Chung Gong, Tsu-Te Wu, and Allen C. Smith
Westinghouse Savannah River Company
Aiken, South Carolina 29808

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

This report has been reproduced directly from the best available copy.

Available for sale to the public, in paper, from:  U.S. Department of Commerce, National Technical Information Service, 5285 Port Royal Road, Springfield, VA 22161,  phone: (800) 553-6847,  fax: (703) 605-6900,  email:  orders@ntis.fedworld.gov   online ordering:  http://www.ntis.gov/support/ordering.htm

Available electronically at  http://www.osti.gov/bridge/

Available for a processing fee to U.S. Department of Energy and its contractors, in paper, from: U.S. Department of Energy, Office of Scientific and Technical Information, P.O. Box 62, Oak Ridge, TN 37831-0062,  phone: (865 ) 576-8401,  fax: (865) 576-5728,  email:  reports@adonis.osti.gov

The purpose of this analysis of the CelotexÔ implanted with glue plates is two-fold, first is to identify the cause of the initial stress peak in the pseudo engineering stress-strain curve in the dynamic impact test that the impact is loaded in the orientation parallel to the plane of the glue. Secondly, from the existing static mechanical properties to derive the true constitutive properties of the CelotexÔ under dynamic impact and other environmental conditions, such as warm (250°F), wet (100% relative humidity), cold (-40°F), and desiccated.

From a series of experimental data, a mathematical model for crushable materials is established for the CelotexÔ . The computed dynamic impact results are closely comparable with the test data.

A series of parametric analyses of the influence from the glue plates implanted in the CelotexÔ indicate that the relative strength of the glue plates effectively controls the amplitude and period of the initial stress peak in the pseudo engineering stress-strain curve.

The mechanical properties of CelotexÔ at room temperature and at normal humidity conditions are usually used for the structural analyses of shipping packages. In the report "CelotexÔ Structural Properties Tests" [Smith, Vormelker, 2000], indicates that the mechanical properties of CelotexÔ are function of:

• Temperature
• Humidity
• Desiccation
• Implanted Glue plates
• Dynamic Impact
• Orientation of load application

In the compression tests, a weight of 580-pound is dropped upon a 4-inch cube of CelotexÔ from a height of 60 inches.

The mechanical behavior of the CelotexÔ implanted with glue plates varies drastically as the orientation of the applied dynamic load changes. The pseudo stress-strain curve has an initial stress peak when the load is applied in the direction parallel the plane of the glue plates. Whereas the dynamic load is applied in the direction normal to the glue plates, the pseudo stress-strain curve is similar to the stress-strain curves of crushable materials. The stress and strain are not computed from the deformation of the element in the CelotexÔ cube, rather, the stress is derived from the deceleration of the dropping weight and the strain is computed by dividing the displacement of the dropping weight with the undeformed height, ‘4-inch’, of the CelotexÔ cube. A slight increase of the strength of the CelotexÔ probably is due to dynamic effects.

The purpose of this analysis is to model and simulate the dynamic mechanical behavior of the CelotexÔ with glue plates that join the half-inch thick CelotexÔ boards, such that the physics of the test results can be understood.

Since the pseudo stress-strain curve for the normal (to the glue plates) loaded CelotexÔ is similar to the majority of the test results [Smith, Vormelker, 2000], this paper will discuss only the case with dynamic load parallel to the glue plates in the CelotexÔ .

A finite element analysis technique is adopted as a numerical computation method for this parametric study. The software, ABAQUS / Explicit version 5.8-1N [HKS, 1998] is used for the numerical calculation.

The finite element mesh of the steel weight and the glue implanted CelotexÔ cube is developed by using the graphic software, MSC / PATRAN, version 8.5 [MSC, 1999].

The dynamic impact weight is a large steel block that weighs 580 pounds. As compared with the CelotexÔ and the glue plates, the strength of the steel block is almost like a rigid body. The steel block is modeled with a plate of dimensions . The mass density of the plate is configured to match the mass of the steel block.

The CelotexÔ is modeled, in 3-D, exactly as the test specimen, a four-inch cube. The geometry and the applied dynamic loading imply plane symmetry in this kinematical system. However, in a highly nonlinear deformation a slight mechanical or material anomaly will obliterate the symmetry. In the numerical calculation, truncation errors from the millions of computer operations in the nonlinear finite element analysis will destroy the geometrical symmetry. Therefore, this model simulates the physical uncertainties with the numerical uncertainties. The physical world may have more randomness in material properties than that in the numerical calculation.

The dynamic impact load will be applied in the negative direction along the z-axis. The glue plates are parallel to the z-axis. The glue plates are modeled in thin shells that are parallel to the y-z-planes. The thickness of each of the plate is 1.0 millimeter (0.03937 inches). The glue plates are implanted inside the CelotexÔ cube at 0.5-inch along the direction parallel to the x-axis. There are seven glue plates.

The steel block drops from 60 inches on the top of the CelotexÔ cube implanted with glue plates. As the steel block touches the CelotexÔ cube, the initial velocity of the mass is:

The corresponding kinetic energy is 580*60 = 34,800 inch-pounds.

In the experiment setup, the bottom supporting foundation is a flat smooth surface. The steel impact block in contact with the CelotexÔ cube may have insignificant friction. Practically, the whole system has no exterior contact friction. Surrounding the four lateral surfaces of the CelotexÔ cube the conditions are traction free. The bottom of the CelotexÔ cube is prevented from vertical movement. Since the calculation is stopped at 0.02 seconds, rebound of the deformed CelotexÔ cube is unlikely. Even though the impact weight plate starts to rebound in the neighborhood of this moment. The pseudo stress strain curve derived from the motion of the weight plate implies the rebound of the plate at the second stress peak. Also the explanation of the rebounding of the CelotexÔ cube is beyond the scope of this analysis.

The primary materials in this analysis are the CelotexÔ and the glue. For the CelotexÔ , a few experimental data are available at room temperature and under normal conditions [Leader, November 1993, December 1993, Gong and Gupta, 1994, Walker, 1991]. The mechanical properties of the CelotexÔ with temperature dependence are not available. The strength of the CelotexÔ under dynamic impact is not well understood. The strain rate hardening in the CelotexÔ involves the collapse of the micro-structures in the fibers, the air flow in the cavities and the strain rate hardening of the cohesive filling materials. The function of this type of material behaviors is complicated. The strain rate dependence behavior may be a function of a polynomial of deformation rates of a wide range of orders. Without sufficient experimental data, the strain rate dependence behaviors of the CelotexÔ cannot be correctly formulated. In the drop test the deformation rate of the CelotexÔ varies during the history of the experiment.

Consequently, for the dynamic impact analysis, the physically reasonable assumption is that the strain rate dependence hardening of the CelotexÔ is linearly proportional to the strength of the material. The volumetric hardening strength of the CelotexÔ during dynamic impact test is about four times of the strength for the CelotexÔ subjected to very slow rate test [Smith, Vormelker, 2000]. The volumetric strain hardening curves are plotted in Figure 1.

The volumetric stress-strain curve indicates that the CelotexÔ is a crushable material. When the microstructures and air in the matrix of the CelotexÔ are gradually crushed and squeezed out, the strength of the compressed material will be steadily increased as the deformation progresses. Of course, the pre-conditioning of the CelotexÔ may significantly influence the mechanical behaviors.

The yield surface. The model uses a yield surface with an elliptical dependence of deviatoric stress on pressure stress. Yielding in the P -plane is assumed to depend on the third invariant of deviatoric stress. The evolution of the yield surface is controlled by the inelastic volume strain experienced by the material: compactive inelastic strains produce hardening while dilatant inelastic volume strains lead to softening.

The yield surface is defined as

M is computed from a uniaxial compression test as

The values of and K are derived from experimental data to define the shape of the yield surface. K=1, is used in this analysis.

Hardening. The yield surface intersects the p-axis at and . The position of is assumed to remain fixed throughout any plastic process. By contrast, the compressive strength, , evolves as a result of compaction (increase in density) or dilation (reduction in density) of material.

The piecewise linear relationship is entered in tabular form. The hardening law defines the value of the yield surface size on the hydrostatic stress axis, , as a function of volumetric compacting plastic strain, . The dynamic impact test curve, in Figure 1,is the hydrostatic pressure stress and volumetric plastic strain curve being used in this analysis.

In this analysis, the initial parameters of mechanical properties for CelotexÔ are derived as shown in the table:

The glue plates are implanted inside the CelotexÔ matrix. The displacement degrees of freedom of the glue plates are completely constrained by the CelotexÔ fibers. Whereas the rotational degrees of freedom of each of the glue plates are completely free. Since the glue plates are sufficiently constrained, the free rotational degrees of freedom will not pose mathematical singularity in the numerical calculation.

Even though the CelotexÔ fibers provide lateral supports to the glue plates, the initial volumetric pressure strength of the CelotexÔ is negligibly small. Particularly, at the first moment of vertical dynamic impact, the CelotexÔ fibers have little (if any) lateral deformation. Therefore, the glue plates essentially stand-alone by themselves until they buckle and bend over the CelotexÔ . As soon as the glue plates buckle, the glue plates comparatively (with the CelotexÔ ) have no strength at all. Consequently, the glue plates can be considered as pure elastic material in this parametric study. In other structural analysis, the glue plates may sustain tensile stresses, the true stress-strain relations should be considered.

The stiffness of the impact weight is virtually rigid as compared to the strength of the CelotexÔ and the glue plates. Using the mechanical properties of stainless steel type 304L for the impact weight is a proper choice.

Based on the weight density (0.283 lbf/in³) of the stainless steel, the volume of a 580-pound block will be 2,050 cubic inches or 1.2 cubic feet. For the purpose of this analysis, the block of the dynamic impact weight is modeled with a plate. The volume of the plate is only 12.5 cubic inches. To match the equivalent weight of real steel block, the weight density of the small plate should be increased by 164 times. As a result, the weight density of the small plate will be 46.4 lbf/in³, and accordingly, the mass density of the small plate in the model will be 0.12008282 lbf-sec²/in⁴

This is a transient dynamic analysis, wave propagation is an important factor. In order to maintain the same wave speed in the small plate as the speed in the real stainless steel, the modulus of elasticity of the material in the small plate is also proportional increased.

Since the fictitious material for the small plate in the model is so strong and it is virtually rigid, the plastic behavior of the small plate will not be discussed in this paper.

The mechanical properties of the equivalent weight, the small plate are listed in the table.

The CelotexÔ is a man-made material that consists of fragmented sugar cane peels and a glue. Since the CelotexÔ fibers are randomly distributed, within the CelotexÔ body, the mechanical properties are approximately constitutionally isotropic. The microstructures in the CelotexÔ are uniformly distributed. The collapse of the microstructures during dynamic impact may display stiffening of the material. With the combined effects of air-squeezing and glue (the glue in the CelotexÔ mix) viscosity the response of the CelotexÔ to the dynamic impact is gentle and smooth as the test results indicated [Smith, Vormelker, 2000].

The glue plates implanted in the CelotexÔ matrix are relatively stronger. The distance between a pair of plates is 0.5 inches that is considerable large as compared with the dimensions of the four-inch cube of the CelotexÔ . The CelotexÔ cube implanted with glue plates is essentially a structure that is composed of two types of structural members, the thick CelotexÔ plates and the thin glue plates. In an event of dynamic impact, the aggregate response of the composite cube divulges not only the mechanical behaviors of each individual structural component but also the interactions between the structural components.

The mechanical effects of the relative stiffness between the CelotexÔ matrix and the glue plates can be demonstrated by varying the strength of the glue plates in the dynamic impact analysis. With the mechanical properties of the CelotexÔ remaining constant, a series of calculations are performed for the modulus of elasticity of the glue plates varies from 5,000 psi to 500,000 psi.

The results indicate that as the strength of the glue plates is very weak (e.g., E=5,000 psi), the glue plates implanted in the CelotexÔ will deform with the CelotexÔ before the rebound of the dynamic impact loading plate. The deformation configuration of the CelotexÔ cube implanted with glue plates shows no vestige of the existence of the glue plates in the cube. As the modulus of elasticity of the glue plates gradually increases, the deformation patterns of the CelotexÔ cube exhibit the structural buckling activities of the glue plates.

With the strength of the glue plates being relatively stronger than the CelotexÔ , as the 580-pound weight drops upon the CelotexÔ cube, the same amount of small vertical strain will produce higher stress in the glue plates than that in the CelotexÔ . As the dynamic impact persists, the glue plates will structurally buckle one by one inside CelotexÔ matrix. When the glue plates start to buckle, the plates possess no mechanical strength at all. The CelotexÔ will carry the remaining kinetic energy of the dynamic impact alone.

During the dynamic impact test, the CelotexÔ cube is devastatingly deformed. It is difficult to measure the interfacial stress and strain at the interface of the CelotexÔ cube and the weight block. The pseudo stress and strain are measured in the experiments. Accordingly pseudo stress and strain are calculated in this analysis.

The procedure of data process and computation of the pseudo stress and strain are described as follows:

• Collect the time history of the vertical deceleration of all the finite element nodes at the bottom of the small weight plate in the model;
• Collect the time history of the vertical displacement of all the finite element nodes at the bottom of the small weight plate in the model;
• Compute the average vertical deceleration time history of the weight plate.
• Compute the average vertical displacement time history of the weight plate.
• Compute the average pseudo stress time history:
• Multiply the average vertical deceleration time history by the total mass of the 580-pound weight;
• Divide the resulting time history by the undeformed area of the contact surface of the CelotexÔ (i.e., 16 square inches).
• Compute the average pseudo strain time history:
• Divide the average vertical displacement time history by the undeformed height of the CelotexÔ cube (i.e., 4 inches).

The time histories of the pseudo stress and pseudo strain are cross-plotted to produce pseudo engineering stress-strain curve that can be compared with the experimental results.

The pseudo stress strain curves for the glue plates with modulus of elasticity = 5,000 psi to 500,000psi are depicted in Figure 2 to Figure 12.

Through close examination of these pseudo stress strain curves, the effects of the variation of the relative (to CelotexÔ ) mechanical strength of the glue plates to the mechanical behaviors of the CelotexÔ cube under dynamic impact are conspicuous. The obvious mechanical consequences are the variation of the magnitude and the strain width of the first stress peak in each of the stress-strain curves as the strength of the glue steadily increases. The fluctuation of the magnitude of the second stress peak has more mechanical implications. The magnitudes of the first and second stress peaks as well as the first stress trough are plotted against the modulus of elasticity of the glue plates in Figure 13. The corresponding magnitudes of strains at the first, second stress peaks and the first stress trough are depicted with respect to the modulus of elasticity of the glue plates in Figure 14.

The magnitude of the first peak of the pseudo engineering stress (Figure 13) increases steadily with the increase in strength of the glue plate implanted in the CelotexÔ under the dynamic impact load. As the impact load touches the top of the CelotexÔ cube with implanted glue plates, the whole cube starts to deflect vertically. When the glue plates are comparatively weak, the plates will deform with the CelotexÔ fibers. However, when the strength of the glue plates is appreciably stronger than the surrounding fibers of the CelotexÔ , a stress peak appears in the stress-strain curve.

With the same amount of vertical strain, a significant amount of stresses will be developed in the relatively stronger glue plates. The glue plates are constrained by the CelotexÔ fibers in lateral displacement. At small strain, there is little or no stresses developed in the CelotexÔ . The rotational degrees of freedom of the glue plates are not constrained by the CelotexÔ and the lateral confinements are practically insignificant. At the moment, when sufficient stresses are developed in the glue plates, the plates become structurally unstable. The detailed mode shape of the plate buckling varies in each plate. The mode shape of the buckling is a function of the location of the plate, in addition to the relative strength of the plate.

The buckling modes of the glue plates determine the amount of kinetic energy being transformed into elastic strain energy and the consecutive deformation of the CelotexÔ . The pattern of the deformation of the CelotexÔ will influence the energy dissipation in the mechanical system.

The broad fluctuation of the magnitude of the second stress peak (Figure 13) is obviously due to the variation of the order of the plate buckling mode shape as a function of the strength of the glue plates.

The magnitude of vertical pseudo strain that induces the buckling of the glue plates gradually decreases as the modulus of elasticity of the glue plates gets stronger (Figure 14). As soon as the glue plates start to buckle, the CelotexÔ fibers begin to interact with the plates. The initial vertical strains and the ensuing plates and the CelotexÔ fibers mechanical interactions determine the mode shapes of buckling of the plates and the consequent deformation of the whole system.

The time history of the three principal energies in the CelotexÔ cube implanted with glue plates will provide more information about the mechanical behaviors of the composite material. In Figures 15 and 16, the time history of the kinetic energy, the dissipated plastic energy and the elastic strain energy are plotted with respect to time for two cases. The initial time=0.0, is the moment when the dropping weight plate in contact with the CelotexÔ cube. The initial vertical velocity is 215.33 inches per second. The end time=0.02 seconds is the time the calculation stopped. At this moment all the compressive impact upon the CelotexÔ cube is completed and the impact weight has already rebounded and moved away from the top of the crushed cube.

The initial kinetic energy in the CelotexÔ cube implanted with glue plates is equal to the total potential energy for the 580-pound weight dropped from 60 inches. The initial kinetic energy in the system equals 580 * 60 = 34,800 inch-pounds. The CelotexÔ fibers are the only energy dissipation material in this model. The elastic strain energy exists only in the glue plates. The glue plates are modeled with elastic materials.

When the modulus of elasticity of the glue plates, E=5,000 psi, is very weak, the plates do not buckle. However, near the end of the calculation, the weight plate rebounds, and the strain energy in the glue plates suddenly releases. The sudden release of the elastic strain energy in the glue plates causes numerical instability. At the end of the calculation, the strain energy turned slightly negative. It is a numerical inaccuracy and it is physically untrue.

When the modulus of elasticity of the glue plates is increased to E=50,000 psi, the glue plates buckle at 0.0054 seconds with a magnitude of maximum elastic strain energy of 8985 inch-pounds. In Figure 15, it clearly indicates that as soon as the glue plates buckle, the elastic strain energy in the plates drops and in the meantime, a big jump in plastic energy dissipation in the CelotexÔ .

The magnitude of the peak elastic strain energy in the glue plates gradually trims down as the modulus of elasticity of the glue plates increases. The peak values fluctuate around 5,500 to 8,000 inch-pounds as a function of both glue plates’ strength and the associated buckling mode shapes.

The deformation configuration of the CelotexÔ cube implanted with glue plates during the dynamic impact can be found in Figures 17 and 18. Figure 17 shows the snapshot of the deformation CelotexÔ with the glue plates. Figures 18 shows the glue plates alone. The deformation plots are for the case in which the modulus of elasticity of the glue plates is set to 360,000 psi. The final deformation configuration is very similar to the test result.

1. Gong, C. and Gupta, N. K., 1994, Numerical Modeling of the Compression Test of a Celotex Cube, SRT-EMS-94-0006, Inter-Office Memorandum, January 17, 1994.
2. HKS, 1998, ABAQUS Theory manual, ABAQUS / Explicit User’s Manual, version 5.8-1, Hibbitt, Karlsson & Sorensen, Inc., 1080 Main Street, Pawtucket, RI 02860-4847, Telephone: 401-727-4200.
3. Leader, D. R., November 1993, Packing Material Compression Tests (U), SRT-MTS-93-3119, Inter-Office Memorandum, November 29, 1993.
4. Leader, D. R., December 1993, Packing Material Dimensional Recovery Tests (U), SRT-MTS-93-3129, Inter-Office Memorandum, December 22, 1993.
5. MSC, 1996, MSC / PATRAN, version 8.5, The MacNeal-Schwendler Corporation, 815 Colorado Boulevard, Los Angeles, Telephone: 213-258-9111.
6. Smith, A. C., Vormelker, P. R., 2000, Celotex Structural Properties Tests (U), WSRC-TR-2000-000444, Technical Report, SRTC, WSRC, December, 2000.
7. Walker, M. S., 1991, Packaging Materials Properties Data, Oak Ridge Report Y/EN-4120, Jan. 19