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A VISCOPLASTIC MODEL OF EXPANDING CYLINDRICAL SH!&LS SUBJECT TO IiVTERNAL EXPLOSIVE DETONATIONS

Rick Martineau, Chuck Anderson, Fred Smith

Int'l Ccmputagional Engineering Science Atlanta, GA October 1998

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A VISCOPLASTIC MODEL OF EXPANDING CYLINDRICAL SHELLS SUBJECTED TO INTERNAL EXPLOSIVE DETONATIONS

Rick L. Martineau, Los Alamos National Laboratory Charles A. Anderson, Los Alamos National Laboratory

Los Alamos, NM 87545, USA

Fred W. Smith, Professor Colorado State University

Fort Collins, CO 80523, USA

Summary

Thin cylindrical shells subjected to internal explosive detonations expand outwardly at strain-rates on the order 10 s . At approximately 150% strain, multiple plastic instabilities appear on the surface of these shells in a quasi-periodic pattern. These instabilities continue to develop into bands of localized shear and eventually form cracks that progress in a way that causes the shell to break into fragments. The entire process takes less than 100 microseconds from detonation to complete fragmentation. Modeling this high strain-rate expansion and generation of instabilities prior to fragmentation is the primary focus of this paper.

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Introduction and Background

The elastic/viscoplastic behavior of materials is an area of study that encompasses a variety of scientific disciplines with industrial and military applications. In either application, the response of structures subjected to rapidly changing loads is considerably different from those under quasi-static conditions. Applications for this research include hypervelocity accelerators, flux compression generators, and explosive containment vessels for terrorist threats and power plants.

When a thin-walled circular cylinder is subjected to an internal explosion, the walls of the cylinder expand radially outward. For ductile materials this radial expansion occurs at very high velocities prior to failure by fragmentation. In 1943, Gurney [I] derived a widely used model for predicting the terminal velocities of fragments from shells subjected to internal explosive detonations. The failure of cylindrical structures was first examined in 1944 as a fragmentation problem by G. I. Taylor [2]. Taylor proposed that initially the hoop stress throughout thickness of the shell is compressive. Eventually longitudinal cracks form on the outer surface where the hoop stress first becomes tensile, but will not penetrate to the inner surface until the hoop stress at the inner surface is also tensile.

About the same time, Mott [3], attempted to predict the distribution and size of fragments from a tubular structure based on the assumptions of a perfectly plastic material model and probability theory. Mott observed two types of fractures; shear fractures and cup and cone type fractures. In 1967, Slate and others [4] concluded that the thicker the shell, the more evident the surface cracking and in addition, the higher the strain to rupture. Hoggatt and Recht [5] furthered the study of fragmenting cylinders and developed a mathematical model assuming the fractures occur along lines of maximum shear. In addition, Hoggatt and Recht observed different types of fractures based on the amount of HE and the detonation pressures. At low detonation pressures, deep cracks formed on the outer surface before unstable shear zones began to develop. At high detonation pressures, the compressive hoop stress from the detonation retards the growth of cracks and the unstable shear zones form earlier and as a result, larger shear zones are observed on the fragments. Al-Hassani and others [6] provided an analytical expression for the hoop and

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radial stress in the vessel wall assuming a perfectly plastic material and confirmed the behavior observed by Taylor. Later Al-Hassani and Johnson [7] concluded that the strain-rate, strain hardening, and deformation induced temperature are important to the yield behavior and that they influence both the fracture radius and velocity, although they did not include them in their analysis.

In 1978, Taylor and others [SI developed a formulation for modeling dynamic plastic instabilities in a thin sheet. Their approach is based on hydrodynamic principles. They introduced a thickness perturbation in thin sheets and demonstrated that the size and appearance of the instabilities are dependent on the material's strain-rate and work hardening. In 1983, Johnson [9] examined the one-dimensional ductile failure of rapidly expanding rings. Johnson describes the time dependent heterogeneous plastic deformation in terms of the differential equations of thermoplasticity, conservation of mass, and conservation of momentum. Johnson's model uses a small perturbation in the wall thickness or porosity to create the instability and examines the influence of work hardening and thermal softening. Finally, in 1985 Anderson, Predebon, and Karpp [ 101 developed a two-dimensional finite difference code to model expanding cylinders. They felt typical hydrodynamic codes over-predicted the fragment velocities and attempted to obtain a more complete solution. Their model includes gas leakage and assumes an elastic- perfectly-plastic material but does not consider quasi-periodic instabilities

In the early experimental literature, the material used in experiments was not carefully characterized or documented and as a result, little is known about the grain size and hardness of the materials making it difficult to duplicate experimental results. Several authors have examined instabilities associated with both uniaxial and bi-axial stress under quasi-static conditions but most have not considered materials subjected to multiaxial stress states at strain rates on the order of 10 s . In some cases, researchers have assumed a perfectly-plastic material or neglected the strength of the material, which could be significant for predicting the expansion and failure of rapidly expanding shells. In addition, very few authors included the high pressure hydrodynamic effects and those that have, examined situations with strain rates much higher than lo4 sec'' or provided formulations that are limited to one-dimensional calculations. Considering the results of the literature, it is clear that strain-rate, material density, temperature, and inertial effects are important in large strain plastic deformation and development of instabilities when materials are subjected to large strain-rates. These effects are included in the numerical model described in the next section.

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Numerical Model

The copper cylinder considered in this study is filled with high explosives and is detonated on one end with a plane wave type of detonation. The conversion of chemical energy in the explosive causes the cylinders to expand at a high rate of strain. A viscoplastic constitutive model, which includes the high pressure shock effects, the rate-dependent strength effects, and the microdefects in the form of microvoids was formulated to model the high strain-rate expansion and provide insight into the development of thermoplastic instabilities. As the material strains, thermal energy is deposited in the material as a result of shock loading and plastic work. This thermal energy has an effect on the yield surface and resultant stress state of the material. The current state of stress and equivalent plastic strain is then used to determine the rate of microvoid nucleation, growth, and coalescence.

The ABAQUS [ 1 11 code was selected to carry out the calculations because it easily accommodates a user written constitutive subroutine through what it calls a VUMAT subroutine. The ABAQUS code also includes a high explosive burn model. The high explosive burn model uses the Jones-Wilkens-Lee (JWL) equation of state to model the pressures generated by the release of chemical energy in the explosive. This model is implemented in the form of a program burn that essentially detonates the high explosive in the

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cylinder based on a detonation wave speed and the geometric distance of high explosive from the detonation point. The conversion of the chemical energy in the high explosive to dynamic pressures causes the cylinder to expand and provides an increment in strain. This increment in strain is then passed into the VUMAT subroutine.

Initially, the main VUMAT subroutine is called. Assuming the EOS subroutine is activated, the first calculation the VUMAT subroutine performs is to calculate the modified bulk modulus and the material temperature as a result of the shock loading. The formulation for this part of the constitutive model follows the work of Johnson [9] and uses the material’s equation of state and current volumetric strain.

These results are then returned to the main VUMAT subroutine and the GTN subroutine is called to determine the new stress state of the material. Once inside, the GTN subroutine immediately calculates the current strain-rate and passes it along with current plastic strain and material temperature to the Johnson- Cook subroutine. The Johnson-Cook subroutine returns the allowable yield stress to the GTN subroutine. The yield surface implemented in this research is a function of the allowable yield stress, the void volume fraction, the equivalent stress, and the pressure and is a combination of the work by Gurson [12] and Tvergaard [13]. Next the code iterates to converge to a point on the yield surface. The iteration technique adopted for this dissertation is called the cutting-plane algorithm and is described by Ortiz and Popov [14]. This algorithm is based on linearization of the plastic consistency condition for the current iteration and satisfaction of the plastic consistency for the new iteration. The cutting plane algorithm is very efficient and demonstrates reasonable accuracy. At this point, the stresses, energies, and temperatures are updated and passed back to the main subroutine.

Next the main VUMAT subroutine calls the void subroutine. The void subroutine determines the nucleation, growth, and coalescence rate for the microvoids and passes back the new volumetric void concentration in the material. Finally, at the user’s discretion, the code enters the element remove subroutine. This subroutine eliminates elements from the computation based on user specified parameters like volumetric void concentrations or equivalent plastic strain. In practice, this subroutine only affects the first couple of elements and does not adversely effect the global solution. A flow chart of the VUMAT subroutine, which is composed of approximately 2300 lines of Fortran source code, is shown in Figure 3. The circled numbers in Figure 3 indicate the calling order for each subroutine.

The mechanical properties used in the constitutive model were taken from the current literature and were not adjusted for this work. A detailed listing of the VUMAT subroutine and the mechanical properties used in this analysis can be found in Martineau [ 151.

ABAQUS Explicit

-) =OS a Subroutine

Subroutine Subroutine @

Subroutine

Remove Subroutine

GTN @ .......... JOhnson-Cook ............................ Main

Subroutine -b VUMAT

I ..............

Figure 1: Flow Chart of VUMAT Subroutine

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Model Verification

An experiment was designed to benchmark the numerical model and to add credence to the formulation of the constitutive model. The experiment used a 406.4 mm long oxygen-free-electronic grade copper tube filled with a high explosive. The copper had a 102.06 mm inner diameter and a wall thickness of 2.54 mm. The copper tube was annealed after the machining process, which resulted in a Rockwell F hardness of 23 and a grain size of approximately 40 pm. A solid circular cylinder PBX-9501 with a diameter of 101.8 mm was used as the high explosive inside of the copper tube. A plane wave lens was bonded to the end of each PBX-9501 cylinder and detonated at its apex with a SE-1 detonator. This caused the walls of the copper cylinder to expand radially outward in a conical form resulting in a hoop strain variation in the longitudinal direction.

Two forms of diagnostics were used in the experiments, fast framing camera and Fabry-Perot. Fabry-Perot is a type of laser or visar interferometer, which is capable of recording velocity information for a single point on the surface of the expanding shell. Ideally, the fast framing camera diagnostic provides 25 photographic images of the expanding cylinder at predetermined increments in time. The hoop strain and the development of the instabilities observed on the surface of the expanding shell can then be extracted from the photographs and plotted as a function of time. In addition, a futile effort was made to ‘soft-catch’ the fragments from the expanding shell.

The numerical analysis consisted of an axisymmetric model of the solid cylinder of HE and the copper shell. The numerical model did not include the plane wave lens or the SE-1 detonator. In the numerical model, four node, linear, axisymmetric elements were used throughout the analysis for both the HE and copper cylinder and five elements were maintained through the thickness of the copper cylinder shells. The size ratio of the HE elements to the copper cylinder elements was maintained at 2:l with a uniform aspect ration of nearly 1: 1. Therefore, one HE element loaded two cylinder elements. This resulted in a total of 25,550 elements. The HE and the copper were separated by a 0.127 mm gap. A sliding contact surface was prescribed at this gap to model the interactions between the HE and the cylinder. Comparison plots of the results from the numerical model and experimental data are shown in Figure 2.

Deformed Geometry at lime t=20.3 mcroseconds

400 360 320 280 240 200 160 120 83

Axial Position From Btmated End (mm) 40 0

Deformed Geometryat Time t49.65 microseconds

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-Numerical Results II

-.- ExDerimental Data 120

100

I - -

80 g 0)

8 60 5

n 40 3

-

B 20

0 c 400 360 320 280 240 200 160 120 80 40 0

Axial Position From Detonated End (mm)

Figure 2: Deformed Geometry for 2.54 mm Thick Cylinder at 20.3 and 49.65 microseconds

The outward radial displacement of the copper shell is plotted with respect to time in Figure 3. The displacement curves shown in this figure are taken at eight different longitudinal locations on the cylinder

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wall. The longitudinal location is shown in the text on the right side of the figures. In this figure, the plots near the longitudinal center of the cylinder illustrate good agreement between the numerical results and the experimental data. However, the plots near the ends of the cylinder show a small discrepancy. This is the

rezoning of the cylinder mesh.

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I same discrepancy observed in Figure 2 and may be reduced with mesh optimization, reduced viscosity, and

I I Numerical Results

120 1 ................ - Experimental Data

1 0 0

80

60

4 0

20

0

. 5 0 . 8

.--= 1

0 1 0 20 30 40 50

T i m e (m i c r o s e c o n d s )

Figure 3: Radial Displacement as a Function of Time at Eight Locations Along the Longitudinal Axis of the 2.54 mm Thick Cylinder

The experiments documented as part of this work were designed to verify the numerical model and good agreement is shown in Figures 2 and 3. The comparisons shown here illustrate that the user supplied constitutive parameters for the numerical model are reasonable. Good agreement is shown for large displacements thus indicating that the distortion of the elements in the numerical model is not affecting the solution. Each photograph from the fast framing camera was digitized and enlarged to determine the number of instabilities for a small characteristic length on the surface of the cylinder. This length and the quantity of instabilities were then used to determine the total number of instabilities on the entire circumference of the shell.

Instabilities were observed on the surface of the expanding shell in the experiment.

The instabilities observed on the surface of the expanding cylinder appear to occur and propagate down the longitudinal axis of the cylinder. The axisymmetric geometry is not capable of predicting a circumferential pattern of quasi-periodic instabilities, which occurs in the longitudinal orientation. As a result, a plane strain model consisting of 20 elements through the thickness of the shell was constructed to examine the development of the instabilities. In this model a dynamic pressure loading was applied uniformly to the entire inner surface. The formation of an oscillatory pattern of equivalent plastic strain was observed in the theta direction as early as 7.6 microseconds. This pattern became more pronounced with time and by 34 microseconds, localized zones of intense plastic strain connected the inner and outer surfaces of the expanding shell as shown in Figure 4. These zones of plastic strain are considered to be related to the instabilities observed in the photographs obtained from the fast framing camera. The number of predicted and observed instabilities for each cylinder is given below in Table 1.

Figure 4: Numerical Prediction of Quasi-Periodic Instabilities

Table 1: Number of Instabilities as Determined from the Fast Framing Camera Photographs

The simulations shown here illustrate the onset and development of plastic instabilities that are associated with rapidly expanding shells. The onset of a periodic pattern is observed as early as 7.6 microseconds and localized areas of intense plastic strain, which extend from the inner to the outer surfaces, are observed later at 34 microseconds. The total number of instabilities predicted by the numerical model is fairly close to the number of instabilities observed from the experimental data. The experimentally verified constitutive model developed here provides a useful tool for further analytical work on the expansion of explosively expanding shells.

In conclusion, the numerical simulations presented here are in good agreement with the experimental data. The experimentally verified constitutive model developed in this dissertation provides a useful tool for further analytical work on the expansion of explosively expanding shells. In addition, the comparisons presented provide insight into the development and growth of the quasi-periodic instabilities observed on the surfaces of the expanding shells.

References

1. Report 405. 2. 44., Cambridge University Press., pp. 387-390. 3. 4. High Strain Rates,” J. Inst. Metah, Vol. 95, pp. 244-25 1. 5.

6. Spherical Shells Containing Explosives,” Int. J. Mech. Sci., Vol. 11, pp. 81 1-823.

Gurney, R. W., 1943, ‘‘The Initial Velocities of Fragments from Bombs, Shells, and Grenades,” BRL

Taylor, G. I., 1963, “Fragmentation of Tubular Bombs,” Scientific Papers of G. I. Taylor, Vol. 3, No.

Mott, N. R. 1947, “Fragmentation of Shell Cases,” Proc. R. SOC. London, Vol. 189, pp. 300-308. Slate, P. M. B, Billings, M. J. W., and Fuller, P. J. A., 1967, “The Rupture Behavior of Metals at

Hoggatt, C. R. and Recht, R. F., 1968, “Fracture Behavior of Tubular Bombs,” J. Appl. Phys.,

Al-Hassani, S. T. S. and Johnson, W., 1969a, “The Dynamics of the Fragmentation Process for Vol. 39, pp. 1856-1862.

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4 . .’ 7. Al-Hassani, S. T. S., Hopkins, H. G., Johnson, W., 1969b, “Note on the Fragmentation of Tubular Bombs,” Znt. J. Mech. Sci., Vol. 11, pp. 545-549. 8. Taylor, J. W., Harlow, F. H., Amsden, A. A., 1978, “Dynamic Plastic Instabilities in Stretching Plates and Shells,” J. Appl. Mech., Vol. 45, pp. 105-1 10. 9. Johnson, J. N., 1983, “Ductile Fracture of Rapidly Expanding Rings,” J. Appl. Mech., Vol. 50,

10. Anderson, C. E., Predebon, W. W., and Karpp, R. R., 1985, “Computational Modeling of Explosive- Filled Cylinders,” Znt. J. Eng. Sci., Vol. 23, pp. 1317-1330. 11. ABAQUS, 1997, Version 5.6, Hibbit, Karlsson, and Sorenson Inc. 12. Gurson, A. L., 1977, “Continuum Theory of Ductile Rupture by Void Nucleation and Growth: Part I- Yield Criteria and Flow Rules for Porous Ductile Materials,” J. Eng. Muter. & Technol., Vol. 99, pp. 2-15. 13. Tvergarrd, V., 1981, “Influence of Voids on Shear Band Instabilities under Plane Strain Condition, ” Znt. J. Fruct. Mech., Vol. 17, pp. 389-407. 14. Ortiz, M. and Popov, E. P., 1985, “Accuracy and Stability of Integration Algorithms for Elastioplastic Constitutive Equations,” Int. J. Numer. Methods Eng., Vol. 21, pp. 1561-1576. 15. Martineau, R. L., 1998, “A Viscoplastic Model of Expanding Cylindrical Shells Subjected to Internal Explosive Detonations,” PHD Dissertation, Colorado State University.

pp. 593-600.

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