FAILURE OF AN IMPULSIVELY-LOADED COMPOSITE STEEL/POLYMER PLATE. JongMin Shim and Tomasz Wierzbicki

Transcription

1 Proceedings of IMECE ASME International Mechanical Engineering Congress and Exposition November 5-1, 26, Chicago, IL USA IMECE FAILURE OF AN IMPULSIVELY-LOADED COMPOSITE STEEL/POLYMER PLATE JongMin Shim and Tomasz Wierzbicki Impact and Crashworthiness Laboratory Massachusetts Institute of Technology Cambridge, MA2139 address ABSTRACT The concept of spraying thick layer of polymer material onto metal plate has recently received considerable interest in many civilian and military applications. There are numerous analytical and numerical solutions for single thin plates membrane) made of either a steel or an elastomer. However, solutions for composite plate made of both of the above constituents are lacking. The objective of the present paper is to formulate a model for composite steel/elastomer plate, derive an analytical solution of the impulsive loading problem and compare it with a more exact numerical solution. It is assumed that the circular plate is fully clamped around its peripheral and it is loaded by uniformly distributed transverse pressure of high intensity and short duration. The pressure is imparting initial impulse which is proportional to initial transverse velocity of the plate. As an example, DH-36 is used for steel backing plate while polyurea is chosen to represent a typical polymer coating. In the analytical model, an iterative method is developed in which steel layer treated as a rigid perfectly-plastic material with magnitude of flow stress adjusted according to calculated magnitude of average strain. A linear elastic material is assumed with elastic modulus in the tensile range calculated from the Arruda-Boyce model for an specific type of polyurea. It was found that the magnitude of the average strain rate is relatively low, about 1 sec 1. Therefore, the effect of strain rate is not considered in this paper. A comprehensive parametric study was performed by varying 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 that the polyurea coating could improve the failure resistance of the composite plate by some 2 % provided the thickness of the coating is 5 1 times larger than the plate. INTRODUCTION Response of a thin metal plate subjected to explosive loading was the subject of numerous experimental, theoretical and numerical studies. Bodner and Symonds [1] reported on an experimental study where thin clamped steel membrane were subjected to an impulsive loading distributed uniformly over the central portion of the structure. Symonds and Wierzbicki [2] developed a closed form solution of this problem using the mode approximation technique and included the effect of the strain hardening and strain rate sensitivity in an iterative way. Good correlation was observed between theory and experiments regarding central deflection of the plate. More recently, very comprehensive series of tests on explosively loaded plates all the way to fracture was conducted in the University of Cape Town by Nurick and his team [3] [4] [5]. It was observed that for a sufficient large impulse, failure occurs either at the clamp edge of the plate through a combination of shear and tension or in the central part in the so-called petaling mode. It was found that the mode solution is able to predict the onset of fracture of plate with uniformly distributed impulse. Attempt to extend this type of approximate method to a plate loaded by more concentrated pressure loading will not successful. Wierzbicki and Hoo Fatt [6], Wierzbicki and Nurick [7], 1 Copyright c 26 by ASME

2 and Mihailescu-Suliciu and Wierzbicki. [8] succeed in deriving closed form solution for this class of problem using the method of wave propagation. Nemat-Nasser [9] reported on experimental and numerical study of small plates made of DH-36 steel without or with polyurea backing subjected to uniformly distributed initial velocity. The ratio of polyurea thickness to steel thickness was approximately 5 1 and the initial impact velocity was in the range of 6 to 7 m/sec. The tested steel plats were fully clamped whereas the layer of polyurea was simply placed in front of or behind with no connection to the clamping ring. It was found that the elastomer coating reduced the central deflection of the plate for impact velocity below the critical value. For a larger initial velocities causing failure of the system typically through the petaling mode) the polyura enhanced or mitigated the extent of failure of system. The same team also performed the numerical simulation of the test which qualitatively agreed with the experimental results. The objective of the present paper to derive a closed form solution for steel/polyurea combination and perform a thorough parametric study to understand the effect of thickness of the polyurea coating on the response and failure of this complex structure. In order to make the problem mathematically tractable, the membrane plate theory is assumed with both steel and polyurea constituents firmly clamped around peripheral. The steel plate is assumed to be made of DH-36 steel and the Johnson-Cook model for this material is calibrated from experiments provide by Nemat-Nasser and Guo [1]. The polyurea coating is described by Arruda-Boyce model which was again calibrated from compressive test performed by Nemat-Nasser [11]. Numerical simulation of this problem is generated by using axisymmetric solid element in ABAQUS/Explicit. Good correlation is obtained form numerical and analytical solution. It was found that the polyurea coating could improve the failure resistance of the composite plate by some 2 % provide the thickness of the coating is 5 1 times larger than the plate. Calibration of Material Model In this section, the constitutive equations of both constituents of the composite plate are defined. The steel model is described by Johnson-Cook model which has been successfully used in many practical applications. Several general constitutive models of elastomers are available in the literature such as Mooney-Rivlin, Ogden, Arruda-Boyce, etc. For the purpose of the present study, Arruda-Boyce model is chosen. term: σ = A+B ε pl) n where σ is the equivalent stress, ε pl is the equivalent plastic strain, A is the yield stress σ y, B and n are the material parameters relating to strain hardening hardening. Equation 1) assumes the isotropic hardening of the material. The analytical solution, to be developed in the next section, is based on the concept of the rigid perfectly-plastic approximation defined by a constant stress with respect to the strain. This constant stress is adjusted to represent in the best possible way the actual hardening properties of the material, according to: 1) ) n σ = σ ε ave pl = A+B ε ave) pl 2) where ε pl ave is the spatial and temporal average value for the equivalent plastic strain. Nemat-Nasser and Guo [1] performed comprehensive study of a constutive behavior of DH-36 steel under static and dynamic loading at different temperatures. In the present study, the temperature effect is not considered and only roomtemperature data were taken as basis for calibration. From the logarithmic representation of the quasi-static stress-strain data ε =.1 sec 1 ), three coefficients A, B and n) in Eq. 1) were determined. The numerical values of all parameters are gathered in Table 1, and they are used for the finite element simulations of DH-36 steel. Comparison between J-C model and experimental data is shown in Fig. 1. Material Model for DH-36 Steel The constitutive model for a steel is assumed to follow a simplified Johnson-Cook [J-C] equation, without the strain rate Table 1. A B n 392 MPa 55 MPa.45 J-C model parameters calibrated for DH-36 steel. 2 Copyright c 26 by ASME

3 True stress [Pa] 1 x Figure 1. Test:.1 sec 1 J C Model:.1 sec True plastic strain Comparison of the true stress versus true plastic strain curve from uniaxial compression test at room temperature [1] and the corresponding J-C model. Material Model for Polyurea The polyurea, as all elastomer materials, behaves differently under compression and tension. The testing of polyurea is typically done under compressive load. At the same time, the material in thin plate undergoing large deformation is subjected to predominantly tensile loads. Therefore, there is a need to extrapolate the compressive data points into the tensile regime. Several models of hyperelastic material have been developed in the literature. For the purpose of the present study, the well-known Arruda-Boyce [A-B] hyperelastic model is chosen [12] [13]. The strain energy potential U in this formulation for an isothermal condition is: { 1 U = µ 2 Ī 1 3)+ 1 Ī2 2λ ) + 11 Ī3 m 15λ ) 3) m + 19 Ī ) 519 ) + Ī5 } ) + 1 D 7λ 6 m J ) lnj 67375λ 8 m where µ, λ m, and D are temperature-dependent material parameters; Ī 1 is the first deviatoric strain invariant; and J is the total volume ratio. The unknown material constants for A-B model µ, λ m, D) can be determined from a number of different tests such as uniaxial, biaxial or planar test combined with volumetric test data. The advantage of working with ABAQUS is that it 5) automatically extrapolates the data into the tensile region from the given compression data, and it provides numerical values for three material model parameters. Test data of polyurea under uniaxial compression and volumetric compression i.e. confined uniaxial compression) were reported by Nemat-Nasser and Guo [11] at different strain rates. The quasi-static test data ε =.1 sec 1 ) is shown in Figs. 2 and 3. The material parameters of the A-B hyperelastic model determined on the basis of above data are summarized in Table 2. Using the calculated parameters, the A-B model predicts the uniaxial stress strain data in the tensile region. The complete stress-strain curve both in compression and tension quadrant is shown in Figs. 2-3 for three different values of strain rate. Engineering Stress [Pa] Table Figure 2. 2 x 17 µ Pa λ m D m 2 /N µ MPa K 2.76 MPa ν.4986 A-B model parameters calibrated for polyurea Test:.1 sec 1 A B Model:.1 sec Engineering Strain Comparison of the engineering stress-strain curve from uniaxial compression tests at room temperature [11] and the corresponding A-B model. 3 Copyright c 26 by ASME

4 1 x x 16 A B Model:.1 sec 1 Elastic Model:.1 sec Pressure [Pa] Engineering Stress [Pa] Test:.1 sec 1 A B Model:.1 sec Volumetric Ratio Engineering Strain Figure 3. Comparison of the pressure versus volumetric ratio V V)/V curve from uniaxial compression tests at room temperature [11] and the corresponding A-B model. Figure 4. Comparison of the engineering stress-strain curve form A-B model and the simplified elastic model, Eqn. 6). It is seen that in the tension quadrant the stress-strain curve can be approximated by straight line defined by a secant modulus E s =.75 E where E is Young s modulus at the zero strain. σ e = E s ε e 6) where σ e and ε e are, respectively, the engineering stress and strain; E s is the secant Young s modulus in the reference quasistatic test. It should be noted that in the problem of the large deformation of composite plate the stress state is essentially uniaxial tension. Therefore, Eqn. 6) is thought to be a convient and good approximation to the actual behavior of polyurea. From the best fit of the tensile region of Fig. 2, one gets E s =.75E = 2.69 MPa, which is a good approximation of the actual A-B constitutive model see Fig. 4). In the general case of finite element simulations, the polyurea response should be describe by a nonlinear hyperelastic model with viscoelastic effect and hysteresis loop. The present simplified model captures the material characteristic in the active loading process without unloading. Such an approach is valid only for the polyurea plate working in combination with the steel backing plate. The irreversible character of plastic deformation in the steel plate will prohibit the polyurea layer to undergo elastic unloading. In this sense, both components of composite plate would undergo monotonic loading without unloading. Problem Formulation Consider a circular composite plate composed of two layers, one representing polyurea coating and the other one steel backing plate see Fig. 5). The analysis will be performed with any combination of geometry and material properties of both plates. A transverse impulse of the intensity I is applied to the upper layer. Within the assumption of the present theory where through-thickiness deformation is not considered, it does not matter which layer polyurea or steel plate) is on the top. Figure 5. Geometry of composite plate. 4 Copyright c 26 by ASME

5 The steel plate is characterized by three parameters: plate thickness h 1, mass density ρ 1, and average flow stress σ. The flow stress σ depends in turn on the magnitude of the average strain see Section ). The polyurea layer follows the constitutive equation 6) with the secant Young s modulus E s and is characterized by its thickness h 2 and mass density ρ 2. It is assumed that polyurea is on the top, and subscript 2 will refer to all relevant parameters. Likewise, the parameter describing the steel plate will be distinguished by subscript 1. The following ratios are introduced: λ = h 2 h 1 7) µ = ρ 2 ρ 1 8) The response of the composite plate consists of two distinct phases. In the first phase, the high pressure acting on a unit surface area is generating plain stress wave in the top layer. A part of the wave is transmitted to the steel plate while the other part is reflected and is interacting with incoming incident waves. A complicated pattern of the wave interaction exists in this transient stage, and eventually a common velocity of the composite plate is attained. In the second phase of the response, the pressure term vanishes and the composite plate continues to deform due to the uniform initial velocity acquired at the end of the first phase. A detailed numerical analysis of this first phase was performed to see what part of energy remains as a transverse vibration of elastomer and what part goes into the initial kinetic energy of the plate [14]. The conclusion from this numerical analysis was that the initial velocity of the plate V ) could be effectively calculated from the momentum conservation of a unit surface element: h 1 ρ 1 + h 2 ρ 2 )V = τ pt)dt = I 9) where I is the total impulse imparted to the structure, p is the applied pressure and τ is the loading duration. In the second phase of the response, the governing equation of the membrane response of the circular plate is: mrẅ = N rr rw ) 1) where w is the transverse displacement, r is the radial coordinate, m is the mass per unit area, N rr is the membrane force per unit length and dot ) and prime ) denote respectively differentiation with respect to time and radial coordinate. In this equation, the pressure term is set up to zero. The expression for m and N rr should now be specified for both the steel and the polyurea layers. As discussed earlier, the rigid perfectly-plastic material idealization is assumed for steel. The radial stress in the steel plate is constant and equal to the plane strain yield stress for the von-mises yield condition: N rr ) steel = 2 3 σ h 1 11) The value of the constant flow stress of a material is not specified at this point. Expression for σ will be different for the first and the second iterative solutions, as explained in what follows. From Eq. 6), the membrane force for polyurea layer is: N rr ) polyurea = σ e h 2 = E s ε rr h 2 12) In the present approximate solution, it is assumed that the hoop strain is zero. In the theory of moderate large deflection of thin plate, the radial strain is ε rr = 1 2 w ) 2 13) Now, the governing equations for the steel and the polyurea plate are given respectively by: 2 ρ 1 h 1 )rẅ 1 3 σ h 1 rw 1) = 14) 1 ρ 2 h 2 )rẅ 2 2 E sh 2 r w ) ) 3 = 15) where w 1 is the transverse deflection of the middle suffice of the steel. It is assumed that both steel and polyurea plates deform with no separation which means that no delamination or spalling would occurs at the interface. Then, the condition of displacement continuity at the interface would apply: w 1 = w 2 = wr,t) 16) By adding Eq. 14) and 15) with Eq. 16), a single governing equation is obtained for the response of the composite plate: ρ 1 h 1 + ρ 2 h 2 )rẅ 2 3 σ h 1 rw ) 1 2 E sh 2 r w ) 3 ) = 17) 5 Copyright c 26 by ASME

6 This is a nonlinear partial differential equation in r and t. For the second order spatial derivative, two boundary conditions are needed. The first boundary condition is vanishing of the displacement the outer perimeter, wr,t) =, where R is the plate radius. In the theory of elastic plates, the slope should also vanish at the clamped edge. However, in the theory of plastic plates, the slope discontinuity is allowed at the clamped edge. In the numerical simulation, where axisymmetric solid element are used, the above contradiction is absent because only the displacement boundary conditions have to be prescribed. One way to overcome the above contradiction for the present plate/membrane problem would be to consider a class of shape function that satisfies the condition, w R,t) =. The initial conditions in the present initial-boundary value problem are wr,) = ẇr,) = V Note that there is an initial discontinuity in the velocity field at r = R, and as a result there will be slope discontinuity propagating from the clamped edge toward center. Closed Form Analytical Solution In the limiting case of h 2 =, Eq. 17) reduces to a linear wave equation. The wave type solution which admits the slope discontinuity was derived by Mihailescu-Suliciu and Wierzbicki [8]. With an addition of polyurea layer, the governing equation becomes nonlinear, and the Riemann wave solution is no longer possible. Instead, a mode type solution will be developed as an extension of the method previously used by Symonds and Wierzbicki [2] for thin metal plates. When the separation of variables is used to solve this partial differential equation, the deflection of the plate over time and space can be written: wr,t) = T t)φr) 18) where T t) is the time variable amplitude, and Φr) is the normalized deformed shape. Note that all the formulations are derived in the cylindrical coordinate system. In this study, the deformed shape is assumed to have the following conical shape: Φr) = 1 R) r m 19) solution [8], it appears that the plate assumes the conical shape with m = 1. The conical shape violates the slope boundary condition at the support. Any exponent slightly greater than one, for example, m = 1.1, will satisfy the slope boundary conditions. In the present approximate solution, the calculations will be run for m = 1. The mode-form solution satisfy the first initial condition, but not the second initial condition. Martin and Symonds [15] developed a kinetic energy difference minimization technique so that the mode-form solution will satisfy in the best possible way the actual solution of the initial-boundary value problem. Initial velocity in the mode solution is: ẇr,) = Ṫ )Φr) = V max Φr) 2) where the unknown amplitude V max can be determined by using the the minimization of the kinetic energy difference: K E ) V max = V max { m 2 R For conical mode shape, 21) gives: V max = 2V = } [V V max Φr)] 2 rdr 21) 2I ρh 1 1+λµ) 22) The unknown time variation of the central amplitude of the plate T t) can be determined by the principle of virtual velocity or equivalently by the Galerkin method. Let us define a residual function R : R =ρ 1 h 1 + ρ 2 h 2 )rẅ 2 σ h 1 rw ) E sh 2 r w ) ) 3 23) After the substitution of 18) into 23), the Galerkin method gives: R R Φr)rdr = = T + α c2 y R 2 ) T + β c2 y R 4 where the parameters α, β and c y are defined by: α = a 1+λµ ; β = aλ 1+λµ ; ) T 3 24) c2 y = σ y ρ 1 25) where m is a fractional exponent that should be chosen to best represent the actual shape of a deformed plate. From the numerical solution, the experimental observation [3] and the wave type in which a = 1 σ ; b = 5 3 σ y 2 E s σ y ; 26) 6 Copyright c 26 by ASME

7 The nonlinear ordinary differential equation 24) can be solved by treating the displacement T as an independent variable instead of time t. The solution satisfying the initial conditions is With σ /σ y =1.5, E s /σ y =.5, µ=.19, I/R c y ρ 1 )=.114, h 1 /R=.1 Ṫ = c y R 4 I c y ρh 1 1+λµ) ) 2 T α R ) 2 β ) T 4 27) 2 R T f ) composite /T f ) steel The velocity Ṫ is seen to be a diminishing function of the displacement amplitude T. The maximum displacement corresponding to vanishing of the velocity is denoted by T f and is given by: Figure λ steel only plate. Ratio of the maximum displacement for the composite and the T f ) composite = R 1 ) α β 2 I 2 + 8β α c y ρh 1 1+λµ) 28) In the limiting case of steel plate alone h 2 =, E s =, λ =, µ= ), Eqn. 28) reduced to: Integrating Eq. 27) by separation of variables, the response time corresponding to the maximum displacement becomes: T f ) steel = 2RI acy ρh 1 29) Tf /R t f = dt 4V ) 2 αc 2 y T/R) 2 βc 2 y T/R) 4 /2 31) The gain of using the polyurea coating can be accessed by studying the ratio: This integral can be easy evaluated numerically for a given set of parameters. A closed form approximation of the above equation was developed in the form: T f ) composite T f ) steel = acy ρh 1 1 α 2I β 2 + 8β I c y ρh 1 1+λµ) ) 2 α 3) The plot of the above function versus the thickness ratio for all other parameter corresponding to DH-36 steel shown in Fig. 6. t f ) approximate = π 2 1+λµe.4b a 32) Comparison between the exact and the approximate solution is given in Fig. 7 showing a good agreement. That completes the derivation of the closed solution of the current problem. Note that, the parameter σ has not been yet specified. 7 Copyright c 26 by ASME

8 The average strain is defined by: ε ave = 1 Tf [ 1 R ] T f πr 2 ε 2πrdr dt 33) where the temporal averaging is performed using T as a timelike parameter. From Eq. 13), the radial strain can be seen to be independent of r, and proportional to square of the displacement amplitude: ε = 1 2 ) T 2 34) R Inserting Eq. 34) into 33), the integration could be easily performed to give: ε ave = 1 6 ) 2 Tf 35) R Using the above solution for the first and the second iterations, simple calculations gave the values of various quantities of interest valid for DH-36 steel see Table 3). A fast convergence of the iterative method was achieved because of a right choice of the stress level in the first iteratioin. Figure 7. Comparison between the exact and the approximate solution of t f with respect to: a) σ /σ y and b) λµ. First Iteration According to Eq. 1), the stress is a nonlinear function of strain. In the first iteration, it is assumed that the flow stress is constant and equal to σ = σ y σ u, where σ y and σ u denote respectively the yield and the ultimate stresses. Note that the parameters a and α depends linearly on the value of σ, and no other parameter depends on the level of yield stress. Second Iteration Now, the magnitude of elevated but constant flow stress will be calculated from Eq. 2), knowing the spatial and temporal variation of strain from the first iteration. 1 st iteration 2 nd iteration 3 rd iteration σ [MPa] T f [m] t f [msec] ε ave [%] Table 3. Iteration procedure under the following conditions, I = 5kPa sec, R = 1m, h 1 =.1m, h 2 =.1m. Critical Impulse to Fracture Fracture of a thin plate occurs inside of a localized neck which is in the state of so-called transverse plane strain, as explained in Fig Copyright c 26 by ASME

9 The gain of using the polyurea coating can be accessed by studying the ratio of the critical impulse to fracture of the composite plate to that of the steel plate: { I c ) composite = 1 } 1 I c ) steel 2 β [2βε f 1+λµ)+ a)] 2 a 2 4) The plot of the function I c ) composite /I c ) steel versus thickness ratio λ is shown in Fig With σ /σ y =1.5, E s /σ y =.5, µ=.19, ε f =.3 Figure 8. Assumed strain path at the center of flat tensile specimen up to the point of fracture initiation. A plane strain tension α c = ) deformation is assumed after the onset of localized necking [16]. Inside the neck, the stress triaxiality σ m / σ is constant and equal to 1/ 3. Therefore, in order to describe fracture initiation in thin plates, it is sufficient to use one fracture parameter, transverse plane strain to fracture ε f rather than the entire dependence of material ductility on stress triaxiality. The critical set of parameters for fracture initiation are determined from: ε max = ε f 36) The maximum radial strain corresponds to the maximum value of the central defection and is given by: ε max = 1 2 ) 2 Tf 37) R Substituting the present solution for T f into Eqns. 36)-37), the following expression is obtained for the critical impulse to fracture: I c ) composite = ρ 1c y h λµ) 1 ] [2βε f + α) 2 α β 2 38) In the case of steel plate alone h 2 =, E s =, λ =, µ= ), Eq. 38) becomes: I c ) steel = ρ 1 c y h 1 aε f 2 39) I c ) composite /I c ) steel Figure λ the steel only plate Ratio of the critical impulse to fracture for the composite and Numerical Solution for a Composite Plate and Comparison In order to check the applicability of the analytical solution for a composite plate, a numerical example is considered with the overall dimensions of the steel plate identical to the cases already studied in Sections and. Four values of the ratio of polyurea to steel thickness were considered, λ = h 2 /h 1 =, 2, 5, and 1. The material for both steel and polyurea layer have been fully characterized in Section. The calculation were run using ABAQUS/Explicit with axisymmetric square solid element of the size 5 5 mm. There were 2 elements throuhg the plate radius and two element through the thcikness of the steel plate, which was held constant in all calculations. In the numerical calculations, the ideal impulse loading was replaced by a pulse loading of an approximately rectangular shape with a steep ramping. The duration of the pressure pulse was constant and equalt to τ 1 =.9 msec. The rumping time was τ 2 =.1 msec. 9 Copyright c 26 by ASME

10 The magnitude of the impulse was adjusted by varing the pressure amplitude p. Therefore, the total impulse impared to the structure is equal to I = τ 1 + τ 2 ) p 41) For example, pressure of p = 5 MPa correspondes to the total impulse I = 5 kpa sec. All the calculation presented below were run by assuming impulse equal to 25% of the vaule. The case with h 2 /h 1 = corresponds to the steel alone case, which was taken as a baseline solution. The transient deformed shape of the steel alone plate for four different times is shown in Fig. 1. Diflection Amplitude [m] t= msec t=.5 msec t=1. msec t=1.5 msec.2 t=2. msec t=2.5 msec t=3. msec t=3.5 msec Nondimensional Radius Figure 11. Three stages of deformed shape of the composite plate with h 2 /h 1 = 1. The lowest shape corresponds to the permanent deformation. From the above calcualtions, it is seen that the final shape of the plate is indeed close to the conical shape which was assumed in the analytical solution. Both simulations shown in Figs provided an evidence of a wave character of the solution with traveling region of slope discontinuity. Such a behavior was well captured in the solution for a homogeneous plate presented in Ref. [8]. The governig equation of present composite plate is nonlinear and there is not an easy way to develp a close form wave type solution. Therefore the mode solution is the only option. This approach predicts correctly the final shape of the plate and the magnitude of permanent central deflection. Time histories of the deflection amplitude for four values of thickness ratio are compared in Fig Steel Only Plate Composite plate with λ=2 Composite plate with λ=5 Composite plate with λ=1 Figure 1. Transient and final deflection shapes of the steel only plate Central Deflection [m] The maximum permanent displacement of the steel plate was calculated to be.24 m which compares favorably with the analytical results of.22 m. A typical history of transient and permanet deflection shape of the composite plate with λ = 1 shown in Figure Time [sec] 6 8 x 1 3 Figure 12. Time history of deflection amplitude for increasing values of thickness ratio λ 1 Copyright c 26 by ASME

11 The effect of thickness of polyurea coating on the final central plate amplitude follows the trend predicted by the analytical solutions see Figure 13)..3 Discussion and Conclusion It transpires from the present analytical and numerical study that the polyurea coating can reduce the maximum deflection of explosively loaded plate and thereby increase the critical impulse to fracture by some 2 %. However, this would require putting the thickness of elastomer 5 to 1 times larger than that of the backing steel plate. In the present formulation, the effect on strain rate on the constitutive behavior of polyurea was not considered. However, it can be shown through simple calculation and also through numerical simulation that the average strain rate in the present explosive loading is of an order of 1 sec 1. The strain rate would increase inversely proportional with the radius of the plate. In either case, the gain in the strength of polyurea would be not more than of a factor of 2 to 3. The effect of strain rate on the response on the composite plate will be reported in a separate publication [14]. The detailed ABAQUS simulation revealed the existence of short flexural type of wave on the free surface of the polyurea. Those wave are generated by the discontinuity in the velocity field at the clamped edge of plate. The amplitude of the wave is increasing with a intensitity of the pressure loading, but it would appear that those waves do not effect much the overall response of the composite plate and its failure pattern. The conical shape of the plate in the analytical solution with a constant slop suggested that fracture could be initiated anywhere in the plate. From numerical solutions, one can see that the maximum slope and thereby the maximum radial strain is attained in the central region of around r/r =.15 see Figure 14). Such a local concentrain of strain will lead to the disking failure followed by petaling which actually was observed in the tests reported by Nemat-Nasser [9]. It should be noted that the boundary condition of the above tests were different from the present ones, and therefore no direct comparison could be made..25 Permanent Displacement [m] Analytical Solution Numerical Simulation Thickness ratio, λ Figure 13. Comparison of analytical and numerical permanent central deflection of a composite plate as a function of the thickness ratio Figure 14. Comparison of final deflected shape of composite plate with different thickness ratios showing maximum slope in the central region of the plate ACKNOWLEDGMENT The authors would like to thank their colleague, Dr. X. Teng, for many helpful discussion on this topic. This work was supported by ONR MURI project subcontracted to MIT through Harvard University. REFERENCES [1] S.R. Bodner and P.S. Symonds. Experiements on viscoplstic response of circular plates to impulsive loading. Technical Report N14-86/6 of Brown University under Grant NSF and Contract N14-75-C-86, [2] P.S. Symonds and T. Wierzbicki. Membrane mode solution for impulsively loaded circular plates. Journal of Applied Mechanics, 46 1): 58-64, [3] Gelman M.E. Nurick, G.N. and N.S. Marshall. Tearing of blast loaded plates with clamped boundary conditions. International Journal of Impact Engineering, ): , [4] S.C.K. Yuen and G.N. Nurick. Experimental and numerical studies on the response of quadrangular stiffened plates. Part I: Subjected to uniform blast load. International Journal of Impact Engineering, 31 1): 55-83, Copyright c 26 by ASME

12 [5] Nurick G.N. Cloete, T.J. and R.N. Palmer. TThe deformation and shear failure of peripherally clamped centrally supported blast loaded circular plates. International Journal of Impact Engineering, ): , 25. [6] T. Wierzbicki and M.S. Hoo Fatt. Impact response of a string-on-plastic foundation. International Journal of Impact Engineering, 12 1): 21-36, [7] T. Wierzbicki and G.N. Nurick. Large deformation of thin plates under localised impulsive loading. International Journal of Impact Engineering, ): , [8] M. Mihailescu-Suliciu and T. Wierzbicki. Wave solution for an impulsively loaded rigid-plastic circular membrane. Archanives of Mechanics, ): , 22. [9] S. Nemat-Nasser. Polyurea project summary of USCD s contributions: Experiments, modeling and simulations. A presentation in Airlie Center, Warrenton, VA., 25. [1] S. Nemat-Nasser and W.G. Guo. Thermomechanical response of DH-36 structural steel over a wide range of strain-rates and temperatures. Mechanics of Materials, 35: , 23. [11] S. Nemat-Nasser. Summary of CEAM s Experimental and Analytical Contributions for Modeling Properties of Polyurea. A document for ONR, University of California, San Diego, CA, 25. [12] E.M. Arruda and M.C. Boyce. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of Mechanics and Physics of Solids, 41 2): , [13] L. Anand. A constitutive model for compressible elastomeric solids. Computational Mechancis, 18 5): , [14] Wierzbicki T. Shim, J. and X. Teng. Effect of throughthickness wave propagation and strain rate on deformation and failure of an impulsively-loaded composite steel/polyurea plates. ICL Report No. 149, Massachusetts Institute of Technology, Cambridge, MA., 26. [15] J.B. Martin and P.S. Symonds. Mode approximations for impulsively loaded rigid-plastic structures. Proceedings of the ASCE, 92: 43-66, [16] Y.W. Lee. Fracture prediction in metal sheets. Ph.D. desertation, Massachusetts Institute of Technology,, Cambridge, MA., Copyright c 26 by ASME