Identification of the model describing viscoplastic behaviour of high strength metals

October 23, 2009 14:42 Inverse Problems in Science and Engineering Rzeszow˙09˙Invers4

Inverse Problems in Science and EngineeringVol. 00, No. 00, January 2008, 1–12

RESEARCH ARTICLE

Identification of the model describing viscoplastic behaviour of

high strength metals

T. Fras, Z. Nowak∗, P. Perzyna, and R.B. Pecherski

5B Pawinskiego, Warsaw, 02-106, POLAND; Institute of Fundamental Technological

Research, Polish Academy of Sciences

(v3.4 released May 2008)

Ultra fine grained (UFG) and nanocrystalline metals (nc-metals) are studied. Experimentalinvestigations of the behaviour of such materials under quasistatic as well as dynamic load-ing conditions related with microscopic observations show that in many cases the dominantmechanism of plastic strain is multiscale development of shear deformation modes. The com-prehensive discussion of these phenomena in UFG and nc-metals is given in [2] where it hasbeen shown that the deformation mode of nanocrystalline materials changes as the grain sizedecreases into the ultrafine region. For smaller grain sizes (d < 300 nm) shear band devel-opment occurs immediately after the onset of plastic flow. Significant strain-rate dependenceof the flow stress, particularly at high strain rates was also emphasized. Our objective is toidentify the parameters of Perzyna constitutive model, a new description of viscoplastic defor-mation, which accounts for the observed shear banding. The viscoplasticity model proposedearlier by Perzyna [5] was extended in order to describe the shear banding contribution in[1]. The shear banding contribution function, which was introduced formerly by Pecherski[8] and applied in continuum plasticity accounting for shear banding in [9] plays pivotal rolein the viscoplasticity model. The derived constitutive equations were identified and verifiedwith application of experimental data provided in paper [3], where quasistatic and dynamiccompression tests of UFG and nanocrystalline iron specimens of a wide range of mean grainsize were reported. Numerical simulation of the compression of the prismatic specimen wasmade by a ABAQUS FEM program with UMAT subroutine. Comparison with experimentalresults proved the validity of the identified parameters and the possibilities of the applicationof the proposed description for other high strength metals.

Keywords: ultra fine grained metals; nanocrystalline metals; viscoplastic deformation;shear banding; shear banding contribution function; numerical simulation of compressiontest; identification of the model parameters

AMS Subject Classification: 74C20; 74S05; 74P99; 74H05

1. Introduction

The subject of the study is concerned with ultra fine grained (UFG) and nanocrys-talline metals (nc-metals). Experimental investigations of the behaviour of suchmaterials under quasistatic as well as dynamic loading conditions related with mi-croscopic observations presented in [2–4] show that in many cases the dominantmechanism of plastic strain is multiscale development of shear deformation modes- called shear banding. The comprehensive discussion of these phenomena in UFGand nc-metals is given in [2] and [3] where it has been shown that the deformationmode of nanocrystalline materials changes as the mean grain size decreases into the

∗Corresponding author. Email: [email protected]

ISSN: 1741-5977 print/ISSN 1741-5985 onlinec© 2008 Taylor & FrancisDOI: 10.1080/1741597YYxxxxxxxxhttp://www.informaworld.com


2 T. Fras, Z. Nowak, P. Perzyna and R.B. Pecherski

ultrafine region. For smaller grain sizes (d < 300 nm) shear band development oc-curs immediately after the onset of plastic flow. Significant strain-rate dependenceof the flow stress, particularly at high strain rates was also emphasized.

The aim of the paper is to propose a new description of viscoplastic deforma-tion, which accounts for the observed shear banding. Viscoplasticity model withan overstress function proposed earlier by Perzyna [5–7] was extended. The theo-retical description of multiscale hierarchy of shear localization modes presented byPecherski [8], [9], [10] and the new concept of shear banding contribution functionintroduced in [8], [10] and identified for polycrystalline Cu in [11], [14] were ap-plied by that. The derived constitutive equations were identified with applicationof experimental data of quasistatic and dynamic compression tests, made for theUFG and nanocrystalline iron specimens of different mean grain sizes described in[3].

The identification procedure was made under the assumption that the Huber-Mises yield condition obeys. This is rather big oversimplification in the situationswhen the strength differential effect (strength asymmetry) is observed. Such aneffect is observed particulary in the case of nanocrystalline metals, where the ratioof yield strengths in compression versus yield strength in tension depends on grainsize and can reach 1.6 [22]. More detail and comprehensive study of the strengthasymmetry and related pressure sensitivity effect on yield strength in amorphousalloys is given in [24]. Some attempts of the application of Coulomb-Mohr andDrucker-Prager criteria to account for the mentioned effects were also discussedby the authors in [23], [22] and [24]. Although our analysis is based on the Huber-Mises criterion, the paper is concluded with discussion of the application of theparaboloidal yield criterion proposed originally by W. Burzynski [25] and rediscov-ered many times by others [27].

2. Physical motivation

Experimental investigations discussed, e.g. by Meyers et al. [2] show that nanocrys-talline materials exhibit very high yield strength. A conventional soft metal canacquire a ten-fold increase in strength when the mean grain size approaches thenanoscale, presumably due to the grain-boundary strengthening known as the Hall-Petch effect. For example, strengths as high as 1.0 GPa in nc-copper and 2.5 GPain nc-iron have been reported. Constitutive models for metallic materials must ac-count for the effects of the rate of deformation. Significant strain-rate dependenceof the flow stress has been observed in many pure metals, particularly at highstrain rates. However, very few results have been reported on the high-strain-ratebehavior of nc- or UFG materials. The work of Jia, Ramesh and Ma [3] on an80 nm-Fe found little strain-rate dependence in the strain rate range from 1×10−4

to 3×10+3s−1. Localized deformation has been reported for nc-metals by severalgroups; typically this is associated with macroscopic perfect plasticity or even ap-parent strain softening. Localization of plastic deformation into shear bands wasobserved in nc-Fe-10%Cu [4] and nc-Fe [3]. Malow et al. [18] observed shear bandsin nc-Fe samples after micro hardness tests. The reported in [2] and in the workof Jia et al. [3] observations indicate that there is a transition from uniform tonon-uniform deformation as the grain size decreases down to the nanoscale, ac-companied possibly by a reduction of strain hardening.


Inverse Problems in Science and Engineering 3

3. Experimental observations of deformation behaviour of nanomaterials

3.1. Quasi-static compression

In the paper of Jia, Ramesh and Ma [3] compression tests were performed to obtainfull stress-strain curves over a wide range of strain rates. The typical specimendimensions for the low and high strain rate tests were 2.2 × 2.2 × 3.5 mm (length)and 1.6 × 1.6 × 1.4 mm (length), respectively. The quasi-static compression tests atstrain rates of 1 – 2 × 10−3 s−1 were performed using a screw-driven ATS machine.From the tests it is apparent, that the yield strength increases with decreasing grainsize. Compared with the 20 µm-Fe, the strength of the nano-Fe (80 nm) is increasedby an order of magnitude. The strain hardening rate changes with the grain size.In the range of grain size from 20 µm to 980 nm, there is no marked change inthe slope of the curves. However, there is a transition from strain hardening toapparent strain softening as the grain size changes from ∼1 µm to ∼300 nm. Forgrain sizes below 300 nm, apparent strain softening appears at a very low plasticstrain. Ductile behaviour is observed at relatively large grain sizes, the samples withmean grain sizes smaller than 200 nm fail relatively early. It was demonstrated byMeyers et al. in [2] and by Jia et al. [3] that for bcc metals we do not expect tohave a significant effect of mean grain size on the strain hardening, and so theseobservations indicate that a change of deformation mechanism has occurred at thesmaller grain sizes. In the work of Jia et al. [3] the measured flow stresses (at afixed strain of 4%) and the yield strengths are observed to satisfy the well-knownHall-Petch relationship (σy= σ0+Kd−1/2 ).

3.2. High strain rate compression

Experimental results of Jia, Ramesh and Ma [3] also show that the little influence ofthe strain rate on the strain hardening is observed, which is typical for bcc metals.The termination of the high-rate stress-strain curves for 80 nm and 138 nm grainsizes represent specimen failure rather than unloading, see Fig. 3. The influence ofthe rate of deformation on the flow stress is also investigated by Jia, Ramesh andMa [3] for the entire range of grain sizes and strain rates, with the flow stressesplotted corresponding to a fixed strain of 4%. In [3] the authors observed that thesmaller grain size materials are much stronger at low rates, but show less relativestrengthening at high strain rates. An approach to understanding mean grain sizeeffects on the viscoplastic deformations and on the flow stress for bcc Fe is presentedin the following subsections.

3.3. Phenomenology of shear bands

The deformation mode of nc- and UFG materials changes dramatically as the grainsize is decreased into the ultra-fine-grain range. In the 20µm-Fe and 980nm-Fe, thecompressive deformations were uniform at all strain rates and no shear bandswere evident under either the optical microscope or SEM. However, for all smallergrain sizes (d < 300 nm) shear band development was observed in [3] to occurimmediately after the onset of plastic deformation, correlating to the observedchange in apparent strain hardening at those grain sizes. Shear bands were observedduring both quasistatic and high-rate deformations for these grain sizes.

It was demonstrated by Meyers et al. in [2] that additional shear bands appearwith increasing strain and that the newly generated shear bands have similar orien-tations (in the four possible shearing planes for these cuboidal specimens). Largenumbers of shear bands are observed, rather than a single dominant band that



lead to failure. Shear bands have been observed by Wei et al. [4] and [20] in bothlow and high strain-rate tests. In [3] it was observed that under dynamic loading,conventional polycrystalline iron did not exhibit localized deformation. Shear bandpopulations were observed in all specimens with grain sizes d < 300 nm. It can beconcluded that the shear bands play an important role in plastic deformation ofultrafine grained Fe.

4. Constitutive modelling

4.1. Viscoplasticity model of Perzyna

If the dependence on strain rate comes into play, the associated viscoplasticity flowlaw can be applied, cf. Perzyna [5, 6]:

Dvp = γvpµ , µ =1√2k

τ′

, (1)

where Dvp is the rate of viscoplastic deformation, τ′

denotes deviatoric Kirchhoffstress and γvp is the viscoplastic shear strain rate, while k is the correspondingquasistatic yield shear strength.

For Huber-Mises yield condition: J2 − k = 0, J2 =√

12σ

′ : σ′ ,

γvp = γvp0

[

J2

k− 1

]

1D

for J2 − k > 0 (2)

and

γvp = 0 for J2 − k ≤ 0 (3)

where σ′

is the deviatoric Cauchy stress and k is the shear yield strength whileγvp

0 and D denote material constants. The shear strain rate (2) is controlled byan overstress function to be specified for a particular material. In our case thepower-like overstress function is assumed.

4.2. Multiscale system of shear bands contribution in plastic flow

The description of a multiscale system of shear bands contribution in plastic flowwas given in papers by Pecherski [8] and [9]). The rate of inelastic deformation wasassumed in the form

Dvp = Dvps + Dvp

SB. (4)

The scalar shear banding contribution function was defined:

fSB =

∥

∥DvpSB

∥

∥

‖Dvp‖ , ‖A‖ =√

A · A =√

AijAij (5)



where A is a symmetric second order tensor, while Dvps is the rate of viscoplastic

deformation produced by dislocation mediated crystallographic slips and DvpSB de-

notes the rate of viscoplastic deformation produced by shear banding.As discussed in [10], among many possible realizations of shear banding, one cansingle out the group of processes characterizing with the same contribution of two

symmetric shear banding systems f(1)SB = f

(2)SB. In the case of proportional loading

paths the total viscoplastic shear strain rate can be expressed as follows:

γvp = γvps + γvp

SB (6)

and due to fSB = γvpSB/γvp we have

γvp(1 − fSB) = γvps , fSB ∈ [0, 1) , (7)

what leads according to (2) to the following constitutive relation for the viscoplasticshear strain rate controlled by the discussed mechanisms of crystallographic slipand shear banding:

γvp =γvp

0

(1 − fSB)

[

J2

ks− 1

]

1D

for J2 − ks > 0 . (8)

Inverting (8) gives the relation for the dynamic yield condition

J2 = kds = ks

{

1 +

[

(

γvp (1 − fSB)

γvp0

)D]}

. (9)

For the compression test considered in [3] the following specification of quasi-static yield strength can be proposed

σY s = A(d) + B(d)(εvp)n (10)

where σY s =√

3ks and εvp is the equivalent plastic strain εvp =√

33 γvp. The sym-

bols A(d) and B(d) denote the values which are dependent on mean grain diameterd, e.g. according to Hall-Petch relation. A is the quasi-static initial yield strength,B(d)(εvp)n corresponds to a plastic hardening function and n corresponds to plastichardening parameter. Furthermore, the viscosity parameter γvp

0 and the contribu-tion function fSB are assumed to be dependent on the mean grain diameter d.

The shape of the contribution function fSB is proposed, accounting to the studiesin [10] and [11] supported by the numerical identification in [14] 1, in the form oflogistic function:

1The proposed specification of the contribution function fSB is suitable for the description of proportionalloading processes. More general formula, applicable for arbitrary states of deformation and loading paths,was proposed in [12].



fSB =fSB0

1 + exp(a − b(d)εvp), (11)

where fSB0, a, b(d) are material parameters to be specified.

The dynamic yield condition J2 = kd is obtained by inverting Eq. (9) and thefinal dynamic yield strength in uniaxial compression σY d reads:

σY d = (1 − fSB(d))2 [A(d) + B(εvp)n]

[

1 +

(

(1 − fSB(d)) εvp

ε0(d)

)D]

. (12)

5. Identification of the model

The identification of the constitutive models parameters is obtained by an inversemethod. A parametric identification program is developed, based on a conjugategradients algorithm (cf. Nowak and Stachurski, [15, 16]). Our objective is to iden-tify the parameters of the Perzyna viscoplastic constitutive model parameters:β = (A, B, n, a, b, fSB0

, εvp0 , D). The identification of these constants is car-

ried out by means of compression true stress-strain diagrams. These curves stemfrom the iron experimental tests performed at various strain rates εvp and variousaverage grain sizes d (cf. Jia et al. [3]).

5.1. Criterion of minimization

The criterion or cost function is the part of the program where the parameter vec-tor β appears. The criterion chosen is the quadratic sum of errors. The method ofleast squares requires the residual sum in stress between the experimental observa-tions and model results to be minimised. This sum is made with every experimentalpoint. This implies that the proposed model of the grain size dependent viscoplas-ticity can be completely calibrated by minimising the residual:

F (x) = minNα∑

α=1

{

σexpeq (εvp

α , εvp, d) − σcaleq (εvp

α , εvp, d, β)}2

σexpeq (εvp

α , εvp, d)2 (13)

where F (x) refers to the residual of the constitutive model and theexperimental data with the number of experimental points α andβ = (A, B, n, a, b, fSB0

, εvp0 , D) denotes a vector of unknown material

parameters to be determined.Furthermore, εvp

α are discrete values of the strains εvp. The symbols σexpeq and

σcaleq denote the experimental and calculated stresses for the same strain level εvp

α ,

Nα is the number of stress-strain data for the test with given strain rate and grainsize. For our constitutive equation Eq. (12) we have



σcaleq = σY d =

(

1 − fSB0

1+exp(a−b(d)εvpα )

)2(A(d) + B(εvp

α )n)

1 +

(

1−fSB0


)

εvpα

εvp0 (d)

D

(14)

5.2. Gradient vector of the cost function

The gradient vector computation is performed with the central derivative formu-lation. The expression of the numerical gradient vector is:

g =[

∇F (βj)]N

j=1=

[

F (βj + dβj) − F (βj − dβj)

2dβj

]N

j=1

(15)

The application of the standard derivative analytical formulation of a multi-variable function gives the theoretical gradient vector. In the case of the Perzynamodel, the theoretical gradient vector is given as follows

g =

[

∂F (β)

∂βj

]

(a, b, fSB0, A, B, n, εvp

0 , D)

(16)

=

[

∂F (β)

∂a,∂F (β)

∂b,∂F (β)

∂fSB0

,∂F (β)

∂A,∂F (β)

∂B,∂F (β)

∂n,∂F (β)

∂εvp0

,∂F (β)

∂D

]

.

It implies that for one material point for Np measurements of (εvp)i and (σexp)i

with known (εvp, d) the gradient vector g is given.

5.3. Termination test

As for the criterion, there are numerous formulations to stop the conjugate gradientalgorithm. The termination test employed for the identification program consistsin computing the following relation for each experimental point:

χ(i) =

∣

∣

∣

∣

∣

(σexpeq )i − (σcal

eq )i

(σexpeq )i

∣

∣

∣

∣

∣

(17)

and for each parameter:

ηi =

∣

∣

∣

∣

∣

βji − β

j−1i

βj−1i

∣

∣

∣

∣

∣

(18)

with βi, the parameter under consideration, and j, the iteration number.The termination test is satisfied when every point verifies:



χ(i) ≤ χref (19)

and when every parameter verifies:

ηi ≤ ηref (20)

with χref , the point accuracy, ηref , the parameters accuracy. The identificationerror limits are given by the equations (17) and (19). The second set of equations(18) and (20), gives parameter stability relation.

5.4. Identification of quasi-static and dynamic compression tests

To evaluate the quality of the identified parameters, a simulation of compressiontest is performed for the prismatic sample and compared with the correspondingtest. The ABAQUS FEM code was employed to realize this simulation. Simulationleads to plastic strain and strain rates in the same range than experimentation. Anexample of the application of the proposed constitutive description for modellingof the behaviour of polycrystalline iron under quasistatic and dynamic compressiontests for experimental data of Jia, Ramesh and Ma (2003) is depicted in Fig. 2 andFig. 3 (cf. [3]), where the compressive yield strength σY d =

√3kd.

In case of the quasistatic and dynamic compression we use the following formsof equation (14):when d > 300 nm, fSB = 0, β = (A, B, n, εvp

0 , D)

σcaleq = [A(d) + B(εvp)n]

[

1 +

(

εvp

ε0(d)

)D]

(21)

and when d < 300 nm, fSB > 0, B = 0, fSB0=0.95, a = 5, D = 0.08 and n = 0

(no hardening) and β = (A, b, εvp0 , D)

σcaleq = (1 − fSB)2 [A(d)]

1 +

(

1 − fSB0


)

εvp

ε0(d)

D

(22)

Perzyna model parameters are determined for each kind of specimen with differ-ent average grain size. In each case we have started our computations assuming atthe beginning a broad range of feasible parameters. For instance the initial param-eters values in cases for different average grain size: d=20µm, d=980nm, d=268nm,d=138nm and d=80nm for dynamic tests we have taken are presented in Table 1.

At least, the identification leads to one set of model parameters available forquasi-static strain rates and dynamic strain rates. In the numerical simulations ofcompression tests by ABAQUS [21] the specimen is completely modelled with 226991 nodes and 216 004 solid elements (type C3D8R in Explicit or C3D8 in Stan-dard). The specimen is supported by a fixed rigid wall and is impacted by a secondmoving one with an imposed velocity of 5 m/s. Rigid walls are chosen as infiniteplanes with infinite mass and finite friction (fc=0.005). The undeformed sample



Table 1. The initial parameters values for dynamic tests.

Cased=20µm: 100.0 ≤ A ≤ 1000.0, 50.0 ≤ B ≤ 1000.0, 0.01 ≤ n ≤ 1.5

1.0x 10+3 ≤ εvp0

≤ 1.0x 10+5, 0.01 ≤ D ≤ 0.3,d=980nm: 100.0 ≤ A ≤ 1000.0, 100.0 ≤ B ≤ 1000.0, 0.01 ≤ n ≤ 1.5

1.0x 10+4 ≤ εvp0

≤ 1.0x 10+7, 0.01 ≤ D ≤ 0.3,d=268nm: 500.0 ≤ A ≤ 2000.0, 10. ≤ b ≤ 100.0,

1.0x 10+5 ≤ εvp0

≤ 1.0x 10+7, 0.01 ≤ D ≤ 0.3d=138nm: 500.0 ≤ A ≤ 3000.0, 50.0 ≤ b ≤ 500.0,

1.0x 10+5 ≤ εvp0

≤ 1.0x 10+7, 0.01 ≤ D ≤ 0.3d=80nm: 1000.0 ≤ A ≤ 5000.0, 50.0 ≤ b ≤ 500.0,

1.0x 10+5 ≤ εvp0

≤ 1.0x 10+7, 0.01 ≤ D ≤ 0.3

mesh and the Mises stress distribution for dynamical compression are presented onFig. 1a and Fig. 1b.

(a)

(Avg: 75%)S, Mises

+1.528e+03+1.531e+03+1.533e+03+1.536e+03+1.539e+03+1.542e+03+1.545e+03+1.548e+03+1.551e+03+1.554e+03+1.557e+03

(b)

Fig. 1. a) The mesh used in numerical simulations and b) Mises stress distri-bution for dynamic compression test of iron for εvp = 0.08 and d = 268nm.

Simulation leads to plastic strain and strain rates in the same range than experi-mentation. Fig. 2 and Fig. 3 compares the experimental and simulated stress–straindiagrams for a 5 m/s impact test. This results shows that identification of consti-tutive models for nano-iron with shear bands can be performed even with a plasticrange of about 0.1.

Finally, the following constitutive parameters are found for the quasi-static anddynamic compression tests when d < 300 nm, fSB > 0, B = 0, a=5, fSB0

=0.95,D = 0.08 and n = 0 (no hardening) and β = (A, b, εvp

0 ) for quasi-static compres-sion and β = (A, b, εvp

0 ) for dynamic compression:

Table 2. The identified constitutive parameters when d < 300 nm.

quasi-static dynamic

d εvp0

(d)[s–1]A(d)[MPa] b(d) A(d)[MPa] b(d)

80 nm 2363.53 50.24 2086.57 56.49 145.0 x 10+6

138 nm 2047.83 52.85 1882.40 55.11 50.5 x 10+6

268 nm 1160.0 27.46 1237.09 30.01 50.1 x 10+6



Fig. 2. True stress – true strain for quasistatic compression test for polycrys-talline iron. Solid lines represent curves obtained from viscoplasticity modelaccounting for shear bands according to equation (21) for d > 300nm andaccording to equation (22) for d < 300nm, symbols � correspond to the qua-sistatic experimental data for iron of purity 99.9% obtained in two-step con-solidation procedure to form bulk Fe with desired grain size from Jia et al. [3].

Fig. 3. True stress – true strain for dynamic compression test for polycrys-talline iron. Solid lines represent curves obtained from viscoplasticity modelaccounting for shear bands according to equations (21) for d > 300nm and ac-cording to equation (22) for d < 300nm, symbols � correspond to the dynamicexperimental data for iron of purity 99.9% obtained in two-step consolidationprocedure to form bulk Fe with desired grain size from Jia et al. [3].

6. Conclusions

The proposed description of viscoplastic behaviour of high strength metals, inparticular UFG and nc-metals, can be extended accounting for the application ofmore adequate yield criterion, which in the case of associated flow law, provides


REFERENCES 11

also the appropriate potential function. According to our studies [29], the yieldcriterion, which is adequate to strength materials and multiphase materials, e.g.metal matrix and ceramic matrix composites reinforced by particles of nano andmicro size should be pressure dependent, as it was discussed in early papers of W.Burzynski (1929) [25], [26]. Such a criteria take, in the principal axes, the shape of arotationally symmetric paraboloid for isotropic materials and ellipsoidal paraboloidfor materials having orthotropic symmetry [28], [29], Fig. 4a and Fig. 4b.

(a) (b)

Fig. 4. a) Burzynski yield limit for plane stress approximating the resultspresented in [23]; b) Burzynski yield surface in the principal axes coordinatescalculated for the data in plane stress given in Fig. 4. a).

Acknowledgement(s)

The part of this work was made within the framework of the Research Project0364/B/T02/2008/35 of the Ministry of Higher Education and Science of Poland.

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Identification of the model describing viscoplastic behaviour of high strength metals