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Constitutive modeling of

Constitutive modeling of high-strain-rate deformation in metals based on the evolution of an e?ective

microstructural length

A.Molinari a ,G.Ravichandran

b,*

a

Laboratoire de Physique et Me

′canique des Mate ′riaux,ISGMP,Universite ′de Metz,Ile du Saulcy,Metz 57000,Cedex 1,France b

Graduate Aeronautical Laboratories,California Institute of Technology,Aeronautics and Mechanical Engineering,

Mail Stop 105-50,Pasadena,CA 91125,USA

Received 21November 2003;received in revised form 21May 2004

Abstract

A phenomenological constitutive model is postulated to represent the response of metals over a wide range of load-ing conditions.The model is based on a single internal variable that can be viewed as being related to a characteristic length scale of the microstructure that develops in the metal during deformation.A scaling law for the evolution of this characteristic (or e?ective)length based on experimental observations is proposed.The constitutive model is illustrated by studying the response of copper and comparing with available experimental results over a very wide range of strain rates (10à2–106s à1),strains (up to 1)and temperatures (77–473K).ó2004Elsevier Ltd.All rights reserved.

Keywords:Constitutive equations;Modeling;Metals;High-strain-rate;Microstructural evolution;Mechanical properties

1.Introduction

Considerable progress has been made in the re-cent years towards developing constitutive models for high-strain-rate deformation of metals.These include both phenomenological and physically

based models (Meyers,1994).Examples of the phenomenological models include,multiplicative,(e.g.,Johnson and Cook,1985)and power law (e.g.,Estrin et al.,1997)models.Examples of the physically based models include dislocation (e.g.,Klepaczko and Chiem,1986;Zerilli and Arm-strong,1987),slip (Nemat-Nasser et al.,1998)and internal state variable (e.g.,Bodner and Par-tom,1975;Estrin and Mecking,1984;Follansbee and Kocks,1988)models.The physically based

0167-6636/$-see front matter ó2004Elsevier Ltd.All rights reserved.doi:10.1016/j.mechmat.2004.07.005

*Corresponding author.Tel.:+16263954525;fax:+16264496359.

E-mail address:ravi@http://www.wendangku.net/doc/af5ff14779563c1ec4da7134.html (G.

Constitutive modeling of

Ravichandran).

Mechanics of Materials 37(2005)

Constitutive modeling of

Constitutive modeling of

737–752

http://www.wendangku.net/doc/af5ff14779563c1ec4da7134.html/locate/mechmat

and internal variable models have been proposed to account for the physics of plastic deformation (thermal activation,dislocations,etc.).Both these types of models have had success in?tting the available experimental data over a certain range of strain,strain rates and temperatures.However, these models have met with only limited success in their ability to understand available constitutive data over a wide range of strain rates or in accu-rately predicting change in stress for abrupt changes in strain rate(e.g.,Follansbee and Kocks, 1988).Many of these models also have not been able to satisfactorily explain,for example in cop-per,the strain rate dependence of strain hardening (Tong et al.,1992;Bodner and Rubin,1994)and the apparent strengthening during high strain rate deformation at elevated temperatures(Frutschy and Clifton,1998).

Recent years also have witnessed substantial development in modeling the physics of plastic deformation as well as models for the dependence of stress on the microscopic parameters such as the dislocation density,cell size and misorientation of cells.These models have been primarily developed to explain the various stages of strain hardening (II through V)observed during large strain defor-mation of metals(e.g.,Nes,1998).Link or gauge length models have been proposed where the?ow stress is inversely proportional to the cell size and is based on long range screening of mobile disloca-tions(e.g.,Kuhlmann-Wilsdorf,1989).It has been also proposed to include the inverse square root dependence of the subgrain size on strength(e.g., Hansen and Hughes,1995).These models in gen-eral do not contain evolution laws for the relevant length scales or other microscopic structural para-meters.Recent developments in materials modeling have focused on incorporating e?ects of substruc-ture evolution on macroscopic behavior of materi-als(VanderGiessen and Needleman,1995;Ortiz and Repetto,1999;McDowell,2000;Ashmawi and Zikry,2003).

Based on microstructural observations of defor-mation processing experiments,it has been recog-nized that one of the key features of deformation of metals is the reduction of cell size with increas-ing strain.It has been observed that even starting with early stages of deformation,the cells emerge relatively devoid of dislocations in the cell interiors as schematically shown in Fig.1.Numerous re-search groups have made these observations over a range of strain and temperature for many metals including fcc(aluminum,copper,nickel)and bcc (a-iron,chromium,niobium),for a detailed re-view,see Sevillano et al.(1980).For such a wide range of metals,the normalized average cell size (d c)as a function of equivalent strain falls within a narrow band of scatter(Nes,1998).It has also been noted that the initial rate of reduction in cell size with increasing strain is very high and eventu-ally the cell size saturates at very large strains. Temperature and strain rate could also in?uence the rate of reduction of cell size and the saturation cell size(e.g.,Staker and Holt,1972;Shankara-narayan and Varma,1995).

Based on the experimental observations,evolu-tion laws for cell size as well as the immobile dislo-cation density and misorientation of cells have been proposed(e.g.,Nes,1998).It has also been noted that the dislocation density in the cell interi-ors and the cell orientation saturate at relatively small strains(stages II and III of hardening)and provide only a minor contribution to the internal resistance stress.By performing detailed

Constitutive modeling of

micro-Fig.1.Idealized schematic of microstructural evolution within a grain during deformation(d is initial grain size and d c is the cell size).GNB denote the geometrically necessary boundaries that divide the cell blocks at large strains.

738 A.Molinari,G.Ravichandran/Mechanics of Materials37(2005)737–752

structural characterization it has been shown that the grains are divided into cell blocks separated by geometrically necessary boundaries(GNB)that are hardly penetrable by dislocations and contrib-ute to internal stress(e.g.,Hansen and Hughes, 1995).The cell boundaries are more permeable to dislocations and contribute to the hardening of the metal.The evolution equations suggested for quantities such as cell size and dislocation density are based on the observed dependence of these quantities on strain(e.g.,Shercli?and Lovatt, 1999).The evolution equations are along the lines of the internal state variable model proposed for deformation processing of metals(Sample et al., 1992)and have features common with earlier inter-nal variable model based on dislocations(Rice, 1971).

In Section2,a phenomenological constitutive model is proposed to represent the response of metals over a wide range of loading conditions. The model is based on a single internal variable d which can be viewed as being related to an e?ec-tive or characteristic microstructural length scale. The cell size is certainly an important characteris-tic length in a metal during deformation.However, the proposed model should be viewed as being phenomenological;therefore no quantitative rela-tionship should be sought between the real cell size measured in a metal and the e?ective microstruc-tural length d introduced in the model.Rather than aiming to capture quantitative information on microscopic features,the evolution law for the internal variable d is aimed to mimic the essen-tial trends with regard to the evolution of the microstructure.It is debatable whether a single e?ective length parameter is su?cient for charac-terizing the microstructural evolution.The present view is that a dominant length scale can be distin-guished and its evolution controls most of the material strain hardening.However,this e?ective length scale is not necessarily directly related to the cell size during the entire deformation history.

E?ects of the loading conditions(rate of strain-ing,temperature)on the cell size evolution provide universal trends that have been characterized by experimental observations.These trends are in-cluded in the proposed phenomenological model. An evolution law for the characteristic length d as a function of deformation is introduced which has a similar form as the universal scaling relation governing the observed cell size evolution for a broad range of metals(Sevillano et al.,1980).It is generally assumed that the rate of cell size re?ne-ment and the reference cell size depend on temper-ature and strain rate.Similarly,in the present model,the rate of re?nement of the characteristic length d is assumed to be dependent upon strain rate and temperature through relations that are either empirical or based on thermal activation theory.Although the present model is phenomeno-logical in nature,it has certain similarities with the dislocation based constitutive models(e.g., Mecking and Kocks,1981;Kocks and Mecking, 2003).Results from the predictions of the constitu-tive model are presented in Section3and compared with experimental results for OFHC copper over a wide range of strain rates including jump in strain rate experiments.In Section4,conclusions for the present study are summarized.

2.Modeling

2.1.Hardening and the e?ective microstructural length

The proposed phenomenological constitutive model is based on the physically observed phe-nomena that any characteristic length associated to the microstructure of a metal(for instance the cell size)undergoes a reduction with increasing strain for temperatures which are not too high in relation to its melting point.The process of emer-gence of cells and reduction of cell size during deformation is schematically depicted in Fig.1. During the early stages of plastic deformation of metals,long range obstacles such as the grain boundaries and hence the grain size(d)control the onset of yielding,i.e.,the Hall-Petch e?ect. Upon yielding,in particular for fcc metals with high stacking fault energies,due to multiplication of dislocations within the grains,cell bounda-ries begin to emerge even at fairly small strains (<0.05)(Sevillano et al.,1980;Ananthan et al., 1991).As the deformation proceeds,the cells begin to organize themselves within the grain with fairly

A.Molinari,G.Ravichandran/Mechanics of Materials37(2005)737–752739

low density of immobile dislocations in the cell interiors and with very high density of dislocations at the cell walls.As the deformation proceeds fur-ther the number of cells increase to accommodate increasing dislocation density,thus leading to the reduction of the average cell size while the grain size itself remains fairly constant.At large defor-mations,the cell size does not change appreciably leading to very low hardening regime (stage IV).Evidence for the formation of cells in materials shock loaded or subjected to very high rates of deformation (>106s à1)has been documented (Fol-lansbee and Gray,1991).It has been observed that the cell size for a given strain decreases with increasing strain rate.

It has been established through a careful and thorough compilation of data (Sevillano et al.,1980)that for numerous metals including copper and aluminum,the cell size (d c )reduction follows a universal relation up to very large strains when normalized by an appropriate length scale,as shown in Fig.2.Considering the errors present in experimental measurements,it is remarkable to note that the data lies within a fairly narrow band of scatter (see also,Nes,1998).The rate of change in cell size with strain is dramatic at small

strains (stage II)and then the cell size changes more gradually with applied strain.It has been also observed that the speci?c shape of the cell size reduction curve depends on the strain rate and temperature.It has been noted that in general,for a given strain,the cell size decreases with increasing strain rate and increases with increasing temperature (e.g.,Staker and Holt,1972;Shanka-ranarayan and Varma,1995).

Based on the link or gage length theory,the ?ow stress (r )is proportional to the inverse of the cell size,which is generally agreed upon for modeling ?ow stress of metals (Kuhlmann-Wils-dorf,1989).It has been noted that the grains are further subdivided into cell blocks separated by geometrically necessary boundaries (GNBs)and contribute to the strength through a Hall-Petch type relationship (e.g.,Hansen and Hughes,1995).Numerous other factors can contribute to the strength of metals including the short range barriers such as the immobile dislocations in the cell interior and the relative rotation between adjoining cells.A review of such models and ap-proach can be found elsewhere (e.g.,Nes,1998;Shercli?and Lovatt,1999).It has been noted that the rotation between the cells and the dislocation density in the interior of the cells saturate at cer-tain level of deformation (stage II)while the cell size continues to evolve with further deformation.An alternative approach to constitutive modeling is based on dislocation density where the ?ow stress is assumed to be proportional to the square root of the dislocation density,q (e.g.,Mecking and Kocks,1981;Prinz and Argon,1984;Nix et al.,1985;Klepaczko and Chiem,1986).Typi-cally,in this approach,thermally activated disloca-tion theory is invoked to incorporate the e?ects of strain rate and temperature.For such theories,the dislocation density as a function of strain is needed for computing the ?ow stress.In general,such data are not readily available and hence,based on reasonable values for parameters in the model for evolution law of dislocation density,stress–strain data is ?tted and constitutive models are developed.More recently,a cell-based model has been developed for the evolution of the dislocation density to explain the various stages of hardening (Estrin et al.,1998

Constitutive modeling of

).

Fig. 2.Normalized average cell size (d c /d q )in copper as a function of equivalent strain (e ).Solid squares (j )are exper-imental data points from Sevillano et al.(1980)and the solid line is the best ?t of Eq.(3).The values of the ?tting parameters used are d p =0.23,d q =0.1l m,d c0=0.5l m.d q is the reference saturation cell size and d c0is the initial cell size at e =0.

740 A.Molinari,G.Ravichandran /Mechanics of Materials 37(2005)737–752

It has been proposed that the cell size(d c)and the average dislocation density(q)are related through a simple similitude relationship(e.g., Kuhlmann-Wilsdorf,1989;Kubin,1993),

d c?K

???q p;e1T

where K is a material constant on the order of10–20(Kuhlmann-Wilsdorf,1989)and for copper K=16(Staker and Holt,1972).

The e?ective length scale d introduced in the present phenomenological model is aimed to mimic the essential trends observed in the evolution of the microstructure of metals under quasi-static and dynamic loading for a large range of deforma-tion and temperature.However,in view of the complexity of the physics of deformation and the variety of mechanisms for hardening involved in such a wide range of loading conditions,the e?ec-tive length d shall not be quantitatively related to the cell size(d c)or to any precise microstructural characteristic length.The interest of the phenome-nological model relies on the possibility to include some basic trends observed in the microstructural evolution,without being constrained by a precise quantitative link to the complex microstructure pattern developed in metals.The characteristic length d can be viewed as an e?ective parameter scaling the microstructure evolution.

2.2.Evolution of the characteristic length,d

Since the cell size d c is an important microstruc-ture parameter in metals,some attention should be paid to the analysis of experimental results con-cerning the cell size evolution.The reduction in cell size d c with increasing equivalent strain(e)is shown for copper in Fig.2,which was originally collected and presented for a wide range of metals (fcc and bcc)and experimental conditions(Sevil-lano et al.,1980).This data for cell size can be ?tted using the following phenomenological evolu-tion equation:

d d c d

e ?à

d p

d q

?d2

c

àd q d c ;e2T

where d p is a dimensionless cell re?nement rate parameter and d q is the saturation cell size at rela-tively large strains,which could be functions of both strain rate and temperature.The?rst term within the brackets corresponds to the driving force for decrease in cell size or storage of cells, contributing to strain hardening;the second term corresponds to the driving force for increase in cell size or annihilation of cells,promoting strain sof-tening.Such competition(between storage and annihilation)is inherent in many thermomechani-cal processes including the evolution of dislocation density during deformation.At large strains,the expression within the brackets tends to zero caus-ing the cell size to saturate at d c=d q.The experi-mental data(Sevillano et al.,1980)presented in Fig.2for copper were obtained from rolling or plane strain compression(constant strain rate and temperature)experiments at low strain rates. Assuming d p and d q to be constant(i.e.,constant strain rate and temperature),the analytical solu-tion of the di?erential equation(2)can be ex-pressed in the following form:

d q

d c

?1à1à

d q

d c0

&'

expeàd p eT

!

e3T

where d c0is the initial cell size.The goodness of the ?t of(3)and hence the applicability of(2)in describing the evolution of cell size is shown by the solid curve in Fig.2.The?t su?ciently cap-tures the cell size reduction in copper for small to moderate strains reaching a steady state satura-tion value at large strain.Eq.(2)allows for only a monotonic decrease of cell size with deformation, assuming that d c0is always greater than d q.The proposed evolution law(2)should be used for pro-portional loading histories and not for complex loading histories during which cell structures can be destroyed and regenerated.The evolution law (2)may be used only at temperatures(T)well below the melting point(T M),say,T<0.3T M,where the cell growth due to thermal e?ects is negligible.At present,(2)is not applicable for di?usional creep or other very high temperature deformation proc-esses where considerable growth of cell size(recov-ery)during deformation could occur.

Except for general trends noted for the e?ect of strain rate and temperature on d p and d q,very little data is available in the literature(Staker and Holt, 1972;Shankaranarayan and Varma,1995).One

A.Molinari,G.Ravichandran/Mechanics of Materials37(2005)737–752741

could postulate that the rate of cell re?nement,d p, increases with increasing strain rates and decreases with increasing temperature.The saturation cell size,d q decreases with increasing strain rate and increases with increasing temperature.

In accordance with the objective of this paper, which is to propose a phenomenological model based on a single internal parameter,it is appeal-ing to consider an evolution law for the e?ective length d which mimics the evolution law of such an important structure parameter as the cell size d c.Therefore,it is postulated that the evolution of d is controlled by a law similar to(2),

d d d

e ?à

d r

d s

?d2àd s d e4T

with d r is a dimensionless characteristic length scale re?nement rate parameter and d s is the satu-ration size of the characteristic length scale at relatively large strains,which could be functions of both strain rate and temperature.

Our primary interest in the present work is in modeling high-strain-rate deformation processes and hence,one can pursue one of the following two approaches to incorporate the e?ects of rate and temperature into the evolution law(4).The ?rst approach is to use empirical relations such as, d r?^d re_e;TTand d s?~d se_e;TTe5Twhere_e is the strain rate and T is the absolute tem-perature.Examples of such relations include the power law or other forms of empirical relations. The second approach is to use the theory of ther-mally activated processes to develop expressions of the following form for d r and d s:

d r d r0?1àk r

T

T r0

log

_e r0

_e

p r!q r

d s d s0?

1

1àk s T

T s0

log_e r0

_e

àá

p s

h i q s

e6T

where d r0and d s0are the reference values of d r and d s respectively.k r,p r and q r,and k s,p s and q s are constants to describe the dependence of micro-structure re?nement and steady state characteristic length on strain rate and temperature.T r0,T s0and _e r0,_e s0,are reference temperatures and strain rates for modeling the observed trends in microstruc-ture re?nement and steady state characteristic length.

2.3.Constitutive model

In the proposed constitutive model,we assume that the measured or applied?ow stress(r)is a function of the intrinsic resistance r0of the mate-rial and the strain rate(_e),

r?r0

_e

e0

1m

r0?^redT

d0

e7a;bT

where_e0is a reference strain rate and m is the instantaneous material rate sensitivity which is as-sumed to be a function of the temperature of the form,A/T where A is a material constant(Estrin et al.,1997).The parameter m accounts for part of the thermal softening behavior of material due to the change in ambient temperature and also the temperature rise during adiabatic deformation (another part of the thermal softening is controlled by the temperature dependence of the parameters d r and d s).The foregoing relation for m has been shown to be e?ective for fcc metals(Estrin et al., 1997).

It is important to note that the laws(4)and(7) can be expressed in terms of the normalized non-dimensional quantities,d/d s0and d s/d s0.Therefore, no attention should be paid to the precise quanti-tative values assigned to d0(initial value of d)and to d s0(reference saturation value).Hence,the value of the ratio d0/d s0is far more important.This ratio scales to the range of the relative variation of d,the parameter which plays the role of the hard-ening variable in the present model.Note that the value of d0/d s0is large,but,remains comparable to the value resulting from the consideration of the cell size evolution shown in Fig.2.This is in keep-ing with the spirit of the present phenomenological approach which is aimed to mimic the experimen-tally observed trends in a wide range of loading conditions.

The intrinsic resistance(r0)is a function of the internal variable or the dominant structure para-meter(d).^redTis a strength coe?cient which is

742 A.Molinari,G.Ravichandran/Mechanics of Materials37(2005)737–752

assumed to be a constant for a given material of grain size(d)and can be viewed as re?ecting the strength of the long range obstacles.The depend-ence of the?ow stress to be inversely proportional to d in(7b)is consistent with familiar models based on the cell size parameter d c(Kuhlmann-Wilsdorf,1989;Hansen and Hughes,1995;Nes, 1998).Remarkably,the?nal?ow stress from experiments on copper over nine decades of strain rates(10à3–106sà1)seems to corroborate such a relation(Staker and Holt,1972;Clifton,1988; Tong et al.,1992).The evolution of d as a function of strain is obtained by integration of(4)with the strain rate and temperature e?ects incorporated using(5)or(6)and together with the initial condi-tion,d=d0at e=0.In the present one parameter model,the characteristic microstructural length d is assumed to be independent of the grain size and is purely controlled by deformation.Hence, the evolution of d alone governs the strain harden-ing behavior of the metal.

In quasi-static deformations,the temperature is taken to be the ambient temperature(T a),while during adiabatic deformations,the temperature rise(D T)is computed using the Taylor–Quinney parameter,b which is assumed to be a constant in our modeling,

T?T atD T?T atb

Z e

0r

d ee8T

where q and c are the mass density and the speci?c heat(temperature corrected)for the material respectively.The variation of b with strain and strain rate has been recently reported for metals (Hodowany et al.,2000)and can be easily incorpo-rated in the present model,if necessary.

2.4.Relationship between dislocation based models and the present model

The dislocation based constitutive models using the Kocks–Mecking approach(Kocks,1976;Mec-king and Kocks,1981;Kocks and Mecking,2003) propose an evolution law for average dislocation density,q:

d q d

e ?k1

???q pàk

2

qe9T

where k1and k2are material parameters,k1is con-

sidered to be rate insensitive,since the?rst term on

the right hand side of(9)corresponds to athermal

dislocation storage,while k2depends on strain rate

and temperature,since the second term corre-

sponds to dislocation annihilation(Kocks,1976).

If one assumes the similitude relationship be-

tween the dislocation density,q,and the cell size,

d c,given by(1),and,since,

d d c

d e

K

2

qà3=2

d q

d e

e10T

the evolution of cell size can be written as,

d d c

d e

1

2K

?k1d2

c

àKk2d c :e11T

Comparing(2)and(11)that were obtained

through entirely di?erent approaches(dislocation

density vs.cell size),there are some striking simi-

larities in terms of the dependence of evolution

of cell size on the linear and quadratic terms,d c

and d2

c

.The phenomenological cell size evolution

law appears to be closely related to the evolution

law for dislocation density if one subscribes to

the similitude relationship given in(1).If one as-

sumes(2)and(11)to be equivalent,then,

k1?2K

d p

d q

;k2?2d p:e12T

In(2),both the parameters d p and d q can be

considered to be rate dependent,as suggested by

(5)or(6).However,for(2)and(11)to be identical

in form,(12)would suggest that the parameters d p

and d q should have same dependence on rate and

temperature,i.e.,proportional to each other.This

appears to be highly restrictive on the nature of

evolution of the cell size.

The similarity in form between(2)and(11)has

been obtained via the assumption that the scaling

law(1)can be used.While this law seems plausible

near equilibrium,its validity is questionable for the

highly transient regimes associated to high strain

rate loading.The di?culty to combine and com-

pare the di?erent approaches based on the use of

various length scales such as the cell size or the

characteristic length1=

???q p,supports the use of

the present phenomenological approach where

these di?culties are avoided,while keeping the

A.Molinari,G.Ravichandran/Mechanics of Materials37(2005)737–752743

fundamental features of the experimentally ob-served microstructural re?nement.In addition,it has been recognized that the dislocation evolution within the cells is di?erent from that associated with statistically stored dislocations.Hence,it ap-pears that it is appropriate to build a constitutive approach based on the‘‘cell’’or a characteristic length scale rather than one based on the average dislocation density and the associated evolution equation(9)with the apparent restriction that the parameter k1is rate and temperature insensi-tive as suggested by dislocation based models. The scaling law which is employed here(1)cannot be used to distinguish between dislocation evolu-tion in the interior of the cells and the cell walls.

A micromechanical model for the evolution of the cell wall dislocation density that is similar to(9)has recently been proposed(Estrin et al., 1998).

3.Results

Results from model simulations as well as their comparison with experimental data are presented in this section.At strain rates above103sà1,adia-batic conditions were assumed to prevail,and the e?ect of increase in temperature due to the conver-sion of plastic work to heat is accounted for in the model,via(8).Comparisons pertaining to jump tests and loading-unloading-reloading experiments are also presented.Model predictions are com-pared with results from uniaxial compression and pressure/shear experiments on OFHC copper. 3.1.Material

The material for which the applicability and e?ectiveness of the model is illustrated is pure OFHC copper,since data from experiments is available over a wide range of strain,strain rates and temperatures.The main input to the model is the evolution law(4)of the e?ective microstruc-tural length d as a function of strain at some refer-ence strain rate and temperature and the parameters necessary to describe the strain rate and temperature dependence of d r(microstructure re?nement rate)and of d s(saturation value of d).These parameters may be viewed currently as?t-ting parameters and yet it is important to empha-size that the same set of material parameters are used in all the simulations described here.For the present simulations,d r and d s are assumed to be dependent on strain rate and temperature as follows:

d r?d r01ta r

_e

e s0

n r T

T0

àm r

"#

e13Td s?d s01àa s

_e

s0

n s T

àm s

"#

e14T

d s and d s0ar

e respectively the saturation value and the reference value(at zero strain rate)o

f the e?ec-tive microstructure length,d;a s,n s,m s are positive material parameters controllin

g the strain rate and the temperature dependence of d s.The phenome-nological relationship(14)is aimed to mimic experimental trends showing that the higher is the strain rate the?ner is the microstructural pat-tern,for a given strain and a given temperature. The opposite e?ect is observed when the tempera-ture is increased(at a given strain rate and strain). The parameter d r quanti?es the rate of evolution of d from its initial value d0to the saturation value d s.Experimental results show that the evolution rate is strain rate and temperature dependent. The strain rate and the temperature dependencies are controlled by the positive exponents n r and m r,respectively.The coe?cient a r is also positive, so as to induce a higher evolution rate of the microstructure for increasing strain rates.In other words,at a given strain,and a given temperature, a?ner microstructure will be obtained when higher strain rates are considered.The relevant physical properties for OFHC copper and the parameters for the model used in the simulations are given in Table1.A systematic and deductive methodo-logy for obtaining these material parameters from the experimental data is described in Appendix A.

3.2.Model predictions and comparison to experiments

In the present model there are inherently two types of rate sensitivities,(i)instantaneous rate

744 A.Molinari,G.Ravichandran/Mechanics of Materials37(2005)737–752

sensitivity of the material which is characterized by

the temperature dependent exponent m in (7a)and (ii)strain rate sensitivity of strain hardening which enters the model through the strain rate sensitivity of the re?nement coe?cient,d r and that of the saturation characteristic length,d s (Eqs.(13)and (14)).The model predictions and experimental re-sults (Follansbee and Kocks,1988;Tong et al.,1992)for annealed OFHC copper at three di?erent strain rates,1.5·10à2,8.5·103and 6.4·105s à1are shown in Fig.3(a).The ambient temperature for these experiments is T =295K.The experiment at _e

?1:5?10à2s à1is isothermal,while adiabatic conditions prevail for the two others.For high strain rate adiabatic heating is accounted for as shown in Fig.3(b)where adiabatic and isothermal

responses are compared for the strain rates _e

?8:5?103s à1and 6.4·105s à1

.

The agreement between the model predictions and the experiments is fairly good considering that the results cover a range of strain rates over seven http://www.wendangku.net/doc/af5ff14779563c1ec4da7134.htmling other constitutive models such as the mechanical threshold stress (MTS)model,this has not been possible to have such robust predic-tions especially at the very high strain rate regime (Follansbee and Kocks,1988).The corresponding evolution of d with strain for the three strain rates is shown in Fig.4.The characteristic length d has been normalized with respect to the reference value,d s0(Table 1).The saturation value of the e?ective length d s decreases with increasing strain rate which is an essential feature of the present model resulting in strain rate hardening of the material.The decrease of the normalized e?ective length d /d s0with increasing strain is aimed to mimic the re?nement of the microstructure during the deformation history.Two experimental trends are reproduced here:(i)the evolution is faster for high strain rates,(ii)the saturation value d s is a decreas-ing function of the strain rate.This strain rate dependence is an essential feature,which brings

into the model a strain rate sensitivity of strain hardening.

The yielding and strain hardening characteris-tics of shock loaded (pre-strained)copper have been investigated (Follansbee and Gray,1991).

Table 1

Values of parameters used in the constitutive model for OFHC copper q

(kg/m 3)

c

(J/kg/K)

b

^

r (MPa)

_e 0(s à1)

A (K)

d 0(l m)d r0a r n r m r _

e r0(s à1)d s0(l m)a s n s m s _e s0(s à1)89403860.955

107

40·103

0.5

4.3

50

0.8

107

0.06

0.377

0.24

0.5

10

Constitutive modeling of

7

Fig.3.(a)Model predictions compared to experimental data of Follansbee and Kocks (1988)for annealed OFHC copper at three di?erent strain rates,1.5·10à2,8.5·103and 6.4·105s à1(two experimental data sets for the largest strain rate);(b)comparison of adiabatic and isothermal responses for the high strain rates.

A.Molinari,G.Ravichandran /Mechanics of Materials 37(2005)737–752

745

Initially,annealed copper was shock loaded to 10GPa pressure for 1l s duration and recovered.Subsequently,the shock-loaded material was sub-jected to range of strain rates at di?erent tempera-tures.The comparison between the present model predictions and the experiments (Follansbee and Gray,1991)is shown in Fig.5at strain rates of 10à3and 4·103s à1at 295K and at a strain rate of 10à3s à1at 77K.During shock loading the state variable d evolves towards the ?nal value,d *,

which is the initial value to consider for comput-ing the material response during the subsequent reloading.Since the details of the strain rate his-tory during shock loading are not known with en-ough precision,it is hard to precisely determine d *from simulations.Therefore,in Fig.5,d *has been calibrated to the value 0.115,so as to obtain the correct level of the ?ow stress for reloading.Simu-lations shows that this value of the state variable,would be reached for a plastic strain rate of 1.5·105s à1at the plastic deformation of 0.04,conditions which are comparable to those ob-served in the shock loading (note however that the strain rate sustained by a material particle dur-ing shock loading is time dependent).The results in Fig.5show a good agreement between the model predictions and the experimental data.The initial response is illustrated by the curve with open circles,corresponding to compression load-ing at the rate _e

?0:001s à1and the temperature T =295K.After shock loading,the material is

reloaded at _e

?0:001s à1at two di?erent tempera-tures T =295K (squares)and T =77K (triangles).Predictions of the modeling are the continuous curves with dots.The initial value d *=0.115has been used to simulate all the reloading curves.This value is supposed to correspond to the material state reached at the end of the shock loading.Dy-namic reloading has also been made on a split

Hopkinson pressure bar at the rate _e

?4000s à1and the ambient temperature T =295K (stars).The response predicted by the model using the same initial value d *=0.115,is the lower dotted curve.The upper dotted curve corresponds to

simulations at the strain rate _e

?9000s à1.Experi-mental results obtained for that loading rate (Follansbee and Gray,1991)(not shown to make the ?gure more readable)are also very well de-scribed by the modeling.

The ?ow stress as a function of strain rate at the strains (0.10)and (0.15)is shown in Fig.6for both annealed and shock loaded (10GPa)copper.In Fig.6(a),the experimental result at the strain rate _e

?0:64?106s à1is from Tong et al.(1992),other experimental results are taken from Fig.1of Follansbee and Kocks (1988).The model predic-tion is shown by the solid line.In Fig.6(b),circles correspond to experimental data at e =0.10for

Constitutive modeling of

the

Fig.4.Evolution for di?erent strain rates of the normalized microstructural length,d /d s0(hardening parameter of the model)towards the saturation value,d s /d s0

Constitutive modeling of

.

Fig.5.Model predictions compared to the experimental results for shock loaded and reloaded copper (Follansbee and Gray,1991).

746 A.Molinari,G.Ravichandran /Mechanics of Materials 37(2005)737–752

shocked copper,Figs.3and 9of Follansbee and Gray (1991).Model predictions for the shocked copper are represented by the solid line for e =0.15and the dotted line for e =0.10.Both the materials exhibit relatively small rate sensitivity for strain rates up to 104s à1and beyond this rate,there is a substantial increase in the ?ow stress.The model predicts the trend in the strain rate dependence of the ?ow stress for annealed copper and compares extremely well with experimental data (Follansbee and Kocks,1988).The sharp in-

crease of the ?ow stress around _e

?104s à1is well captured by the model.It can be observed that the strain rate dependence of strain hardening is quite high.It is also clearly seen that the shock loaded material exhibits far less rate dependence of ?ow stress in comparison to the annealed mate-rial and the model predictions compare well with the limited experimental data that is available (Follansbee and Gray,1991).

It has been asserted that when comparison is made at constant structure rather than at constant strain,no increase in strain rate sensitivity is ob-served in polycrystalline copper and other fcc metals at strain rates exceeding 103s à1(Follansbee and Kocks,1988).To assess the concept of con-stant structure,the ?ow stress for annealed copper is plotted in Fig.7as a function of strain rate for three di?erent values of structure parameter,nor-malized e?ective microstructural length (d /d s0),2,1.67and 1.33.In each of these cases it is observed that the ?ow stress is nearly independent of the strain rate.The small rate sensitivity one observes is due to the temperature dependent instantaneous strain rate sensitivity (1/m )in (7).Hence,if the structure evolution (i.e.,the characteristic length d )is not rate dependent,then one would expect that material behavior to be nearly rate indepen-dent since the instantaneous rate sensitivity of all materials is very small (m is typically 100or

Constitutive modeling of

more;

Fig.6.(a)Flow stress as a function of strain rate at the strain e =0.15for annealed copper.Model prediction (solid line)is compared with experimental data (stars);(b)in addition to the results for annealed copper displayed in (a),results are shown here for the pre-shocked copper at two di?erent levels of strains e =0.10and

Constitutive modeling of

0.15.

Fig.7.Flow stress for annealed copper as a function of strain rate for three di?erent values of structure parameter,(d /d s0)=2,1.67,and 1.33.

A.Molinari,G.Ravichandran /Mechanics of Materials 37(2005)737–752747

in the present model m has the value m =137at T =293K and m =85at T =473K).

Fig.8shows the stress strain response in a com-pression test at the strain rate _e

?5000s à1and the initial temperature T =473K.Adiabatic condi-tions have been used in the simulation but they have a minor e?ect for the level of deformation considered here.The predictions of the model compare well with the experimental results,Fig.11of Follansbee and Kocks (1988).

An important aspect of constitutive modeling is the ability to predict the transition in ?ow stress behavior when the strain rate is abruptly changed from one strain rate to another during jump tests.This phenomenon of transition behavior is widely reported in copper (Senseny et al.,1978;Follans-bee and Kocks,1988;Bodner and Rubin,1994).The comparison between the model prediction and the experimental data for annealed copper for a jump in strain rate from 1.5·10à3to 2.5·103s à1at a strain of 0.1625and from 2.5·103to 1.5·10à3s à1at a strain of 0.52are shown in Figs.9and 10and were conducted at an ambient tem-perature of T =295K.The agreement between the model and the experimental results are good and the transition behavior of stress from one strain rate to the other is well represented.In both these cases,the MTS model was not able to cap-

ture this transition behavior accurately (Follans-bee and Kocks,1988).They attributed this shortcoming to the sensitivity of one of their model parameters to strain rate.In Fig.10,the experiments at the dynamic strain rate were per-formed in two steps,the ?rst step being from e =0to 0.52.Accordingly,the simulations were conducted in two steps under adiabatic conditions.The restart of the second step is at room

Constitutive modeling of

tempera-

Fig.8.Prediction of the model (solid line)compared with experimental results (circles)for a compression test at the strain

rate _e

?5000s à1and the ambient temperature T =473K (Follansbee and Kocks,1988

Constitutive modeling of

).Fig.9.Predictions of the model for a strain rate change from _e

?0:0015s à1to 2500s à1at the strain of e =0.1625.Experi-mental results from the jump test are represented by circles (Follansbee and Kocks,1988

Constitutive modeling of

).

http://www.wendangku.net/doc/af5ff14779563c1ec4da7134.htmlparison between experimental data (Follansbee and Kocks,1988)and predictions of the model for a jump in strain rate from 2500s à1to 0.0015s à1at the strain of 0.52.Solid lines show model predictions.

748 A.Molinari,G.Ravichandran /Mechanics of Materials 37(2005)737–752

ture,producing a small bump in the stress strain curve at e=0.52.

It is interesting to note that the e?ective micro-structural length at the end of the dynamic loading (_e?2500sà1,e=0.52)has in the simulation the value d*=0.0622l m.This value is close to the saturation length d s=0.0599l m associated to the subsequent loading at the strain rate _e?0:0015sà1.Therefore the capacity of hardening is considerably reduced as illustrated by the nearly ?at response observed in the subsequent loading. Here one has to note an interesting feature con-tained in the modeling.Suppose that the value d*is a little bit smaller than d s=0.0599l m,then during the subsequent loading d should increase to reach the saturation value d s.This would be a manifestation of strain softening.It is worth not-ing that the experimental data seem to re?ect such softening.The present simulation does not predict such strain softening,but the value of the state variable d*is not far from having material softening.

4.Conclusions and discussion

A phenomenological constitutive model based on the evolution of a single internal structure parameter,the e?ective microstructural length (d),has been postulated,Eq.(7).From the availa-ble data in the literature(Sevillano et al.,1980),it appears that the change in d is purely function of deformation and temperature history.A scaling law for describing the evolution with strain of the normalized e?ective microstructural length has been proposed,Eq.(4).The evolution law (4)has three terms,the?rst term which promotes the reduction of d(strain hardening)and is domi-nant in the very early stages of deformation and the second which inhibits the reduction of d(strain softening)and is dominant at large strains.A third term can be introduced,leading to a constant athermal hardening rate at very large strains(stage IV),but this additional e?ect was not explored in the present simulations.The actual rate of micro-structure re?nement with strain d r and the refer-ence e?ective length d s are material properties that are functions of strain rate and temperature.In the present model,the structure evolution parameter alone governs the strain hardening char-acteristics of the material.The scaling law(Eq.(4)) governing the evolution of the e?ective microstruc-tural length d serves as the input for the constitutive modeling of materials.

The model has been used to study the constitu-tive response of copper over a very wide range of strain rates(10à3–106sà1),strains(up to1)and temperatures(77–473K).For such a wide range of conditions,the input model parameters were kept?xed in all the simulations.The model pro-vided excellent predictions in most cases including the highest strain rates and temperatures as well as for transition behavior during strain rate jump experiments.Under these conditions,the existing constitutive models have considerable di?culties in making predictions that compare well with the experimental data.The present model is simple and yet robust in modeling material behavior over a broad range of deformations.The present evalu-ation of the data for copper in the literature vali-dates the use of the evolution law(4)which was inspired from consideration of the universal scal-ing law(2)governing the cell size evolution. Acknowledgements

GR gratefully acknowledges the support of the O?ce of Naval Research(Dr.J.Christodoulou, Program Manager)for his research on Dynamic Behavior of Metals.He wishes to express his appreciation for the hospitality provided by Labo-ratoire de Physique et Me′canique des Mate′riaux (LPMM)during his visits to the Universite′de Metz.AM gratefully acknowledges the support of Direction Ge′ne′rale de l?Armement(France) and the hospitality provided by the Graduate Aeronautical Laboratories(GALCIT)at Caltech. Appendix A.A systematic method for determining material model parameters

The constitutive law is entirely determined by a set of independent material parameters which are identi?ed as follows.In Eq.(7)^r is considered as

A.Molinari,G.Ravichandran/Mechanics of Materials37(2005)737–752749

a material constant,and the inverse of the instan-taneous strain rate sensitivity m=A/T is charac-terized by the value of A;_e0and d0are not independent parameters,since they can be embed-ded in^r.In(13)and(14)the independent para-meters are respectively,d r0,a r,n r,m r and d s0,a s, n s,m s;while_e r0and_e s0are not independent para-meters,since they can be embedded in a r and a s respectively.In the present paper,because of insuf-?ciency in experimental information,the tempera-ture dependence of the re?nement rate d r is ignored,thus we set m r=0.Also,parametric stud-ies have shown that m r has negligible e?ect on material response for the range of experimental data considered here for OFHC copper.Therefore, to summarize,the model is characterized by a total of nine independent parameters,^r,A,d r0,a r,n r, d s0,a s,n s,m s.These parameters can be identi?ed in a systematic way.Other(dependent)parameters can be chosen in an arbitrary way;the following values are assumed,_e0?107sà1,_e r0?107sà1, _e s0?107sà1,d0=0.5l m.Note that the character-istic lengths d,d s,d s0entering into the constitutive formulation can be normalized by the initial value for the characteristic length,d0.Therefore,from the physical point of view,the relevant quanti-ties are the non-dimensional lengths,d/d0,d s/d0, d s0/d0,and the precise value of d0is not so important.

A?rst characterization of A can be made by calibrating the value of the rate sensitivity para-meter m=A/T0at room temperature T0=293K. For A=40,000,it follows that m=137,the instan-taneous strain rate sensitivity being1/137.The parameter^r has the dimension of stress and scales the material resistance;^r is chosen so that the sim-ulated stress strain curves matches the experimen-tal curves for small deformations.

To determine d r0,a r,n r,d s0,a s and n s,three experimental stress–strain curves(C)are used. Data can be obtained from tensile(or shear)tests performed at three di?erent strain rates at room temperature.The procedure is illustrated for OFHC copper(Follansbee and Kocks,1988). Here,the tests(Fig.3)performed at the strain rates_e1?0:015sà1,_e2?8500sà1and_e3?6:4?105sà1are used to select the parameters.For a given strain rate,say,_e1?0:015sà1,and for the ?xed temperature T0=293K,the values d r1and d s1of the microstructure re?nement rate and of the saturation microstructural length are deter-mined so that the numerical simulation provides the best?t of the corresponding stress strain curve C1.Similarly the values d r2,d s2and d r3,d s3are determined to?t the curves C2and C3correspond-ing to_e2?8500sà1and_e3?6:4?105sà1.At this stage the simulations are made under isothermal conditions.An adiabatic correction accounting for the material self heating due to the dissipation of part of plastic work,should be made later for the highest values of the strain rate_e2and_e3. The adiabatic correction might be anticipated by calibrating the simulated isothermal strain-stress curves so as to be a little bit above the adiabatic experimental curves C2and C3.

Once d r1,d r2,d r3,d s1,d s2,d s3have been esti-mated,the material parameters d r0,a r,n r and d s0, a s,n s are determined in a unique way.Considering Eq.(14)for T=T0and for the strains rates_e1,_e2 and_e3,it follows that,

d s1?d s0e1àa se_e1=_

e s0Tn sTeA:1T

d s2?d s0e1àa se_e2=_

e s0Tn sTeA:2T

d s3?d s0e1àa se_e3=_

e s0Tn sTeA:3T

For a given value of n s,(A.1)and(A.3)provide a system of two equations for the unknowns a s and d s0(dividing(A.1)by(A.3)to get a s,then using (A.1)to calculate d s0).Actually,a s and d s0are ob-tained as functions of n s.Then(A.2)can be consid-ered as a non-linear equation for the unknown n s, d s2?d s0en sTe1àa sen sTe_e2=_e s0Tn sTeA:4TSimilarly d r0,a r,n r can be determined from d r1,d r2, d r3by using Eq.(13).

After obtaining a?rst estimate of these quanti-ties by simulations under isothermal conditions,an adiabatic correction is made for the largest rates of straining.In this correction the temperature in-crease predicted by Eq.(8)is taken into account. These later simulations allow the estimation of the value m s=0.5by getting the best?t of the adi-abatic strain-stress curves C2and C3for OFHC copper(Follansbee and Kocks,1988).Small cor-rections of d r2,d r3,d s2,d s3have to be simulta-

750 A.Molinari,G.Ravichandran/Mechanics of Materials37(2005)737–752

neously made to have a better correspondence between numerical simulations and experimental results when adiabatic e?ects are accounted for.

Finally,an adjustment of A can be made to better account for the test at the low temperature T=77K(Follansbee and Gray,1991).The?nal value,A=40,000was obtained by combining this information with a good match of the instantane-ous strain rate sensitivity at room temperature.

In summary,a total of four stress–strain curves (e.g.,three at di?erent strain rates at room temper-ature and another at one of these three strain rates at a di?erent temperature to re?ne the estimate of the instantaneous rate sensitivity parameter A)are needed to extract the necessary material parame-ters of the model.The procedure outlined above has been used to obtain the material parameters using the experimental data in Fig.3for OFHC copper and are shown in Table1.Once the para-meters are obtained,they are then?xed to perform simulations of response at other strain rates (including jump-tests)and as well as at other temperatures.

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