Fe

Finite Elements in Analysis and Design 36 (2000) 99}133

Penetration simulation for uncontained engine debris impact on fuselage-like panels using LS-DYNA Norman F. Knight Jr *, Navin Jaunky
, Robin E. Lawson, Damodar R. Ambur Veridian MRJ Engineering, Yorktown, VA 23693-2619, USA
Eagle Aeronautics, Inc., Newport News, VA 23606, USA Newport News Ship Building and Dry Dock Company, Newport News, VA 23607, USA NASA Langley Research Center, Hampton, VA 23681-0001, USA

Abstract Modeling and simulation requirements for uncontained engine debris impact on fuselage skins are proposed and assessed using the tied-nodes-with-failure modeling approach for penetration. A "nite element analysis is used to study the penetration of aluminum plates impacted by titanium impactors in order to simulate the e!ect of such uncontained engine debris impacts on aircraft fuselage-like skin panels. LS-DYNA is used in the simulations to model the impactor, test "xture frame and target barrier plate. The e!ects of mesh re"nement, contact modeling, and impactor initial velocity and orientation are studied using a con"guration for which limited test data are available for comparison.  2000 Elsevier Science B.V. All rights reserved. Keywords: Impact; Penetration; LS-DYNA; Finite elements

1. Introduction Prediction of the elasto-plastic, large-deformation, transient dynamic behavior involving impact of multiple deformable bodies continues to provide new insights into the response of complex structural systems subjected to extreme loading conditions or exposed to extreme environments (e.g., Refs. [1}3]). Much of the computational mechanics technology necessary for simulating this behavior evolved over decades of research sponsored in part by government laboratories which also have had access to large supercomputer facilities. The rapid development of a!ordable computer technology with high-speed processors, large memories, and large, fast secondary storage devices has contributed to the integration of these analysis tools within design and analysis groups * Corresponding author. E-mail address: [email protected] (N.F. Knight Jr.) 0168-874X/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X ( 0 0 ) 0 0 0 1 1 - 1

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in industry. This technology transfer has provided methods and software that can be used to improve designs, reduce uncertainties, and increase product safety. One such application involves simulating the response of a fuselage skin when impacted by uncontained aircraft turbine-engine debris. Developing accurate "nite element models and analysis strategies for this event has the potential of signi"cantly improving the design, reliability, and safety of engines and primary aircraft structures, especially for commercial transport applications. Two potential hazards involving the turbine-engine debris are the subject of ongoing research e!orts. One event involves containing failed engine debris within the engine housing } contained failure. Examples of research in this area include Refs. [4}13]. The other event involves the potential impact of uncontained failed engine debris on other parts of the aircraft } uncontained failure. Examples of research in this area include Refs. [11}17]. The potential hazard resulting from an uncontained turbine engine failure has been a long-term concern of the Federal Aviation Administration (FAA), National Aeronautical and Space Administration (NASA), and the aircraft industry (e.g., see Refs. [4}17]). For the purpose of airplane evaluations, the FAA de"nes an uncontained failure of a turbine engine as any failure which results in the escape of rotor fragments from the engine or Auxiliary Power Unit (APU) that could result in a hazard (see Ref. [7]). A contained failure is one where no fragments are released through the engine structure; however, fragments may be ejected out the engine air inlet or exhaust. Rotor failures which are of concern are those where released fragments have su$cient energy to create a hazard to the airplane and its passengers. Failed rotating components can release high-energy fragments which are capable of penetrating the engine cowling and damaging the fuel tank, hydraulic lines, auxiliary power units, and other accessories [11]. The penetration capability of the material released is a!ected by its shape, orientation of impact, material properties, and kinetic energy. The high-energy fragments are dispersed circumferentially in all directions at very high velocities. When the fragments escape or penetrate the engine casing, the consequences can range from minor damage to catastrophic failures. These fragments, released during engine failure, a!ect the #ying performance of the aircraft in a number of direct or indirect ways, since they can impact and damage surrounding structures and equipment. Behaving as projectiles, these fragments have damaged surrounding runways, residences and vehicles [18,19]. Uncontained failure of engine rotating components in turbine engine aircraft is considered a serious safety hazard to occupants as well as to the aircraft itself. On August 22, 1985, a Boeing 737 operated by British Airtours, su!ered an uncontained failure in the left engine about 36 s into take-o! [20]. As the airplane approached an airspeed of 125 knots, a fragment punctured a wing fuel tank access panel. Fuel leaking from the wing ignited and burnt as a large plume of "re trailing directly behind the engine. As the aircraft turned o!, wind carried the "re onto and around the fuselage. Subsequently "re developed within the cabin. Despite the prompt attendance of the airport "re service, the aircraft was destroyed and "fty-"ve persons on-board lost their lives. The cause of the accident was attributed to an uncontained failure of the left engine, initiated by a failure of the No. 9 combustor can which had been the subject of a repair. A section of the combustor can, which was ejected forcibly from the engine, struck and fractured an underwing fuel tank access panel [20]. On June 8, 1995, as ValuJet Flight 597 began its take-o! roll, shrapnel from the right engine penetrated the fuselage and the right engine main fuel line, and a cabin "re erupted [21]. A #ight

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attendant received serious puncture wounds from shrapnel and thermal injuries; another #ight attendant and "ve passengers received minor injuries. An investigation revealed that an uncontained rotor failure in the right engine had occurred due to fatigue of its 7th stage high-compressor disc. According to National Transportation Safety Board (NTSB) "ndings, the maintenance and inspection personnel failed to perform a proper inspection of the disc, thus allowing the detectable crack to grow to a length at which the disc ruptured under normal operating conditions, propelling engine fragments into the fuselage; the fragments severed the right engine main fuel line, which resulted in a "re that rapidly engulfed the cabin area [21]. On July 6, 1996 in Pensacola, FL, Delta Airlines Flight 1288 experienced an uncontained failure of the left engine during the beginning take-o! roll about 1400 ft down the runway [23,24]. The 125 lb. hub fractured as the Pratt & Whitney engine approached maximum take-o! power, turning at 8000 rpm. Pieces of the hub and the attached fan section were #ung out at violent speed, tearing into the fuselage and hurling pieces to the right and left of the rolling aircraft (see Fig. 1). A mother and her 12-yr-old son died instantly as blades and fragments penetrated the cabin. The injuries involved two other children in the family. A total of seven passengers su!ered injury in the incident and during evacuation. A large fragment, which included a two-thirds chunk of the hub and a section of attached blades, gouged the runway and tumbled to a point 620 ft west of the runway. A second hub section, without blades attached, was found 2400 ft away, outside the airport perimeter. This site is east of the runway, on the opposite side of the aircraft from where the large piece was found. After investigation, the fan hub for the left engine was found fractured. The 1 in long crack, suspected to be of fatigue origin, was located inside a tierod hole through which one of twenty-four 0.5 in bolts attach the fan hub to the engine's low-pressure compressor [23]. According to the Aerospace Information Report 4003 [9], a total of 315 uncontained rotor failures occurred from 1976 to 1983 in commercial, general, and rotorcraft aviation. While the

Fig. 1. Delta Flight 1288 uncontained engine failure (from Ref. [24]).

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probability of failure is already low, turbine engine and APU manufacturers are making e!orts to reduce the probability of uncontained rotor failures; however, service experience shows that uncontained compressor and turbine rotor failures continue to occur. In 1997, statistics summarizing twenty-eight years of service experience for "xed-wing airplanes with over one billion engineoperating hours on commercial transports are presented [7]. A total of 676 uncontained engine failure events includes 93 events classi"ed in Category 3 and 15 events classi"ed in Category 4 damage to the airplane. Category 3 is de"ned as signi"cant airplane damage with the airplane capable of continuing #ight and making a safe landing. Category 4 damage is de"ned as severe airplane damage involving a crash landing, critical injuries, fatalities or hull loss. The events were caused by a wide variety of in#uences classi"ed as environmental (bird ingestion, corrosion/erosion), foreign object damage, manufacturing and material defects, mechanical, and human factors (i.e., maintenance and overhaul, inspection error, and operational procedures). The FAA has taken a three prong approach to the protection of aircraft from uncontained engine rotor fragments. First, the design and test requirements are imposed on engines to determine the root causes of failures and attempt to eliminate them from the design. Second, because the possibility exists that the rotor may fail, design and test requirements are imposed on the engine to ensure some containment capability. Finally, because complete containment has yet to be realized, design requirements are imposed on transport aircraft to minimize the hazard from fragments of an uncontained engine failure [4]. Statistics given in Ref. [7] indicate the existence of many di!erent causes of failure not readily apparent or predictable by failure analysis methods. For example, since the uncontained failure of the JT8D engine on Delta Flight 1288 [23] and the crash of a United DC-19 in Sioux City, IA, in 1989 [22], industry has redoubled its e!orts to investigate the integrity of titanium forgings and rotating titanium components [23]. Citing an unsafe condition in the high-pressure compressor rotor of General Electric CF6 turbofans, the FAA is requiring an improved #uorescent penetrant inspection procedure to detect cracks in the rotor spool [23]. Because of the variety of causes of uncontained rotor failures, it is di$cult to anticipate all possible causes of failure and provide protection to all areas. Accepting that the failures will continue to occur at a higher rate than is tolerable, attempts are made to contain all debris within a strengthened structure (e.g., see Refs. [11}15]). Design and test requirements are imposed on the engine to ensure some containment capability. Engine design and test requirements are covered in the United States Code of Federal Regulations, Title 14, Aeronautics and Space, Part 33, Airworthiness Standards; Aircraft Engines [4]. Part 33 of the Federal Aviation Regulations (FAR) has always required the engine to be designed to contain damage resulting from rotor blade failure. The containment of failed rotor blades is a complex process which involves high energy, high-speed interactions of numerous locally and remotely located engine components (such as failed blade, other blades, containment structure, adjacent cases, bearings, bearing supports, shafts, vanes and externally mounted components). Once failure begins, secondary events of a random nature may occur whose course cannot be precisely predicted [8]. Therefore, assuming that uncontained debris will continue to be generated, design considerations outlined in the AC 20-128A [7] provide guidelines for achieving the desired objective of minimizing the hazard to an airplane from uncontained rotor failure. These guidelines assume a rotor failure will occur and that analysis of the e!ects of this failure is necessary. The designs

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intend to make the aircraft invulnerable to the debris by such means as de#ection, the judicious location of critical parts, hydraulic lines, and structure, suitable duplication where appropriate, etc. Given that the damage is uncontained, developing an understanding of the impact event of the engine fragments or other parts of the structure is needed. FAA Advisory Circular AC 20-128A [7] provides speci"cations for fragment sizes to be used in the safety analysis models. The fragment size includes a single one-third disc fragment with one-third blade height and one-third the mass of the bladed disc, intermediate fragments with one-third of the bladed disc radius with a mass of 1/30th of the bladed disc, and small fragments (shrapnel) ranging in size up to a maximum dimension corresponding to the tip half of the blade airfoil. Experimental studies and numerical simulations continue to provide insight into this impact and penetration problem. The numerical simulations are also being used to improve the design and mitigate its e!ect on the structure and improve safety. Through this understanding, improved designs can be made in selected regions with high susceptibility to engine debris strike on the aircraft. The objective of this paper is to simulate the penetration of aluminum plates impacted by titanium fragments using the LS-DYNA nonlinear transient dynamic "nite element code [26]. These simulations are related to the impact and penetration scenario that would result from uncontained engine debris striking the fuselage skin. The con"guration studied herein has previously been studied by Shockey et al. [14] of SRI and is referred to herein as the SRI con"guration. The present investigation assesses the spatial discretization needs in the vicinity of the impact, material models for tearing and failure, and contact modeling for impact and penetration. Selected parametric studies related to impactor initial speed and orientation will be performed. Detailed discussions of the simulation tools, modeling strategies, computational approach, and additional simulation results are presented by Lawson [25]. Related simulations for the NASA Langley gas-actuated penetration device are presented by Ambur et al. [17].

2. Con5guration studied The basic con"guration considered in this study is de"ned in Ref. [14] and referred to as the SRI con"guration. It is studied to provide some level of veri"cation for the present simulations using LS-DYNA by comparing with the test results given by Shockey et al. [14]. The SRI con"guration, shown in Fig. 2 (taken from that report), was tested using the SRI gas-gun facility and analyzed using DYNA3D by SRI (see Ref. [14]). The gas-gun facility was used to "re fragments or fragment simulators having a maximum transverse dimension of 1.75 in to velocities ranging to 8400 in/s (700 ft/s). Square target barriers of either 6 in or 12 in were held in place by large test frames. Numerous tests were performed by SRI along with selected DYNA3D "nite element simulations. Ref. [14] presents DYNA3D "nite element results for a single "nite element mesh (referred to herein as Mesh 4). Limited analysis results were presented for a rebound case and a penetration case. Two test con"gurations from Ref. [14] are also analyzed in the present work where the emphasis is on studying the modeling, mesh re"nement, solution parameters, and computational requirements. Both con"gurations use a small 0.06 lb titanium fragment with truncated corners and a 6 in square aluminum target barrier with a 0.040 in thickness. Two values of the initial speed of the fragment are considered: 2328 in/s (194 ft/s) for the rebound simulation with partial perforation and

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Fig. 2. Sketch of SRI Gas-Gun facility (Fig. 6 from [14]).

3744 in/s (312 ft/s) for the complete penetration simulation. The orientation of the impactor is somewhat di!erent in the two cases relative to its orientation to the target plate based on measured data from the test. The impactor orientation is de"ned by its pitch (nose up or down) and roll about the axial direction. For the rebound simulation (Test 3), the pitch angle is !13.33 (nose down) and the roll angle is !10.03. For the penetration simulation (Test 6), the pitch angle is !9.33 and the roll angle is !9.53. It should be pointed out that the simulations reported herein and in Ref. [14] as Test 3 have the same characteristics.

3. Simulation tools Finite element simulations of structural problems involve three basic steps: the pre-processing step; the analysis step; and the post-processing step. For the impact and penetration simulations, the pre-processing step requires more preparation than other analyses because of the need to develop appropriate "nite element models for contact, failure, and penetration. The INGRID computer code [28] has been used in the present study for this step. The analysis step involves the nonlinear transient dynamic response prediction for the dynamic behavior prior to impact, the nonlinear impact/penetration event itself, and the subsequent dynamic behavior. The LS-DYNA computer code [26] has been used herein. Finally the post-processing step involves a signi"cant amount of e!ort (computational and manpower) as a result of the volume of computed results to interrogate. A typical simulation could involve 10,000 elements marched forward in time for 10,000 time steps. The LS-TAURUS computer code [26] is used for post-processing. Its post-processing

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features signi"cantly aid the analyst in interrogating the computed solution and assessing its validity. LS-DYNA computer code [26,27] performs nonlinear transient dynamic analysis of threedimensional structures. Originally developed as the DYNA3D-family of computer codes at the Lawrence Livermore National Laboratory, LS-DYNA represents the commercial version of these codes and is available through the Livermore Software Technology Corporation. LS-DYNA has a wide variety of analysis capabilities including a large number of material models, a variety of contact modeling options, a large library of beam, plate, shell, and solid elements and robust algorithms for adaptively controlling the solution process. The transient analysis is performed using an explicit direct-time-integration procedure and thereby avoids the need for matrix evaluation, assembly and decomposition at each time step as required by many implicit time-integration algorithms. As such, LS-DYNA requires minimal computer memory to execute even very large problems. The algorithm is CPU intensive, and as with any transient dynamic simulation, a large amount of secondary storage or disk space is required for subsequent post-processing of the computed solution. LS-DYNA automatically examines the "nite element mesh and material properties in order to determine an appropriate time step size for numerical stability. This time step size is then automatically adjusted throughout the transient analysis to account for contact and local material and geometric nonlinearities. Contact/impact algorithms have always been an important capability in the DYNA3D-family of codes. Contact may occur along surfaces of a single body undergoing large deformations, between two or more deformable bodies, or between a deformable body and a rigid barrier. In the present study, the sliding interface with friction and separation approach (Interface Type 3) is used to model the impact event between the impactor and the barrier target plate. The bounding surface of three-dimensional impactor is treated as the slave surface, and the target plate is treated as the master surface. Contact is treated using a penalty approach (see Refs. [27,29]). Once contact is determined, then an interface restoring force is imposed on the slave node and distributed across the nodes on the master segment. This force is computed based on the bulk modulus, volume, and face area of the element that contains the master segment. In addition, local and global penalty factors are available. One factor is the local penalty scale factor on default slave or master penalty sti!ness which has default value of one. Another factor is the global sliding interface penalty factor or SIPF (i.e., SLSFAC on the *CONTROL}CONTACT record) which has a default value of 0.1. The global factor scales the local factors to form a total penalty factor for contact sti!ness. As SIPF increases, the interface force required to move the slave node to the surface increases which causes an increase in the system frequency, locally. This local increase in slide-surface penalty sti!ness results in a high frequency and hence a much smaller time step size, locally. As a result, the explicit scheme may become unstable unless the computed time step is reduced. If SIPF is decreased, then excessive interpenetration may occur. Penetration of the target plate can be simulated in two ways depending on the modeling approach used for the target plate itself. One way is the tied-nodes-with-failure (TNWF) approach, and the other way is the element-erosion (EE) approach. Using the tied-nodes-with-failure approach, coincident nodes are generated in selected regions and then tied together with a constraint relation. In LS-DYNA, these tied nodes remain together until the volume-weighted e!ective plastic strain, averaged over all elements connected to the coincident nodes in a given constraint, exceeds

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a speci"ed value. Once this value is exceeded, all nodes associated with that constraint are released to simulate the initiation of a crack, fracture or penetration. Using the element erosion approach, the "nite element model is generated in the standard manner without requiring duplicate or coincident nodes. Once the e!ective plastic strain in an element reaches a speci"ed critical value, the element is removed from the computations (see Refs. [27,30}32]). In the EE approach, elements do not tear o! or separate from the initial "nite element model in the same manner as in the TNWF approach. Elements may erode away and thereby `freea other elements or groups of elements and tracking the rigid-body motion of these newly created fragments is necessary. While some mass is lost, conservation of momentum is insured. When an element is eroded, the internal energy of that element is no longer considered in the energy balance. If a node is deleted due to erosion its kinetic energy is lost. Thus, the modeling guidelines using the EE approach are di!erent from those for the TNWF approach. For the EE approach, the analyst needs to monitor the sliding interface energy. If the sliding interface energy is a negative number, there could be a large initial penetration or an `over-fasta penetration. Under these circumstances, a node penetrates in one time step far into the surface. This can occur in high-velocity penetration problems in which case the SIPF must be increased and the computed time step scale factor must be decreased. As the mesh is re"ned in these highly strained areas, the initial penetration and `over-fasta penetration becomes more of a concern. Contact modeling issues have been discussed by Reid [33] and the need for evaluating modeling techniques and code features advocated.

4. Modeling Finite element modeling of the spatial domain has to be su$cient to represent the geometry of the structure and to capture the structural response characteristics anticipated in the response. For the problem considered, the structural response has three distinct classes. The "rst class involves rebound of the impactor with minimal damage to the target plate. In this case, the response is more of a standard structural dynamics problem involving contact possibly with local regions of high plastic strain. The response is more global in that the entire target plate participates in the motion. The second class involves rebound of the impactor with partial perforation of the target plate. Here the response is primarily like the rebound case with the exception that severe localized damage occurs. These localized e!ects need to be accounted for in the "nite element modeling. However, the simulation is still very close to a standard structural dynamics problem with contact and localized damage. The third case involves the penetration of the impactor through the target plate. The contact interaction event now involves more than just the normal face of the impactor. The in#uence of the sliding contact along the bounding surfaces of the impactor as it penetrates the plate a!ects the results. As the contact interaction between the impactor and the target plate increases, the importance of the selection of simulation parameters also increases. Simulation of the response for speeds just above the threshold speed for penetration are perhaps the more di$cult simulations to perform and to ensure accuracy and robustness in the solution. The "nite element modeling strategy used herein is essentially the same as that used by Shockey et al. [14]. The present study examines mesh re"nement as well as the contact modeling for impact and penetration. In all cases, the simulation model involves three components: impactor, test "xture frame, and target plate. Finite element re"nement studies are performed wherein di!erent

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levels of mesh re"nement are used for the barrier target plate region. Re"nement in the barrier target plate region naturally extends into the test "xture frame in order to have compatible meshes. The "nite element models use the standard 8-node solid elements and the 4-node Belytschko}Lin}Tsai shell elements, both with single-point integration and default settings for hourglass control (see Refs. [26,27]). Modeling details for the three components are described next. 4.1. Impactor The impactor, shown in the top left of Fig. 2, is modeled using 8-node solid elements with an elastic-plastic strain-hardening material model (Material Type 3) and given an initial speed in the x-direction. The impactor is initially positioned slightly away from the target, and its orientation (pitch and roll) are speci"ed. The impactor is made of titanium. The mechanical properties for titanium (Ti-6Al-4V) are: Young's modulus, 16 Msi; Poisson's ratio, 0.30; yield strength, 120 ksi; hardening or tangent modulus, 0.3 Msi; hardening parameter, 0.2; and weight density, 0.16 lb/in. For the SRI con"guration shown in Fig. 2, the impactor is roughly a rectangular fragment (1.435 in long, 0.25 in thick, and 1.0 in wide) with tapered edges so that the #at contact area is 0.5 in;0.25 in. The mass of this impactor is 0.06 lb The spatial discretization of the impactor is held constant regardless of the mesh re"nement used for the other components. The element size on the normal contact face of the impactor is 0.0625 in. Also the initial location of the impactor is 0.46 in from the target plate so that contact does not initially occur (i.e., some simulation time is spent moving the impactor to the target plate). 4.2. Test xxture frame The test "xture frame is also modeled using 8-node solid elements with an elastic-plastic strain-hardening material model (Material Type 3). The frame has a lower part and an upper part that are assembled together to clamp down on the target plate. For the SRI con"guration, the test "xture frame parts are made of titanium. The outer border of the square frame is 6.0 in on each side with a clamping border 0.375 in wide. The combined thickness of the lower and upper parts is 0.20 in. The spatial discretization of the test frame through its thickness involves two eight-node solid elements. The spatial discretization in its plane is determined by the spatial discretization used for the target plate. Boundary conditions are imposed on the nodes in the test frame model to prevent any motion (i.e., rigid test frame). 4.3. Target plate The target plate is modeled using four-node Belytschko}Lin}Tsay shell elements with an elastic-plastic strain-hardening material model. The barrier target material is aluminum. The mechanical properties for the aluminum (2024-T3) target barrier are: Young's modulus, 10 Msi; Poisson's ratio, 0.33; yield strength, 50 ksi; hardening or tangent modulus, 0.1 Msi; hardening parameter, 0.2; weight density, 0.10 lb/in, and ultimate failure strain, 0.20 in/in This value for the failure strain is used as the constraint value to release any tied nodes once the element strain reaches this level. Strain rate e!ects are not accounted for in these simulations (nor in those

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reported in Ref. [14]). Additional material data are required. Strain rate sensitivity should be assessed. The "nite element modeling strategy used for the target plate involves three separate regions: the boundary region restrained by the test "xture clamping, the outer region which is in the test area and assumed not to be penetrated by the impactor, and the inner region of the test area for which penetration may occur. Elements in the boundary region of the target plate overlap elements in the test "xture frame parts. The "nite element models of the upper and lower test frame parts and the target plate are then joined together as in the test con"guration. The test area of the target is modeled by two regions: an outer region which extends from the frame boundary to the inner region. The inner region which is the area of probable impact and possible penetration. This inner region of the target plate is called the `shell-break area.a An alternative model for the target was also analyzed using eight-node solid elements rather than shell elements for the target plate. For isotropic materials, the number of elements needed through the thickness depends on the amount of bending occurring in the simulation. In these studies, the bending is localized and complete penetration occurs with limited bending. Hence good results should be obtained with only three elements through the target plate thickness. However, in cases with substantial bending, the number of elements needed through the thickness will need to be increased to capture this behavior. This three-dimensional model is studied as a "rst step in developing models for laminated structures and to study the e!ect of partial through-the-thickness damage of the isotropic target plate. For the SRI con"guration, the shell-break area is a 2 in;2 in area in the center of the target plate. In this area, elements are individually created with their own independent nodes. Coincident nodes are then identi"ed and tied together until the average volume-weighted plastic strain exceeds a speci"ed value (i.e., 0.20 in/in in these simulations). Spatial discretization in the shell-break area determines the element distribution throughout the target plate for the most part. Di!erent discretizations were considered (see Table 1) and are de"ned based on the element edge length in the shell-break area. Mesh 1, shown in Fig. 3, has an element edge length of 0.25 in in the shell-break area and represents the coarsest mesh considered. Mesh 2, shown in Fig. 4, has an element edge length of 0.10 in in this area. Mesh 3, shown in Fig. 5, has an element edge length of 0.0625 in in this area. Mesh 4, shown in Fig. 6, has an element edge length of 0.05 in and represents the most re"ned mesh considered herein and is the same and only discretization reported in Ref. [14]. Note that the element size in the shell-break area for Mesh 3 is the same as the element size on the impactor contact face. Table 1 Summary of "nite element models Model

Mesh Mesh Mesh Mesh

1 2 3 4

Impactor

Frame

Target

No. of Nodes No. of Elements No. of Nodes

No. of Elements No. of Nodes

No. of Elements

1275 1275 1275 1275

792 1368 1944 2328

1296 3600 7056 10,000

896 896 896 896

1584 2736 3888 4656

1516 7372 17,836 27,372

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Fig. 3. Mesh 1 "nite element model for the TNWF approach.

Fig. 4. Mesh 2 "nite element model for the TNWF approach.

109

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Fig. 5. Mesh 3 "nite element model for the TNWF approach.

Fig. 6. Mesh 4 "nite element model for the TNWF approach.

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5. Results using tied nodes with failure Simulation results obtained using TNWF approach are reported here for the SRI con"guration, and comparison with test results is made using available data. Simulation results are presented to identify the impactor initial speed which results in rebound and complete penetration. A summary of the "nite element models used in the various spatial discretization studies is given in Table 1. Simulation results for several "nite element meshes considered are summarized in Table 2 along with the test results [14]. Convergence of the "nite element results is assessed by comparing global quantities such as the internal and kinetic energies and point quantities such as displacements and velocities of selected points. The time variation of the interface contact forces in the axial direction for the two interfaces (on the impactor and on the target plate) are also given for the rebound case and the penetration case. The time variations of these selected response parameters for the di!erent "nite element meshes are shown for the rebound case and for the penetration case. These results provide a basis to assess convergence and to aid in developing modeling guidelines for future simulations. 5.1. Rebound case The "rst case considered has an initial impactor speed of 2328 in/s (194 ft/s) and an orientation given by a !13.33 pitch angle and a !10.03 roll angle. This simulation is similar to the simulation reported as Test 3 in Ref. [14]. Typical execution times for the rebound case on a SGI O2 Unix workstation with an R10000 processor ranged from 720 CPU seconds to 7400 CPU seconds for 1 ms of simulation time. It is interesting to note that approximately one third of the execution time is spent in the contact algorithm.

Table 2 Comparison of test and analysis results for the SRI con"guration Test No. or Mesh No.

Before impact

After impact

Impact result

Velocity (in/s)

KE (in lbf)

Velocity (in/s)

KE (in lbf)

Test 3H Mesh 1 Mesh 2 Mesh 3 Mesh 4

#2328 #2328 #2328 #2328 #2328

389 389 389 389 389

!787 !1196 !1138 !1118 !1020

* 110 101 98 83

Deformation only Deformation only Deformation only Deformation only Partial perforation

Test 6 Mesh 1 Mesh 2 Mesh 3 Mesh 4

#3744 #3744 #3744 #3744 #3744

991 1018 1018 1018 1018

#2434 !1106 #1098 #1894 #2078

425 135 147 330 345

Complete penetration Deformation only Complete penetration Complete penetration Complete penetration

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For the rebound case, initial contact between the impactor and the plate occurs at approximately t"0.16 ms. The impactor has rebounded and lost contact with the plate at approximately t"0.72 ms. A front view of the deformed geometry is shown in Fig. 7 for Mesh 4 just after rebound. A close-up view of the back surface is shown in Fig. 8 with contours of the plastic strain. A well-de"ned permanent `denta due to the plastic strain in the vicinity of impact is clearly visible. From the simulation using Mesh 4, partial perforation of the target plate is evident in Fig. 8 even though the impactor does rebound. The time variations of selected response parameters for the di!erent "nite element meshes are shown in Figs. 9}12 for the rebound case. A comparison of the axial (x-direction) displacement of Node 675 at the center of the impactor for the "nite element models considered is given in Fig. 9 for the rebound case (initial impactor velocity of 2328 in/s). During the earlier stages of the simulation, all meshes give the same result. After the positive peak value of the displacement for this point, di!erences begin to develop between the coarser meshes and Mesh 4. The results for Meshes 1, 2 and 3 are very close after rebound occurs (t*0.5 ms). Maximum displacements are nearly the same except for Mesh 4. Because of the partial perforation and slower rebound, Mesh 4 continues to predict a larger axial displacement for the impactor (i.e., slower rebound, hence it takes longer to move to a certain location compared to those with higher rebound velocities).

Fig. 7. Test panel deformed geometry plot after impactor rebound (Mesh 4).

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Fig. 8. Close-up view of the plastic strain distribution near the impact site for the rebound case (Mesh 4).

Fig. 9. Comparison of axial displacement of the impactor for di!erent "nite element discretizations } rebound case.

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Fig. 10. Comparison of axial velocity of the impactor for di!erent "nite element discretizations } rebound case.

Fig. 11. Comparison of internal and kinetic energies for di!erent "nite element discretizations } rebound case.

A comparison of the axial (x-direction) velocity component of Node 675 at the center of the impactor for the "nite element models considered is given in Fig. 10 for the rebound case. Slight di!erences in velocity begin to occur just as the impactor starts to rebound (when the velocity component is zero, at t+0.3 ms). The di!erence between Meshes 1, 2 and 3 and the most re"ned mesh (Mesh 4) increases after the impactor rebounds (t*0.5 ms). The more re"ned model captures the elasto-plastic deformation pattern in the vicinity of the impact site better and also predicts partial perforation of the impactor through the thin plate. From the results in Table 2 and Fig. 10,

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Fig. 12. Comparison of the interface contact force in the axial direction for di!erent "nite element discretizations } rebound case.

the rebound velocity component in the x-direction for Meshes 2 and 3 appears to be converging to a value of !1128 in/s (!94 ft/s). However, the lower rebound velocity predicted by Mesh 4 (!1020 in/s or !85 ft/s) appears to be due to the partial perforation of the plate and the associated loss of energy. Further re"nement of the mesh will permit a higher resolution of the local damage near the impact site; however, complete penetration is not anticipated. A comparison of the internal and kinetic energies for the "nite element models considered is given in Fig. 11 for the rebound case. Prior to rebound, the kinetic energies agree fairly well with one another. After rebound, the kinetic energy from the coarser meshes (Meshes 1}3) are close to each other but di!er from that predicted by the most re"ned mesh (Mesh 4) results. Mesh 4 has less kinetic energy after impact due to extensive plastic deformation and partial perforation of the plate that the other meshes did not predict. The internal energies follow the same overall trends after impact and prior to rebound. Following rebound, the energies for each mesh are all within approximately 12% of each other. Mesh 4 results exhibit an overall higher internal energy following rebound. This is attributed to the overall higher levels of deformation predicted using Mesh 4 and associated with the partial perforation of the plate. A comparison of the interface contact force in the axial direction for the "nite element models considered is given in Fig. 12 for the rebound case. Interface 1, denoted by the "lled circles in the "gure) is de"ned as the contact interface surface on the impactor, while Interface 2 is the contact interface surface on the target plate. The duration of the impact event is approximately 0.56 ms with a peak force of approximately 1400 lbf. The overall distribution of the interface force in the x-direction is basically the same for all "nite element models considered. Mesh re"nement tends to cause the interface force pro"le to become a somewhat smoother distribution in time. Di!erences between the results for Mesh 3 and those for Mesh 4 are attributed to the partial perforation predicted using Mesh 4. The peak value of the interface contact force is smaller for the "ner mesh.

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5.2. Penetration case The second case considered has an initial impactor velocity of 3744 in/s (312 ft/s) and an orientation given by a !9.33 pitch angle and a !9.53 roll angle. This velocity corresponds to the condition used in the SRI penetration simulation [14]. Typical execution times for the penetration case on a SGI O2 Unix workstation with an R10000 processor ranged from 1800 CPU seconds to 14,000 CPU seconds for 1 ms of simulation time } approximately twice the execution times for the rebound case. The deformed geometry for Mesh 4 just prior to complete penetration is shown in Fig. 13. From the tied-nodes-with-failure model, a damage area corresponding to the large hole `puncheda by the impactor is clearly evident as are additional small fragments generated by the `tearinga of this hole. The tearing of the tied nodes occurs once the volume-weighted plastic strain reaches a value of 0.2 in/in. A close-up view of the impact site is shown in Fig. 14 with contours of the plastic strain. A large region of plastic strain is predicted in the vicinity of the impact, and the overall transient dynamic response is somewhat localized. The time variations of the same selected response parameters for the di!erent "nite element meshes are shown in Figs. 15}18 for the penetration case. A comparison of the axial (x-direction) displacement of Node 675 at the center of the impactor for the "nite element models considered is

Fig. 13. Test panel deformed geometry plot after complete penetration (Mesh 4).

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Fig. 14. Close-up view of plastic strain distribution near the impact site after complete penetration (Mesh 4).

Fig. 15. Comparison of axial displacement of the impactor for di!erent "nite element discretizations } penetration case.

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Fig. 16. Comparison of axial velocity of the impactor for di!erent "nite element discretizations } penetration case.

Fig. 17. Comparison of internal and kinetic energies for di!erent "nite element discretizations } penetration case.

given in Fig. 15 for the penetration case (initial impactor velocity of 3744 in/s). During the earlier stages of the simulation (t)0.2 ms), all meshes give the same result. For t*0.2 ms, the axial displacement time histories become di!erent. The displacements predicted using Mesh 1 decrease indicating possible rebound. The displacements from Meshes 2}4 both continue to increase indicating penetration of the target plate. It should also be pointed out that this response parameter is more sensitive to mesh re"nement for the penetration case than for the rebound case (compare Fig. 9 with Fig. 15).

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Fig. 18. Comparison of the interface contact force in the axial direction for di!erent "nite element discretizations } penetration case.

A comparison of the axial (x-direction) velocity component of Node 675 at the center of the impactor for the "nite element models considered is given in Fig. 16 for the penetration case. The Mesh 1 simulation indicated that at approximately 0.4 ms, the impactor has clearly rebounded after impact. The more re"ned models predict penetration of the plate. The residual speed of the impactor for Mesh 4 is higher than for Mesh 2 by nearly a factor of two. The residual speed of the impactor (i.e., velocity in the x-direction after penetrating the plate) is 2080 in/s (173 ft/s) from Mesh 4 which is approximately 14% lower than the test result of 2433 in/s (203 ft/s) from Test 6 given in Ref. [14]. This velocity component varies over the impactor "nite element by approximately 10}15% [25]. Additional information and data from the experiment are needed; however, the simulation results appear to correlate qualitatively with the test and analysis data given in Ref. [14]. A comparison of the internal and kinetic energies for the "nite element models considered is given in Fig. 17 for the penetration case. Mesh 1 results exhibit similar trends to those computed for the rebound case shown in Fig. 11. The results for the other more re"ned meshes have a di!erent behavior. As a result of the penetration and the tearing of nodes and elements in the `shell-breaka area, signi"cant di!erences occur for these models. Mesh 4 exhibits higher kinetic energy and lower internal energy apparently due to the "ner resolution of the plastic zone in the vicinity of impact, leading to early tearing of nodes and elements, and a more localized deformation pattern. Following penetration, the energy values level o! with Meshes 3 and 4 having nearly the same kinetic energies and their internal energies are within approximately 16%. A comparison of the interface contact force in the axial direction for the "nite element models considered is given in Fig. 18 for the penetration case. The results obtained using Mesh 1, which predicts rebound even for this higher speed impact, are similar to the distribution predicted for the rebound case (see Fig. 12) and not included on this "gure. The results predicted for the other meshes indicate that the contact force magnitude and duration of the contact event both decrease

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as the "nite element model is re"ned. This behavior is due to the improved representation of the localized plastic strain in the vicinity of the impact and better resolution of the penetration process. Following penetration (0.3 ms)t)0.6 ms), small variations of the interface contact force are observed as a result of the sliding contact along the side faces of the impactor and the edges of the `puncheda hole in the plate. Peak values of the contact force are smaller and occur sooner for the "ner mesh. The duration of the impact event for Mesh 4 is approximately 0.28 ms with a peak force of approximately 1700 lbf. The e!ect of the sliding interface penalty factor (SIPF) on predicting the response of the plate during the penetration case is shown in Fig. 19 in terms of the total energy as a function of time. Using Mesh 3, the penetration simulation is attempted using various values for SIPF. For values of 1.0, 0.1 and 0.01, the initial portion of the simulation appears correct, and then the total energy suddenly increased. Associated with each jump in total energy shown in Fig. 19 are out-of-range values for the velocities. These values thereby caused the kinetic energy to grow without bound, and hence the total energy grew without bound until the simulation stops. When SIPF equals 0.001, the impactor penetrated the target by about 7% of its length before any `tearinga of the target elements occurred (i.e., `no damagea penetration). This appreciable `no-damagea initial penetration led to a `softa impact where the impactor traveled 1.79 in after 1 ms of simulation time. The residual velocity of the impactor after penetration is also about 1068 in/s (89 ft/s). This is evident from the total energy plot given in Fig. 19 which indicates that the total energy is below the initial energy for almost the entire simulation. Also the maximum ratio of the sliding interface energy to initial energy is about 0.1%. `No-damagea penetration tends to occur when the ratio of the maximum sliding interface energy to initial energy ratio is very small. In such cases, the SIPF value is normally too small for the impact simulation considered and needs to be increased.

Fig. 19. E!ect of sliding interface penalty factor on total energy for Mesh 3 } penetration case.

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Increasing SIPF to 0.006 gives the results used in the earlier "gures. When SIPF equals 0.006, the impactor traveled 2.4 in after 1 ms of simulation time (see Mesh 3, Fig. 15). The residual velocity of the impactor after penetration is about 192 ft/s when SIPF equals 0.006 or about twice the value predicted when SIPF equals 0.001. At approximately 0.56 ms, the impactor has completely penetrated and passed through the plate. This is evident in Fig. 19 by the slight increase in total energy as the sliding interface energy goes to zero. If the ratio of the interface energy to the initial energy (all kinetic energy of the impactor) is larger than 10%, then this penalty factor should be adjusted. As a goal, this ratio should be under 10% of the initial energy (smaller the better). A threshold impactor speed can be determined above which the impactor will penetrate rather than rebound. At this threshold speed, the impactor will penetrate the target plate but the response may still display a signi"cant structural dynamic response globally. As the speed is increased, the response becomes increasingly localized near the impact site. This is demonstrated by simulating the response for an initial impactor speed equal to twice the speed of the penetration case (7488 in/s or 624 ft/s). A close-up view of the plastic strain contours after complete penetration is shown in Fig. 20. The damage is more localized than for the penetration case for an initial impactor speed of 3744 in/s (312 ft/s) shown in Fig. 14. By comparing Figs. 8, 14 and 20 (which are all shown at the same scale and contour levels), the damage area is seen to become increasingly localized as the initial impactor speed increases above the threshold speed for penetration.

Fig. 20. Close-up view of plastic strain distribution near the impact site after complete penetration for an initial impactor speed of 624 ft/sec (Mesh 4).

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6. Results using element erosion Simulation results obtained using EE approach are reported here for the SRI con"guration and comparison with test results is made using the available data. Simulation results are presented for an impactor initial velocity of 3744 in/s (312 ft/s) resulting in complete penetration. Simulation results for the seven "nite element meshes considered are summarized in Table 5 along with the test data [14]. Convergence of "nite element results is assessed by comparing global quantities such as internal, kinetic and total energies and point quantities such as displacements and velocities of selected nodes. The time variations of these selected response parameters for the di!erent "nite element meshes are shown for the penetration case. As with the TNWF studies, these results provide a basis to assess convergence and to aid in developing modeling guidelines using the EE approach for future simulations. Finite element modeling for penetration using the element-erosion (EE) approach is presented in this section. Element erosion is the process used to eliminate elements during the computation that no longer contribute to the overall response determination. Eroding elements are elements which are destroyed during the course of the computation because of very high strains. They represent elements of the target that have ceased to play a signi"cant role in the physics of the problem [30]. Elements may still `break o!a from the target and need to be tracked as rigid bodies. However, the mechanism to `break o!a elements is when eroded or eliminated elements free an adjacent element. This is in contrast to the TNWF approach that releases a constraint tying coincident nodes together. Modeling for the element erosion approach does not require coincident nodes or constraints. Finite element meshes are created in the usual manner with a material failure model based on e!ective plastic strain or minimum time step value for the element. The EE approach may be used with either shell or solid elements. Shell element models more readily capture the bending behavior, while the solid element models have the potential to capture through the thickness damage. While shell elements can be developed with a thickness thinning capability, solid element modeling is attractive for layered structures (e.g., composite laminates and sandwich structures). For this reason, the EE approach used herein is based on the solid element modeling approach. In the EE modeling approach used for this study, both the target plate and impactor are modeled using three-dimensional eight-node solid elements, and only the penetration case from Ref. [14] is studied. The designs for the titanium impactor fragment, test "xture frame and target plate are the same as in the simulations using the TNWF approach. The initial velocity of the fragment is 3744 in/s (312 ft/s) for the penetration simulation. The orientation of the impactor fragment is the same as de"ned previously. The impactor "nite element model for the element-erosion approach is identical to the model description given for the TNWF approach. The test "xture frame "nite element model for the element erosion approach is similar to the model description given for the TNWF approach except that the target plate thickness is now explicitly present. The material properties are the same as given previously as are the overall dimensions. In the element erosion case, the spatial discretization of the frame is varied through the thickness as a result of di!erent through-the-thickness modeling of the target plate (i.e., modeling of the test frame was the same in all cases). The barrier target plate is now modeled using eight-node solid elements with an elastic-plastic strain-hardening material model. As in the TNWF approach, the target plate material is aluminum,

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and the same mechanical properties are used. The "nite element modeling strategy used for the barrier target plate component involves a boundary region and a target region. The boundary region of the target plate overlaps elements in the frame model. The two are then joined together as in the test con"guration. In the EE approach, the eroding surface-to-surface approach (Interface Type 14) is used to model the contact event between the impactor and the barrier target plate. As noted earlier, the bounding surfaces of the three-dimensional impactor are treated as slave surfaces, and the target is treated as the master surface. The interaction of the contact surfaces is handled by operations on the slave nodes of the impactor and the master elements of the barrier target plate. Once the e!ective plastic strain in an element reaches 0.2 in/in, the element is removed from the computations. Three di!erent in-plane mesh discretizations are considered based on the in-plane element edge length. Meshes 1 and 2 (see Figs. 21 and 22, respectively) have the same uniform in-plane discretization as their counterparts in the TNWF approach. Mesh 3, shown in Fig. 23, has a non-uniform in-plane discretization with re"nement near the impact site. Four di!erent through-the-thickness mesh discretizations are considered for the target plate based on the number of elements used in the thickness direction. A summary of the "nite element models used in the various spatial discretization studies is given in Table 3. In the notation Mesh i : j, the `i a denotes the in-plane spatial discretization mesh and the `j a denotes the number of elements through the thickness. The computational equipment used varied between two SGI Unix workstations. Table 4 gives the impactor initial velocity, the number of time steps computed, the disk space required, the computer system used, the CPU execution time and the wall-clock time factor for 1 ms of simulation time. Mesh 1, shown in Fig. 21, has an element edge length of 0.25 in in the target barrier plate area and represents the coarsest mesh. Since Mesh 1 was a coarse mesh, only two elements through the

Fig. 21. SRI con"guration } Mesh 1: EE.

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Fig. 22. SRI con"guration } Mesh 2: EE.

Fig. 23. SRI con"guration } Mesh 3: EE.

thickness were used. Mesh 2, shown in Fig. 22, has an element edge length of 0.097 in in the target barrier plate area. Additional studies were performed using mesh discretizations of two, three, four and "ve elements through the thickness, with element edge thicknesses of 0.020, 0.013, 0.010, and 0.008 in, respectively. An example of the through-the-thickness discretization with two elements through the target plate thickness is depicted in Fig. 24. Mesh 3, shown in Fig. 23, is the most

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Table 3 Summary of "nite element models for the EE approach Model

Mesh Mesh Mesh Mesh Mesh Mesh Mesh

1:2 2:2 2:3 2:4 2:5 3:3 3:4

Impactor

Frame

Target

Nodes

Elements

Nodes

Elements

Nodes

Elements

1275 1275 1275 1275 1275 1275 1275

896 896 896 896 896 896 896

2640 4640 5568 6496 7424 6240 7280

1584 2784 3480 4176 4872 3900 4680

4107 11,529 15,372 19,215 23,058 18,020 22,525

2592 7440 11,160 14,880 18,600 13,104 17,472

Table 4 Computational requirements for the EE approach Model

Number of time steps

Disk space required (MB)

Computer system name

CPU execution time (s)

Factor for wall-clock time

Mesh Mesh Mesh Mesh Mesh Mesh Mesh Mesh Mesh

24,958 25,994 39,163 39,188 50,176 63,280 39,869 36,778 53,505

49.2 112.5 145.9 115.0 185.2 222.4 173.3 196.9 165.4

nova pumbaa nova nova pumbaa pumbaa nova nova pumbaa

4413 18,263 21,493 35,244 56,452 79,934 26,075 47,285 75,523

8.4 1.0 1.2 1.3 1.0 1.0 2.4 3.3 1.0

1:2 2:2 2:3 2 : 3a 2:4 2:5 3:3 3 : 3a 3:4

Table 5 Control parameters for the EE approach Model

Sliding interface penalty factor

Computed time step factor

E /E 1 2 (%)

E /E 2 ' (%)

Mesh Mesh Mesh Mesh Mesh Mesh Mesh

0.0025 0.005 0.005 0.005 0.005 0.005 0.005

0.6 0.6 0.6 0.6 0.6 0.6 0.6

5.4 2.0 4.9 2.2 1.7 1.4 10.2

66 75 69 74 75 80 73

1:2 2:2 2:3 2:4 2:5 3:3 3:4

re"ned mesh and has an element edge length (in the most re"ned area in the center of the target plate) in the y-direction of 0.0667 in and in the z-direction of 0.0625 in. For Mesh 3, two studies were performed with three and four elements through the target plate thickness.

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Fig. 24. Side view of an inside slice showing a two-element through-the-thickness discretization.

6.1. Penetration case The penetration simulation performed in this study has an initial impactor velocity of 3744 in/s (312 ft/s) and an orientation given by a !9.33 pitch angle and a !9.53 roll angle, identical to the TNWF penetration case. Typical execution times for the penetration case on a SGI O2 Unix workstation ranged from 4400 CPU seconds to 79,900 CPU seconds for 1 ms of simulation time. For the EE penetration case, initial contact between the impactor and the plate occurs at approximately 0.13 ms. A close-up view of the deformed geometry and plastic strain distribution for Mesh 3 : 4 after penetration are shown in Fig. 25. Complete penetration of the target plate by the impactor and a damaged area `broken o!a by the impactor is evident. Elements are removed once the plastic strain within an element reaches 0.2 in/in. The overall transient dynamic response is much more localized than the damage predicted using the TNWF approach (compare Fig. 25 with Fig. 14). Petaling of the target plate is also clearly visible in these "gures. At approximately t"0.17 ms, elements begin to `faila and are eliminated from the simulation. By t"0.35 ms, 182 elements have been removed. This corresponds to a hole size of approximately 0.267 in;1.125 in;0.040 in. The hole size predicted using the EE approach is approximately 26% smaller than that predicted in the TNWF approach. Note that the element size of Mesh 3 using the EE approach is somewhat di!erent than the element size of Mesh 4 using the TNWF approach. Although Mesh 3 using the EE approach is a coarser mesh than Mesh 4 using the TNWF approach, the three-dimensional modeling of through-the-thickness failures helps to localize the failure near the impact site. That is, the solid elements erode in the thickness direction while the shell elements account for throughthe-thickness e!ects in the kinematics.

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Fig. 25. Close-up back view after penetration: Mesh 3:4 deformed geometry and plastic strain distribution.

The time variations of selected response parameters for the di!erent "nite element meshes are shown in Figs. 26}31. In the axial displacement and velocity time variation plots, the results obtained for Meshes 2 : 2, 2 : 3, 2 : 4 and 2 : 5 appear to be converging towards results obtained using the shell element target plate model (Mesh 4 with TNWF approach). In the total energy time variation plot shown in Fig. 31, the results obtained for Mesh 3 : 3 and 3 : 4 indicate less of an energy loss. Mesh 3 : 4 is considered as the `referencea solution in the EE simulations. These results also indicate that three elements throughout the thickness of the target plate are su$cient. Results for all seven mesh discretizations are presented in these "gures. A comparison of the axial (x-direction) displacement of Node 675 of the impactor for the "nite element model considered is given in Fig. 26. During the earlier stages of the simulation (t)0.20 ms), all meshes give the same result. For t*0.20 ms, the axial displacement time histories di!er. The axial displacements predicted using Mesh 1 : 2 decrease indicating rebound of the impactor. The displacements for all other EE meshes continue to increase indicating penetration through the plate. The four Mesh 2 models give similar results and appear to be converging towards the Mesh 4 TNWF results. Comparing these Mesh 2 EE results with the Mesh 2 TNWF results in Fig. 15 indicates a larger displacement for the EE approach. Similarly the Mesh 3 EE results shown in Fig. 26 are higher than the Mesh 3 TNWF results in Fig. 15. Because the EE results explicitly models the thickness of the target plate, portions of the target may be `erodeda prior to penetration. In the TNWF approach, the penetration event happens for the entire thickness at once. As a result, the EE approach predicts larger displacements. Also as the in-plane

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Fig. 26. Comparison of axial displacement of the impactor for di!erent discretizations of the target plate } EE: penetration case.

Fig. 27. Comparison of axial velocities of the impactor for di!erent discretizations of the target plate } EE: penetration case.

spatial discretization increases (i.e., going from Mesh 2 to Mesh 3), better resolution of the local stress gradient in the vicinity of impact occurs. Element erosion occurs faster, and the impactor moves further for a given point in time. A comparison of the axial (x-direction) velocity component of Node 675 on the impactor for the "nite element models considered is given in Fig. 27. At 0.5 ms, the results for Mesh 1 : 2 indicate that the impactor has clearly rebounded after impact. The more re"ned models predict the complete

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Fig. 28. Comparison of axial velocity of the impactor for di!erent nodes with Mesh 3:4 } TNWF: rebound case.

Fig. 29. Comparison of the system internal energy of the impactor for di!erent discretizations of the target plate } EE: penetration case.

penetration of the plate. The residual velocity of the impactor (i.e., velocity in the x-direction after penetrating the plate) is 2758 in/s (230 ft/s) from Mesh 3 : 4 which is approximately 13% higher than the test results for 2433 in/s (203 ft/s) velocity case from Test 6 given in Ref. [14]. A comparison of the axial (x-direction) velocity component of the three nodes on the impactor (located at the center back, centroid and center front) is shown in Fig. 28 and veri"es that the velocity remains consistent throughout the impactor implying little or no change in orientation. Again the Mesh 3 EE results

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Fig. 30. Comparison of the system kinetic energy of the impactor for di!erent discretizations of the target plate } EE: penetration case.

Fig. 31. Comparison of the system total energy of the impactor for di!erent discretizations of the target plate } EE: penetration case.

give a higher residual velocity for the impactor than the models using the TNWF approach. Since element erosion occurs through the thickness of the target plate, the impactor has not slowed down as much as in the models using the TNWF approach. With a higher residual velocity, the axial displacements are naturally larger as indicated in Fig. 26. A comparison of the internal, kinetic and total energies for the "nite element models considered are given in Figs. 29}31, respectively. Mesh 1 : 2 results exhibit similar trends as those predicted in the TNWF Mesh 1 case (i.e., rebound behavior). As in the TNWF approach, the other meshes have

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a di!erent behavior. As Mesh 2 is re"ned through the thickness, the responses appear to `convergea, yet still indicate a large energy loss overall. Mesh 3 : 3 and 3 : 4 mirror each other in their response predictions. Notice that at the approximate time the elements begin to erode (t*0.17 ms), the total energy loss signi"cantly increases (see Fig. 31). Although, with each in-plane mesh re"nement, the total energy loss decreases. In the EE approach, elements `erodea once the strain level reaches 0.2 in/in. When an element is `erodeda, its internal energy is lost as well as its kinetic energy. The mass loss is redistributed so that momentum is conserved [27]. As a result, signi"cant energy loss may occur for the EE cases. The TNWF cases `break o!a elements but do not eliminate them, hence their energies are still included in the energy calculation. As in the TNWF studies, the e!ect of the sliding interface penalty factor (SIPF) on predicting the response of the plate during the penetration case is considered. Table 5 gives the initial velocity of the impactor, the sliding interface penalty factor, the computed time step factor, the percentage of sliding interface energy to the total energy, and the percentage of total energy to initial energy for the EE penetration case. Using Mesh 2 : 2, the penetration simulation is attempted using two SIPF values; 0.0025 and 0.005. For SIPF of 0.0025, the impactor penetrated the target barrier plate with no damage resulting to the plate (i.e., the simulation was too `softa). By increasing the SIPF to 0.005, the simulation appears correct with elements of the plate reaching the maximum value of the plastic strain and element erosion occurs.

7. Conclusions Modeling and computational requirements for the penetration simulation of thin aluminum plates by a titanium impactor have been investigated. These simulations were performed to assist in the development of the modeling requirements for simulating uncontained engine debris impact on fuselage skins. The simulations used the LS-DYNA nonlinear transient dynamics "nite element analysis code, and comparisons with available test data were made. The con"guration studied by Shockey et al. [14] and by Ambur et al. [17] are representative of fuselage-like panels from transport aircraft which have large radii and internal rings and stringers supporting the shallow panel segments. These simulations indicate that a highly re"ned model is needed in the vicinity of the impactor. Element size in this region should be 20% to 25% of the smallest dimension of the contact surface on the impactor. For the cases considered, this equates to four or "ve elements in the target plate along the smallest dimension of the impactor's contact surface projected onto the target plate. Modeling the penetration process using the tied-nodes-with-failure approach provides a capability to tear nodes and elements during the event. This approach, while requiring additional modeling e!ort and computing, provides a way to simulate the `petalinga observed in some penetration tests. The element-erosion approach is an alternative penetration modeling approach that does not require any additional modeling e!ort. For the cases considered, through-the-thickness e!ects were modeled using solid elements; however, shell elements may also be used as in Ref. [17]. These LS-DYNA simulations provide reasonably good overall correlation with the test and analysis results reported in Ref. [14]. The contact interaction event initially involves the target plate and the impactor. As penetration occurs, subsequent sliding contact along the faces of the impactor as it penetrates the target plate.

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In addition, contact between the `petalsa near the penetration site and the back surface of the target plate or between `fragmentsa and the back surface needs to be accounted for in the model. For speeds below and near the threshold speed for penetration, the response of the target plate is more of a global structural dynamic response with severe local gradients near the impact site. For speeds above the threshold speed, the response is more localized with limited structural dynamic excitement in the target plate beyond the impact site. Hence, it is suggested that the modeling and simulation parameters required for the threshold speed case pose more severe challenges to determine. Once determined, these parameter values and the associated "nite element models may be used for other impact speeds with reasonable reliability. However, this must always be veri"ed by close examination of the computed results. Nonlinear transient dynamics simulations involving large deformations, nonlinear elastic-plastic material response, multi-body contact, and penetration place high demands and responsibilities on the analyst to verify the computed results and to insure the predicted behavior is physically consistent and accurate. Multiple executions of the model and possible "nite element remodeling are necessary and should be anticipated, expected and performed in order to validate the simulation results.

Acknowledgements The work of the "rst author before he left ODU and the work of the second author were supported by NASA Cooperative Agreement NCC-1-284. The work of the third author was supported by the NASA Graduate Student Researcher Program Grant No. NGT-1-52173 and by a Virginia Space Grant Fellowship while at ODU. This support is gratefully acknowledged. The authors also acknowledge the support provided by LSTC during this study.

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