Failure of an Impulsively-loaded Composite Steel/polymer Plate

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 100 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 20 % provided the thickness of the coating is 5−10 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 © 2006 by ASME 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 60 to 70 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 [10]. 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 20 % provide the thickness of the coating is 5−10 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. 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 term: