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

Figure 1shows the optimum fit achieved by the four equations 5–8for the flow curves of an austenitic steel at different temperatures in therange of relatively small strain up to 0.2.. In

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

was suggested by Hollomon [2] Because of its simplicity, it is till now themost common relation applied for the description of the flow curve How-ever, no yield point is considered by this relation as sY¼ 0 for e ¼ 0 Espe-cially 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 smallstrains A more adequate description is achieved by the Swift relation [3]

For e¼ 0, a yield point is considered with a value of sY¼ KBn Analternative description

was introduced by Voce [4] and is well applicable for the range of small strains

Figure 1shows the optimum fit achieved by the four equations (5–8)for the flow curves of an austenitic steel at different temperatures in therange of relatively small strain up to 0.2 The figure shows that the Swiftrelation and the Voce-relation describe well the flow curves in the relative

If the parametersk1 and k2 are considered to be constants, the flowcurves follow by:

s¼ s0þ ðs1 s0Þ 1  exp e=e½ ð Þ ð12ÞThis equation is identical with the empirical Voce relation In therange of relatively small strains, it fits the experimental data very well How-ever, it fails to describe the flow curves in the range of high strains becausethe experimental results for the flow stress do not asymptotically approach adefinite value [6]

The following modification can be suggested, to yield an tion equation that describes well the strain hardening in the range of highstrains The parameter k1¼ 1=ðl ffiffiffirp Þ, where l is the dislocation free path.This parameter can be considered as a function of strain and may beexpressed as k1 ¼ kð1 þ ceÞ The evolution equation of the flow stressbecomes

The solution of this differential equation is

s¼ C1þ C2eþ C3½1 expðC4eÞ ð14ÞwhereC1 is the yield stress,C2¼ kc=ðk2aGbÞ, C3¼ kð1 þ 2c=k2Þ=ðk2aGbÞ;andC4 ¼ k2=2 This equation is identical with the empirical relation intro-duced in Ref [7] It is found to give the optimum fit for the experimentalresults of several materials(Fig 2).However the determination of its para-meter needs some more effort It should be mentioned that also the empiri-cal Swift relation given by Eq (5) fits well the experimental data in thisstrain range

B Influence of Strain Rate and Temperature

Figure 3ashows an example for the influence of increasing temperature onthe flow stress for given values of strain and strain rate [8] Considering theslope ds=dT, three different temperature ranges can be defined: (A) range oflow temperatures, between absolute zero and about 0.2 of the absolute melt-ing point, where the influence of the temperature on the flow stress is great.The material behavior is governed by thermally activated glide, (B) range ofintermediate temperatures between 0.2 and 0.5 of the absolute melting tem-perature Only a slight influence of strain rate and temperature on the flowstress is usually observed in this range, and (C) range of temperatures higherthan 0.5Tm in which the flow stress depends highly on the temperaturesbecause 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_ee ¼ 3  104sec1[8] (b) Influence of stain rate on shear yield stress of mild steel [9].

variation of@s=@ ln _ee: (I) range of low strain rates with only a slight ence of the strain rate due to athermal glide processes, (II) range of inter-mediate and high strain rates with relatively high strain rate sensitivitydue to thermal activated glide mechanisms, and (III) range of very highstrain rates where internal damping processes dominate and a very highstrain rate sensitivity is observed The boundary between the ranges (I)and (II) depends on the temperature Overviews concerning the mechanicalbehavior under high strain rates are represented, e.g in Refs [10,11].

influ-To estimate the mechanical behavior over wide ranges of strain rateand temperature, constitutive equations must be established taking the timedependent material behavior into consideration A visco-plastic behavior isoften assumed by using, for example, the Perzyna equation [13]

_eeij¼ _SSij

2mþ1 2v2E _sskkdijþ 2ghFðFÞi @f

@sij

ð15Þ

where m is the shear modulus, f is square root of the second invariant of thestress deviator Sijand F¼ (f=k) 1 is the relative difference between f andthe shear flow stressk ¼ sF=pffiffiffi3 The function FðFÞ is often estimated usingsimple rheological models assuming FðFÞ ¼ F and leading to linear relation

of the type s¼ sFðeÞ þ Z_ee which is acceptable for metals only at strain rates

>103

sec1

1 Empirical Relations

Different empirical relations could be implemented in Eq (15) With

FðFÞ ¼ expðF=aÞ  1 or FðFÞ ¼ F1=m, the corresponding relations betweenstress and stress rate in the uniaxial case are identical with the empirical rela-tions introduced 1909 by Ludwik [14]

The influence of temperature on the flow stress is also described by ferent relations of the type s¼ sðe; _eeÞf ðT=TmÞ where Tm is the absolutemelting point of the material, such as

or according to Ref [15]

s¼ s0ðe; _eeÞ 1  ðT=TmÞv

On applying such empirical relations, the flow stress is usuallyrepresented by s¼ ftðeÞf2ð_eeÞf3ðTÞ as a product of three separate func-tions of strain, strain rate and temperature This is a rough approxima-tion especially in the case of moderate strain rates of _ee < 103sec1.However, the basic problem is that nearly all the parameters of theseempirical equations can only be regarded as constants only within rela-tively small ranges of e, _ee, and T The determination of the functionalbehavior of the parameters requires a great number of experiments.Therefore, constitutive equations based on structure-mechanical modelsare gaining increasing interest as they can improve the description ofthe mechanical behavior in wider ranges of strain rates and temperatureand may, if carefully used, allow for the extrapolation of the determinedrelations.

2 Structure-Mechanical Models

The macroscopic plastic strain rate of a metal that results from the lation of sub-microscopic slip events caused by the dislocation motion isgiven by

In this equation, the Burger vector b and the Taylor factor MT areconstants for a given material whereas the mobile dislocation density rm

is mainly a function of strain The relation between the dislocation velocity

vand the stress was experimentally determined for several materials [16] Itcan be represented in the range of low stresses by a power law

v¼ v0ðs=s0ÞN

At very high stresses, the dislocation velocity approachesasymptotically the shear wave velocity cTand s¼ avn=qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1  ðv=cTÞ2

3 Athermal Deformation Processes

In the range of intermediate temperatures and low strain rates (combinedranges B and I), and at relatively low temperatures, i.e., less than 0.3 ofthe absolute melting point Tm, the influence of strain rate and temperaturedepends on the _ee-range of the deformation process

Below a specific value of the strain rate, that depends on ture, only a slight influence of strain rate and temperature on the flowstress is observed In this region I, athermal deformation processes aredominant, in which the dislocation motion is influenced by internal longrange stress fields induced by such barriers as grain boundaries, precipi-tations, and second phases The flow stress varies with temperature in the

tempera-same way as the modulus of elasticity The influence of strain rate can bedescribed by

s¼ C EðTÞ

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 rangestress fields induced by barriers like forest dislocations and solute atomgroups in fcc-materials or by the periodic lattice potential (Peierls-stress)

in bcc materials If the applied stress is high enough, these barriers canimmediately be overcome At lower stresses, a waiting time Dtwis requireduntil the thermal fluctuations can help to overcome the barrier A part of thedislocation line becomes free to run, in the average, a distance s until itreaches the next barrier within an additional time interval Dtm The meandislocation velocity is given by v¼ s=ðDtwþ DtmÞ

The waiting time Dtw equals the reciprocal value of the frequency

n of the overcoming attempts If the strain rate is lower than ca 103sec1, it can be assumed that Dtw4 Dtm The relation between strain rateand stress is then given by _ee ¼ _ee0ðeÞ exp DG=kT½  where _ee0¼

brmn0s=MT The activated free enthalpy DG depends on the difference

s¼ s  sa between the applied stress and the athermal stress according

to kT lnð_ee0=_eeÞ ¼ DG ¼ DG0R

Vds where V¼ bls=MT is thereduced activation volume

For given stress and strain, the value ofT lnð_ee=_eeÞ is constant for alltemperatures and also for all strain rate values between _ee0exp½DG0=ðkTÞ and _ee0 This means that the increase of stress at constant strain withdecreasing temperature or with increasing strain rates is the same, as long

as the values of DG ¼ kT lnð_ee=_eeÞ are equal in both cases

Depending upon the formulation of the function VðsÞ; differentrelations for _ee ¼ _ee0ðsÞ were proposed in Refs [17–21] The mostcommon are the relation introduced by Vo¨hringer [19,20] and by Kocks

and that by Zerilli and Armstrong [22,23]

s sa¼DG0

V0 exp  b0þDGk

0

ln _ee0_ee

5 Transition to Linear Viscous Behavior

At strain rates higher than some 103sec1, the stress is high enough to theextent that Dtwvanishes Only the motion time Dtmis to be considered Thedislocation run with high velocity throughout the lattice and dampingeffects dominate The dislocation velocityv ¼ s=Dtmcan then be given by

v ¼ bðt  thÞ=B according to Ref [24] The flow stress follows the relation

with Z¼ MTB=ðb2NmÞ This relation is validated experimentally in Ref [9]

as well as by Sakino and Shiori [25], as shown inFig 4a.A continuous sition takes place, when the strain rate is increased from the thermal activa-tion range (II) to the damping range (III) This can be described in twodifferent ways: regarding the dislocation velocity to be equal to

tran-v ¼ s=ðDtwþ DtmÞ, the strain rate can be represented by

_ee ¼ _ee0 exp DG0

where x is a function of strain Alternatively, the continuous transition can

be described by an additive approximation The stress is regarded to be thesum of the athermal, the thermal activated and the drag stress components.According to this approximation, s saþ sthþ Z_ee where sthis the thermalactivated component of stress determined from Eq (22) or (23)

nexp Q1RT

C Material Laws for Wide Ranges of Temperatures

and Strain Rates

Material laws that describe the flow behavior over very wide ranges of peratures and strain rates are needed for the simulation of several deforma-tion processes, such as high-speed metal cutting In this case, differentphysical mechanisms have to be coupled by a transition function Fig 5

tem-shows the dependence on the stress with the strain rate at different tures for a constant strain Three main mechanisms can be distinguished: (a)diffusion-controlled creep processes with_eecr/ sNðTÞin the region (1) of lowstrain rates and high temperatures, (b) dislocation glide plasticity with

tempera-s/_eemðTÞ

pl in the region (2) of intermediate temperatures and strain rates,and (c) viscous damping mechanism with s¼ sGþ Zð_ee  _eeGÞ in the region

of very high strain rates _ee > 1000 sec1 in the region (3).

1 Visco-plastic Material Law

For a continuous description over the different ranges, the strain rates have

with_ee ¼ 1 sec1

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 sGand_eeGare determined requiring that thederivative @s=@_ee follows a continuous function in the transition region:

sG¼ sHð_eeG=_eeÞm and _eeG¼ ðmsH=ZÞ1=ð1mÞ The values of the parameterused are given in Ref [28]

An exception of the rule of the reduction of flow stress with increasingtemperature is the influence of dynamic strain hardening observed in ferriticsteel at temperatures between 2008C and 4008C, where the flow stressincreases towards a local maximum It is caused by the interaction betweenmoving dislocations and diffusing interstitial atoms The additional stresscan be described by Ds¼ að_eeÞ exp½fðT  bð_eeÞÞ=cð_eeÞg2, With this addi-tional term, the dependence of flow stress of steel Ck45 (AISI 1045) on tem-perature and strain rate is determined [28] and represented inFig 6

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 majorpart of the deformation energy is transformed to heat while the rest isconsumed 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

dT¼ k0:9

where the factor 0.9 is the fraction of the deformation work transformed toheat, s is the current value of the flow stress which is already influenced bythe previous temperature rise and k is the fraction of energy remaining in thedeformation zone At low strain rate, there is enough time for heat transfersout of the deformation zone and the temperature increase is negligible Inthis case, k¼ 0 On the other hand, the deformation process is almost adia-batic at high strain rate and k¼ 1 A continuous transition from the isother-mal deformation under quasi-static loading to the adiabatic behavior underdynamic loading can be achieved considering k as a function of strain rate inthe form

Figure 7 Quasi-static and adiabatic flow curves of unalloyed fine grained steel

kð_eeÞ ¼1

3þ 43parctan

_ee_eead

An overview of different criteria for the thermally induced mechanicalinstability is presented in Ref [29] The adiabatic flow curve can be deter-mined numerically for an arbitrary function sðe; _ee; TÞ for the shear stresswhich has been determined in isothermal deformation tests In order toobtain a closed-form analytical solution demonstrating the adiabatic flowbehavior, the simple stress–temperature relation s¼ sisoðe; _eeÞCðDTÞ can

be used [30,31] In this case, the change of temperature can simply be mined by separation of variables and integration For example,

deter-s¼ sisoðe; _eeÞ 1  mT T0

Tmis the absolute melting point of the material,
rr and
cc are the mean values

of density and specific heat in the temperature range considered Aroundroom 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 usingTmin K

Many experimental investigations e.g Ref [32] were carried out inorder to determine the temperature dependence of the flow stress Up to ahomologous temperature of 0.6, the stress–temperature relation can bedescribed 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 sion If the isothermal stress can be simply described by

the flow stress, determined in an adiabatic test with constant _ee, is thengiven by

is of the order of magnitude of 1 K=MPa The flow curve shows a maximum

smax at the critical strain ec, where

K¼1þ n

Fð_eeÞ

smax

en c

exis-by two slices, a reference one and another slice with slight deviations instrength 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 ofcompression test at high strain rates

Due to friction between the cylindrical specimen and the loadingtools, a compression specimen becomes a barrel form during the test In

an etched cross-section of a quasi-statically tested specimen, two conicalzones of restricted deformations can often be recognized after quasi-staticupsetting The deformed geometry is symmetrical about the midplane

(Fig 8) An FE-simulation is carried out for a compression test with _ee¼0.001 sec1 considering stain hardening according to Eq (8) and friction

at the upper and lower surfaces by a coefficient m ¼ 0.1 The tional results indicate that the maximum values of equivalent stress as well