An Empirically Validated Topology Optimization ...

An Empirically Validated Topology Optimization Based Design Methodology for Additively Manufactured Penetrating Warheads William T. Graves Jr.a , David Liub , and Anthony N. Palazottoc Air Force Institute of Technology, Wright-Patterson AFB, OH, 45433

A penetrating warhead design featuring a thin outer case compared to current technology was produced to improve blast characteristics following impact. To ensure material survival of the thin warhead, a topology-optimized interior structure capable of providing necessary stiffness to the warhead was incorporated. Loads applied to the topology optimization were resolved using an impact simulation calibrated to match test results of previous research in the field. Two warhead designs were created by this method, and the survivability of each design was evaluated using finite element analysis (FEA) to determine the stress responses generated against a load simulating impact. Scaled test articles consisting of control and optimized designs were manufactured via Direct Metal Laser Sintering (DMLS) in 15-5 stainless steel and live-fire tested against normal and oblique monolithic concrete targets. Test results validated the project concept, the design methodology used to determine structural layout, and the static FEA analysis conducted of the completed designs. Results of this research prove thin-walled, internally-supported, additively manufactured penetrators are capable of surviving high energy impacts with hard targets. Also, this work shows the potential for design and production of warheads tailored uniquely to their intended targets, which are additively manufactured as needed by operational military forces.

a b c

MS Student, Aeronautical Engineering. Student Member AIAA. Assistant Professor, Aerospace Engineering, Senior Member AIAA. Distinguished Professor, Aerospace Engineering, Fellow AIAA.

Nomenclature

V0 M C √ 2E σi∗ σf∗ D ρ G HEL PHEL A B N C ε˙0 ∗ σf,max T D1, D2, ... K1 K2, K3 β σ a b n c ε˙ ε˙∗ T∗ m Tmelt Tref ∆ε εf U F K ue ke xmin Nel p V (x) Vvol,0 f

= = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =

Initial velocity of ejected fragment Mass of metallic outer shell of warhead Mass of high explosive charge contained within warhead An empirically derived constant Intact strength of brittle material for Johnson-Holmquist model Fracture strength of brittle material for Johnson-Holmquist model Damage for brittle material fracture for Johnson-Holmquist model and for Johnson-Cook failure model kg Density, mm 3 Shear modulus, GP a Hugoniot elastic limit, GP a Pressure at the Hugoniot elastic limit, GP a Intact normalized strength constant for Johnson-Holmquist model Fractured normalized strength constant for Johnson-Holmquist model Intact strength pressure exponent for Johnson-Holmquist model Strain rate coefficient for Johnson-Holmquist model 1 Reference strain rate ms Maximum normalized fractured strength for Johnson-Holmquist model Maximum tensile hydrostatic pressure for Johnson-Holmquist model, GP a Damage constants for various fracture models Bulk modulus for Johnson-Holmquist model, GP a Pressure coefficients for Johnson-Holmquist model, GP a Bulking pressure coefficient for Johnson-Holmquist model vonMises flow stress for Johnson-Cook material model, GP a Plastic yield stress for Johnson-Cook material model, GP a Plastic hardening parameter for Johnson-Cook material model Plastic hardening exponent for Johnson-Cook material model Strain rate coefficient for Johnson-Cook material model Strain rate Dimensionless plastic strain rate for Johnson-Cook material and fracture model Homologous temperature for Johnson-Cook material and fracture model Temperature exponent for Johnson-Cook material model Melting temperature, K Reference temperature, K Small increment of equivalent plastic strain for Johnson-Cook fracture model Equivalent strain to fracture for Johnson-Cook fracture model Global displacement vector Global force vector Global stiffness matrix Element displacement vector Element stiffness matrix Vector of minimum relative densities The number of elements discretizing a design domain The penalization power for topology optimization Design domain material volume Design domain initial material volume Design space volume fraction

2

I.

Introduction

As long as humans have built fortified structures to protect valuable assets, they have also designed projectiles to defeat those structures. On the modern battlefield, precision guided munitions have vastly increased the lethality of air-delivered weapons since their widespread proliferation during the First Gulf War. Despite increases in guidance capability, however, the design of penetrating warheads themselves has remained unchanged over the same time period. Generally speaking, the currently fielded engineering solution to creating a hard target penetrating warhead is simply to increase case thickness, without adding any internal structure [1]. While this approach is effective in getting an explosive charge through thick barriers, it severely hinders the lethality of the weapon since most of the explosive energy released upon detonation is expended in simply breaking up the outer case. The relationship between case mass, explosive mass, and initial fragment velocity (as a measure of lethality) is given by the Gurney model, presented as equation (1) [2].

√ V0 =

 2E

M 1 + C 2

 −1 2 (1)

In this model, V0 is the initial velocity of the ejected fragments, M is the mass of the outer shell of the warhead, C is the mass of high explosive within the warhead, and

√

2E is an empirically derived

constant unique to the explosive used. The goal of this research is to create a warhead design which increases V0 by decreasing M , without sacrificing penetrative performance of the weapon. To improve upon this thick-walled warhead design, topology optimization is used to determine the appropriate structural layout to enable a thin-walled warhead to maintain comparable survivability to a thick-walled design. Loading conditions for optimization are determined from live fire test data [3] by creating dynamic, nonlinear FEA simulations which are designed to match known test results. Additive manufacturing is utilized in the production of the topology optimized penetrator design, as the internal structural layout of such designs is prohibitively difficult to manufacture by traditional (subtractive) manufacturing methods. As an example of the complexity of structures recently researched in this field, a penetrator model designed by Richards and Liu is included as Figure 1 [4]. 3

Fig. 1 An internal view of a recently tested penetrating warhead design [3].

II.

Impact Simulation and Force Distribution Determination

Previous work in the field of optimized warhead design produced live-fire test data of both standard and optimized penetrating warheads [3]. Additionally, other research conducted by Teng et al. [5] and Tai and Tang [6] explored the numerical simulation of concrete penetration events by steel projectiles using live-fire test data as a baseline for accuracy evaluation. For the simulations conducted in this study, the explicit FEA solver Altair RADIOSS was used. The present section details the material constitutive and erosion models used, the FEA setup of the impact simulation, and the results obtained. A method for determining static load distributions from the dynamic impact simulation is presented to enable application of simulation data to a topology optimization.

A.

Impact Simulation Setup and Results

Three material models are defined within the impact simulation. Those models represent the materials making up the un-reinforced concrete target, the stainless steel warhead, and the sand used to simulate explosive filler during live-fire testing. Only a brief discussion of each material constitutive model is presented, along with the pertinent parameters used. The reader is directed to the appropriate references for a full development of each model. For the unreinforced concrete target, the brittle material model of Johnson and Holmquist was used [7]. This model is appropriate for brittle materials subjected to large strains, high strain rates and high pressures, and is also applicable to both Lagrangian and Eulerian formulations [7]. The model determines the equivalent strength (σ ∗ ) of the brittle material as a function of its intact strength (σi∗ ), its fracture strength (σf∗ ), and damage (D). The equivalent strength expression is presented as equation (2). 4

σ ∗ = σi∗ − D(σi∗ − σf∗ )

(2)

In order to fully describe the above parameters, several material constants are required. These constants consist of strength, damage, and equation of state (EOS) constants. Parameters used in this research describing 7 ksi concrete are presented in Table 1. Although concrete targets used for testing in this research had a stated compressive strength of 5 ksi, all were aged at least seven years and their resulting aged compressive strength was assumed to more closely resemble 7 ksi.

Table 1 Material parameters describing the Johnson-Holmquist constitutive model.

Strength Parameters kg Density, ρ ( mm 3) Shear Modulus, G (GPa) HEL (GPa) PHEL (GPa) A B n C 1 ) ε˙0 ( ms ∗ σf,max T (GPa)

2.24×10 13.567 2.79 1.46 0.75 1.65 0.76 0.007 0.001 0.048 0.004

−6

Damage Parameters D1 D2

0.03 1

EOS Parameters K1 (GPa) K2 (GPa) K3 (GPa) β

17.4 38.8 29.8 1

In Table 1, HEL represents the Hugoniot elastic limit of the concrete, and PHEL represents the pressure in the material at the Hugoniot elastic limit. A, B, N and C are non-dimensional ∗ constants, ε˙0 is the reference strain rate, and σf,max is the maximum normalized fractured strength

of the material. T is the maximum tensile hydrostatic pressure. D1 is the damage constant, and D2 is the damage exponent. K1 is the bulk modulus of the material, while K2 and K3 are pressure coefficients of the EOS model. β is the fraction of elastic energy loss converted to potential hydrostatic energy, referred to in RADIOSS as the bulking pressure coefficient. Values in Table 1 were taken from Tai and Tang where appropriate [6]. Values not available in Tai and Tang were taken as the generic concrete model available in RADIOSS. The elasto-plastic material model of Johnson and Cook (J-C) was used to describe the stainless 5

steel warhead material [8]. This model includes strain rate and temperature effects widely used to model elasto-plastic material behavior in explicit FEA codes. In the J-C model, materials behave as linear-elastic when the equivalent stress is below the plastic yield stress. Beyond the plastic yield stress, vonMises flow stress, σ, is calculated using equation (3) [8].

  ε˙ σ = (a + bεnp ) 1 + c ln (1 − (T ∗ )m ) ε˙0

(3)

In equation (3), a is the plastic yield stress (GPa), b is the plastic hardening parameter, n is the plastic hardening exponent, εp is the equivalent plastic strain, c is the strain rate coefficient, ε˙ is the strain rate, ε˙0 is the reference strain rate, T ∗ is the homologous temperature, and m is the temperature exponent. The argument of the natural logarithm in equation (3) is redefined as ε˙∗ , the dimensionless plastic strain rate. Values for this model are obtained from torsion tests over a range of strain rates and Hopkinson bar tests over a range of temperatures [8]. The test data gathered by Richards and Liu utilized a warhead fabricated from 15-5 precipitation hardening (PH) stainless steel. J-C parameters for this alloy determined by Mondelin et al. are used in this research, and are presented in Table 2 [9]. Generic values for precipitation hardening martensitic stainless steel were used where Mondelin et al. did not provide values [9]. The constants given in Table 2 represent wrought 15-5PH stainless steel, as parameters for additively manufactured material were not available at the time of this writing.

Table 2 J-C material model parameters for the steel warhead.

J-C Parameters kg Density, ρ ( mm 3) Young’s Modulus, E (GPa) Poisson’s Ratio, ν a (GPa) b (GPa) n c 1 ε˙0 ( ms )

Temperature Parameters Tmelt (K) Tref (K) m

7.85×10−6 212 0.291 0.855 0.448 0.14 0.014 0.001

1713 298 0.63

For the sand filling the warhead, a linear elastic material law was used. The density of the 6

material is known as 1.6×10−6 kg/mm3 . For this simple material law, only Young’s Modulus (E) was needed in addition to density to fully describe the material. According to Berney and Smith, the Young’s Modulus of isotropically confined sand varies as a function of the effective mean stress of the sand [10]. Since this value is not known a priori, and since it will vary throughout the penetration event, an estimate of an appropriate value for this property was made. A reasonable value of E was determined as 1.5 GPa through iterative numeric simulation and comparison with live-fire warhead plastic deformation. For the concrete and steel material models, erosion (or failure) criteria were incorporated in order to allow for penetration of the target, as well as to allow for the removal of material considered to have failed during the simulation. As noted by Teng et al., erosion criteria, specifically for the target elements, are derived for the numerical simulation and do not necessarily represent experimental data [5]. For the concrete material model, a simple tensile strain failure criteria was incorporated. This criteria removes any concrete element experiencing a tensile strain of 0.5. When this criteria is met, the element is considered to no longer contribute to the penetration process and is removed from the simulation. The strain value of 0.5 was chosen as it allows the warhead penetration depth to match live-fire test data. For the steel material model, the failure criteria of Johnson and Cook was used [11]. In this model, damage, D, is defined by equation (4), where ∆ε is a small increment of equivalent plastic strain and εf is the equivalent strain to fracture under the material’s given conditions. Fracture occurs when D = 1.0. The calculation of εf is presented as equation (5), where numbered D coefficients are empirically derived constants for the given material [13].

D=Σ

∆ε εf

  εf = [D1 + D2 exp(D3 σ ∗ )] 1 + D4 ln ε˙∗ [1 + D5 T ∗ ]

(4)

(5)

In equation (5), σ ∗ is the dimensionless pressure ratio defined as the ratio of the average of the 7

three normal stresses and the vonMises equivalent stress, and ε˙∗ and T ∗ are the same as previously defined in the J-C elasto-plastic material model. The values of constants used in this research are taken from Johnson and Holmquist [12], though constants used resemble those found for 4340 steel in reference [11]. The explicit impact simulation was modeled in two-dimensions using a plane strain formulation as in Teng et al. and Tai and Tang [5, 6]. As the goal was to develop a warhead capable of surviving a penetration event, the test scenario of Richards and Liu considered as the worst-case for the warhead was reproduced in the finite element model. Critical aspects of this case were the firing velocity, target angle of obliquity (AoO) and warhead angle of attack (AoA) at impact, which combine to generate a dynamic load difficult for a thin-walled warhead to survive. In total, 104,892 elements were used to discretize the warhead, its internal sand, and the concrete target. Reduced integration elements with physical hourglass control were used to improve computational efficiency of the model. The final result of the model, showing a standard warhead impacting the concrete target, is shown as Figure 2. With the parameters of the impact simulation tuned to match test results, the fully-thick standard warhead mesh was replaced with a half-thick warhead mesh, and the simulation was ran again. Though the results of the thin-walled simulation are not physically meaningful, they allow the determination of force distributions for direct application to topology optimization. Results of the thin-walled simulation are presented as Figure 3.

B.

Force Distribution and Constraint Determination

Relevant loading conditions for application to a topology optimization problem were next found using the results of the impact simulation. As RADIOSS geometrically distributes the mass of all elements to their attached nodes in its solution process, node acceleration vector components output by the solver enables the solution of Newton’s Second Law at any moment in time throughout the simulation, across the entire body of the warhead. To generate useful results for topology optimization, force distributions corresponding to a few critical time periods throughout the penetration were needed. To determine which moments in time were most critical to consider, both qualitative and quantitative approaches were taken. First, 8

Fig. 2 Final standard warhead penetrating simulation result.

qualitatively, it is intuitive the force distribution corresponding to the moment of impact between the warhead and the target is critical to consider. The inertia of the warhead is at its maximum at this time, and the magnitude of the resulting external force is large, even if it only acts over a small portion of the warhead. Second, more quantitatively, the moment in time corresponding to the largest average magnitude of nodal acceleration was found, and a load distribution was derived representing that moment. Lastly, two further critical times were determined by examination of accelerometer output generated by RADIOSS. Within RADIOSS, the user may attach an “accelerometer" to any node within a model. As the solver runs, it then applies a four-pole Butterworth low-pass filter with a user-defined cutoff frequency to the raw acceleration output at the node to which the accelerometer is attached [13]. At the completion of the solver run, the user is able to view accelerometer output as a function of time throughout the penetration event. Using a distribution of these accelerometers across the length of the warhead, two further critical time frames were 9

Fig. 3 Simulation result using thin-walled warhead geometry.

chosen. These times correspond to a time where the magnitude of accelerometer output near the nose of the warhead was at its greatest, and to a time where output along the body of the warhead was at its greatest. Figure 4 shows the approximate penetration depths corresponding to each one of these four significant moments in time. A load distribution consisting of vector force components at each node across the thin-walled warhead’s body was resolved at each of the four time steps shown in Figure 4. At impact, unfiltered acceleration data was used. For each of the remaining three critical times, nodal acceleration data was filtered in MATLAB using a four-pole Butterworth low-pass filter with a cutoff frequency of 1650 Hz to mimic the accelerometer function of RADIOSS. Filtered acceleration data was then used to resolve force distributions at the remaining three critical times. In determining how best to constrain the model, test specimens from Richards and Liu were 10

Fig. 4 Penetration depths corresponding to the four critical times during warhead penetration.

Fig. 5 Test specimens with center of gravity location noted in red.[3]

examined [3]. Two of these warheads are shown in Figure 5, where the red line indicates the approximate location of the warhead’s center of gravity. Inspection of the test articles shown in Figure 5 shows these warheads either deformed or failed near their respective centers of gravity. This indicates the inertia carried by the aft end of the warhead generated a large bending moment when combined with the accelerations imparted on the fore of the warhead by contact forces with the target. This bending moment seems to have concentrated stresses at or near the center of gravity of the test warheads. Consequently, it was determined the best means of constraining the warhead for analysis and topology optimization was to apply a clamp at the location where the center of gravity was likely to exist in the finished design, slightly forward of the warhead’s midpoint. Applying a constraint at this point causes stresses to concentrate there, and consequently causes the solver to commit material to the area to reduce 11

compliance. For this reason, the center of gravity clamp was chosen to aid in resisting the type of failures seen in Figure 5. This constraint was applied to all four load distributions used in the topology optimization of the warhead.

III.

Topology Optimization

Topology optimization is a concept-level design tool allowing an engineer to determine proper material distribution for a structure in response to given loading conditions [14]. Topology optimizations conducted in the course of this research utilize the Altair products HyperMesh and OptiStruct for pre-processing and solving, respectively. Topology optimization within OptiStruct utilizes the power-law approach known as the simple isotropic material with penalization (SIMP) method. Within this method, all material properties within a defined design space are held constant and materials are considered as linear-elastic only. The resulting stiffness of each element is scaled by multiplication of the element’s relative density, a value between 0 and 1, raised to a penalization power p. For reference in subsequent discussion, the term “compliance”, whether applied locally to an element or globally to a domain, is defined as the inverse of stiffness. When the objective of the optimization is to minimize the compliance of a domain (c(x)), as it is in this research, the topology optimization problem is written as  Nel X   min : c(x) = U KU = (xe )p uTe ke ue    x   e=1      V (x)   subject to : =f Vvol,0       : KU = F         : 0 < xmin ≤ x ≤ 1 T

(6)

where U is the global displacement vector, F is the global force vector, K is the global stiffness matrix, ue is the element displacement vector, ke is the element stiffness matrix, x is the vector of design variables (element densities in this case), xmin is a vector of minimum (but non-zero) relative densities, Nel is the number of elements discretizing the design domain, p is the penalization power (equal to 3 in this research), V (x) is the material volume, Vvol,0 is the design domain volume, and f is the volume fraction defined by the user [15]. This approach “penalizes" elements of low density, 12

as the effect of the exponent p is magnified at those elements by comparison to elements with a relative density value close to 1.

A.

Weighted Compliance Considerations

Weighted compliance is a method used to consider multiple load and constraint combinations simultaneously in a topology optimization [14]. When conducting a weighted compliance optimization, the user defines a relative weight for each load considered in the optimization setup. For the present project, weighting values between 0 and 1 were used. For a given design iteration, the solver determines the structure’s compliance to each of the given loads, multiplies the compliance by the respective load’s weight, and then sums the compliance for all loads. Design iterations are conducted until a minimum sum of weighted compliances is achieved within a set tolerance. As a simple example, if the user defines the weight for each of their given loads as 1, each load is considered equally in the final optimization result. A test matrix was carried out to determine the appropriate weights to apply to each of the four load distributions determined previously. This process allowed the final design to appropriately represent each loading condition to ensure survival throughout the penetration event. Table 3 shows pertinent information for the discussion of the weighted compliance optimization test matrix. The four loads generated previously are labeled for ease of discussion, and the compliances resulting from warhead optimizations considering those loads individually are presented. This information provides a performance baseline against which the weighted compliance solutions are subsequently compared. Table 3 Optimized warhead compliances considering each load distribution individually.

Load Description

Load Label

Time of Impact Maximum Average Acceleration Magnitude Maximum Nose Acceleration Magnitude Maximum Body Acceleration Magnitude

(a) (b) (c) (d)

Optimized Warhead Compliance to Single Load Only (mm/kN ) 0.922 0.0422 0.0343 0.0261

Beginning with an equally weighted optimization, subsequent tests were run whereby loads generating lesser compliance values for their individual optimizations were successively de-weighted. 13

Four total test cases were run, and the parameters and results of each are presented as Table 4. Table 4 Weighted compliance test matrix and results.

Load (a) (b) (c) (d)

Test 1 Compliance Test 2 Compliance Test 3 Compliance Test 4 Compliance Weight Increase Weight Increase Weight Increase Weight Increase 1 1.80% 1 1.02% 1 0.99% 1 1.25% 1 32.66% 0.9 34.91% 0.8 36.37% 0.7 41.19% 1 33.98% 0.7 36.33% 0.6 37.83% 0.5 43.15% 1 34.33% 0.5 35.56% 0.4 36.51% 0.3 39.17%

Table 4 generally demonstrates the expected trend as loads (b)-(d) are de-weighted. Compliance to load (a) decreases and the compliance to loads (b)-(d) increases as the relative weighting values associated with loads (b)-(d) are decreased. In Test 4, a deviation from this trend arises as compliance against load (a) increases as compared to Test 3. EXPLAIN THIS AND JUSTIFY THE USE OF TEST 3 WEIGHTS

B.

Member Size Control

The warhead design arising from the previous determination of ideal weighted compliance parameters led to the mathematically optimum truss panel layout for use in design of the full, threedimensional (3D) warhead. Examination of the solution vector plotted as a contour, where blue represents elements of very low relative density and red represents fully dense elements, shows the difficulty in manufacturing the design. This contour is presented as Figure 6a. A large amount of partially dense elements and very small truss members makes this solution difficult to translate to a design consisting of only fully dense material and void space. Such a transition would necessarily deviate from the optimization solution, and an unknown increase in compliance would arise. To alleviate this issue, member size control is implemented in the topology optimization. Within OptiStruct, the user may bound minimum and maximum member dimensions, as well as the minimum gap between members. Appropriate scales were chosen for each of these dimensions, and again a test matrix was carried out to determine the impact of implementing these parameters. The results of the member size control test matrix are presented in Table 5. As expected, the implementation of any member size control parameter forces the solver away from the mathematically optimum topology, and therefore a compliance increase is noted for each 14

Table 5 Member size control test matrix and results.

Test Case No Member Size Control Minimum Member Size Only Minimum and Maximum Member Size Minimum, Maximum and Minimum Gap

Compliance Increase 10.77% 78.10% 104.24%

test case. When implementing only minimum member size control, however, a relatively small increase of only 10.77% is suffered. Comparison of the optimization solutions presented in Figure 6 justifies the use of this parameter and its known compliance increase. By accepting a relatively small, but known, increase in compliance, a great improvement in manufacturability is achieved. For this reason, the solution shown as Figure 6b was selected as the final topology optimization solution.

(a) Optimization solution without member size control

(b) Optimization with minimum member size control

Fig. 6 Illustration of the effect of minimum member size control.

IV.

Three Dimensional Design Interpretation and Analysis

To this point, design considerations have focused only on a two-dimensional (2D) warhead truss panel. The final design must then translate this optimized 2D solution into a 3D warhead. To begin this process, applicable constraints of the DMLS process were consulted. Most notably, the use of sacrificial support material inside the warhead’s case is not desired as its removal is not possible 15

Fig. 7 Truss overhang constraints for warhead manufacturing.

after warhead fabrication, and as its presence occupies volume intended for explosive filler. When built in a vertical orientation, truss angles must therefore allow each member to support itself during the build process. In practice, this requirement translates to a maximum truss overhang angle of no more than 45 degrees, as shown in Figure 7. In addition to the overhang constraint, the final mass of the optimized warhead design must match the control design mass, and the center of gravity must lie in the forward half of the warhead for aerodynamic stability. Even given the 2D topology optimization solution and the constraints listed above, an infinite number of design interpretations exist when developing a 3D design. Two such solutions are presented in Section IV A.

A.

Three Dimensional Designs

Figure 8 presents the fist design developed from the 2D topology optimization solution. The figure shows the topology optimization solution, reflected on itself for symmetry and with very low density elements isolated. The 2D sketch used to create truss panels within the warhead is overlaid on the optimization solution. The sketch lines deviate from the optimization only where needed in order to satisfy the overhang angle constraint of the DMLS process. Once rotated to form a solid, truss panels were formed in the design space using a series of extruded and lofted cuts. The selection of four full truss panels and eight symmetrical channels was made to form a “wagon-wheel” design 16

Fig. 8 The first 3D warhead design interpretation.

capable of surviving any impact orientation, while maintaining reasonable truss thickness. The 3D design presented in Figure 8 is referred to as the “waisted” design for the remainder of this article. The waisted design represents the most strict adherence to the 2D optimization as was achievable within the constraints of the project. Several key features highlighted in Figure 8, however, bear further discussion. The most glaring feature is the solid waist from which the design bears its name. This waist was created in response to the thick walls of the optimization solution in the forward half of the warhead. By keeping this region as a solid waist, it was hoped the warhead could resist the bending loads apparently responsible for the failure of Richards and Liu’s optimized warhead against an oblique target [3]. It does, however, create a sharp transition area where the thick and thin wall regions meet, which could result in strong stress concentrations. Additionally, the thick waist removes a good deal of the thin-walled surface area, whose creation was the original purpose of the project. Given the apparent limitations of the waisted warhead design, a second design interpretation was created. The goals of this design were to eliminate any sharp stiffness transitions while maintaining as much thin-walled surface area as possible. Figure 9 presents the isolated design space, a fully cased cutaway and highlighted critical design features of this second design interpretation. The design shown in Figure 9 is referred to as the “ribbed” design for the remainder of this report. 17

Fig. 9 The second 3D warhead design interpretation.

Note the transition area seen in the waisted warhead design is eliminated in the ribbed design, and significantly more thin-walled surface area is preserved. Also, the ribbed design represents a 40.2% increase in thin-walled surface area as compared to the waisted design. Lastly, the CG position is significantly farther forward in the ribbed design, though the waisted design meets the forward CG criteria stated at the outset of the project.

B.

Design Analysis

Though a full 3D impact simulation was beyond the computational capability of this project, a meaningful design comparison between the waisted and ribbed designs was conducted using a simple static analysis. To conduct this analysis, a similar approach as Richards and Liu’s “bodyangle" load was used. To apply this load, the nose of each warhead was clamped in six degrees of freedom, and the tail was constrained to only longitudinal translation. A force angled at the expected obliquity of target impact was then applied to all nodes on the external surface of the warheads. Force magnitudes were determined by a simple impulse and momentum balance, using high-speed video as well as mass and velocity data taken by Richards and Liu to determine the proper inputs in determining the force to apply. This force was then distributed over all external surface nodes of each warhead, and the resulting vonMises stresses were resolved. Resulting contour 18

(a) Ribbed warhead stress response to angled body load.

(b) Waisted warhead stress response to angled body load.

Fig. 10 Static analysis stress comparison in response to the angled body force load. Gray elements shown within dashed red box have vonMises stress values in excess of 1.17 GPa, the ultimate tensile strength of traditionally manufactured 15-5PH H925 stainless steel [16]

plots are bounded between 0 GigaPascals (GPa) and 1.17 GPa, the ultimate tensile strength of traditionally manufactured 15-5PH H925 stainless steel [16], and presented as Figure 10.

Examining the static analysis results, the ribbed design clearly outperforms the waisted design. Also, the angled body load developed from Richards and Liu’s work provides a compelling argument as to how the waisted warhead could potentially fail during a live-fire test with target obliquity. Though an extreme simplification of the dynamic event, when force magnitudes applied to this body force load are resolved by solving an impulse and momentum balance, the resulting stress distributions predict the ribbed design to survive the impact and the waisted design to fail at its transition region.

Examining the foregoing results qualitatively, the conclusion is stress must not experience an abrupt material transition when translating along the body of the warhead. The transition at the end of the thick waist in the waisted design presents an area where concentrated stresses are not efficiently transmitted by the truss structure alone, and therefore represents the area where the warhead is likely to fail at impact. 19

V.

Experimental Results

Test warheads for this project were fabricated by i3D Manufacturing in Bend, Oregon using an EOS M290 DMLS machine. Eight total warheads were built, consisting of three designs. Two control designs were produced, featuring a thick exterior wall and no internal trussing. Also, two waisted and four ribbed warhead designs were built. Minimal machining was conducted on the warheads, consisting of surface smoothing and exterior support material removal. Each warhead also received a threaded, traditionally manufactured end-cap to allow kiln-dried sand, as a surrogate for explosive filler, to fill each warhead during testing. Prior to firing, each warhead was heat-treated to an H925 condition by the Air Force Research Laboratory, Munitions Directorate (AFRL/RW) at Eglin Air Force Base, FL. The warheads were then filled with sand, capped, and marked with an alphanumeric designator and orientation lines intended to aid analysis of post-mortem test articles. The test matrix was designed to provide a comparison between the control design and the two optimized designs at operationally relevant impact velocities and angles of obliquity. As neither the external geometry nor the mass of the optimized designs varied from the control design, no increase in penetration depth was expected. Rather, the test was designed to determine if the thin-walled warheads could survive the penetration event intact, while maintaining similar penetration depth to the control design. The results of the control warhead impacts are presented as Figure 11. Control warheads

(a) Control warhead impact at normal obliquity

(b) Control warhead impact with target obliquity

Fig. 11 Control warhead design impacts and post-mortem articles.

penetrated approximately their full body length, and both survived the penetration event in tact. While the normal target impact article suffered very little plastic deformation, the oblique target 20

article showed a crushing deformation at the tail of the warhead. This crushing deformation seemed to result from the rotation of the warhead’s flight path during the impact, and the resulting “tailslap” with the fractured face of the target as seen in the top right of Figure 11b. No perceptible plastic deformation occurred as a result of bending at the center of gravity of the control design during the oblique target impact test. With the control designs as a performance baseline, ribbed warheads were next fired at normal and oblique targets at the same velocity as the control design tests. Results of these tests are presented as Figure 12, and post-mortem test articles are presented next to cutaways revealing their internal structure for reference. Note the similarity of the post-mortem test articles between

(a) Ribbed warhead impact at normal obliquity

(b) Ribbed warhead impact with target obliquity

Fig. 12 Ribbed warhead design impacts and post-mortem articles.

Figures 11 and 12. The thin-walled ribbed warhead design survived impact with both normal and oblique targets, and the recovered test articles exhibit nearly identical deformation to the control designs. The penetration depth of the ribbed warheads was also comparable to the control design, though the normal impact penetration depth was approximately 10% less for the ribbed design than the control design. This discrepancy is likely do to small variability in actual firing velocity (though powder mass used for each shot was held constant) and warhead angle of attack at impact between these two tests. These shots demonstrated the validity of the project’s concept, as well as the design methodology used to generate the ribbed warhead design. To determine where potential weaknesses exist in the ribbed design, the remaining two ribbed warheads were fired at a 20% increased velocity as compared to the control designs. Results of these 21

tests are presented as Figure 13. As shown in Figure 13, the ribbed design survived impact with

(a) Ribbed warhead impact at normal obliquity with increased velocity

(b) Ribbed warhead impact with target obliquity with increased velocity

Fig. 13 Ribbed warhead design impacts and post-mortem articles after increased velocity tests.

both normal and oblique targets. Though the effect of impact yaw is apparent in Figure 13a, and the tail-slap phenomena is magnified in Figure 13b, both of these warheads again survived intact. This result seems to indicate the increased velocity at impact affected the magnitude of forces experienced by the warhead, but not their relative location or orientation. As topology optimization considers relative forces, this indicates the optimization solution would not significantly change were simulation velocities leading to load distribution determination increased. Operationally, this conclusion carries considerable significance, as application of a single topology optimized design across a wide range of impact velocities appears possible. The final test shots were conducted using the waisted design warheads. Results of these tests validated the simple static FEA presented in Section IV B, and are presented as Figure 14. Waisted warheads were fired at the same velocity as the control warheads. The waisted warhead designs showed significantly less survivability to impact than the ribbed designs. This is a positive result, as it validates the boundary conditions and loads used to evaluate the final designs in static FEA. Though the static analysis was conceptually and computationally simple to execute, it properly predicted material behavior during the dynamic impact event and is therefore a valuable tool for analysis of future warhead designs. The waisted warhead design was stated in Section IV A as the most strict representation of 22

(a) Waisted warhead impact at normal obliquity

(b) Waisted warhead impact with target obliquity

Fig. 14 Waisted warhead design impacts and post-mortem articles.

the topology optimization solution attainable within the manufacturing constraints of the project. The failures observed by the waisted warheads could therefore lead one to conclude the boundary conditions and loads applied to the topology optimization were not well-posed. This is not a correct conclusion, however, as the dominant design choices leading to the failure of the waisted design occurred during the transition from the 2D optimization solution to full, 3D warhead design. When the same optimization solution was applied to the ribbed design, the longitudinal ribs provided the load paths required for the optimized truss layouts to transmit stresses as designed allowing the warhead to survive. Two-dimensional topology optimization solutions are therefore valid as guide in the development of warhead designs, but engineering judgment is also critical to the success of the final design. When transitioning from 2D to 3D design, the critical conclusion from these results is to avoid sharp stiffness transitions leading to stress concentrations and material failure at impact. The waisted warhead design featured a pronounced such transition, and as a result the warhead failed catastrophically at impact with an oblique target.

VI.

Conclusion

The most conceptually challenging question answered through this research was how to resolve a meaningful static load distribution out of an extremely dynamic penetration event. Even with this 23

solved, the question of how to constrain the analysis still remained. By placing the static constraint at the CG of the warhead, a constraint was created which in practice concentrated stresses in the structure at this critical region. Topology optimization solutions, therefore, responded to this stress concentration by allocating more material to the area supporting the warhead’s CG. The survivability of the ribbed warhead design observed during testing illustrates the validity of the foregoing approach. Though subsequent decisions were made which also impacted the final design of the warhead, no single phase of the project carried as much influence on its structural design as this transition from dynamic event to static loading. In producing the 3D design, significant deviations from the optimization solution were required in order to meet manufacturing constraints. Additionally, some areas of partially-dense material existed in the optimization which required interpretation into a design consisting of only fully-dense material and void space. Consequently, this phase of the project is where the greatest amount of engineering judgment was required. As discussed in Section IV A, the waisted warhead design actually represented the closest adherence to the optimization solution, though intuition immediately showed it was not likely to survive the oblique target penetration. This does not indicate the topology optimization was not valid, but rather highlights the requirement for judgment in transitioning from 2D to 3D design. The ribbed design, while more of a departure from the optimization solution, provided better survivability during live-fire testing and also offered significantly more thin-walled surface area than the waisted design. The conclusion, therefore, is topology optimization, though a valuable conceptual design guide, does not provide the “final answer” for part design. Prudent engineering demands analysis of parts derived from topology optimization solutions, and the present project proved no exception. The angled body force load corresponding to Richards and Liu’s work was meaningful in identifying the failure area of the waisted warhead. In live-fire testing, the waisted warhead failed cleanly against the oblique target at the precise region highlighted in the FEA. This accuracy, coupled with its relative computational simplicity, makes the angled body force load distribution a valuable tool for any future 3D warhead analyses. Through this research, a survivable, thin-walled penetrating warhead was designed, built and 24

live-fire tested. The test results proved both the validity of the approach used to achieve the design, and also the concept of the project. Additive manufacturing via DMLS provided a rapid, costeffective and accurate means by which to produce the warheads in 15-5PH stainless steel. These warheads highlight the capabilities of weapons of the future, which are more lethal than current designs, are tailored for specific target sets, and are additively manufactured as needed by operational military forces.

Acknowledgments

The authors would like to thank the project sponsors, AFRL/RW as well as the Joint Aircraft Survivability Program Office, for their support of this project.

References

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