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Proceedings of IMECE 20062006 ASME International Mechanical Engineering Congress and Exposition

November 5-10, 2006, Chicago, IL USA

IMECE2006-16224

FAILURE OF AN IMPULSIVELY-LOADED COMPOSITE STEEL/POLYMER PLATE

JongMin Shim and Tomasz Wierzbicki∗

Impact and Crashworthiness LaboratoryMassachusetts Institute of Technology

Cambridge, MA02139Email address ([email protected])

ABSTRACTThe concept of spraying thick layer of polymer material onto

metal plate has recently received considerable interest inmanycivilian and military applications. There are numerous analyticaland numerical solutions for single thin plates (membrane) madeof either a steel or an elastomer. However, solutions for com-posite plate made of both of the above constituents are lacking.The objective of the present paper is to formulate a model forcomposite steel/elastomer plate, derive an analytical solution ofthe impulsive loading problem and compare it with a more exactnumerical solution. It is assumed that the circular plate isfullyclamped around its peripheral and it is loaded by uniformly dis-tributed transverse pressure of high intensity and short duration.The pressure is imparting initial impulse which is proportional toinitial transverse velocity of the plate.

As an example, DH-36 is used for steel backing plate whilepolyurea is chosen to represent a typical polymer coating. In theanalytical model, an iterative method is developed in whichsteellayer treated as a rigid perfectly-plastic material with magnitudeof flow stress adjusted according to calculated magnitude ofav-erage strain. A linear elastic material is assumed with elasticmodulus in the tensile range calculated from the Arruda-Boycemodel for an specific type of polyurea. It was found that themagnitude of the average strain rate is relatively low, about 100sec−1. Therefore, the effect of strain rate is not considered inthis paper. A comprehensive parametric study was performedbyvarying various material and structural parameters of the model.A closed form solution was compared with the results of de-

∗Address all correspondence to this author.

tailed FE simulations of composite plates. It was found thatthepolyurea coating could improve the failure resistance of the com-posite plate by some 20 % provided the thickness of the coatingis 5−10 times larger than the plate.

INTRODUCTIONResponse of a thin metal plate subjected to explosive loading

was the subject of numerous experimental, theoretical and nu-merical studies. Bodner and Symonds [1] reported on an experi-mental study where thin clamped steel membrane were subjectedto an impulsive loading distributed uniformly over the centralportion of the structure. Symonds and Wierzbicki [2] developeda closed form solution of this problem using the mode approxi-mation technique and included the effect of the strain hardeningand strain rate sensitivity in an iterative way. Good correlationwas observed between theory and experiments regarding centraldeflection of the plate. More recently, very comprehensive se-ries of tests on explosively loaded plates all the way to fracturewas conducted in the University of Cape Town by Nurick and histeam [3] [4] [5]. It was observed that for a sufficient large im-pulse, failure occurs either at the clamp edge of the plate througha combination of shear and tension or in the central part in theso-called petaling mode.

It was found that the mode solution is able to predict theonset of fracture of plate with uniformly distributed impulse.Attempt to extend this type of approximate method to a plateloaded by more concentrated pressure loading will not success-ful. Wierzbicki and Hoo Fatt [6], Wierzbicki and Nurick [7],

1 Copyright c© 2006 by ASME

and Mihailescu-Suliciu and Wierzbicki. [8] succeed in derivingclosed form solution for this class of problem using the methodof wave propagation.

Nemat-Nasser [9] reported on experimental and numeri-cal study of small plates made of DH-36 steel without or withpolyurea backing subjected to uniformly distributed initial ve-locity. The ratio of polyurea thickness to steel thickness was ap-proximately 5−1 and the initial impact velocity was in the rangeof 60 to 70m/sec. The tested steel plats were fully clampedwhereas the layer of polyurea was simply placed in front of orbehind with no connection to the clamping ring. It was foundthat the elastomer coating reduced the central deflection oftheplate for impact velocity below the critical value. For a largerinitial velocities causing failure of the system (typically throughthe petaling mode) the polyura enhanced or mitigated the extentof failure of system. The same team also performed the nu-merical simulation of the test which qualitatively agreed with theexperimental results.

The objective of the present paper to derive a closed formsolution for steel/polyurea combination and perform a thor-ough parametric study to understand the effect of thicknessofthe polyurea coating on the response and failure of this com-plex structure. In order to make the problem mathematicallytractable, the membrane plate theory is assumed with both steeland polyurea constituents firmly clamped around peripheral.The steel plate is assumed to be made of DH-36 steel and theJohnson-Cook model for this material is calibrated from exper-iments provide by Nemat-Nasser and Guo [10]. The polyureacoating is described by Arruda-Boyce model which was againcalibrated from compressive test performed by Nemat-Nasser[11]. Numerical simulation of this problem is generated by usingaxisymmetric solid element in ABAQUS/Explicit. Good corre-lation is obtained form numerical and analytical solution.It wasfound that the polyurea coating could improve the failure resis-tance of the composite plate by some 20 % provide the thicknessof the coating is 5−10 times larger than the plate.

Calibration of Material ModelIn this section, the constitutive equations of both con-

stituents of the composite plate are defined. The steel modelisdescribed by Johnson-Cook model which has been successfullyused in many practical applications. Several general constitu-tive models of elastomers are available in the literature such asMooney-Rivlin, Ogden, Arruda-Boyce, etc. For the purpose ofthe present study, Arruda-Boyce model is chosen.

Material Model for DH-36 SteelThe constitutive model for a steel is assumed to follow a

simplified Johnson-Cook [J-C] equation, without the strainrate

term:

σ = A+B(

εpl)n

(1)

where σ is the equivalent stress,εpl is the equivalent plasticstrain,A is the yield stressσy, B andn are the material parametersrelating to strain hardening hardening. Equation (1) assumes theisotropic hardening of the material.

The analytical solution, to be developed in the next section,is based on the concept of the rigid perfectly-plastic approxima-tion defined by a constant stress with respect to the strain. Thisconstant stress is adjusted to represent in the best possible waythe actual hardening properties of the material, accordingto:

σ0 = σ(

εplave

)

= A+B(

εplave

)n(2)

whereεplave is the spatial and temporal average value for the equiv-

alent plastic strain.

Nemat-Nasser and Guo [10] performed comprehensivestudy of a constutive behavior of DH-36 steel under static anddynamic loading at different temperatures. In the presentstudy, the temperature effect is not considered and only room-temperature data were taken as basis for calibration. From thelogarithmic representation of the quasi-static stress-strain data(ε = 0.001 sec−1), three coefficients (A, B andn) in Eq. (1) weredetermined. The numerical values of all parameters are gatheredin Table 1, and they are used for the finite element simulations ofDH-36 steel. Comparison between J-C model and experimentaldata is shown in Fig. 1.

A B n

392MPa 550MPa 0.45

Table 1. J-C model parameters calibrated for DH-36 steel.


0 0.2 0.4 0.60

2

4

6

8

10x 10

8

True plastic strain

Tru

e st

ress

[P

a]

Test:0.001 sec−1

J−C Model:0.001 sec−1

Figure 1. Comparison of the true stress versus true plastic strain curve

from uniaxial compression test at room temperature [10] and the corre-

sponding J-C model.

Material Model for PolyureaThe polyurea, as all elastomer materials, behaves differ-

ently under compression and tension. The testing of polyureais typically done under compressive load. At the same time,the material in thin plate undergoing large deformation is sub-jected to predominantly tensile loads. Therefore, there isaneed to extrapolate the compressive data points into the tensileregime. Several models of hyperelastic material have been de-veloped in the literature. For the purpose of the present study,the well-known Arruda-Boyce [A-B] hyperelastic model is cho-sen [12] [13]. The strain energy potentialU in this formulationfor an isothermal condition is:

U = µ

{

12

(I1−3)+1

20λ2m

(

I21 −9

)

+11

1050λ4m

(

I31 −27

)

(3)

+19

7000λ6m

(

I41 −81

)

+519

673750λ8m

(

I51 −243

)

}

(4)

+1D

(

J2−12

− lnJ

)

(5)

whereµ, λm, andD are temperature-dependent material param-eters;I1 is the first deviatoric strain invariant; andJ is the totalvolume ratio. The unknown material constants for A-B model(µ, λm, D) can be determined from a number of different testssuch as uniaxial, biaxial or planar test combined with volumetrictest data. The advantage of working with ABAQUS is that it

automatically extrapolates the data into the tensile region fromthe given compression data, and it provides numerical values forthree material model parameters.

Test data of polyurea under uniaxial compression and vol-umetric compression (i.e. confined uniaxial compression) werereported by Nemat-Nasser and Guo [11] at different strain rates.The quasi-static test data (ε = 0.01 sec−1) is shown in Figs. 2and 3. The material parameters of the A-B hyperelastic modeldetermined on the basis of above data are summarized in Table2. Using the calculated parameters, the A-B model predicts theuniaxial stress strain data in the tensile region. The completestress-strain curve both in compression and tension quadrant isshown in Figs. 2-3 for three different values of strain rate.

µ 7.972×106 Pa

λm 6.874×103

D 7.246×10−10 m2/N

µ0 7.246MPa

K0 2.760MPa

ν 0.4986

Table 2. A-B model parameters calibrated for polyurea

−0.6 −0.4 −0.2 0 0.2 0.4 0.6−6

−4

−2

0

2x 10

7

Engineering Strain

Eng

inee

ring

Str

ess

[Pa]

Test: 0.01 sec−1

A−B Model: 0.01 sec−1

Figure 2. Comparison of the engineering stress-strain curve from uni-

axial compression tests at room temperature [11] and the corresponding

A-B model.


0.97 0.98 0.99 1 1.01 1.02 1.03−1

−0.5

0

0.5

1x 10

8

Volumetric Ratio

Pre

ssur

e [P

a]

Test: 0.01 sec−1


Figure 3. Comparison of the pressure versus volumetric ratio (V0 −∆V)/V0 curve from uniaxial compression tests at room temperature [11]

and the corresponding A-B model.

It is seen that in the tension quadrant the stress-strain curvecan be approximated by straight line defined by a secant modulusEs = 0.75E0 whereE0 is Young’s modulus at the zero strain.

σe = Esεe (6)

where σe and εe are, respectively, the engineering stress andstrain;Es is the secant Young’s modulus in the reference quasi-static test. It should be noted that in the problem of the largedeformation of composite plate the stress state is essentially uni-axial tension. Therefore, Eqn. (6) is thought to be a convient andgood approximation to the actual behavior of polyurea. Fromthebest fit of the tensile region of Fig. 2, one getsEs = 0.75E0 =2.069 MPa, which is a good approximation of the actual A-Bconstitutive model (see Fig. 4).

0 0.2 0.4 0.60

3

6

9

12x 10

6

Engineering Strain

Eng

inee

ring

Str

ess

[Pa]


Elastic Model: 0.01 sec−1

Figure 4. Comparison of the engineering stress-strain curve form A-B

model and the simplified elastic model, Eqn. (6).

In the general case of finite element simulations, thepolyurea response should be describe by a nonlinear hyperelasticmodel with viscoelastic effect and hysteresis loop. The presentsimplified model captures the material characteristic in the activeloading process without unloading. Such an approach is validonly for the polyurea plate working in combination with the steelbacking plate. The irreversible character of plastic deformationin the steel plate will prohibit the polyurea layer to undergo elas-tic unloading. In this sense, both components of composite platewould undergo monotonic loading without unloading.

Problem FormulationConsider a circular composite plate composed of two lay-

ers, one representing polyurea coating and the other one steelbacking plate (see Fig. 5). The analysis will be performedwith any combination of geometry and material properties ofboth plates. A transverse impulse of the intensityI is appliedto the upper layer. Within the assumption of the present theorywhere through-thickiness deformation is not considered, it doesnot matter which layer (polyurea or steel plate) is on the top.

Figure 5. Geometry of composite plate.


The steel plate is characterized by three parameters: platethicknessh1, mass densityρ1, and average flow stressσ0. Theflow stressσ0 depends in turn on the magnitude of the averagestrain (see Section ). The polyurea layer follows the constitutiveequation (6) with the secant Young’s modulusEs and is charac-terized by its thicknessh2 and mass densityρ2. It is assumed thatpolyurea is on the top, and subscript 2 will refer to all relevantparameters. Likewise, the parameter describing the steel platewill be distinguished by subscript 1. The following ratios areintroduced:

λ =h2

h1(7)

µ =ρ2

ρ1(8)

The response of the composite plate consists of two distinctphases. In the first phase, the high pressure acting on a unit sur-face area is generating plain stress wave in the top layer. A partof the wave is transmitted to the steel plate while the other partis reflected and is interacting with incoming incident waves. Acomplicated pattern of the wave interaction exists in this transientstage, and eventually a common velocity of the composite plate isattained. In the second phase of the response, the pressure termvanishes and the composite plate continues to deform due to theuniform initial velocity acquired at the end of the first phase. Adetailed numerical analysis of this first phase was performed tosee what part of energy remains as a transverse vibration of elas-tomer and what part goes into the initial kinetic energy of theplate [14]. The conclusion from this numerical analysis wasthatthe initial velocity of the plate (V0) could be effectively calculatedfrom the momentum conservation of a unit surface element:

(h1ρ1 +h2ρ2)V0 =

Z τ

0p(t)dt = I (9)

whereI is the total impulse imparted to the structure,p is theapplied pressure andτ is the loading duration. In the secondphase of the response, the governing equation of the membraneresponse of the circular plate is:

mrw =(

Nrr rw′)′ (10)

wherew is the transverse displacement,r is the radial coordinate,m is the mass per unit area,Nrr is the membrane force per unitlength and dot (·) and prime (′) denote respectively differentia-tion with respect to time and radial coordinate. In this equation,the pressure term is set up to zero.

The expression form andNrr should now be specified forboth the steel and the polyurea layers. As discussed earlier, therigid perfectly-plastic material idealization is assumedfor steel.The radial stress in the steel plate is constant and equal to theplane strain yield stress for the von-Mises yield condition:

(Nrr )steel=2√3

σ0h1 (11)

The value of the constant flow stress of a material is notspecified at this point. Expression forσ0 will be different forthe first and the second iterative solutions, as explained inwhatfollows. From Eq. (6), the membrane force for polyurea layeris:

(Nrr )polyurea= σeh2 = Esεrr h2 (12)

In the present approximate solution, it is assumed that the hoopstrain is zero. In the theory of moderate large deflection of thinplate, the radial strain is

εrr =12

(

w′)2(13)

Now, the governing equations for the steel and the polyureaplate are given respectively by:

(ρ1h1) rw1−(

2√3

σ0h1rw′1

)′= 0 (14)

(ρ2h2) rw2−(

12

Esh2r(

w′)3)′

= 0 (15)

wherew1 is the transverse deflection of the middle suffice of thesteel. It is assumed that both steel and polyurea plates deformwith no separation which means that no delamination or spallingwould occurs at the interface. Then, the condition of displace-ment continuity at the interface would apply:

w1 = w2 = w(r,t) (16)

By adding Eq. (14) and (15) with Eq. (16), a single govern-ing equation is obtained for the response of the composite plate:

(ρ1h1 + ρ2h2) rw− 2√3

σ0h1(

rw′)′− 12

Esh2

(

r(

w′)3)′

= 0

(17)


This is a nonlinear partial differential equation inr andt. Forthe second order spatial derivative, two boundary conditions areneeded. The first boundary condition is vanishing of the dis-placement the outer perimeter,w(R, t) = 0, whereR is the plateradius. In the theory of elastic plates, the slope should also van-ish at the clamped edge. However, in the theory of plastic plates,the slope discontinuity is allowed at the clamped edge. In the nu-merical simulation, where axisymmetric solid element are used,the above contradiction is absent because only the displacementboundary conditions have to be prescribed. One way to over-come the above contradiction for the present plate/membraneproblem would be to consider a class of shape function that sat-isfies the condition,w′ (R, t) = 0.

The initial conditions in the present initial-boundary valueproblem are

w(r,0) = 0

w(r,0) = V0

Note that there is an initial discontinuity in the velocity field atr = R, and as a result there will be slope discontinuity propagat-ing from the clamped edge toward center.

Closed Form Analytical SolutionIn the limiting case ofh2 = 0, Eq. (17) reduces to a linear

wave equation. The wave type solution which admits the slopediscontinuity was derived by Mihailescu-Suliciu and Wierzbicki[8]. With an addition of polyurea layer, the governing equa-tion becomes nonlinear, and the Riemann wave solution is nolonger possible. Instead, a mode type solution will be devel-oped as an extension of the method previously used by Symondsand Wierzbicki [2] for thin metal plates. When the separationof variables is used to solve this partial differential equation, thedeflection of the plate over time and space can be written:

w(r, t) = T (t)Φ(r) (18)

whereT (t) is the time variable amplitude, andΦ(r) is the nor-malized deformed shape. Note that all the formulations are de-rived in the cylindrical coordinate system.

In this study, the deformed shape is assumed to have thefollowing conical shape:

Φ(r) =(

1− rR

)m(19)

wherem is a fractional exponent that should be chosen to bestrepresent the actual shape of a deformed plate. From the numer-ical solution, the experimental observation [3] and the wave type

solution [8], it appears that the plate assumes the conical shapewith m= 1. The conical shape violates the slope boundary con-dition at the support. Any exponent slightly greater than one,for example,m = 1.01, will satisfy the slope boundary condi-tions. In the present approximate solution, the calculations willbe run form= 1.

The mode-form solution satisfy the first initial condition,butnot the second initial condition. Martin and Symonds [15] de-veloped a kinetic energy difference minimization technique sothat the mode-form solution will satisfy in the best possible waythe actual solution of the initial-boundary value problem.Initialvelocity in the mode solution is:

w(r,0) = T (0)Φ(r) = VmaxΦ(r) (20)

where the unknown amplitudeVmax can be determined by usingthe the minimization of the kinetic energy difference:

∂(∆KE)

∂Vmax=

∂∂Vmax

{

m2

Z R

0[V0−VmaxΦ(r)]2 rdr

}

≡ 0 (21)

For conical mode shape, (21) gives:

Vmax= 2V0 =2I

ρh1 (1+ λµ)(22)

The unknown time variation of the central amplitude of theplateT (t) can be determined by the principle of virtual velocityor equivalently by the Galerkin method. Let us define a residualfunctionR :

R =(ρ1h1 + ρ2h2) rw− 2√3

σ0h1(

rw′)′− 12

Esh2

(

r(

w′)3)′

(23)After the substitution of (18) into (23), the Galerkin methodgives:Z R

0R Φ(r) rdr = 0 =⇒ T +

(

αc2

y

R2

)

T +

(

βc2

y

R4

)

T3 ≡ 0 (24)

where the parametersα, β andcy are defined by:

α =a

1+ λµ; β =

aλ1+ λµ

; c2y =

σy

ρ1(25)

in which

a =10√

3

σσy

; b =52

Es

σy; (26)


The nonlinear ordinary differential equation (24) can besolved by treating the displacementT as an independent variableinstead of timet. The solution satisfying the initial conditions is

T =cy

R

√

4

(

Icyρh1(1+ λµ)

)2

−α(

TR

)2

− β2

(

TR

)4

(27)

The velocityT is seen to be a diminishing function of the dis-placement amplitudeT. The maximum displacement corre-sponding to vanishing of the velocity is denoted byTf and isgiven by:

(Tf )composite= R

√

√

√

√

√

1β

√

α2 +8β(

Icyρh1(1+ λµ)

)2

−α

(28)In the limiting case of steel plate alone (h2 = 0, Es = 0, λ = 0,µ= 0), Eqn. (28) reduced to:

(Tf )steel=2RI√acyρh1

(29)

The gain of using the polyurea coating can be accessed by study-ing the ratio:

(Tf )composite

(Tf )steel

=

√acyρh1

2I

√

√

√

√

√

1β

√

α2 +8β(

Icyρh1 (1+ λµ)

)2

−α

(30)The plot of the above function versus the thickness ratio forallother parameter corresponding to DH-36 steel shown in Fig. 6.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

λ

(Tf) co

mpo

site

/(T

f) stee

l

With σ0/σ

y=1.5, E

s/σ

y=0.005, µ=0.109, I/(R c

y ρ

1)=0.0114, h

1/R=0.01

Figure 6. Ratio of the maximum displacement for the composite and the

steel only plate.

Integrating Eq. (27) by separation of variables, the responsetime corresponding to the maximum displacement becomes:

t f =

Z Tf /R

0

dT√

4(V0)2−αc2

y (T/R)2−βc2y (T/R)4/2

(31)

This integral can be easy evaluated numerically for a given set ofparameters. A closed form approximation of the above equationwas developed in the form:

(t f )approximate=π

2√

a

√

1+ λµe−0.04b (32)

Comparison between the exact and the approximate solution isgiven in Fig. 7 showing a good agreement. That completes thederivation of the closed solution of the current problem. Notethat, the parameterσ0 has not been yet specified.


Figure 7. Comparison between the exact and the approximate solution

of t f with respect to: (a) σ0/σy and (b) λµ.

First Iteration According to Eq. (1), the stress is a nonlin-ear function of strain. In the first iteration, it is assumed that theflow stress is constant and equal toσ0 =

√σyσu, whereσy andσu denote respectively the yield and the ultimate stresses. Notethat the parametersa andα depends linearly on the value ofσ0,and no other parameter depends on the level of yield stress.

Second Iteration Now, the magnitude of elevated but con-stant flow stress will be calculated from Eq. (2), knowing thespatial and temporal variation of strain from the first iteration.

The average strain is defined by:

εave=1Tf

Z Tf

0

[

1πR2

Z R

0ε 2πrdr

]

dT (33)

where the temporal averaging is performed usingT as a time-like parameter. From Eq. (13), the radial strain can be seen to beindependent ofr, and proportional to square of the displacementamplitude:

ε =12

(

TR

)2

(34)

Inserting Eq. (34) into (33), the integration could be easily per-formed to give:

εave=16

(

Tf

R

)2

(35)

Using the above solution for the first and the second itera-tions, simple calculations gave the values of various quantities ofinterest valid for DH-36 steel (see Table 3). A fast convergenceof the iterative method was achieved because of a right choice ofthe stress level in the first iteratioin.

1st iteration 2nd iteration 3rd iteration

σ0 [MPa] 517 435 438

Tf [m] 0.143 0.156 0.156

t f [msec] 3.67 4.00 4.00

εave [%] 0.342 0.407 0.403

Table 3. Iteration procedure under the following conditions,

I = 5kPa·sec, R= 1m, h1 = 0.01m, h2 = 0.1m.

Critical Impulse to FractureFracture of a thin plate occurs inside of a localized neck

which is in the state of so-called transverse plane strain, as ex-plained in Fig. 8.


Figure 8. Assumed strain path at the center of flat tensile specimen up to

the point of fracture initiation. A plane strain tension (αc = 0) deformation

is assumed after the onset of localized necking [16].

Inside the neck, the stress triaxialityσm/σ is constant andequal to 1/

√3. Therefore, in order to describe fracture initia-

tion in thin plates, it is sufficient to use one fracture parameter,transverse plane strain to fractureε f rather than the entire depen-dence of material ductility on stress triaxiality. The critical setof parameters for fracture initiation are determined from:

εmax = ε f (36)

The maximum radial strain corresponds to the maximum valueof the central defection and is given by:

εmax =12

(

Tf

R

)2

(37)

Substituting the present solution forTf into Eqns. (36)-(37),the following expression is obtained for the critical impulse tofracture:

(Ic)composite=ρ1cyh1

2√

2(1+ λµ)

√

1β

[

(2βε f + α)2−α2]

(38)

In the case of steel plate alone (h2 = 0,Es = 0, λ = 0, µ= 0), Eq.(38) becomes:

(Ic)steel= ρ1cyh1

√

aε f

2(39)

The gain of using the polyurea coating can be accessed by study-ing the ratio of the critical impulse to fracture of the compositeplate to that of the steel plate:

(Ic)composite

(Ic)steel=

12

√

{

1β

[(2βε f (1+ λµ)+a)]2−a2

}

(40)

The plot of the function(Ic)composite/(Ic)steel versus thicknessratio λ is shown in Fig. 9.

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

λ

(Ic) co

mpo

site

/(I c) st

eel

With σ0/σ

y=1.5, E

s/σ

y=0.005, µ=0.109, ε

f=0.3

Figure 9. Ratio of the critical impulse to fracture for the composite and

the steel only plate

Numerical Solution for a Composite Plate and Compar-ison

In order to check the applicability of the analytical solutionfor a composite plate, a numerical example is considered withthe overall dimensions of the steel plate identical to the casesalready studied in Sections and . Four values of the ratio ofpolyurea to steel thickness were considered,λ = h2/h1 = 0, 2, 5,and 10. The material for both steel and polyurea layer have beenfully characterized in Section . The calculation were run us-ing ABAQUS/Explicit with axisymmetric square solid elementof the size 5× 5 mm. There were 200 elements throuhg theplate radius and two element through the thcikness of the steelplate, which was held constant in all calculations. In the nu-merical calculations, the ideal impulse loading was replaced by apulse loading of an approximately rectangular shape with a steepramping. The duration of the pressure pulse was constant andequalt toτ1 = 0.09msec. The rumping time wasτ2 = 0.01msec.


The magnitude of the impulse was adjusted by varing the pres-sure amplitudep. Therefore, the total impulse impared to thestructure is equal to

I = (τ1 + τ2) p (41)

For example, pressure ofp = 50 MPa correspondes to the totalimpulseI = 5 kPa·sec. All the calculation presented below wererun by assuming impulse equal to 25% of the vaule.

The case withh2/h1 = 0 corresponds to the steel alone case,which was taken as a baseline solution. The transient deformedshape of the steel alone plate for four different times is shown inFig. 10.

0 0.2 0.4 0.6 0.8 1−0.25

−0.2

−0.15

−0.1

−0.05

0

Nondimensional Radius

Difl

ectio

n A

mpl

itude

[m]

t=0 msect=0.5 msect=1.0 msect=1.5 msect=2.0 msect=2.5 msect=3.0 msect=3.5 msec

Figure 10. Transient and final deflection shapes of the steel only plate

The maximum permanent displacement of the steel platewas calculated to be 0.24 m which compares favorably with theanalytical results of 0.22 m. A typical history of transient andpermanet deflection shape of the composite plate withλ = 10shown in Figure. 11.

Figure 11. Three stages of deformed shape of the composite plate with

h2/h1 = 10. The lowest shape corresponds to the permanent deforma-

tion.

From the above calcualtions, it is seen that the final shape ofthe plate is indeed close to the conical shape which was assumedin the analytical solution. Both simulations shown in Figs.10-11 provided an evidence of a wave character of the solution withtraveling region of slope discontinuity. Such a behavior waswell captured in the solution for a homogeneous plate presentedin Ref. [8]. The governig equation of present composite plateis nonlinear and there is not an easy way to develp a close formwave type solution. Therefore the mode solution is the onlyoption. This approach predicts correctly the final shape of theplate and the magnitude of permanent central deflection. Timehistories of the deflection amplitude for four values of thicknessratio are compared in Fig. 12.

0 2 4 6 8

x 10−3

−0.25

−0.2

−0.15

−0.1

−0.05

0

Time [sec]

Cen

tral

Def

lect

ion

[m]

Steel Only PlateComposite plate with λ=2Composite plate with λ=5Composite plate with λ=10

Figure 12. Time history of deflection amplitude for increasing values of

thickness ratio λ


The effect of thickness of polyurea coating on the final cen-tral plate amplitude follows the trend predicted by the analyticalsolutions (see Figure 13).

0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

Thickness ratio, λ

Per

man

ent D

ispl

acem

ent [

m]

Analytical SolutionNumerical Simulation

Figure 13. Comparison of analytical and numerical permanent central

deflection of a composite plate as a function of the thickness ratio

Discussion and ConclusionIt transpires from the present analytical and numerical study

that the polyurea coating can reduce the maximum deflection ofexplosively loaded plate and thereby increase the criticalimpulseto fracture by some 20 %. However, this would require puttingthe thickness of elastomer 5 to 10 times larger than that of thebacking steel plate. In the present formulation, the effectonstrain rate on the constitutive behavior of polyurea was notcon-sidered. However, it can be shown through simple calculationand also through numerical simulation that the average strain ratein the present explosive loading is of an order of 100sec−1. Thestrain rate would increase inversely proportional with theradiusof the plate. In either case, the gain in the strength of polyureawould be not more than of a factor of 2 to 3. The effect of strainrate on the response on the composite plate will be reported in aseparate publication [14].

The detailed ABAQUS simulation revealed the existence ofshort flexural type of wave on the free surface of the polyurea.Those wave are generated by the discontinuity in the velocityfield at the clamped edge of plate. The amplitude of the wave isincreasing with a intensitity of the pressure loading, but it wouldappear that those waves do not effect much the overall responseof the composite plate and its failure pattern.

The conical shape of the plate in the analytical solution witha constant slop suggested that fracture could be initiated any-

where in the plate. From numerical solutions, one can see thatthe maximum slope and thereby the maximum radial strain isattained in the central region of aroundr/R = 0.15 (see Figure14). Such a local concentrain of strain will lead to the diskingfailure followed by petaling which actually was observed inthetests reported by Nemat-Nasser [9]. It should be noted that theboundary condition of the above tests were different from thepresent ones, and therefore no direct comparison could be made.

Figure 14. Comparison of final deflected shape of composite plate with

different thickness ratios showing maximum slope in the central region of

the plate

ACKNOWLEDGMENTThe authors would like to thank their colleague, Dr. X.

Teng, for many helpful discussion on this topic. This work wassupported by ONR MURI project subcontracted to MIT throughHarvard University.

REFERENCES[1] S.R. Bodner and P.S. Symonds.Experiements on vis-

coplstic response of circular plates to impulsive loading.Technical Report N00014-0860/6 of Brown University un-der Grant NSF 74-21258 and Contract N00014-75-C-0860,1977.

[2] P.S. Symonds and T. Wierzbicki.Membrane mode solutionfor impulsively loaded circular plates. Journal of AppliedMechanics, 46 (1): 58-64, 1979.

[3] Gelman M.E. Nurick, G.N. and N.S. Marshall.Tearing ofblast loaded plates with clamped boundary conditions. In-ternational Journal of Impact Engineering, 18 (7-8): 803-827, 1996.

[4] S.C.K. Yuen and G.N. Nurick.Experimental and numericalstudies on the response of quadrangular stiffened plates.Part I: Subjected to uniform blast load. International Jour-nal of Impact Engineering, 31 (1): 55-83, 2005.


[5] Nurick G.N. Cloete, T.J. and R.N. Palmer.TThe defor-mation and shear failure of peripherally clamped centrallysupported blast loaded circular plates. International Jour-nal of Impact Engineering, 32 (1-4): 92-117, 2005.

[6] T. Wierzbicki and M.S. Hoo Fatt.Impact response of astring-on-plastic foundation. International Journal of Im-pact Engineering, 12 (1): 21-36, 1992.

[7] T. Wierzbicki and G.N. Nurick.Large deformation of thinplates under localised impulsive loading. InternationalJournal of Impact Engineering, 18 (7-8): 899-918, 1996.

[8] M. Mihailescu-Suliciu and T. Wierzbicki.Wave solutionfor an impulsively loaded rigid-plastic circular membrane.Archanives of Mechanics, 54 (5-6): 553-576, 2002.

[9] S. Nemat-Nasser.Polyurea project summary of USCD’scontributions: Experiments, modeling and simulations. Apresentation in Airlie Center, Warrenton, VA., 2005.

[10] S. Nemat-Nasser and W.G. Guo.Thermomechanical re-sponse of DH-36 structural steel over a wide range ofstrain-rates and temperatures. Mechanics of Materials, 35:1023-1047, 2003.

[11] S. Nemat-Nasser. Summary of CEAM’s Experimentaland Analytical Contributions for Modeling Properties ofPolyurea. A document for ONR, University of California,San Diego, CA, 2005.

[12] E.M. Arruda and M.C. Boyce.A three-dimensional consti-tutive model for the large stretch behavior of rubber elasticmaterials. Journal of Mechanics and Physics of Solids, 41(2):389-412, 1993.

[13] L. Anand. A constitutive model for compressible elas-tomeric solids. Computational Mechancis, 18 (5): 339-355,1996.

[14] Wierzbicki T. Shim, J. and X. Teng.Effect of through-thickness wave propagation and strain rate on defor-mation and failure of an impulsively-loaded compositesteel/polyurea plates. ICL Report No. 149, MassachusettsInstitute of Technology, Cambridge, MA., 2006.

[15] J.B. Martin and P.S. Symonds.Mode approximations forimpulsively loaded rigid-plastic structures. Proceedings ofthe ASCE, 92: 43-66, 1966.

[16] Y.W. Lee.Fracture prediction in metal sheets. Ph.D. deser-tation, Massachusetts Institute of Technology,, Cambridge,MA., 2005.


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