideration: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 0 ðs À X0ij Þðs0ij À X0ij Þ f¼ 2 ij ð48Þ 0 where s0ij is the deviatoric stress tensor and is the Xij deviatoric part of the back stress tensor The associated plastic flow is given by _ ij epl ¼ @F pl 3 ðs0ij À X0ij Þ pl _ _ e e ¼ @sij 2 f ð49Þ _ _ e where epl represents the rate of plastic flow and pl is the equivalent plastic strain rate rffiffiffiffiffiffiffiffiffiffiffiffiffi pl ¼ 2 epl epl _ _ _ e ð50Þ 3 ij ij The size of the elastic range, s0 , is a function of the equivalent plastic strain pl and the temperature For a constant temperature, it is written simie lar to Eq (40) as   à e ð51Þ s0 ¼ s0 þ Q1 1 À expðÀbpl Þ where sj0 is the yield surface size at zero plastic strain, and Q1 and b are additional material parameters that must be determined from cyclic experiment Copyright 2004 by Marcel Dekker, Inc All Rights Reserved The evolution of the kinematic component of the model, when temperature and field variable are neglected, is defined as 2 _ _ Xij ¼ C_ pl À Xijpl ¼ C e e 3 ij ! s0ij À X0ij _ À gXij pl e s0 ð52Þ where C and g are material parameters C Application to Life Assessment The assessment of the fatigue life under cyclic elasto-plastic deformation requires an accurate determination of the strain ranges of the individual loading cycles in the region of maximum local deformation For this reason, FE simulation is often needed especially when the analytical solutions are not available or when they include unacceptable simplifications For example, the fatigue life of notched machine parts is often predicted using approximation formulas [43–45] that have been driven using the Neuber rule [46] However, the accuracy of these methods remains lower than that of the inelastic FE analysis, when adequate materials lows are implemented Figure 21a illustrates the distribution of the axial stress sxx in a notched 3-point bending specimen The mesh is built of three-dimensional continuum solid elements Around the notch, a refined mesh is chosen so that the critical zone at notch root would cover several elements The material considered is the AlZnMgCu alloy AA7075 The material parameters determined in uniaxial cyclic experiments are: sj0 ¼ 310 MPa, Q ¼ 75 MPa, b ¼ 36.6, C ¼ 14,844 MPa and g ¼ 86.3 As a loading condition, a line load with a total compressive force F is applied at the midlength of the upper surface The force follows a sinusoidal time function with Fmin =Fmax ¼ 0:1 The specimen is supported at two parallel lines on the lower surface with a span width of 80 mm The computational results of the local strain components at notch root are represented in Fig 21b as functions of time For an arbitrary loading cycle, the time functions eij ðtÞ are used to determine an equivalent strain range D for the cycle according to different e approaches [47–52] With an additional damage accumulation rule, a representative periodically repeated strain range can be computed which should lead to the same fatigue life Experimentally, fatigue life is determined as the number of cycles, at which a technically detectable crack is initiated Often, a direct current potential drop system [53,54] is used to determine the potential drop across the notch as time function This is then converted to a relation between crack length and number of cycles to initiate cracks Copyright 2004 by Marcel Dekker, Inc All Rights Reserved  ... activation volume For given stress and strain, the value of T lnð_ à =_ Þ is constant for all e e _ temperatures and also for all strain rate values between e0 expẵDG0 = _ kTị and e0 This means... induced mechanical instability can take place leading to a concentration of deformation, a localization of heat and even to the formation of shear bands An overview of different criteria for the... of the material, r and are the mean values of density and specific heat in the temperature range considered Around room temperature, the product rc lies between and MPa=K for most of the materials