Timothy J. Holmquist

Gordon R. Johnson

Southwest Research Institute,

5353 Wayzata Blvd.,

Minneapolis, MN 55416

A Computational ConstitutiveModel for Glass Subjected toLarge Strains, High Strain Ratesand High PressuresThis article presents a computational constitutive model for glass subjected to largestrains, high strain rates and high pressures. The model has similarities to a previouslydeveloped model for brittle materials by Johnson, Holmquist and Beissel (JHB model),but there are significant differences. This new glass model provides a material strengththat is dependent on the location and/or condition of the material. Provisions are madefor the strength to be dependent on whether it is in the interior, on the surface (differentsurface finishes can be accommodated), adjacent to failed material, or if it is failed. Theintact and failed strengths are also dependent on the pressure and the strain rate. Ther-mal softening, damage softening, time-dependent softening, and the effect of the thirdinvariant are also included. The shear modulus can be constant or variable. The pres-sure-volume relationship includes permanent densification and bulking. Damage is accu-mulated based on plastic strain, pressure and strain rate. Simple (single-element)examples are presented to illustrate the capabilities of the model. Computed results formore complex ballistic impact configurations are also presented and compared to experi-mental data. [DOI: 10.1115/1.4004326]

1 Introduction

This article presents a computational constitutive model forglass. Glass has been the focus of extensive research over theyears and much has been documented regarding its unique behav-ior. Bridgman [1,2] provides some of the earliest work where heidentifies three important characteristics of glass; it is very com-pressible, the bulk modulus decreases over a specified pressureregion, and a portion of the volumetric strain is unrecoverrable(which produces permanent densification). Many researchers havecontinued the investigation into the compressibility of glass withsome documenting up to 20% permanent densification [3–6].There is also evidence that densification is dependent on tempera-ture and shear [5,6]. Mackenzie [6] provides data that showsincreased densification when shear stress and/or temperature isadded. One of the more unique characteristics of glass is its veryhigh internal tensile strength. Rosenberg et al. [7] performed ten-sile-spall plate-impact experiments on soda lime glass and wasnot able to produce spall and concluded that the spall strengthexceeded 5 GPa. Brar et al. [8] performed spall experiments onsoda lime and borosilicate glass and was unable to produce spallin either one. Cagnoux [9,10] performed spall experiments on bor-osilicate glass and documented a high spall strength of 1.8 GPa onaverage. Then there is the “failure wave” phenomenon that hasgenerated a considerable amount of research [11,12]. Althoughmuch has been learned regarding this phenomenon, there are stilluncertainties regarding the correlation between opacity, damageand stress state. Glass is also surface finish dependent, where asmooth surface is stronger than a rough surface. Nie et al. [13]demonstrated this effect by performing flexural tests on glass pre-pared with a very rough surface and with a surface etched usinghydrofluoric acid. The rough surface failed at a tensile stress ofapproximately 100 MPa and the etched surface failed at 1200MPa. Glass also exhibits scale effects where smaller samples are

stronger than larger samples. Anderson et al. [14] demonstratedthe effect of scale by performing ballistic impact experimentsonto thin glass plates. The smaller-scale glass plates were stron-ger, providing more resistance to penetration. Sun et al. [15] alsodemonstrated the effect of scale by impacting small metal spheresonto thin glass plates. The smaller-scale plates were less damagedthan the larger-scale plates. Wereszczak et al. [16] demonstratedsize effects by performing indentation tests onto soda lime glass.The tensile strength increased significantly as the scale wasreduced. Wereszczak et al. concluded that this was the effect ofsurface flaws and the statistical nature of their occurrence (largerscales are more likely to have larger flaws and thus fail at a lowerstress). Glass, like other brittle materials, loses strength whendamaged. Chocron et al. [17] performed compression tests onundamaged and predamaged borosilicate glass. The predamagedsamples were weaker than the undamaged samples. Chocron et al.also demonstrated the significant effect that pressure has on thestrength, where higher pressures produce higher strengths. Glassalso appears to lose strength under one-dimensional plate-impactloading [18,19], possibly a result of damage softening and/or ther-mal softening. There is also evidence that the failure process istime dependent. Simha and Gupta [18] performed plate-impactexperiments on soda lime glass and concluded that the failure pro-cess was time dependent. Sunderam [19] performed pressure-shear plate-impact experiments on soda lime glass and alsoproduced, what appeared to be, time-dependent failure. Glass alsoexhibits a softening of the bulk modulus under compression. Thiswas first identified by Bridgeman [1] through quasi-static experi-ments and later confirmed through plate-impact tests [20]. Thereis also evidence that the shear modulus softens. This is evident inthe Hugoniot Elastic Limit (HEL) for borosilicate glass [21] thatcan only be reproduced using a reduced shear modulus. Thestrength of glass is also dependent on the intermediate principalstress (often referred to as the 3rd invariant effect). Handen et al.[22] produced significantly higher strengths in pyrex glass whenthe stress state was on the compressive meridian compared to thetensile meridian. Glass also appears to be strain-rate sensitive. Nieet al. [13] produced higher tensile strengths as the loading rate

Contributed by Applied Mechanics Division of ASME for publication in the JOUR-

NAL OF APPLIED MECHANICS. Manuscript received October 15, 2010; final manuscriptreceived May 6, 2011; published online July 27, 2011. Assoc. Editor: Bo S. G. Janzon.

Journal of Applied Mechanics SEPTEMBER 2011, Vol. 78 / 051003-1Copyright VC 2011 by ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 11/18/2014 Terms of Use: http://asme.org/terms

was increased. Glass is also sensitive to temperature. Chen [23]demonstrated increased ductility and decreased strength in borosi-licate glass when the temperature approached the glass transitiontemperature, Tg. Glass also produces dilatancy, or bulking, duringfailure. Glenn et al. [24] state that glass can bulk as much as 20%or more under impact loading. Lastly, glass can produce dwelland interface defeat as demonstrated by Behner et al. [25] andAnderson et al. [26]. Borosilcate glass defeated a gold rod(impacting at approximately 890 m/s) at the glass surface, produc-ing no significant penetration into the glass target.

Glass is clearly a very complex material and modeling itpresents a significant challenge. The glass model presented hereinattempts to account for many of the aforementioned glass charac-teristics. The remainder of this article presents a description of theglass model, a demonstration of various features of the model, andcomputed results for several ballistic impact configurations thatare compared to experimental data.

2 Description of the Model

A summary of the glass model is presented in Fig. 1. The modelhas three components; one to describe the strength, one todescribe damage, and one to describe the pressure-volume behav-ior. Although the model is similar to a previously developedmodel for brittle materials by Johnson, Holmquist and Beissel(JHB model) [27], there are many differences. This new glassmodel has an intact strength that is dependent on whether the ma-terial is located in the interior, on the surface, or adjacent to failedmaterial. There is also a failed strength which represents thestrength of the material that is fully damaged (D¼ 1.0). Thermalsoftening, damage softening (instantaneous or gradual), time-de-pendent softening, and the effect of the third invariant are also

included. The shear modulus can be constant or variable, and thepressure-volume relationship includes permanent densificationand bulking.

2.1 Strength. The strength portion of the model is shown inthe upper portion of Fig. 1. The available strength (von Misesequivalent stress), r, is dependent on the pressure, P, the dimen-sionless equivalent (total) strain rate, _e� ¼ _e= _eo (for _eo ¼ 1:0s�1),the location of the material (interior, surface or adjacent to failedmaterial), and the damage, D. For undamaged material, D¼ 0; forpartially damaged material, 0<D< 1.0; and for fully damaged(failed) material, D¼ 1.0.

There are three curves that represent the strength of the intactmaterial. The reference intact strength represents the strength formaterial adjacent to failed material, or effectively adjacent to asurface with a very rough finish. This is the minimum intactstrength (when compared to the interior and surface intactstrengths). For a dimensionless strain rate of _e� ¼ 1:0, the refer-ence intact strength goes (linearly) from r ¼ 0 at a tensile pres-sure of P¼� T, to a strength of r ¼ ri at a pressure of P¼Pi. T isthe maximum hydrostatic tension the reference strength can with-stand. For pressures greater than Pi the strength is represented bythe analytic expression

r ¼ ri þ rmax � rið Þ 1:0� exp �ai P� Pið Þ½ �f g (1)

where

ai ¼ ri= rmax � rið Þ Pi þ Tð Þ½ � (2)

This form provides a continuous slope at Pi and it limits the maxi-mum strength to rmax (for _e� ¼ 1:0).

The strength for the surface and the strength for the interiorhave similar forms and are defined by two dimensionless con-stants Dsurf and Dint respectively. Dsurf is a factor ð0:01� Dsurf � 1:0Þ to allow the strength to be increased for materialon the surface that is not adjacent to failed material. This increasebecomes more pronounced as the pressure becomes more tensile.For Dsurf¼ 0.01 (the minimum allowed for numerical considera-tions) the surface strength is slightly below rmax, and forDsurf¼ 1.0 the surface strength is identical to the reference intactstrength. For intermediate values of Dsurf a reference angleðhref < 90oÞ is determined that defines the angle between the hori-zontal line at rmax and the extrapolated line formed for the linearrelationship between the strength at r¼ 0 (at P¼� T) to thestrength at r ¼ ri (at P¼Pi) as illustrated in Fig. 2. The corre-sponding angle for the strength defined by Dsurf ishsurf ¼ Dsurf � href . The two extrapolated lines (defined by href

Fig. 1 Description of the glass model

Fig. 2 Description of the interior, surface and referencestrength

051003-2 / Vol. 78, SEPTEMBER 2011 Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 11/18/2014 Terms of Use: http://asme.org/terms

and hsurf ) meet at a common point at the strength of rmax. Thisalso defines rsurf

i at Pi. For pressures greater than Pi the analyticexpressions presented in Eqs. (1) and (2) are used but ri isreplaced by rsurf

i and T is replaced by Tsurf ¼ tan�1ðhsurf Þ� rsurf

i � Pi. For pressures less than Pi the interior strengthdecreases linearly from rsurf

i to Tsurf.The strength for the interior is defined in a similar manner. Dint

is a factor ð0:01 � Dint � 1:0Þ to allow the strength to beincreased for material in the interior that is not adjacent to failedmaterial. Again, this increase becomes more pronounced as thepressure becomes more tensile. For Dint¼ 0.01 the interiorstrength is slightly below rmax, and for Dint¼ 1.0 the interiorstrength is identical to the reference intact strength. For intermedi-ate values of Dint a reference angle ðhref < 90oÞ is determined, asdefined previously. The corresponding angle for the strengthdefined by Dint is hint ¼ Dint � href . This also defines rint

i at Pi. Forpressures greater than Pi the analytic expressions presented inEqs. (1) and (2) are used but ri is replaced by rint

i and T isreplaced by Tint ¼ tan�1ðhintÞ � rint

i � Pi. For pressures less thanPi the interior strength decreases linearly from rint

i to Tint.The strength for the failed material has a similar form to that of

the reference intact strength. It goes linearly from r ¼ 0 at P¼ 0,to r ¼ rf at P¼Pf. For pressures greater than Pf the strength isrepresented by

r ¼ rf þ rfmax � rf

� �1:0� exp �af P� Pf

� �� �� �(3)

where

af ¼ rf

.Pf rf

max � rf

� �� �(4)

The intact and failed strengths in Eqs. (1)–(4) are for a dimension-less strain rate of _e� ¼ 1:0. If ro is the strength at _e� ¼ 1:0, thenthe strength at other strain rates is

r ¼ ro 1:0þ C ln _e�ð Þ (5)

where C is the dimensionless strain rate constant.Thermal softening is included through the thermal softening

exponent M. If M> 0 the material is softened by the factor(1� T

�M) for both the intact and failed material. The homologoustemperature is �T

� ¼ (Tc�Tr)/(Tg� Tr) where Tc is the currentcomputed temperature, Tr is the room temperature and Tg is theglass transition temperature. This form is used because it isstraightforward to incorporate into the strength model, requiresonly one constant, M, and provides good agreement with the testdata from Chen [23].

Lastly, the effect of the third invariant is described. The imple-mentation is similar to that presented by Frank and Adley [28]. Itapplies to both the intact and failed strengths. When the thirdinvariant is used, the intact strength should be defined for a stressstate in the compressive meridian (r3 < r1 and r1 ¼ r2) wherer1, r2 and r3 are the principal stresses (tension is positive). Thestrength for the tensile meridian is reduced by a dimensionlessconstant J3fact times the strength in the compressive meridianwhere 0.5< J3fact< 1.0. The stress state for the tensile meridian is(r1 > r2 and r2 ¼ r3).

The interpolation between the compressive and tensile meri-dians is provided by the Lode angle (h) which can be defined as afunction of the second and third invariants (J2 and J3)

sinð3hÞ ¼ 3ffiffiffi3p

2� J3

J3=22

(6)

where J2 and J3 are defined in terms of the deviator stresses

J2 ¼1

2ðs2

x þ s2y þ s2

z þ 2s2xy þ 2s2

yz þ 2s2xzÞ (7)

J3 ¼1

3ðs3

x þ s3y þ s3

z þ 3sxs2xy þ 3sxs2

xz þ 3sys2xy þ 3sys2

yz

þ 3szs2xz þ 3szs

2yz þ 6sxysyzsxzÞ (8)

The reduced intact strength is given by

rreducedi

rinputi

¼ J3fact

21þ sinð3hÞ þ ð1=J3factÞð1� sinð3hÞÞ� �

(9)

where rinputi is the input intact strength along the compressive

meridian.Some comments should be made regarding the algorithms used

to determine surface elements, interior elements and elements ad-jacent to failed material. If an element has a least one node on thesurface it is considered a surface element and Dsurf is used todescribe the intact strength. If an element has at least one node incommon with a failed element it is considered adjacent to failedmaterial and the reference intact strength is used to describe thestrength. If an element is not a surface element or an element adja-cent to failed material it is an interior element and Dint is used todescribe the intact strength. Also, the factors Dsurf and Dint are notused for particles (because it is not a straightforward procedure todetermine if a particle is on the surface, in the interior, or adjacentto failed material). This is generally not a problem when conver-sion of elements into particles is used, because the material usu-ally fails before it is converted.

2.2 Damage. The damage for failure is shown in the lowerleft portion of Fig. 1 and it is accumulated as follows:

D ¼ R Dep=efp

(10)

where Dep is the increment of equivalent plastic strain during thecurrent cycle of integration and e f

p is the plastic strain to failureunder the current dimensionless pressure, P� ¼ P=rmax. The gen-eral expression for the failure strain is

e fp ¼ D1 P� þ T�ð ÞN 1þ Cf ln _e�

� �(11)

where D1, N and Cf are dimensionless constants and T*¼T/rmax.The strain rate constant Cf provides an increase in the strain tofailure for elevated strain rates. Tensile hydrostatic pressuresgreater than T will not cause the material to fail (D¼ 1.0) instanta-neously as is the case for the JHB model [27]. Here, very largetensile pressures are possible if the material remains elastic, butany amount of plastic strain will produce failure. This allows thestrength of the interior and surface strengths to be reached at ten-sile pressures greater than T. This damage formulation assumesthat damage does not accumulate when the material is in an elasticstress state. The authors are not aware of any quantitative datawhich indicates a weakening of the material when it is within theyield surface.

When D¼ 1.0 the material fails and the strength is reduced asshown in the top portion of Fig. 1. Two types of softening (dam-age softening or time-dependent softening) are provided to allowfor variations in how the material transitions from the intact sur-face to the failed surface. Damage softening allows the material tosoften gradually, rather than fail instantaneously at D¼ 1.0. Mate-rial begins to soften when D¼ (1.0 – Dsoft) and then linearly soft-ens to the failed strength at D¼ 1.0 where Dsoft is a dimensionlesscoefficient (0<Dsoft< 1.0). In the softening region [D> (1.0 –Dsoft)] the bulking pressure is added incrementally, in a mannersimilar to that used for the JH-2 model for brittle materials [29].For Dsoft¼ 0 there is no gradual softening and the material failsinstantaneously at D¼ 1.0. Alternatively, a time-dependent soft-ening can be used. This option gradually reduces the strength over

Journal of Applied Mechanics SEPTEMBER 2011, Vol. 78 / 051003-3

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 11/18/2014 Terms of Use: http://asme.org/terms

a specified time, tfail. The material begins to soften in a linearmanner when D¼ 1.0.

2.3 Pressure. Glass exhibits permanent densification whensubjected to high pressures [2–6] and bulking when failure occurs[24]. The hydrostatic pressure, with permanent densification andbulking, is shown in the lower right portion of Fig. 1. Permanentdensification begins when l> lelastic and is complete whenl¼ llock. The pressure for loading only (no unloading), no bulk-ing (b¼ 0, defined later) and before complete densification(l< llock), is

P ¼ K1lþ K2l2 þ K3l

3 (12)

where K1, K2 and K3 are constants (K1 is the bulk modulus); andl ¼ Vo=V � 1 ¼q=qo � 1 for current volume and density (V andq), and initial volume and density (Vo and qo). The pressure forl> llock is simply

P ¼ Plock þ K4ðl� llockÞ (13)

where Plock is the pressure at llock, and K4 and llock are constants.The process of unloading is more complex and is a function of

the magnitude of the permanent densification, lperm, and the maxi-mum volumetric strain attained, lmax. When lmax< lelastic thereis no permanent densification and unloading occurs along theloading path described by Eq. (12). When lmax � llock permanentdensification is complete and unloading occurs along the perma-nent densification curve (from llock to lperm). The unloading path(from llock to lperm) is determined by interpolating between theresponse provided by Eq. (12) and the response provided byEq. (13) (when extrapolated down to lo

lock). The interpolation isbased on the ratio lperm=l

olock and produces unloading along a path

described by Eq. (12) when lperm=lolock ¼ 0, and by Eq. (13) when

lperm=lolock ¼ 1.0.

There is also the possibility to unload in the transition zone(lelastic < lmax < llock). The unloading path is interpolatedbetween the path describe by Eq. (12) and the permanent densifi-cation curve previously described. The interpolation is based onthe ratio / ¼ lmax � lelasticð Þ= llock � lelascicð Þ and producesunloading along a path described by Eq. (12) when /¼ 0, andalong the permanent densification curve when /¼ 1.0.

For tensile pressures ð�l < 0Þ, P¼ �K1 �l where K1 ¼ K1 and�l ¼ l when lmax< lelastic, �K1 is the stiffness at lperm and�l ¼ l� lperm when lmax � llock, and in the transition zone(lelastic < lmax < llock) �K1 is interpolated and �l ¼ l� l0 wherel0 ¼ lperm lmax � lelasticð Þ= llock � lelascicð Þ is the permanent volu-metric strain at P¼ 0.

During the softening/failure process (from damage or time-de-pendent softening) bulking (pressure and/or volume increase) canoccur. This requires an additional pressure increment, DP, suchthat

P ¼ K1lþ K2l2 þ K3l

3 þ DP (14)

The pressure increment is determined from energy considerations.Looking back to the strength model in Fig. 1, there is a decreasein strength when the material goes from an intact state (D< 1.0)to a failed state (D¼ 1.0) (for instantaneous failure with Dsoft¼ 0).This represents a decrease in the elastic internal energy of the de-viator stresses. The general expression for this elastic internalenergy (of the deviator stresses) is

U ¼ r2=6G (15)

where r is the von Mises equivalent stress and G is the elasticshear modulus. The corresponding decrease in internal energy canbe expressed as

DU ¼ Ui � Uf (16)

where Ui is the internal energy of the intact material before failure(D< 1.0) and Uf is the internal energy after failure (D¼ 1.0).

This internal energy loss (of the deviator stresses) can be con-verted to potential hydrostatic internal energy by adding DP. Anapproximate equation for the internal energy conservation is

DPlf þ DP2=ð2 �KÞ ¼ bDU (17)

where lf is the current value of l at failure, �K is the effective bulkmodulus, and b is the fraction (0 � b � 1:0Þ of the internal (devi-ator) energy loss converted to potential hydrostatic energy. Thefirst term (DPlf ) is the approximate potential energy for l> 0 andthe second term [DP2= 2 �Kð Þ] is the corresponding potential energyfor l< 0.

Solving for DP gives

DP ¼ � �Klf þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

�Klf

� �2þ2b �KDU

q(18)

The bulking pressure is computed only for failure under compres-sion (lf > 0). Note that DP ¼ 0 for b ¼ 0 and that DP increasesas DU increases and/or lf decreases. When there is gradual dam-age softening (Dsoft> 0), DP is updated during the course of thesoftening process [29].

In addition to a constant shear modulus (which implies a vari-able Poisson’s ratio), there is a provision to use a variable shearmodulus, Gvar, that is proportional to the current bulk modulus,Kcur and is given by

Gvar ¼ Kcur3

2

1� 2�

� þ 1

� �(19)

where � is Poisson’s ratio. This relationship is based on theassumption of a constant Poisson’s ratio.

It should be noted that the pressure does not include energy de-pendence (other than bulking) such as occurs in the Mie-Gru-neisen and other Equations of State. Generally, the energy effectsare more important for very high pressures, and are not significantfor ballistics problems which involve lower pressures. The inclu-sion of energy effects would be very complex when combinedwith permanent densification and bulking, and it is not clear howthese three effects (internal energy, permanent densification, bulk-ing) could be combined. Furthermore, the authors are not aware ofany test data from which the appropriate constants could be deter-mined. The bulking algorithm has been used with the JHB modelfor ceramics [27] and it provides a qualitative description ofbehavior during and after failure. The material rapidly expandswhen failed under low confinement/pressure, and it develops adistinct pressure increase when it is confined. Similar behaviorhas been observed when glass fails. Two desirable features of thisbulking algorithm are that it does not require any additional con-stants (other than the fraction of energy that goes into bulking),and that internal energy is conserved. For penetration problems atballistic velocities, the presence of bulking does not appear tohave a significant effect on the penetration velocities and depths.The effect of strength is generally much more important.

3 Demonstration of the Model

Several features of the model are illustrated by the examplesshown in Figs. 3–6. The examples presented in Figs. 3–5 demon-strate the effect of densification, damage softening and the inter-mediate principal stress (3rd invariant) respectively, and areproduced by axially loading (and unloading) a single element inuniaxial strain. The example presented in Fig. 6 demonstrates theeffect of time-dependent failure and high-internal-tensile strengthusing computed results from spall plate-impact computations.

051003-4 / Vol. 78, SEPTEMBER 2011 Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 11/18/2014 Terms of Use: http://asme.org/terms

Simple, generic, constants are used for these illustrations:qo¼ 2210 kg/m3, G¼ 5 GPa, T¼ 0.1 GPa, ri¼Pi¼ 0.5 GPa,rmax¼ 1.0 GPa, rf¼Pf¼ 0.1 GPa, rf

max¼ 0.2 GPa,Dsurf¼Dint¼ 1.0 (the surface, interior and reference strengths areidentical unless otherwise noted), K1¼ 10 GPa, K2¼� 10 GPa,K3¼ 0, K4¼ 20 GPa, lelastic¼ 0.1, lperm¼ 0.3 and llock¼ 0.5.The effect of bulking, strain rate and thermal softening are notincluded (to simplify the responses), and the strain to failure is setvery high (unless otherwise noted) so the material does not fail.

Figure 3 illustrates the process of developing permanent densi-fication. From point 1 to point 2 the response is elastic and no per-manent densification occurs. From point 2 to point 5 defines thetransition region where permanent densification is produced. Per-manent densification is initiated at point 2 (lelastic¼ 0.1) and iscomplete at point 5 (llock¼ 0.5). If unloading occurs in the transi-tion region, prior to reaching point 5, the permanent densificationprocess is not complete and only partial densification occurs. Thisis demonstrated by partially unloading the pressure from point 3to point 4. If unloading continues beyond point 4, only partial per-manent densification (l¼ 0.17) results. Also, in the transitionregion, unloading and reloading occur along the same path asdemonstrated between points 3 and 4. Finally, loading from point5 to point 6 is on the fully dense curve and unloading in thisregion occurs along the path defined by points 6, 5 and 7.

Damage softening is a feature that allows the material to softengradually when D¼ (1.0 – Dsoft) where Dsoft is a dimensionlesscoefficient. Figure 4 illustrates the process of damage softeningfor Dsoft¼ 0, 0.5 and 1.0 using an approximate strain to failure of,�p

f¼ 0.10 (by setting N¼ 0.0001 and D1¼ 0.10). For all threecases the initial loading is elastic until the reference intact strength

is encountered at point 2. For Dsoft¼ 1.0 damage softening beginsat the onset of plastic deformation (point 2) and continues tosoften until D¼ 1.0 and the failed surface is reached (point 3). ForDsoft¼ 0.5 gradual softening begins at D¼ 0.5 (point 4) and grad-ually softens to the failed surface at D¼ 1.0 (point 5). ForDsoft¼ 0 there is no gradual softening and the material fails instan-taneously at D¼ 1.0 (point 6 to 7). If bulking would be included,a pressure increment (DP) would be added to the pressures inFig. 4.

Figure 5 illustrates the effect of the third invariant. Two illustra-tions are provided, one that includes the third invariant effect,J3fact¼ 0.5, and one that does not, J3fact¼ 0. For both illustrationsthe loading and unloading is identical from points 1 to 5. FromPoints 1 to 2, the loading is elastic until the reference intact sur-face is encountered at point 2. From points 2 to 3, the materialflows plastically along the intact surface. From points 3 to 4 theloading direction is reversed and the material is unloaded elasti-cally. At point 4, the deviator stresses are completely unloadedand the resulting stress is due only to hydrostatic pressure. Thestress state during loading and partial unloading (from point 1 topoint 4) is on the compressive meridian where the net axial and ra-dial stresses are compressive, the axial deviator stress is positiveand the radial deviator stress is negative. From points 4 to 5, elas-tic unloading continues and the axial deviator stress is now in ten-sion and the radial deviator stress is in compression (the stressstate is now on the tensile meridian). For J3fact¼ 0 there is no 3rd

invariant effect and unloading continues to point 6 where the

Fig. 3 Example demonstrating the pressure-volume responseincluding permanent densification

Fig. 6 Example demonstrating the effects of time-dependentfailure and high-internal-tensile strength

Fig. 5 Example demonstrating the effect of the 3rd invariant

Fig. 4 Example demonstrating damage softening

Journal of Applied Mechanics SEPTEMBER 2011, Vol. 78 / 051003-5

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 11/18/2014 Terms of Use: http://asme.org/terms

intact surface is again encountered. Unloading continues plasti-cally from points 6 to 7. At point 7, both the axial deviator stressand the pressure go to zero. For J3fact¼ 0.5 the unloading is dif-ferent because a lower intact surface is used for the tensile merid-ian. Elastic unloading continues from points 4 to 5 where thelower intact strength is encountered. Unloading continues plasti-cally from Points 5 to 7. At point 7, both the axial deviator stressand the pressure go to zero.

Figure 6 presents the effect of time-dependent failure and theability to develop high-internal-tensile strength. Four tensile-spallplate-impact computations are used to demonstrate these effects.The material constants are the same as those presented previouslyexcept the interior strength is increased to Dint¼ 0.7 (this providesa significant increase in the interior tensile strength and corre-sponds to a spall strength of approximately 450 MPa). The plateimpact configuration consists of a 5 mm glass impactor and a 15mm glass target (this configuration will produce a tensile stress 5mm from the rear target surface). The impact velocity is 200 m/sand the particle velocity-time histories are taken at the rear targetsurface. In the upper left quadrant of Fig. 6 is the computed waveprofile for tfail¼ 0.4 ls, in the lower left is the wave profile fortfail¼ 0.2 ls, in the upper right is the wave profile for tfail¼ 0.02ls and in the lower right is the wave profile for tfail¼ 0.

Before the results in Fig. 6 are discussed, a brief overview of atensile-spall plate-impact test is provided. In a spall experiment,an impactor strikes a target producing an elastic compressivewave that travels into the target and into the impactor. The elasticcompressive wave reflects off the rear surface of the target andpropagates back toward the impact surface. Due to wave interac-tions a tensile stress is created in the interior of the target locatedone impactor thickness from the rear target surface (this is truewhen both the impactor and target are made of the same material).If the material can withstand the magnitude of the tensile stress,the material does not fail and no spall is produced. If the materialcannot withstand the tensile stress, the material fails, creating a“pullback signal” which travels back to the rear surface where it ismeasured. The magnitude of the pullback signal is proportional tothe tensile stress at which the material fails (the spall strength,rspall). Since spall occurs inside the material (but is measured atthe rear surface), a spall experiment provides an indirect measureof the internal tensile strength.

Returning to Fig. 6, the computed result for tfail¼ 0.4 ls is dis-cussed first. This response is described with the help of the stress-pressure insert in the lower right quadrant in Fig. 6 (the stress-pressure response is taken at the spall location, 5 mm from therear surface). The impactor strikes the target at 200 m/s and thestress goes from point 1 to point 2. The peak compressive stress iselastic and is maintained until the arrival of the release wave. Therelease wave rapidly takes the material into tension (at the spalllocation). The stress goes from point 2 to point 1 to point 3 wherethe interior intact strength is encountered. The material can de-velop no plastic strain and failure occurs producing the pullbacksignal. The magnitude of the pullback signal is proportional to thespall stress of 450 MPa. Since the tensile stress occurs in the inte-rior of the target the interior intact strength governs the response(point 3) and not the reference intact strength (point 4). Also notethe time delay of approximately 0.4 ls in the pullback signal, dueto the time-dependent failure. The result for tfail¼ 0.2 ls is similarto that of tfail¼ 0.4 ls except the duration of the pullback signal isreduced, which is due to the reduction in the time it takes the ma-terial to fail. The result for tfail¼ 0.02 ls is more complex. Themagnitude of the pullback signal is significantly smaller than pro-duced for tfail¼ 0.4 ls and tfail¼ 0.2 ls. This is due to the veryshort duration of failure, the condition of the material and thepropagation of the pullback signal. When failure occurs at thespall plane a new surface is created. This new surface is adjacentto failed material (created at the spall plane) and is governed bythe reference intact strength. If the time to failure is sufficientlyshort, the magnitude of the pullback signal cannot be maintainedas it propagates to the rear surface. The pullback signal will

continue to decay until either the reference intact strength isreached or the signal arrives at the rear surface. The final exampleis for tfail¼ 0 and produces only a small pullback signal. Eventhough the spall strength of 450 MPa is attained at the spall plane,it cannot be maintained and quickly decays to the stress repre-sented by the reference intact strength at point 4.

4 Example Computations for High-Velocity Impact

Conditions

The following computed results are for more complex, high-velocity-impact conditions. The computed results use the newglass model and preliminary constants for borosilicate glass esti-mated from the literature. The determination of constants is not astraightforward process because some of the constants cannot bedetermined explicitly from the test data. The general approach todetermine constants for the glass model is as follows: The pres-sure-volume response, the maximum intact strength, the timerequired for the material to fail, and the shear modulus (constantor variable) are estimated from compression plate-impact experi-ments where both the longitudinal and lateral stresses aremeasured. The interior strength is estimated from tensile-spallplate-impact experiments. Quasi-static and dynamic four-pointbending flexural tests are used to estimate the reference strength,the surface strength and the strain-rate effect for damage. Quasi-static and dynamic compression tests are used to estimate thestrain-rate effect on the strength. Confined quasi-static compres-sion tests and pressure-shear plate-impact tests are used to esti-mate the strength of the failed material. The damage modelconstants are estimated by providing computed results that are ingood agreement with ballistic experiments that produce dwell,interface defeat and penetration. Finally, although it is clear thatglass undergoes permanent densification [3–6], a procedure todetermine the onset, completion and magnitude of permanent den-sification has not been identified. This will be an area of futureresearch.

The objective of this section is not so much to provide a mea-sure of the accuracy of the model but rather to demonstrate itsability to produce reasonable results for a broad range of impactconditions using one set of constants. All the computations wereperformed with the 2010 version of the EPIC code using one-dimensional (1D) and two-dimensional (2D) Lagrangian finiteelements [30] and 2D meshless particles [31]. For the 2D examplecomputations involving severe distortions, the finite elements areautomatically converted into meshless particles during the courseof the computation [32].

4.1 Plate Impact. Figure 7 shows a comparison of the com-puted results and experimental results for two compressive plate-impact tests (BORO-9 is offset by 1.0 ls for clarity). The com-pression tests (BORO-9 and BORO-10) are provided byAlexander et al. [21] and use a copper impactor and a borosilicatetarget backed by lithium fluoride. The particle velocity is pre-sented as a function of time recorded at the borosilicate-lithiumfluoride interface. For both compressive tests there is reasonableagreement between the computed and experimental results. A fewcomments should be noted regarding the computed wave profiles;the ramping of the initial elastic wave is due to a softening of thebulk modulus, the hugoniot elastic limit (HEL) has a magnitudeof approximately up¼ 0.6 km/s, and the increases in the particlevelocity (at approximately 3 ls for BORO-10 and at 4 ls forBORO-9) are a result of wave reflections at the impactor-targetinterface.

4.2 Interface Defeat and High Velocity Penetration. Figure 8presents computed results for a gold projectile impacting a borosi-licate target with a copper buffer attached to the impact surface.The buffer is included to attenuate the impact shock and producegradual loading on the glass. The computed results produce

051003-6 / Vol. 78, SEPTEMBER 2011 Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 11/18/2014 Terms of Use: http://asme.org/terms

interface defeat (no glass penetration) at Vs¼ 800 m/s and promptpenetration at Vs¼ 900 m/s. This is consistent with the test resultsprovided by Anderson et al. [26].

Behner et al. [25] performed experiments using no buffer (goldrod impacting bare glass) producing penetration at much lowerimpact velocities. For these experiments the penetration velocitywas determined using a series of flash x-rays and are presented asa function of impact velocity as shown in the lower portion ofFig. 9. Also shown are the computed penetration velocities forVs¼ 900 m/s and Vs¼ 2400 m/s. The computed penetration veloc-ity is slightly low compared to the test data at Vs¼ 900 m/s, but isin closer agreement at Vs¼ 2400 m/s. In the top portion of Fig. 9,the computed results show material damage at 30 ls (for Vs¼ 900m/s) and at 10 ls (for Vs¼ 2400 m/s). Behner et al. [25] also pro-vides flash x-ray images and high-speed photographs at similartimes and impact velocities as those presented in the top portionof Fig. 9. A comparison of the computed responses to the x-rayand optical images yield the following observations: The com-puted and experimental cavity sizes were generally in good agree-ment for both impact velocities (as defined by the flow of the high-

density projectile debris). The lower impact velocity produced alarger and less uniform cavity, the higher impact velocity produceda smaller more uniform cavity. Both the computed and experimen-tal results produced a significant amount of glass fragments movingout from the impact surface (although there appeared to be morefragments produced in the experiments than in the computations).Lastly, the outer edge of the experimental targets expand radiallyoutward during penetration, but the computed results did not (thiscould be due to the computations being performed in 2D axisym-metry where radial cracking cannot occur).

4.3 Steel Projectiles Impacting Thin Glass Targets. Andersonet al. [14] performed experiments using a pointed steel projectileimpacting thin plates of borosilicate glass at three different scalesizes. For the largest scale (scale¼ 1.0) the projectile was 12.7mm in diameter and 38.1 mm long (0.50-cal) and the target was21.0 mm thick and 457.2 mm square. For the smaller scales(scale¼ 0.75 and 0.44) both the projectile and target plates werescaled accordingly. The experimental results produced some sig-nificant findings: The targets appeared stronger as the scale wasreduced (particularly at the lower impact velocities); glass is verystrong producing significant projectile erosion; and there appearsto be a change in the glass behavior at approximately Vs¼ 500–600 m/s, which produced a discontinuity in the Vs-Vr curve (this isalso where the maximum projectile erosion occurs). The computa-tions focused primarily on the 0.50-cal experiments (scale¼ 1.0),although the 0.22-cal (scale¼ 0.44) was also used to investigatethe scale effect.

Figures 10 and 11 present computed and experimental resultsfor steel projectiles impacting thin glass targets. Figure 10 showsthe computed results (showing material damage) for three impact

Fig. 8 The initial geometry and computed results for a goldrod impacting a borosilicate target with a copper buffer atV 5 800 m/s and V 5 900 m/s

Fig. 9 Comparison of computed and experimental results for agold rod impacting bare borosilicate glass

Fig. 7 Comparison of the computed results and experimentalresults for two plate-impact tests

Journal of Applied Mechanics SEPTEMBER 2011, Vol. 78 / 051003-7

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 11/18/2014 Terms of Use: http://asme.org/terms

velocities using the 0.50-cal projectile. At Vs¼ 300 m/s the pro-jectile exits the target with a very low velocity of 22 m/s and thereis a large spall cone produced on the rear target surface. Therealso has been some erosion of the projectile. At Vs¼ 400 m/s and500 m/s the projectile exits the target at a much higher exit veloc-ity and with less projectile erosion. The targets appear similar,exhibiting large cone cracks that propagate from the target mid-

section to near the rear surface. The lower right portion of Fig. 10presents a comparison of the residual velocity as a function ofimpact velocity for the computed and experimental results. Thecomputed exit velocities are in good agreement at Vs¼ 400 m/sand for Vs> 600 m/s, but the targets are too strong for Vs< 400m/s. The computed results also do not capture the target responsefor 500 m/s � Vs � 600 m/s where the maximum projectile

Fig. 10 Comparison of computed and experimental results for a steel projectile impacting bor-osilicate glass

Fig. 11 Comparison of computed results for a steel projectile impacting borosilicate glass attwo scales

051003-8 / Vol. 78, SEPTEMBER 2011 Transactions of the ASME

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 11/18/2014 Terms of Use: http://asme.org/terms

erosion occurs. It is not clear why the glass appears stronger inthis velocity regime and then suddenly appears much weaker atVs> 600 m/s. Although the computed results do not capture thisresponse, it may be possible to produce better agreement using adifferent set of constants (specifically T, tfail, Dint and Dsurf). Gen-eral comparisons can also be made regarding the computed andexperimental post-mortem targets. The tested targets consistentlyproduced rear surface spall cones and a significant amount of ra-dial and circumferential cracks emanating from the impact loca-tion (although the cracking was significant, the targets remainedintact). The computed targets also produced rear surface spallcones, but the cracking was much less than what was observedexperimentally (this could be due to the computations being per-formed in 2D axisymmetry where radial cracking cannot occur).

Figure 11 demonstrates the capability to compute size effects.Computed results are presented for the 0.50-cal (scale¼ 1.0) andthe 0.22-cal (scale¼ 0.44) projectile for Vs¼ 300 m/s and 400m/s. At 300 m/s, the 0.50-cal projectile exits the target atVr¼ 22 m/s but the smaller scale 0.22-cal projectile is stopped(there is also significantly more projectile erosion). As theimpact velocity is increased the computed size effect diminishesas shown at Vs¼ 400 m/s. The ability for the computations toproduce size effects is due to the time-dependent features in theglass model. Smaller scale events happen in a shorter time pe-riod resulting in larger strain rates for the smaller scale. Thestrain rate effect in the ductility (Cf), the strain rate effect in thestrength (C), and the finite time-to-fail (tfail), all contribute tomaking smaller scales stronger.

5 Summary and Conclusions

This article has presented a new computational constitutivemodel for glass subjected to large strains, high strain rates andhigh pressures. This new glass model has an intact strength and afailed strength that are dependent on the pressure and the strainrate, damage softening or time-dependent softening that transi-tions the strength from intact to failed, thermal softening, a con-stant or variable shear modulus, the effect of the third invariant, apressure-volume relationship that includes permanent densifica-tion and bulking, and damage that is accumulated from plasticstrain, pressure and strain rate. The intact strength is also depend-ent on the location of the material (whether it is in the interior, onthe surface or adjacent to failed material). Significant features ofthe model are the ability to compute size effects (smaller is stron-ger), surface effects, and high internal tensile strength.

Several features of the model were illustrated using a simple(single-element) problem. Computations were also presented formore complex, high-velocity-impact conditions using preliminaryconstants for borosilicate glass determined from the literature.The computed results demonstrate the ability of the model to pro-duce reaseonable results for a broad range of impact conditions.

Acknowledgment

This work was performed for the U. S. Army RDECOM-TARDEC under Contract No. W56HZV-06-C-0194. The authorswould like to thank D. Templeton and R. Rickert (U. S. ArmyTARDEC) and to C. Gerlach (Southwest Research Institute) fortheir contributions to this work. The authors would like to espe-cially thank C. Anderson, Jr. for his helpful discussions and exper-imental work in support of this effort.

References[1] Bridgman, P. W., 1948, “The Compression of 39 Substances to 100,000 kg/

cm2,” Proceedings of the American Academy of Arts and Sciences, 76(3), pp.55–87.

[2] Bridgman, P. W. and Simon, I., 1953, “Effects of Very High Pressures onGlass,” J. Appl. Phys., 24(4), pp. 405–413.

[3] Uhlmann, D. R., 1973, “Densification of Alkali Silicate Glasses at High Pres-sure,” J. Non-Cryst. Solids, 13, pp. 89–99.

[4] Rouxel, T., Ji, H., Hammouda, T., and Moreac, A., 2008, “Poisson’s Ratio andthe Densification of Glass Under High Pressure,” Phys. Rev. Lett., 100,p. 225501.

[5] Sakka, S. and Mackenzie, J. D., 1969, “High Pressure Effects on Glass,” J.Non-Cryst. Solids, 1, pp. 107–142.

[6] Mackenzie, J. D., 1963, “High-Pressure Effects on Oxide Glasses: I, Densifica-tion in Rigid State,” J. Am. Ceram. Soc., 46(10), pp. 461–470.

[7] Rosenberg, Z., Yaziv, D., and Bless, S., 1985, “Spall Strength of Shock-LoadedGlass,” J. Appl. Phys., 58(8), pp 3249–3251.

[8] Brar, N., Rosenberg, S., and Bless, S., 1991, “Spall Strength and Failure Wavesin Glass,” J. Phys. (Paris), 1, pp. 639–634.

[9] Cagnoux, J., 1985, “Deformation et Ruine d’un Verre Pyrex Soumis a un ChocIntense: Etude Experimentale et Modelisation du Comportement,” Ph.D. thesis,L’Universite de Poitiers.

[10] Cagnoux, J. and Longy, F., 1988, “Spallation and Shock-Wave Behaviour ofSome Ceramics,” J. Phys. (Paris), 49(9), pp. 3–10.

[11] Kanel, G. I., Bogatch, A. A., Razorenov, S. V., and Chen, Z., 2002,“Transformation of Shock Compression Pulses in Glass due to the FailureWave Phenomena,” J. Appl. Phys., 92(9), pp. 5045–5052.

[12] Anderson, C. E., Jr., Orphal, D. L., Behner, T., and Templeton, D. W., 2009,“Failure and Penetration Response of Borosilicate Glass During Short-RodImpact,” International Journal of Impact Engineering, 36, pp. 789–798.

[13] Nie, X., Chen, W., Wereszczak, A., and Templeton, D., 2009, “Effect of Load-ing Rate and Surface Conditions on the Flexural Strength of BorosilicateGlass,” J. Am. Ceram. Soc., 92(6), pp. 1287–1295.

[14] Anderson, C. Jr., Weiss, C., and Chocron, S., 2009, “Impact Experiments intoBorosilicate Glass at Three Scale Sizes,” Southwest Research Institute, SanAntonio, TX, Technical Report No. 18.12544/018.

[15] Sun, X., Khaleel, M., and Davies, R., 2005, “Modeling of Stone-Impact Resist-ance of Monolithic Glass Ply using Continuum Damage Mechanics, Int. J.Damage Mech., 14, pp. 165–178.

[16] Wereszczak, A. A., Kirkland, T. P., Ragan, M. E., Strong, K. T., Jr., and Lin,H., 2010, “Size Scaling of Tensile Failure Stress in a Float Soda-Lime-SilicateGlass,” International Journal of Applied Glass Science, 1(2), pp. 143–150.

[17] Chocron, S., Anderson, C. E., Jr., Nicholls, E., and Dannemann, K. A.,“Characterization of Confined Intact and Damaged Borosilicate Glass,” J. Am.Ceram. Soc. (to be published).

[18] Simha, C. and Gupta, Y., 2004, “Time-Dependent Inelastic Deformation ofShocked Soda-Lime Glass,” J. Appl. Phys., 96(4), pp. 1880–1890.

[19] Sundaram, S., 1993, “Pressure-Shear Plate Impact Studies of Alumina Ceramicsand the Influence of an Intergranular Glassy Phase,” Ph.D. thesis, Brown Uni-versity, Providence, RI.

[20] Cagnoux, J., 1982, “Shock-Wave Compression of a Borosilicate Glass up to170 kbar,” Shock Compression of Condensed Matter-1982, pp. 392–296.

[21] Alexander, C. S., Chhabildas, L. C., Reinhart, W. D., and Templeton, D. W.,2008, “Changes to the Shock Response of Fused Quartz due to Glass Mod-ification,” International Journal of Impact Engineering, 35, pp. 1376–1385.

[22] Handin, J., Heard, H. C., and Magouirk, J. N., 1967, “Effects of the Intermedi-ate Principal Stress on the Failure of Limestone, Dolomite, and Glass at Differ-ent Temperatures and Strain Rates,” J. Geophys. Res., 72(2), pp. 611–640.

[23] Chen, W., 2010, Purdue University, private communication.[24] Glenn, L. A., Moran, B., and Kusubov, A. S., 1990, “Modeling Jet Penetration

in Glass,” Lawrence Livermore National Laboratory, CA., Technical ReportNo. UCRL-JC-103512.

[25] Behner, T., Anderson, C., Jr., Orphal, D., Hohler, V., Moll, M., and Templeton,D., 2008, “Penetration and Failure of Lead and Borosilicate Glass against RodImpact,” International Journal of Impact Engineering, 35, pp. 447–456.

[26] Anderson, C. E., Jr., Behner, Th., Holmquist, T. J., Wickert, M., Hohler, V.,and Templeton, D. W., 2007, “Interface Defeat of Long Rods Impacting Borosi-licate Glass,” Proceedings of the 23rd International Symposium on Ballistics,Tarragona, Spain, pp. 1049–1056.

[27] Johnson, G. R., Holmquist, T. J., and Beissel, S. R., 2003, “Response of Alumi-num Nitride (Including a Phase Change) to Large Strains, High Strain Rates,and High Pressures,” J. Appl. Phys., 94(3), pp. 1639–1646.

[28] Frank, A. and Adley, M., 2007, “On the Importance of a Three-Invariant Modelfor Simulating the Perforation of Concrete Targets,” Proceedings from the 78th

Shock and Vibration Symposium, Philadelphia, PA.[29] Johnson, G. R. and Holmquist, T. J., 1994, “An Improved Computational Con-

stitutive Model for Brittle Materials,” High Pressure Science and Technology–1993, S. C. Schmidt, J. W. Schaner, G. A. Samara, and M. Ross, eds., AIP,New York, pp. 981–984.

[30] Johnson, G. R., Stryk, R. A., Holmquist, T. J., and Beissel, S. R., 1997,“Numerical Algorithms in a Lagrangian Hydrocode,” Wright Laboratory, FL,Technical Report No. WL-TR-1997-7039.

[31] Johnson, G. R., Beissel, S. R., and Stryk, R. A., 2002, “An Improved General-ized Particle Algorithm that Includes Boundaries and Interfaces,” Int. J. Numer.Methods Eng., 53, pp. 875–904.

[32] Johnson, G. R., Stryk, R. A., Beissel, S. R., and Holmquist, T. J., 2002,“Conversion of Finite Elements into Meshless Particles for Penetration Com-putations Involving Ceramic Targets,” Shock Compression of CondensedMatter–2001, M. D. Furnish, N. N. Thadhani, and Y. Horie, eds., AIP, NewYork.

Journal of Applied Mechanics SEPTEMBER 2011, Vol. 78 / 051003-9

Downloaded From: http://appliedmechanics.asmedigitalcollection.asme.org/ on 11/18/2014 Terms of Use: http://asme.org/terms