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International Journal of Impact Engineering 68 (2014) 15e27

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International Journal of Impact Engineering

journal homepage: www.elsevier .com/locate/ i j impeng

The effect of adiabatic thermal softening on specific cavitation energyand ductile plate perforation

Rami MasriMechanical Engineering Department, ORT Braude College of Engineering, Karmiel, Israel

a r t i c l e i n f o

Article history:Received 10 June 2013Received in revised form15 December 2013Accepted 24 December 2013Available online 8 January 2014

Keywords:Cavitation pressureSpecific cavitation energyAdiabatic thermal softeningDuctile hole enlargementBallistic limit

E-mail address: ramim@braude.ac.il.

0734-743X/$ e see front matter � 2014 Elsevier Ltd.http://dx.doi.org/10.1016/j.ijimpeng.2013.12.008

a b s t r a c t

The effect of adiabatic thermal softening on specific cavitation energy of metals is analytically investi-gated with respect to ballistic limit predictions. Explicit stress-plastic strain relation that includes strainhardening response, thermal softening effect and constant strain rate sensitivity is obtained fromJohnsoneCook integral equation under adiabatic conditions and an analogous stress-strain relation issuggested for the Ludwik hardening model. Extensions of these two adiabatic curves for an arbitrarystrain hardening response are derived from generalized integral equations and an example for the Vocehardening model is demonstrated. Adiabatic thermal softening is found to be governed by an exponentialdecay which is controlled by two nondimensional softening parameters and the strain hardening effectwhile increase of yield stress by a constant strain rate response leads to an increase of the thermalsoftening effect. Decrease of spherical and cylindrical, plane-strain and plane-stress, specific cavitationenergies due to adiabatic thermal softening is quantified for several aluminium and Weldox steel alloysand reveals an effect of 2e21% with the greatest impact on aluminium 7075-T651 plates under plane-stress conditions. This effect is reduced by a factor of two in ballistic limit predictions but is intensi-fied in estimations of low residual velocities via striking velocities that are close to the ballistic limit.Comparison of theory predictions with simulation results and experimental data for several aluminiumand Weldox steel alloys demonstrates the validity of the present analytical model.

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1. Introduction

In numerical simulations of perforation and penetration me-chanics strain hardening, strain rate sensitivity and thermal soft-ening of metal targets are mostly modelled by the JohnsoneCook(JC) phenomenological relation [1]

se ¼�Aþ Bεnp

��1þ C ln

�_εp_ε0

���1�

�DT

Tm � Tr

�m�(1)

where the temperature increase under adiabatic conditions due toplastic work dissipation is given by Refs. [2e4]

DT ¼ T � Tr ¼ cWp

rCpWp ¼

Zεp0

sedεp: (2)

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The effective stress se is expressed in terms of the effectiveplastic strain _εp and the effective plastic strain rate εp and(A,B,n,C,m,Tm,c,r,Cp) are material parameters. With _ε0 for thequasi-static strain rate response A denotes the static yield stressY, (B,n) reflect the strain hardening response, C is the strain ratecoefficient, m is the thermal softening exponent, Tm and Tr are themelting and the room temperatures, r denotes the materialdensity, Cp is the specific heat capacity and Th ¼ DT/(Tm�Tr) is thehomologous temperature. The TayloreQuinney coefficient c isthe part of the specific plastic work Wp (plastic work per unitvolume) that is converted into specific heat [5] and usually esti-mated by 0.9. Besides perforation mechanics, the JC model iswidely used in other applications such as machining [6,7] andcrashworthiness [8,9].

Increase of the hardening exponent n reflects decrease of thestrain hardening response for 0 < εp < 1 and increase of this effectfor εp > 1 while increase of the softening exponent m reduces thethermal softening effect. The special case ofm ¼ 1 is used in impactsimulations of steel plates subjected to small-arms [10], in a nu-merical investigation of quasi-brittle fracture during impact of7075-T651 aluminium plates [11] and in further perforation

Fig. 1. Adiabatic (solid line for m ¼ 0.859) and isothermal (dashed line) curves of a25 mm thick 5083-H116 aluminium plate (Table 1) due to JC model (3). The curve form ¼ 1 is identical to the exact solution (8) while dotted and dash-dotted linesrepresent the approximations (10) and (12) respectively. The square markersdemonstrate the accuracy of (14) for the maximum point.

R. Masri / International Journal of Impact Engineering 68 (2014) 15e2716

mechanics researches [12,13]. In other numerical investigations ofperforation mechanics [14e17] the values ofm are ranged between0.859 and 1.131 but it should be mentioned that values outside thisrange are possible. Actually, in Table 1 in Ref. [1] eight differentmetals were modelled bym values equal or very close to one whilefor other four metals m ¼ 0.55,0.55,1.44,1.68.

In Ref. [16] a parametric study, based on finite element simula-tions, is carried out to identify the importance of the different termsof the JC constitutive relation on ductile perforation resistance. Theresults indicate that thermal softening cannot be neglected and anad-hoc procedure has been proposed to include this effect in thecavitation model for ballistic limit predictions [18]. However, themodel in Ref. [18] is limited to cylindrical plane-strain deformationpattern inMisesmedia and to the Ludwik power hardening law andis not developed to take material softening into account. Moreover,unlike quasi-static and dynamic spherical [19e21] and plane-straincylindrical [22e24] cavitation fields, where internal pressure rea-ches a saturation level known as the cavitation pressure (pc), incircular hole enlargement under plane-stress conditions a finiteasymptotic value is exist for the energy input needed to create a unitof nominal expansion volume [25,26]. This energy saturation level,which is denoted in Ref. [26] as the specific cavitation energy (sc), is

Table 1Material properties of aluminium 5083-H116 for different plate thicknesses [16] via JC madiabatic c ¼ 0.9) curves along with thermal softening effect (62). Other material paramkgK]) and Tr ¼ 293 [K]. Numerical predictions (NP) of ballistic limits [16] and eight esadiabatic) curves. Experimental results (ER) from Ref. [39] are also presented in [m/s].

Plate thickness 15 mm 20 mm

Y [MPa] (100Sy) 143 (0.20) 124 (0.18)(n,B [MPa]) (H) (0.216,462) (3.23) (0.252,456) (3.S (c ¼ 0.9) 0.087 0.076pSc (1.49,1.36) (�8.7%) (1.35,1.24) (�8pMc (1.26,1.17) (�7.1%) (1.14,1.06) (�7pTc (1.10,1.03) (�6.4%) (0.99,0.93) (�6sTc (0.94,0.79) (�16.0%) (0.86,0.74) (�1Vb (NP) (251,229) (270,250)Vb (ER) 216.8 249.0VSb (267,255) (293,281)

VMb (246,237) (270,260)

VTb (229,222) (251,244)

VsTcb (212,194) (234,217)

identified with cavitation pressures for plane-strain and sphericaldeformation patterns and captures an essential nature of penetra-tion [27] and perforation [26,28,29] processes.

The main goals of the present research are to shed more light onthe effect of adiabatic thermal softening via the JC formulation andto quantify this effect when connected with ductile plate perfora-tion and ballistic limit predictions. The latter is achieved with thequasi-static spherical and cylindrical (plane-stress and plane-strain) specific cavitation energies for Mises and Tresca solids. Inorder to assess only thermal softening effect we neglect the strainrate sensitivity (C ¼ 0) so that the effective stress se is expressedonly in terms of the effective plastic strain εp by (Ref. [16])

se ¼�Y þ Bεnp

��1�

�cWp

rCpðTm � TrÞ�m

: (3)

Expression (3) represents an integral equation for the adiabaticstress-plastic strain curve se ¼ seðεpÞ that can be solved by asimple numerical procedure. The corresponding isothermal curve(where the generated heat flows away such that DT h 0 andthermal softening is avoided) is obtained with c ¼ 0. In Fig. 1 weshow the adiabatic (solid line) and the isothermal (dashed line)curves of a 25 mm thick 5083-H116 aluminium plate (as demon-strated in Ref. [16] for plastic strain levels up to one). Material pa-rameters from Ref. [16], with m ¼ 0.859, are given in Table 1. Wealso include in Fig. 1 the curve for m ¼ 1 to demonstrate that in-crease in m reduces the thermal softening effect. Adiabatic andisothermal curves of a Weldox 900E steel alloy are shown in Fig. 2and material parameters from Ref. [14], withm ¼ 1.131, are listed inTable 2. The curve for m ¼ 1 demonstrates again that increase in mreduces the thermal softening effect. The solid lines in Fig. 3 showthe temperature increase DT for both metals as function of εp. Thetemperature increase of the steel is about twice the temperatureincrease of the aluminium but the Th profiles are much closer sincethe steel melting temperature is about twice the melting temper-ature of the aluminium. However, the aluminium experiences astronger softening response and one reason is that its m value islower by about 25% while in the next section we will discuss theinfluence of other material parameters.

In Section 2 we first find an explicit JC adiabatic stress-plasticstrain relation as an approximate solution of (3) and then extendthe solution to an arbitrary hardening response with an examplefor the Voce hardeningmodel. In Section 3we repeat the procedurefor the Ludwik stress-strain curve along with connections to Baiand Johnson relation and Bell’s formula [30] and extend the solu-tion to an arbitrary strain hardening response. Besides thermal

odel. Specific cavitation energies sc ¼ ðpSc ; pMc ; pTc ; sTc Þ, in [GPa], for (isothermal c ¼ 0,

eters are (E ¼ 70 [GPa],n ¼ 0.3,m ¼ 0.859,Tm ¼ 893 [K],r ¼ 2700 [kg/m3],Cp ¼ 910 [J/timations from (63) for sc ¼ ðpSc ; pMc ; pTc ; s

Tc Þ are given in [m/s] for the (isothermal,

25 mm 30 mm

59 (0.08) 119 (0.17)68) (0.285,511) (8.66) (0.256,475) (3.99)

0.036 0.073.1%) (1.17,1.07) (�8.6%) (1.36,1.25) (�8.1%).0%) (0.98,0.90) (�8.2%) (1.15,1.06) (�7.8%).1%) (0.84,0.78) (�7.1%) (1.00,0.93) (�7.0%)4.0%) (0.77,0.67) (�13.0%) (0.87,0.75) (�13.8%)

(286,267) (328,308)256.6 309.7(305,292) (361,346)

(280,268) (332,318)

(259,249) (309,298)

(248,231) (289,268)

Fig. 2. Adiabatic (solid line for m ¼ 1.131) and isothermal (dashed line) curves of aWeldox 900E steel alloy (Table 2) due to JC model (3). The curve for m ¼ 1 is identicalto the exact solution (8) while dotted and dash-dotted lines represent the approxi-mations (10) and (12) respectively. The square markers demonstrate the accuracy of(14) for the maximum point. Function pSðεpÞ (divided by three for scaling with othercurves) for spherical cavity expansion (Section 4) is presented under isothermal(dashed line) and adiabatic (dash-dotted line) conditions.

Fig. 3. Temperature increase due to JC model (3) for two metals (Tables 1 and 2) asfunction of the effective plastic strain (solid lines). Dash-dotted and dotted linesrepresent approximations (15) and (16) respectively.

R. Masri / International Journal of Impact Engineering 68 (2014) 15e27 17

softening response, in both sections a constant strain rate effect isalso discussed. Such constant value may represent some averagevalue of the strain rate response or may reflect the assumption thateffects due to strain rate variations are negligible at high strainrates. With the aid of a new unified expression for the cavitationpressures, Section 4 is devoted to examination of thermal softeningeffect on spherical and cylindrical plane-strain and plane-stressspecific cavitation energies. The formulae for these fundamentalmaterial properties were obtained in previous papers [19,20,23,26],where material behaviour is modelled by hypoelastic Mises orTresca flow theories with arbitrary stress-strain curves, accountingfor (elastic) compressibility and large strains (including finitethickness changes under plane-stress conditions). This generalformulation for the specific cavitation energy allows us to include

Table 2Material properties of threeWeldox steel alloys [14] via JC model. Specific cavitationenergies sc ¼ ðpSc ; pMc ; pTc ; s

Tc Þ, in [GPa], for (isothermal c ¼ 0, adiabatic c ¼ 0.9)

curves along with thermal softening effect (62). Other material parameters are(E ¼ 210 [GPa],n ¼ 0.33,Tm ¼ 1800 [K],r ¼ 7850 [kg/m3],Cp ¼ 452 [J/kgK]) andTr ¼ 293 [K]. Numerical predictions (NP) and experimental results (ER) of ballisticlimits for conical nose [14] and eight estimations from (63) for sc ¼ ðpSc ; pMc ; pTc ; s

Tc Þ

for (isothermal, adiabatic) curves are given in [m/s].

Steel alloy Weldox 900E Weldox 700E Weldox 460E

Y [MPa] (100Sy) 992 (0.47) 859 (0.41) 499 (0.24)(n,B [MPa]) (H) (0.568,364) (0.37) (0.579,329) (0.38) (0.458,382) (0.77)m 1.131 1.071 0.893S (c ¼ 0.9) 0.167 0.145 0.084pSc (4.12,3.99) (�3.2%) (3.66,3.55) (�3.0%) (2.58,2.49) (�3.5%)

pMc (3.51,3.42) (�2.6%) (3.12,3.04) (�2.6%) (2.19,2.13) (�2.7%)

pTc (3.11,3.04) (�2.3%) (2.76,2.70) (�2.2%) (1.92,1.87) (�2.6%)

sTc (2.33,2.16) (�7.3%) (2.04,1.90) (�6.9%) (1.43,1.33) (�7.0%)Vb (NP) 330.9 329.5 272.4Vb (ER) 340.1 335.0 290.6

VSb (397,391) (374,369) (314,309)

VMb (367,362) (346,341) (290,286)

VTb (345,341) (325,321) (271,268)

VsTcb (299,288) (279,270) (234,226)

adiabatic thermal softening effect and constant strain rate responsewith the aid of the explicit curves of Sections 2 and 3. However, inSection 4 we have focused on the thermal softening effect forseveral aluminium (5083-H116, 7075-T651, 6070-(0,T4,T6,T7)) andWeldox (460E,700E,900E) steel alloys via JC and Voce formulations.The weakening effect of thermal softening is found to be similar forspherical and plane-strain cylindrical cavitation pressures but isabout 1.5e3.5 times higher for plane-stress specific cavitation en-ergy. Connection of ductile hole enlargement process to ductileplate perforation and ballistic limit predictions is discussed inSection 5. The effect of thermal softening on specific cavitationenergy is found to reduce by a factor of two in ballistic limit cal-culations but is intensified for low residual velocity estimationswhere striking velocities are close to the ballistic limit. Comparisonof theory predictions with simulation results and experimentaldata demonstrates the validity of the present model.

2. Explicit JohnsoneCook adiabatic curve

The nondimensional form of the stress-plastic strain relation (3)is

S

Sy¼�1þ Hεnp

�241�0@SZεp0

S

Sydεp

1Am35 (4)

where S ¼ se/E with E for the elastic modulus, and the yield strainSy the (nondimensional) hardening parameter H and the (nondi-mensional) thermal softening coefficient S are given by

Sy ¼ YE

H ¼ BY

S ¼ cYrCpðTm � TrÞ : (5)

Examples of these nondimensional measures are given inTables 1 and 2. As it stands, expression (4) is an integral equation forthe effective stress as function of the effective plastic strainS ¼ SðεpÞ and it satisfies the boundary condition Sðεp ¼ 0Þ ¼ Sy.Constant effective plastic strain rate ð _εp ¼ const > _ε0Þ may repre-sent some average value of the strain rate response or may reflectthe assumption that effects of strain rate variations are negligible athigh strain rates. The contribution of such constant strain rate dueto JC relation (1) l ¼ 1þ C lnðconst= _ε0Þ is equivalent to an increaseof the static yield stress by a factor of l. In terms of equation (4) it

R. Masri / International Journal of Impact Engineering 68 (2014) 15e2718

means replacing the static yield stress Y with the ‘dynamic’ yieldstress lYonly in the expressions for the yield strain and the thermalsoftening coefficient so that

Sy ¼ ðlYÞE

H ¼ BY

S ¼ cðlYÞrCpðTm � TrÞ : (6)

By differentiating the integral equation (4) with respect to εp

and using (4) to eliminate the integral expressions in the derivativewe arrive with a first order nonlinear ODE for S

dSdεp

¼8<:nHεn�1

p

1þ Hεnp�mS

�1þ Hεnp

�241� S

Sy

�1þ Hεnp

�351�1=m9=;S

(7)

with εp as the independent variable. Exact closed-form solution isexist for the practical case m ¼ 1 under the boundary conditionSðεp ¼ 0Þ ¼ Sy

S ¼ Sy

�1þ Hεnp

�exp

�� Sεp

�1þ H

1þ nεnp

��: (8)

The isothermal curve is obtained for c¼ 0 (0S¼ 0) and thermalsoftening effect, which is governed by an exponential decay like inRef. [30], is increased with increase of S and is found to be coupledwith the strain hardening response. Moreover, while constanteffective plastic strain rate increases thematerial yield strength, thestrain hardening response is reduced due to a stronger thermalsoftening effect. In addition, the exact solution (8) reveals that theeffective stress is approaching asymptotically to zero in the deepplastic zone ðεp/NÞ so that T is approaching asymptotically to themelting temperature.

For understanding the role of parameterm in (7) we suggest thevariable transformation S=Sy ¼ 1þ Hεnp � Z and arrive with anequation for Z

dZdεp

¼ nHεn�1p

1þ HεnpZ þmS

�1þ Hεnp

�1=mZ1�1=m

�1þ Hεnp � Z

�(9)

under the boundary condition Zðεp ¼ 0Þ ¼ 0. From (4) it is clearthat Z � 1 for small values of εp for which SxSy0ZxðSεpÞmð1þ HεnpÞ. Substituting this approximation, instead ofneglecting Z with respect to one in the last term of (9), gives adifferential Bernoulli equation for Z with a closed-form solutionwhich is valid for εp � 1 and leads to

S ¼ Sy

�1þ Hεnp

��1

��Sεp

�1þ H

1þ nεnp �

Sm

1þmεmp � HSm

1þ nþmεnþmp

��m:

(10)

This approximation is presented by dotted lines in Figs. 1 and 2and the much better agreement, even beyond εp ¼ 1, in Fig. 2 isbecause in this case the effective stress remains relatively close tothe yield stress. Expression (10) can be further approximated by anexpansion in powers of εp

S=Sy ¼ 1þ Hεnp � Smεmp � ð1þ nþmÞ1þ n

HSmεnþmp : (11)

This relation suggests a practical boundary condition for nu-merical integration of (7) since the RHS in (7) is unbounded underthe condition Sðεp ¼ 0Þ ¼ Sy.

In the spirit of the exact result (8) for m ¼ 1 and theapproximate solution (11) for any value of m and by recallingthat many practical values of m are identical or near one, wesuggest the following approximation for the stress-plastic straincurve

S ¼ Sy

�1þ Hεnp

�exp

���Sεp

�1þ H

1þ nεnp

��m(12)

which reduces to (8) form¼ 1 and to (11) for small values of εp. Thisexpression is very accurate for practical values ofm as illustrated bythe dash-dotted lines in Figs. 1 and 2. Yet, in the deep plastic zoneapproximation (12) overestimates the JC adiabatic curve for mvalues lower than one and underestimates this curve for m > 1.However, even relatively high differences (via m values not nearone) have only little effect on specific cavitation energies (Section4) and they are with even less effect in ballistic limit predictions(Section 5).

By differentiating (12) with respect to εp and eliminating theexponent term with the aid of (12) we obtain the differentialequation

dSdεp

¼(nHεn�1

p

1þ Hεnp�mSm

�1þ Hεnp

��εp

�1þ H

1þ nεnp

��m�1)S

(13)

which serves as an excellent approximation of (7) and leads to anequation for the maximum point of the stress-plastic strain curveðεpm;SmÞ

εm�npm

�1þ Hεnpm

�2�1þ H

1þ nεnpm

�m�1

¼ nHmSm

: (14)

The accuracy of the results via equation (14) is demonstrated bysquare markers in Figs. 1 and 2. Increase of m or decrease of S(weaker softening response) lead to a maximum point for a higherplastic strain level with the asymptotic result εpm/N forisothermal curves (S ¼ 0). The effect of a constant strain rate isreflected here only through S so that increase of the strain rateresponse (stronger softening effect) leads to a maximum point for alower plastic strain.

With the aid of (4) and (12) and since Th ¼ SZ

εp

0

S

Sydεp a very

accurate expression for the temperature increase is given by

DT ¼ ðTm � TrÞ�1� exp

���Sεp

�1þ H

1þ nεnp

��m�1=m;

(15)

as illustrated in Fig. 3 by the dash-dotted lines, while theapproximation

DT ¼ cYrCp

εp

�1þ H

1þ nεnp

�(16)

for εp � 1 is insensitive to (m,Tm,Tr) and shown by dotted lines inFig. 3. The effect of a constant strain rate is reflected here onlythrough S (or by replacing Y in (16) with lY) so that increase of thestrain rate response leads to an increase of the temperature for thesame plastic strain level. The material temperature is approachingasymptotically to the melting temperature in the deep plastic zoneso that the homologous temperature Th is less than one for anyplastic strain level. The exact result for m ¼ 1 can be simply ob-tained by using the first law of thermodynamics for adiabaticconditions rCpdT ¼ csedεp with the JC model and solving for

Fig. 4. Adiabatic (solid lines) and isothermal (dashed lines) curves of 20 mmaluminium 5083-H116 and 7075-T651 plates (Tables 1 and 3) due to JC model (3).Differences from the dash-dotted lines that illustrate (12) are hardly noticeable.Horizontal dotted lines represent perfectly-plastic response for the given yieldstresses. The square markers demonstrate the accuracy of (14) for the maximumpoint.

Table 3Material properties of aluminium 7075-T651 [11] via JC model. Specificcavitation energies sc ¼ ðpSc ; pMc ; pTc ; s

Tc Þ, in [GPa], for (isothermal c ¼ 0,

adiabatic c ¼ 0.9) curves along with thermal softening effect (62). Otherparameters are identical with those in Table 1. Experimental result (ER)for ballistic limit from Ref. [43] and eight estimations from (63) for sc ¼ðpSc ; pMc ; pTc ; s

Tc Þ for (isothermal, adiabatic) curves are given in [m/s].

Aluminium 7075-T651

Y [MPa] (100Sy) 520 (0.74)(n,B [MPa]) (H) (0.52,477) (0.92)m 1S (c ¼ 0.9) 0.317pSc (2.29,2.05) (�10.5%)

pMc (1.94,1.76) (�9.3%)

pTc (1.71,1.57) (�8.2%)

sTc (1.49,1.18) (�20.8%)Vb (ER) 633

VSb (721,683)

VMb (664,632)

VTb (623,597)

VsTcb (582,518)

R. Masri / International Journal of Impact Engineering 68 (2014) 15e27 19

T ¼ TðεpÞ under the condition Tðεp ¼ 0Þ ¼ Tr while the exactdifferential equation for any value of m is

dTh1� Tmh

¼ S�1þ Hεnp

�dεp: (17)

Same procedure can be performed when replacing JC hardeningterm with an arbitrary strain hardening response Z ¼ ZðεpÞ whereZðεp ¼ 0Þ ¼ 1, namely

S

Sy¼ Z

241�0@SZεp0

S

Sydεp

1Am35 (18)

which leads to the differential equation

S0 ¼"Z0

Z�mSZ

�1� S

SyZ

�1�1=m#S (19)

where superposed prime denotes differentiationwith respect to εp.Function Z defines the relative deviation of the strain hardeningresponse from perfectly-plastic behaviour and takes the form ofZ ¼ 1þ Hεnp for the JC model (4). Again, the effect of a constantstrain rate is included in (18) by replacing the static yield stress Ywith the ‘dynamic’ yield stress lY only in Sy and S.

For m ¼ 1 the exact solution is

S ¼ SyZ expð�SIZÞ IZ ¼Zεp0

Zdεp ¼ Wpðc ¼ 0ÞlY

(20)

with an interpretation of IZ as the normalized isothermal specificplastic work with respect to the ’dynamic’ yield stress. The term SI-includes the contributions of yield stress, strain hardeningresponse, strain rate effect and thermal parameters (c,Cp,r,Tm,Tr).For Z h 1 only a softening response is governed by a separableequation via (19) which, with the aid of 1� S=SyxðSεpÞm, leads tothe approximate solution

S ¼ Sy exp��Smεmp

�(21)

with reduction to perfectly-plastic response by c¼ 0. In the spirit ofthese two solutions and the results for JC model we suggest theapproximation

S ¼ SyZ exp� ðSIZÞm

�(22)

for practical values of m not far from one which for εp � 1 reducesto

S ¼ SyZ1� ðSIZÞm

�xSyZ

�1� Smεmp

�: (23)

Thermal softening effect is stronger for higher values of S andlower values of m and increases with the increase in strainhardening response and strain rate parameter l. In Fig. 4 wecompare the isothermal (dashed lines) and the adiabatic (solidlines) curves of aluminium 5083-H116 and 7075-T651 alloys. TheJC material parameters of aluminium 5083-H116 for plate thick-ness of 20 mm are given in Table 1 and the JC parameters ofaluminium 7075-T651 [11] are listed in Table 3. The yield stress ofaluminium 7075-T651 is about four times higher and leads to athermal softening coefficient which is higher by the same factorwhile the value of m is higher only by about 15%. The effect ofstrain hardening on thermal softening response is demonstratedin Fig. 5 by the area under the curves. These curves represent the

relative deviation of isothermal response from perfectly-plasticbehaviour (dotted lines in Fig. 4) which is much lower for the7075-T651 aluminium alloy. Following all these observations weconclude that the stronger softening effect of the aluminium7075-T651 (Fig. 4) is due to its much higher yield stress so that theeffect of S overcomes the opposite trends due to the values of mand I-. However, by comparing data for the 25 mm aluminium5083-H116 plate (Table 1, Fig. 1) with data of Weldox 900E steelalloy (Table 2, Fig. 2) we see that the effect of a much higher Svalue for the steel does not overcome the opposite trends due tomand I- so that the aluminium experiences a stronger softeningresponse.

By differentiating (22) with respect to εp and eliminating theexponent term with the aid of (22) we obtain

Fig. 5. The curves represent the relative deviation of isothermal response fromperfectly-plastic behaviour for 20 mm aluminium 5083-H116 and 7075-T651 plates(Tables 1 and 3) due to JC model (3). The effect of strain hardening on thermal soft-ening is demonstrated by the area under the curves (I-).

Table 4Material properties of four aluminium 6070 tempers [31] via Voce model. Specificcavitation energies sc ¼ ðpSc ; pMc ; pTc ; s

Tc Þ, in [GPa], for (isothermal c ¼ 0, adiabatic

c ¼ 0.9) curves along with thermal softening effect (62). Parameters (E,n,Tm,Tr,r,Cp)are identical with those in Table 1 while herem ¼ 1. Numerical predictions (NP) andexperimental results (ER) of ballistic limits [31] and eight estimations from (63) forsc ¼ ðpSc ; pMc ; pTc ; s

Tc Þ for (isothermal, adiabatic) curves are given in [m/s].

Temper 0 (Annealed) T4 (naturallyaged)

T7 (Overaged) T6 (peakstrength)

(Q1,Q2) [MPa] (79.5,88.2) (35.6,247.7) (55.3,31.1) (30.1,72.8)(C1,C2) (56.9,4) (80.6,6.5) (317.2,10) (185.9,7.7)Y [MPa] (100Sy) 38.8 (0.055) 172.7 (0.25) 292.5 (0.42) 350.0 (0.50)S (c ¼ 0.9) 0.024 0.105 0.179 0.214pSc (0.51,0.50)

(�2.6%)(1.16,1.09)(�6.0%)

(1.33,1.28)(�3.8%)

(1.49,1.41)(�5.4%)

pMc (0.43,0.42)(�2.3%)

(0.98,0.93)(�5.1%)

(1.14,1.10)(�3.6%)

(1.27,1.22)(�3.9%)

pTc (0.37,0.36)(�2.0%)

(0.85,0.81)(�4.7%)

(1.01,0.98)(�3.0%)

(1.13,1.09)(�3.6%)

sTc (0.32,0.31)(�4.4%)

(0.73,0.66)(�9.6%)

(0.73,0.65)(�11.0%)

(0.85,0.74)(�12.9%)

Vb (NP) 296.0 438.1 438.3 480.6Vb (ER) 348.0 506.2 529.1 562.5

VSb (340,337) (513,498) (550,539) (582,566)

VMb (313,309) (472,460) (509,500) (537,527)

VTb (290,286) (440,429) (479,472) (507,498)

VsTcb (270,265) (407,387) (407,384) (440,410)

Fig. 6. Adiabatic (solid line form ¼ 1) and isothermal (dashed line) curves of a 6070-T4aluminium alloy (Table 4) due to Voce model (28). The adiabatic curves form ¼ 0.8 andm ¼ 1.2 demonstrate the excellent approximation (22) that is given by dash-dottedlines while the curve m ¼ 1 is identical with the exact solution (20). The squaremarkers demonstrate the accuracy of (25) for the maximum point.

R. Masri / International Journal of Impact Engineering 68 (2014) 15e2720

dSdεp

¼�Z0

Z�mSmZIm�1

Z

�S (24)

which serves as an excellent approximation of (19) for practicalvalues of m and leads to an equation for the maximum point

Z0 ¼ mSmZ2Im�1Z : (25)

With the aid of (18) and (22) and since Th ¼ SZ

εp

0

S

Sydεp a very

accurate approximation of the temperature increase is given by

DT ¼ ðTm � TrÞ�1� exp

� ðSIZÞm� 1=m (26)

which is independent of elastic parameters while the approxima-tion for εp � 1

DT ¼ cðlYÞrCp

IZ ¼ cWpðc ¼ 0ÞrCp

(27)

is also insensitive to the values of (m,Tm,Tr). Expression (26) sup-ports the numerical findings in Ref. [16] that DT will increase withincreasing target yield stress, strain hardening response and strainrate effect.

As an example, in Ref. [31] the JC hardening law has beenreplaced by a two-term Voce hardening response, where(Q1,Q2,C1,C2) are material parameters,

Z ¼ 1þ Q1

Y

1� exp

��C1εp��þ Q2

Y

1� exp

��C2εp��

(28)

to describe the stress-plastic strain curves of four different tempersof aluminium 6070. For this strain hardening response

IZ ¼�1þ Q1

Yþ Q2

Y

�εp � Q1

C1Y

1� exp

��C1εp��� Q2

C2Y

1

� exp��C2εp

��(29)

and

Z0 ¼ C1Q1

Yexp

��C1εp�þ C2

Q2

Yexp

��C2εp�

(30)

and the effective stress of the isothermal curve is approachingasymptotically to (Y þ Q1 þ Q2) in the deep plastic zone. In Ref. [31]the value m ¼ 1 was suggested for all four tempers and othermaterial parameters are given in Table 4. In Fig. 6 we present foraluminium 6070-T4 the adiabatic curve with m ¼ 1 and theisothermal curve with the asymptotic value of 456 [MPa] whilecurves for m ¼ 0.8 and m ¼ 1.2 demonstrate the excellentapproximation by (22). The square markers illustrate the accuracyof (25) for the maximum point.

R. Masri / International Journal of Impact Engineering 68 (2014) 15e27 21

3. Explicit Ludwik adiabatic curve

With the modified Ludwik power hardening law the adiabaticstress-plastic strain curve is given by the integral equation

S

Sy¼�Sþ εp

Sy

�n241�

0@SZεp0

S

Sydεp

1Am35 (31)

where here the hardening response (first term in the RHS) is not anexplicit function of εp. Note that the hardening exponent n isdifferent than that of the JC relation (4) but constant strain rateresponse is included, as in Section 2, by replacing Y with lY in theyield strain and the thermal softening coefficient. Differentiation ofthe simpler equation for m ¼ 1 with respect to εp leads to

S0 ¼�nS0 þ 1Sþ εp

� S�Sþ εp

Sy

�n�S (32)

and by introducing the effective elastoplastic (total) strainε ¼ Sþ εp we arrive with the differential relation

d ln S ¼ nd lnε� S�

ε

Sy

�n

dεþ SS�

ε

Sy

�n

d ln S (33)

where the last term can be neglected when compared with the LHS.Integrating the approximate equation under the boundary condi-tion Sðε ¼ SyÞ ¼ Sy leads to a very accurate expression for theadiabatic stress-strain curve

S ¼ Sy

�ε

Sy

�n

exp

(� SSy

1þ n

"�ε

Sy

�1þn

� 1

#)ε � Sy (34)

with the extension for practical values of m not far from one

S ¼ Sy

�ε

Sy

�n

exp

(�

SSy

1þ n

"�ε

Sy

�1þn

� 1

#!m)ε � Sy:

(35)

Fig. 7 presents isothermal and adiabatic Ludwik curves with thematerial parameters of the 25 mm thick 5083-H116 aluminium

Fig. 7. Adiabatic (solid lines) and isothermal (dashed line) curves due to Ludwik model(31) for material parameters of 25 mm 5083-H116 aluminium plate (Table 1). Adiabaticcurves for m ¼ (0.8,1,1.2) demonstrate the excellent approximation (35) which isshown by dash-dotted lines. The square markers demonstrate the accuracy of (38) forthe maximum point.

plate (Table 1) including n ¼ 0.285 but with m ¼ 1 to demonstratethe excellent accuracy of (34) while the curves for m ¼ (0.8,1.2)illustrate the accuracy of (35).

The maximum point ðεm;SmÞ of the stress-strain curve (34) isgiven by

εm ¼ Sy

�n

SSy

�1=ð1þnÞ

Sm ¼ Sy

�n

SSy

�n=ð1þnÞexp

�� SSy

1þ n

�n

SSy� 1�� (36)

with physical result for n � SSy and under the practical assumptionn [ SSy

Sm ¼ Sy

1þ n

�n

SSy

�n=ð1þnÞ(37)

where we have used e�n/(1þn)x1/(1 þ n) for practical values of n.The exact value of εm for the stress-strain curve (35) is obtainedfrom an implicit equation like (14) but for the practical case εm[Sy

an excellent approximation is

εm ¼ Sy

��n

mð1þ nÞ�1=m1þ n

SSy

1=ð1þnÞ(38)

which reduces to the exact result in (36) for m ¼ 1. The accuracy of(38) is demonstrated by the square markers in Fig. 7.

An equivalent form of (34) is

S ¼ Si

�ε

εi

�n

exp

(SSy

1þ n

�εi

Sy

�1þn"1�

�ε

εi

�1þn#)

ε � Sy

(39)

where ðεi;SiÞ is an arbitrary point on the curve (34). Under theassumption that effective shear stress s and effective shear strain g

can be calculated based on the Mises relationsffiffiffi3

ps ¼ se and

g ¼ffiffiffi3

pε we can arrive with an expression for the adiabatic shear

stress-strain curve s ¼ s(g). Now, with (gm,sm) as the maximumpoint of this curve (under the practical assumption SSy � n < 0.3)

gm ¼ffiffiffi3

pSy

�n

SSy

�1=ð1þnÞsm ¼ Y=

ffiffiffi3

p

1þ n

�n

SSy

�n=ð1þnÞ(40)

the following formulation

s ¼ sm�

g

gm

�n

exp

(n

1þ n

"1�

�g

gm

�1þn#)

g �ffiffiffi3

pSy (41)

is identical to Bai and Johnson [30] relation, with (gm,sm,Tr) for(gi,sM,q0) in their notation, but nowwehave expressions for (a,s

*) in

Bell’s formula [30]

a ¼ ��

11� ð1� lcÞTr=Tm

�lc

Tm

s*¼

264 1� ð1� lcÞTr=Tm� ffiffiffi3

pSy

�nð1� Tr=TmÞ

375 Yffiffiffi3

p :

(42)

Without any heat loss (c ¼ 1) and with negligible strain rateeffect (l ¼ 1) we arrive with a ¼ �1/Tm while for a general casej1 � lcjTr/Tm � 1. Formulation (40 and 41) can be extended toinclude the effect of m.

R. Masri / International Journal of Impact Engineering 68 (2014) 15e2722

Same procedure can be performed when replacing the Ludwikhardening expression in (31) with an arbitrary strain hardeningresponse bh ¼ bhðεÞ where bhðε ¼ SyÞ ¼ 1, namely

S

Sy¼ bh

241�0@SZεp0

S

Sydεp

1Am35ε � Sy (43)

which, by differentiating with respect to εp and using (43), leads to

S0 ¼24d lnbh

dεp�mSbh 1� SbhSy

!1�1=m35S: (44)

For the practical case m ¼ 1 we arrive with

d ln S ¼ d lnbh � Sbhdεþ SSbhd ln S (45)

and by neglecting the last term due to SSbh � 1 an excellentapproximation of the stress-strain curve is obtained by a straight-forward integrationwith the expected extension to practical valuesof m near one

S ¼ Sybhexp

264�

0B@SZεSy

bhdε1CA

m375 ε � Sy: (46)

For Ludwik stress-strain curve bh ¼ ðε=SyÞn and for bhh1 onlythermal softening is observed with further reduction to perfectly-plastic response by c ¼ 0. By differentiating (46) with respect to ε

and eliminating the exponent term with the aid of (46) we obtain

dSdε

¼

2664d lnbhdε

�mSmbh0B@ Zε

Sy

bhdε1CA

m�13775S (47)

which leads to an equation for the effective strain at the maximumpoint ðεmÞ

dbhdε

¼ mSmbh2

0B@ ZεmSy

bhdε1CA

m�1

: (48)

Like in the previous section, the homologous temperature isgiven by

Th ¼

8><>:1� exp

264�

0B@SZεSy

bhdε1CA

m3759>=>;

1=m

(49)

and is less than one for any strain level above the yield strain.

Table 5Values of constants (a,b,c) in the unified expression (53) for three cavitation models.

Cavitationmodel

Spherical ðpScÞ Cylindrical-Mises ðpMc Þ

Cylindrical-Tresca ðpTc Þ

a 3/2ffiffiffi3

p2

b (3/2)b 1�2kffiffiffi3

p bx1:123b �2n2

c 2 2ð1�kÞffiffiffi3

p x1:700 1 þ n

4. Thermal softening effect on specific cavitation energy

In previous papers it is shown that specific cavitation energyof metals (sc), which is identified with cavitation pressures (pc)for plane-strain and spherical deformation patterns [26], cap-tures an essential nature of penetration [27] and perforation[26,28,29] processes. While in these papers only strain hard-ening response is considered, here we also include the thermalsoftening effect and compare sc under isothermal and adiabaticconditions for spherical and cylindrical (plane-strain and plane-stress) cavitations in several aluminium and steel alloys. Whilethe effect of a constant strain rate can be considered through

l > 1 here we will focus on thermal softening response and willleave the discussion on strain rate effect to a separate investi-gation on this subject.

As discussed in Ref. [26], when strain softening is present theexact solution of sc for quasi-static spherical cavitation in Mises(Tresca) media [19,20] should be presented with ε as the integra-tion variable by

pSc ¼ EZN0

ð1þ bdS=dεÞSdεexp

�32 ε� b

2S�� 1þ 2bS

: (50)

Here b ¼ 1 � 2n is the (elastic) compressibility parameterwith n for Poisson ratio and S ¼ SðεÞ is the material stress-strain curve with dS=dε for the nondimensional tangentmodulus. The analogous expressions for plane-strain cylindricalcavitation are [23]

pMc ¼ EZN0

�1þ 1�2k

3 bdS=dε�Sdε

exp� ffiffiffi

3p

ε� bffiffiffi3

p S�� 1þ 2ð1�kÞffiffiffi

3p bS

k ¼ �0:4725

(51)

for Mises media and

pTc ¼ EZN0

�1� n2dS=dε

�Sdε

exp½2ε� ð1� nÞS� � 1þ ð1þ nÞbS (52)

for Tresca solids. The value of parameter k has been determined inRef. [23] by an optimization procedure, based on the axially-hydrostatic assumption [32], to include (elastic) compressibilityeffect in the most accurate way.

While these formulae have the same mathematical structurewhich differ through four constants, they all can be written underthe unified expression

pc ¼ EZN0

½1þ ðb=aÞdS=dε�Sdεexp½aε� ðcb� bÞS� � 1þ cbS

(53)

with only three constants (a,b,c) given in Table 5 for each cavitationmodel. The incompressible version, where n ¼ 1/20b ¼ 0, is in-dependent of constant c while constant b is not zero only for pTc(b ¼ �1/2), and leads to simple algebraic expressions for specificstress-strain relations [29] like the Ludwik hardening law which iswidely used in perforation mechanics [16,18,31]. Connection be-tween cavitation pressure and specific elastoplastic work incre-ment in a uniaxial tensile test ðdW ¼ sedεÞ is reflected in (53) withsimpler versions for spherical and Mises cylindrical cavitations inincompressible media

pSc ¼ZN0

dWexp

�32 � 1

pMc ¼ZN0

dW

exp� ffiffiffi

3p

ε

�� 1

: (54)

R. Masri / International Journal of Impact Engineering 68 (2014) 15e27 23

The plane-stress analysis for the Tresca media [26] has led to aclosed-form expression for the specific cavitation energy

sTc ¼ E

8<:1� exp½ � 2εv þ ð1� nÞSv�SvexpðnSvÞ þ

ZNεv

dεSexpð2ε� bSÞ

9=;�1

(55)

where εv ¼ εðS ¼ SvÞ and Sv is the effective stress at the Trescavertex point

Sv ¼Zεv0

Sdεexp½2ε� ð1� nÞS� � 1þ ð1� nÞS: (56)

Specific cavitation energy of Mises media under plane-stressconditions ðsMc Þ is found to be just a few percent higher than theTresca analogue [26] so here we limit the plane-stress discussion tothe Tresca media (sMc ¼ 1:03sTc is an approximation for solidswhich do not experience strong hardening response). In Ref. [23] itis shown that elastic compressibility reduces the cavitation pres-sures ðpSc ; pMc ; pTc Þ by a few percent and in Ref. [26] sTc and sMc arefound to be nearly insensitive to elastic compressibility. These ob-servations remain valid when adiabatic thermal softening responseis included.

Relations (53,55 and 56) are useful when material stress-straincurve S ¼ SðεÞ is known as a table of experimental data or as abest fit expression. The curve (46) with (47) for the derivative dS=dεis one example for which (53) and (56) should be split to elastic andelastoplastic parts. However, for a solid with definite yield strainand known stress-plastic strain relation S ¼ SðεpÞ, as in (22), theelastic branch of (53) can be integrated under the practicalassumption Sy � 1 and the integration variable of the plastic partshould be changed to εp, namely

pc ¼ Yaþ E

ZN0

1þ ðh=aÞdS=dεp

�Sdεp

expaεp þ ðh� cbÞS�� 1þ cbS

h ¼ aþ b:

(57)

In addition, with the practical assumption εv ¼ Sv þ εpv � 1expressions (55 and 56) can be approximated by

sTc ¼ E

8><>:2εpv

Svþ ð1þ nÞ þ

ZNεpv

�1þ dS=dεp

�dεp

S exp2εp þ ð1þ 2nÞS�

9>=>;�1

(58)

2Sv ¼ Sy þZεpv0

�1þ dS=dεp

�dεp

1þ εp=S: (59)

For stress-plastic strain relation (22) the derivative dS=dεp isgiven in (24) and Sv ¼ SyZðεpvÞð1� SmεmpvÞ due to (23). However,the derivative (24) may be singular at εp ¼ 0 due to a singularderivative of Z, like for the JC relation, or when c > 0 andm < 1. Forsuch cases we shall extend the elastic branch slightly above theyield strain where plasticity is very small and numerical calcula-tions of (57) and (59) should be performed with

pc ¼ YaZ�εp*��

1� Smεmp*�

þ EZNεp*

1þ ðh=aÞdS=dεp

�Sdεp

expaεp þ ðh� cbÞS�� 1þ cbS

(60)

2SyZ�εpv��

1� Smεmpv�

¼ SyZ�εp*��

1� Smεmp*�

þZεpvεp*

�1þ dS=dεp

�dεp

1þ εp=S(61)

for different and very small values of εp* in order to locate the re-sults of pc and εpv which are practically insensitive to εp*. Here wehave used (23) to replace the yield stress Y with the slightly highervalue se ¼ seðεp ¼ εp*Þ.

Expressions (60 and 61) together with (58) and (12 and 13) areused to calculate the four specific cavitation energiessc ¼ ðpSc ; pMc ; pTc ; s

Tc Þ of aluminium 5083-H116, aluminium 7075-

T651 and three Weldox steel alloys via the JC model. The resultsfor the (isothermal c ¼ 0, adiabatic c ¼ 0.9) curves for each specificcavitation energy are given in Tables 1e3 along with the thermalsoftening effect on sc which is defined by

dts ¼ scðc ¼ 0:9Þ � scðc ¼ 0Þscðc ¼ 0Þ (62)

and reveals that thermal softening reduces sc by 2%e21% so thatmaterial resistance to cavitation is weakened. The softening effectis similar for all three cavitation pressures, with the highest effectfor spherical model and the lowest effect for Tresca plane-straincylindrical model, but it is 1.5e3.5 times higher for plane-stressspecific cavitation energy, so that thin plates are more affected bythermal softening than thick plates. Note that by ’thin plates’ wemean plates which are thin enough to be deformed in a ductile holeenlargement process under plane-stress conditions but not toomuch thin to develop a dishing deformation pattern [33]. InTables 1e3 jdtsj is found to be higher for the aluminium alloys eventhough the thermal softening coefficients of the aluminium 5083-H116 plates are lower than the values of S for the steel alloys.This is attributed to the much higher values of H and to the lowervalues of (n,m) for the aluminium plates. Actually, the thermalparameters (m,S) are almost identical for 15 mm aluminium 5083-H116 plate andWeldox 460E steel alloy so that the higher value ofHand the lower value of n for the aluminium plate (stronger strainhardening response) are responsible for the higher jdtsj. Moreover,the material parameters (Y,H,n,n) for aluminium 7075-T651 andWeldox 460E (Tables 2 and 3) are close but the elastic modulus ofthe steel is three times larger and its much lower yield strain is themain reason for the higher cavitation pressures ðpSc ; pMc ; pTc Þ of theisothermal curve. However, the values of sTc for the isothermalcurves are very close since cavitation under plane-stress conditionsis less sensitive to the yield strain value [26,29]. This similarityfades by including thermal softening effect due to the very largedifference between the thermal softening coefficients.

Similar trends are observed for the four 6070 aluminiumtempers in Table 4 via the Voce hardening model and it is quiteinteresting that such a relatively large difference between theisothermal and the adiabatic curves (for example, Figs. 2 and 6)translates to a much smaller effect on the cavitation pressures. Theexplanation for this observation is that the integrand in (57) ismonotonically decreasing and asymptotically approaching zero inthe deep plastic zone while the difference between isothermal andadiabatic curves is increasing with the increase of εp. This meansthat most of integral (57) value is obtained for plastic strain levels

Fig. 8. The nondimensional ratio m ¼ s(c ¼ 0)/s(c ¼ 0.9) as function of the cavity walleffective plastic strain εpa for all four cavity expansion models of the 25 mm thick5083-H116 aluminium plate.

Fig. 9. The nondimensional ratio m ¼ s(c ¼ 0)/s(c ¼ 0.9) as function of the cavity walleffective plastic strain εpa for all four cavity expansion models of Weldox 900E.

R. Masri / International Journal of Impact Engineering 68 (2014) 15e2724

where isothermal and adiabatic curves are not so different whilecontribution of deep plastic zone is relatively small. To supportthis explanation we replace the upper limit of integral (57) (or(60)) with εp and arrive with the function p ¼ pðεpÞ whichapproaching asymptotically to pc in the deep plastic field. Thisfunction is presented in Fig. 2 for spherical cavity expansion pS

under isothermal (dashed line) and adiabatic (dash-dotted line)conditions (divided by 3 for scaling with other curves). Followingthese observations we can conclude that the difference betweenapproximation (22) and the actual adiabatic curve (for m s 1),which is noticeable only in the deep plastic zone, is practicallyinsignificant in calculations of specific cavitation energies since(22) is very accurate for any value of mwhen εp � 1. Moreover, foraccurate estimations of sc via experimental data for stress-straincurves it is important that the data will be accurate up to someeffective strain value which above it the contribution to sc ispractically negligible. Note that pðεpÞ also describes the internalpressure which is needed to expand the cavity up to a cavity walleffective plastic strain of εp.

The inequalities pTc < pMc < pSc which are valid for isothermaland adiabatic curves (Tables 1e4) are due to the values of constanta in Table 5 which is the dominant constant in (57). Actually, theratio between the constants for spherical (3/2) and Mises cylin-drical ð

ffiffiffi3

pÞ models is identical to the ratio between the constants

for Mises ðffiffiffi3

pÞ and Tresca (2) cylindrical models and suggests the

ad-hoc estimations pSc=pMc zpMc =pTcz2=

ffiffiffi3

pz1:155 in consistency

with the results in Tables 1e4 and in a previous investigation [23].For most of this data pSc=p

Mc is slightly higher than 1.155 and pMc =pTc

is slightly lower than 1.155 but several exceptions are observed(for example in Table 3 in Ref. [23]). Moreover, the values ofconstant a are the reason that the cylindrical Tresca cavitationpressure is least affected by thermal softening while sphericalcavitation pressure is the most affected cavitation pressure(Tables 1e4).

Johnson and Cook [34] also proposed a fracture criterion tak-ing into account the effects of stress triaxiallity, strain hardening,strain rate sensitivity and thermal softening. The criterion isbased on damage evolution and fracture occurs when the damagevariable reaches a critical value. On the other hand, cavitationtheory is based on the assumption of an infinitely ductileresponse. This is reflected by the cavity wall infinite effectivestrain at the upper limits of the integrals in (53) and (55). Due tocavity expansion theory, for which cavitation phenomenon is alimit state, replacing the upper limit in (53) with εa (or the upperlimits in (57) or (60) with εpa) leads to the internal pressure p (orspecific expansion energy s ¼ p [26]) for each model which isneeded to expand the cavity up to a cavity wall effective strain ofεa. The equivalent expression for the Tresca plane-stress specificexpansion energy is more complicated (Eq. (5.4) in Ref. [26] for εa� εv). All these solutions are universal since to each cavity walleffective strain there corresponds an internal pressure and eachpair corresponds to a deformed radius in the medium [19,26].However, the saturation levels (the specific cavitation energies)are practically obtained for εa values of O(1) and the connectionwith the JC fracture criterion deserves further investigation. InFigs. 8 and 9 we demonstrate, for the 25 mm thick 5083-H116aluminium plate and the Weldox 900E steel alloy, the nondi-mensional ratio m ¼ s(c ¼ 0)/s(c ¼ 0.9) as function of εpa for allfour cavity expansion models. While thermal softening effect on pis increasing with the increase of εpa it has a maximum underplane-stress conditions. However, saturation values of m areobserved in the deep plastic zone for all four deformation modelsin consistency with the results in Tables 1 and 2 under the rela-tion dts ¼ �1þ 1=mðεpa/NÞ. Tresca plane-strain cavity expan-sion model is the least affected model by thermal softening

response while the plane-stress model is the most affected one. Itshould be mentioned that identical εpa values for isothermal andadiabatic curves necessarily not referring to the same deformedradius of the cavity [26].

5. Thermal softening effect on ballistic limit predictions

Consider a ductile plate of uniform thickness h perforated atnormal incidence by a rigid, nose pointed, projectile of mass M andshank diameter D such that h/D is suitable for ductile holeenlargement mode of failure [26,28]. For ballistic limit impact mostof projectile kinetic energy is consumed by creating volume ofperforation. Friction, target material inertia and projectile noseshape effects are of second order in agreement with experimentaldata and simulation results for conical and ogival projectiles[14,16,18,28,31,35e38] and are not included in the following model.Energy conservation leads to a simple relation for the ballistic limitvelocity of a monolithic target [26]

Fig. 10. With the permission of IJIE, Fig. 10 in Ref. [16]. Six velocity-time simulationcurves for various versions of the JC constitutive relation for the 25 mm 5083-H116aluminium plate struck at velocity of Vs ¼ 303 [m/s]. Ballistic limits for curves 1 and 4are given in Fig. 13 in Ref. [16].

R. Masri / International Journal of Impact Engineering 68 (2014) 15e27 25

Vb ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Rt=gp

q; gp ¼ 4M

pD2; Rt ¼ hsc (63)

with an obvious extension to multilayered targets with air gaps[26]. This relation is based on the specific cavitation energy of thetarget material and is governed by two fundamental parameters,one for the target and one for the projectile. Parameter gp is theprojectile areal density which reflects the projectile perforationpotential, while parameter Rt reflects the target resistance toperforation (with sc for material resistance to quasi-static cavita-tion). Ballistic limit velocity is linear with

ffiffiffiffiffiRt

pfor a given projectile,

however, the specific cavitation energy sc is a function of h/D[26,28] and this seems to be the reason that in Ref. [37] the linearscaling law which is based on a plane-strain model requires a bestfit parameter. Moreover, theoretical considerations in Ref. [26] haveled to the conclusion that for ductile plate perforation monolithictarget is stronger than any combination of its air-gapped parts sincesc decreases with every split of the target to separated thinnerplates.

With (63) the projectile residual velocity Vr can be evaluatedfrom RechteIpson formula for nose pointed projectiles [38]

Vr ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiV2s � V2

b

q(64)

where Vs is the projectile striking velocity. Moreover, the squareroot function in (63) reduces the effect of uncertainties or smallcontributions (such as elastic compressibility or thermal softening)jdsj � 1 on the specific cavitation energy value sc ¼ ð1þ dsÞsc by afactor of two since

Vb ¼ Vb

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ds

px

�1þ 1

2ds

�Vb: (65)

The effect of ds on Vr is obtained by substituting (65) in (64)

Vr ¼ Vr

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ds

ðVs=VbÞ2 � 1

s(66)

and is found to be intensified for maxð1;ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ ds

pÞ < Vs=Vb <

ffiffiffi2

p

with decrease of intensification with increase of Vs/Vb so thatVs=Vb >

ffiffiffi2

pleads to lower effect on Vr than on Vb. Note that ds is

negative for elastic compressibility or thermal softening contribu-tions since both effects reduce the specific cavitation energy. Forthermal softening ds ¼ dts so half of these values in Tables 1e4represent the effect on the ballistic limits. In addition, the sug-gested relation sMc ¼ 1:03sTc reflects a negligible effect of 1.5% onthe ballistic limit so that under plane-stress conditions Vb is prac-tically insensitive to yield surface contours due to Hershey-Hosfordyield criterion that lie between Mises and Tresca loci [25].

The finite element simulations in Ref. [16] have ignored the JCfracture criterion in order to be as close as possible to the cavityexpansion model [18]. The simulations were run assuming perfo-ration of h ¼ (15,20,25,30) mm aluminium 5083-H116 plates by arigid conical nose projectile (gp ¼ 0.627 [g/mm2]). The change intarget response by including or neglecting strain hardening, strainrate sensitivity and thermal softeningwas studied byexpressing thevon Mises effective stress with the aid of JC model. The numericalsimulations predictions of the ballistic limits inTable 1 (NP) are fromFig. 13 in Ref. [16]. However, while results for isothermal curves aregiven in this figure (diamond markers), the predictions which arepresented by trianglemarkers include, besides strain hardening andthermal softening responses, also strain rate effect. In Table 1 theorypredictions of ballistic limits with sc ¼ ðpSc ; pMc ; pTc ; s

Tc Þ are given for

isothermal and adiabatic curves and very good agreement betweensimulation results and theory estimations due toMises plane-strain

model ðVMb Þ is observed. This means that numerical deformation

patterns in Ref. [16] for the four plate thicknesses should be close toplane-strain response. Comparisonof experimental results fromRef.[39] for the ballistic limits (ER in Table 1) with the simulation pre-dictions reflects the combined contribution of thermal softeningand strain rate effects to strain hardening response. The experi-mental results are slightly lower than theory results due to pMc underadiabatic conditions and slightly above the results due to sTc underisothermal conditions while they are in better agreement with theresults due to pTc . However, Tresca yield criterion is expected to beconservative for the aluminium 5083-H116 since in Ref. [16] theMises yield surface was adopted for numerical simulations. Theexperimental deformation patterns in Fig. 6 in Ref. [16] for the fourplate thicknesses (under striking velocities which are close to theballistic limits) seem to be closer to a deformation pattern underplane-strain conditions while an example for a deformation patternunder plane-stress conditions is given in Fig. 5 in Ref. [33]. In Ref.[40] the experimental ballistic limit for perforation of 20 mmaluminium 5083-H116 plate by the same rigid projectile(gp¼ 0.627 [g/mm2]) butwith an ogival nose is 244 [m/s], very closeto the equivalent result of 249 [m/s] in Table 1 for conical noseprojectile. This similarity demonstrates the slight effect of projectilenose shape on ductile perforation of aluminium plates.

In Fig. 10 the velocity-time curves from Fig. 10 in Ref. [16] arepresented. These curves show numerical simulations results ofvarious versions of JC constitutive relation for a 25 mm thick 5083-H116 plate struck at velocity of Vs¼ 303 [m/s]. The ratio Vs/Vb for thenumerical simulations when only strain hardening is considered isabout (303/286¼ )1.06 (ballistic limit is fromFig.13 in Ref. [16]). Thisslightly above one value explains, due to (66), the thermal softeningeffect of about 47% on the value of Vr when comparing curves 2 and4,much above the effect on the ballistic limit. Actually,with the ratiofor the thermal softening effectVr=Vr ¼ 159=108wearrive, by (66),with an estimation of this effect on the specific cavitation energyds ¼ �14.3% which translates to an effect of about �7.4% on theballistic limit. Here it should bementioned that theory predicts that8% < jdtsj < 13% for Mises media under deformation patterns be-tween plane-stress and plane-strain conditions (Table 1) and morenumerical investigations such as [16] are needed to verify or rejectthe existence of a slight difference between theory and simulations.A close result of ds ¼ 20.3% is obtained by calculating the effect ofremoving thermal softening response from the full JC model, wherenow Vr=Vr ¼ 80=147 and Vs/Vb ¼ 303/267(¼1.13) due to curves 1and 3 and Fig. 13 in Ref. [16]. However, the ratio of 1.4 between the

R. Masri / International Journal of Impact Engineering 68 (2014) 15e2726

absolute values of ds indicates that including thermal softeningwhen strain rate response is present increases the thermal softeningeffect in agreement with the influence of l > 1 on the value of S.

With the appropriate ratios for the strain rate effectVr=Vr ¼ 80=108 and Vs/Vb ¼ 303/286(¼1.06) due to curves 3 and 4we arrive with an estimation of this effect on the specific cavitationenergy ds ¼ 5.5% which translates to an effect of about 2.7% on theballistic limit. A very close result of ds ¼ �4.9% is obtained bycalculating the effect of removing the strain rate response from thefull JC model, where now Vr=Vr ¼ 159=147 and Vs/Vb ¼ 303/267(¼1.13) due to curves 1 and 2. Note that by removing the strainrate response we decrease the yield stress but increase the strainhardening effect (or decrease the thermal softening response).However, the decrease of the yield stress is found to be more sig-nificant leading to an overall decrease of sc by 4.9%. Now, eventhough the two absolute values of ds are very close, we see fromFig. 10 that the strain rate effect on Vr is much reduced whenthermal softening is present since the ratio Vs/Vb in (66) is higherwhen thermal softening is active. In Refs. [41,42] it is found thatstrain rate effect increases spherical and Mises plane-strain cylin-drical cavitation pressures of aluminium 6061-T6511 by about 10%.However, thermal softening was neglected in the investigationsand the expected coupling between strain rate sensitivity andthermal softening response was not observed.

Ballistic limit predictions with sc ¼ ðpSc ; pMc ; pTc ; sTc Þ are also given

in Table 2 for isothermal and adiabatic curves of three Weldox steelalloys for perforation of 12 mm plates by projectiles with arealdensity of gp ¼ 0.627 [g/mm2]. Numerical simulations predictions(NP) of ballistic limits for conical nose projectiles [14], which takeinto account strain hardening, strain rate and thermal softening dueto JC model along with a failure criterion, are also given in Table 2.These values are between plane-stress and plane-strain theorypredictions forMisesmediawhile theyare in better agreementwithresults due to pTc . However, Tresca yield criterion is expected to beconservative for the Weldox steel alloys since in Ref. [14] the Misesyield surfacewas adopted for numerical simulations. Comparison ofexperimental results from Ref. [14] for the ballistic limits (ER inTable 2) with the simulations predictions reflects slightly higherexperimental results which are closer to theory predictions due tothe Mises plane-strain model. However, in Ref. [14] it wasmentioned that the tip of the conical nose projectiles shatteredduring all impacts with Weldox 700E and Weldox 900E. The noseshape changes fromconical profile to a spherical like shape resultingin ejection of a small plug. Such phenomena were not observed inany of the tests withWeldox 460E and in Ref. [14] the experimentalballistic limit for perforation of 12 mm Weldox 460E plate by thesame rigid projectile (gp¼ 0.627 [g/mm2]) butwith an ogival nose is295.9 [m/s], very close to the equivalent result of 290.6 [m/s] inTable 2 for the conical nose projectile. This similarity demonstratesagain the slight effect of projectile nose shape on ductile plateperforation. Yet, for theseWeldox steel plates the thermal softeningeffect on Vb is also small and is between 1.3% and 3.6% (Table 2).

Table 3 includes ballistic limit predictions withsc ¼ ðpSc ; pMc ; pTc ; s

Tc Þ for isothermal and adiabatic curves of

aluminium 7075-T651 for perforation of 20 mm plates by hardcores (gp ¼ 0.176 [g/mm2]) of 7.62 mm APM2 bullets. Numericalsimulation prediction of Vb is not available in Ref. [43] but theexperimental result (ER) is given in Table 3. Note that we have usedmaterial parameters for the JC model from Ref. [11] to predict theexperimental result in Ref. [43] which is found to be identical withtheory prediction due to pMc under adiabatic conditions. Theorypredictions of ballistic limits are also given in Table 4 for isothermaland adiabatic curves of four aluminium 6070 tempers for theperforation of 20 mm plates by the same steel hard cores. Nu-merical simulations predictions (NP) of the ballistic limits [31],

which take into account strain hardening, strain rate and thermalsoftening along with a failure criterion, are also given in Table 4.These simulations predictions are between plane-stress and plane-strain theory estimations while they are in better agreement withresults due to pTc but Tresca yield criterion is expected to be con-servative [31]. In Ref. [31] it was mentioned that numerical simu-lations which neglect thermal softening effect have led to anincrease of the ballistic limits by about 5e10% which is equivalentto 10% < jdtsj < 20%. Even though thermal softening effect is ex-pected to be smaller when strain rate sensitivity is neglected, itseems that numerical simulations predict slightly higher effectthan the theory predictions in Table 4 (2% < jdtsj < 13%) and morenumerical investigations such as [16] are needed to verify or rejectthis observation. Interesting observation from Table 4 is the prac-tically identical plane-stress ballistic limit predictions ðVsTc

b Þ foraluminium 6070-T4 and 6070-T7 in consistency with the identicalsimulation results. Comparison of experimental results from Ref.[31] for the ballistic limits (ER in Table 4) with the simulationspredictions reflects much higher experimental results which arecloser to theory predictions with spherical cavitation model pSc .However, due to the large differences between experimental andnumerical results it seems that the Voce model in Ref. [31] un-derestimates the material strength and leads to similar un-derestimations by theory predictions.

6. Concluding remarks

The effect of adiabatic thermal softening on specific cavitationenergy of metal plates is investigated with connection to ductileplate perforation and ballistic limit predictions. Explicit stress-plastic strain relation that includes strain hardening responseand thermal softening effect is found for JohnsoneCook model.Extension for arbitrary hardening response is derived by similarlines and demonstrated for the Voce hardening law. Explicit adia-batic stress-strain curve is also found for Ludwik hardening lawwith an extension for arbitrary hardening response and connectionis made with Bell’s formula and Bai and Johnson relation [30].Simple implicit equations or explicit expressions are suggested forthe maximum point of the adiabatic curves in both cases.

Like in Bai and Johnson relation [30], thermal softening is gov-erned by an exponential decay and in the present paper we showthat the decay is controlled by two nondimensional softening pa-rameters (S,m) and the integral of the relative deviation ofisothermal response from perfectly-plastic behaviour. Thermalsoftening effect is stronger for higher values of S and lower values ofm and increases with the increase in strain hardening response.Increase of yield stress due to a constant strain rate response leadsto an increase of the thermal softening effect so that onset of plasticyield is postponed but when plasticity sets in the strain hardeningresponse decreases due to a stronger softening effect. In addition,we show that material temperature is approaching asymptoticallyto the melting temperature in the deep plastic zone but for lowplastic strain levels the temperature is insensitive to (m,Tm).

The effect of adiabatic thermal softening on spherical and cy-lindrical (plane-strain and plane-stress) specific cavitation energiesis demonstrated for several aluminium and steel alloys and revealsthat material resistance to cavitation is weakened via this effect. Inaddition, while softening effect is similar for the cavitation pres-sures ðpSc ; pMc ; pTc Þ it is stronger for the specific cavitation energy sTc ,namely thin plates which deform in a ductile hole formation modeare more affected by thermal softening than thick plates. Thermalsoftening effect on specific cavitation energy is reduced by a factorof two in ballistic limit calculations but is intensified for low re-sidual velocity estimations where striking velocities are close to theballistic limit. Comparison of theory predictions for ballistic limit

R. Masri / International Journal of Impact Engineering 68 (2014) 15e27 27

with simulation results and experimental data demonstrates thevalidity of the analytical model. However, numerical simulations ofcavitation fields, such the investigation in Ref. [44] but for John-soneCook model, and simulations which ignore the fracture cri-terion, such as in Ref. [16], are needed to verify or reject theexistence of a slight difference between theory and simulations forthe thermal softening effect. Improvements of the present modelshould take into account the h/D effect for completing the gapsbetween pSc ; p

Mc and sTc . In addition, the small effect of target inertia

on Vb should be accurately quantified and be compared against thesmall opposite effect of thermal softening.

Finally, we suggest the following relations for stress-plasticstrain curve

se ¼ YZ�εp�g�_εp�exp

24�0@SZεp0

Zdεp

1Am35εp � 0 (67)

and stress-strain curve

se ¼ YbhðεÞg� _εp�exp264�

0B@SZεSy

bhdε1CA

m375 ε � Sy; (68)

with gð_εpÞ for the strain rate contribution (like the relevant term inJC model or other possibilities such as in Ref. [45]), as alternativemodels for numerical simulations. These expressions take into ac-count adiabatic thermal softening, strain rate sensitivity and arbi-trary strain hardening responses in an explicit form. However, theexpected coupling between strain rate and thermal softening ef-fects is not included in these relations. By replacing S with SGð _εpÞ,where Gð _εpÞ is another function of _εp, we can include the couplingbetween strain rate sensitivity and thermal softening effect.Moreover, instead of calculating the thermal softening coefficient Swe may consider it as another material parameter which is deter-mined by fitting the stress-strain curve to experimental data. Thiseliminates the effect of uncertainties in the material parameters(c,Tm,r,Cp) andmay compensate on the assumptions involved in theadiabatic thermal softening model. Physical consistency of the bestfit procedure may be checked by comparing the best fit value of Swith the calculated value.

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