ISSN 1068-7998, Russian Aeronautics (Iz.VUZ), 2015, Vol. 58, No. 2, pp. 137–144. © Allerton Press, Inc., 2015. Original Russian Text © A.V. Gerasimov, S.V. Pashkov, 2015, published in Izvestiya VUZ. Aviatsionnaya Tekhnika, 2015, No. 2, pp. 3–9.

STRUCTURAL MECHANICS AND STRENGTH OF FLIGHT VEHICLES

Numerical Simulation of Fracture in Thin-Walled Structures by a Group of High-Velocity Elements A. V. Gerasimov and S. V. Pashkov National Research Tomsk State University, pr. Lenina 36, Tomsk, 634050 Russia e-mail: [email protected] Received April 25, 2014

Abstract—The objective of the work is to study the fracture pattern for thin-walled structures at collision with a group of spherical projectiles. Consequently, we reveal the special features of projectile and barrier fragmentation and formation of debris fields. DOI: 10.3103/S1068799815020014 Keywords: numerical simulation, high-velocity collision, thin-walled structures, spherical projectiles, fragmentation, probability.

The processes of interaction of the streams of high-velocity elements with thin-walled enclosures at different angles of approach to the latter, as well as the behavior of enclosure fragments and element debris behind the first shell of the construction, are of great practical interest to ensure reliable operation of spacecraft, aircraft and missile technology. The streams of high-velocity elements may have different physical nature. They may be man-made and natural meteoric particles in space, ice particles in the atmosphere, and damaging elements in all media. In this paper, we consider the processes of plate breaking by clusters of spherical projectiles and fragmentation pattern, as well as the formation of debris streams in behind-the barrier space. THE EQUATIONS DESCRIBING THE MOTION OF A COMPRESSIBLE ELASTIC-PLASTIC BODY TAKING INTO ACCOUNT PROBABILISTIC NATURE OF FRACTURE The equations describing the spatial adiabatic motion of a solid compressible medium are differential consequences of the fundamental laws of mass, pulse and energy conservation. In general, they have the following form [1–3]: —the continuity equation 1 dρ ∂vi + = 0; (1) ρ dt ∂xi

—the equation of motion ρ

dvi ∂P ∂Sij ; = ρFi − + dt ∂xi ∂x j

(2)

—the energy equation

dE P dρ = Sij εij + , (3) dt ρ dt where xi are the coordinates; t is the time; ρ is the current density of the medium; vi are the components of the velocity vector; Fi are the components of mass force vector; Sij are the components of stress tensor deviator; E is the specific internal energy; εij are the components of strain rate tensor deviator; P is the pressure. ρ

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The equations that take into account the relevant thermodynamic effects associated with adiabatic compression and strength of the medium should be added to Eqs. (1)–(3). In general case, under the influence of the forces on the solid-deformable body, both volume (density) and shape of the body are changed by different dependencies. Therefore, the stress tensor is presented as a sum of spherical tensor and the stress tensor deviator σij = Sij − Pδij , i, j = 1,2,3; δij = 1, i = j; δij = 0, i ≠ j, where δij is the Kronecker delta. To describe shear strength of the body, the following relations are used 1 ⎛ ⎞ DSij 2μ ⎜ eij − ekk δij ⎟ = + λSij ; 3 ⎝ ⎠ Dt DSij dSij = − Sik ω jk − S jk ωik ; Dt dt ∂v ∂v j 2ωij = i − ; ∂x j ∂xi 2eij =

(4) (5) (6)

∂vi ∂v j , + ∂x j ∂xi

(7)

as well as the condition of plasticity 1 1 J 2 = Sij Sij = σ2 , (8) 2 3 where eij are the components of the strain rate tensor; μ is the shear modulus; σ is the dynamic yield

stress; D Dt is the Yauman derivative. The equation of a solid state was chosen in the Mie–Grüneisen form K (1 − Γ 0 ξ 2 ) P= ξ + ρ0 Γ 0 E , 2 (1 − cξ )

(9)

where Γ 0 is the Grüneisen coefficient; c, K are the material constants; ρ0 is the initial density of the medium; ξ = 1 − ρ0 ρ . As a shear failure criterion we used the criterion of limiting equivalent plastic strain [4] ε p = ε∗p . In this case, when ε p reaches the limit value ε*p , the calculated cell is considered to be destroyed. The system of equations (1)–(9) is written in general form for the spatial motion of a deformable body. The process of destructing the real materials is largely determined by the internal structure of the medium, the presence of heterogeneities, usually caused by a different orientation of the grains in the polycrystalline material or heterogeneities in the composition of composite materials, the difference in the micro-strength inside the grain and on the intergrain or interface boundaries. Therefore, to improve compliance of a numerically simulated process with the experimental data, it is necessary to generate disturbances in the physical-mechanical characteristics of the medium being destroyed, i.e. to set a random distribution of the factors determining strength properties of the material. The introduction of information about polycrystalline structure of the material into the computational technique requires a large amount of experimental data and increased requirements for computer power that limits the ability to implement and apply this approach. In view of this,, we used a simplified version

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of probabilistic modeling of fracture mechanism. Physical and mechanical characteristics of the medium responsible for strength are assumed to be randomly distributed over the material volume. The distribution probability density of these parameters is taken in the form of various distribution laws, which are generally dependent on table (average) value of the parameter being distributed, dispersion of the parameter distribution being varied, and other characteristics of the medium. Such parameters as yield strength, tensile strength, maximum strains, and other constants that define the moment of destruction in various theories of strength and fracture criteria, are directly dependent on the number and size of defects and should be randomly distributed over the volume with dispersion depending on material homogeneity. Natural fragmentation of projectiles and barrier is calculated by introducing a probabilistic mechanism for distribution of initial defects of the material structure to describe tear and shear cracks. As a criterion of failure under intense shear strains we used the achievement of limiting values by equivalent plastic deformation. The initial heterogeneities were simulated so that the maximum equivalent plastic strain was distributed into membrane cells using a modified random number generator, which gave out a random variable obeying to the distribution law selected. The system of basic equations is added with necessary initial and boundary conditions. At the initial moment of time all points of the projectile have the axial velocity V0 in view of its sign and the barrier state is assumed to be unperturbed. The boundary conditions are as follows, namely, the conditions σ n = τn = 0 are satisfied on borders free from stress. Conditions of ideal sliding of one material relative to another along the tangent and impermeability along the normal are set on contact sites between the bodies: σ n1 = σ n 2 , vn1 = vn 2 , τn1 = τn 2 = 0, where σ n , τn are the normal and tangent components of the stress vector; vn is the normal component of the velocity vector at the point of contact; subscripts 1 and 2 refer to the bodies being in contact. To calculate the elastic-plastic flows, use is made of a technique implemented on tetrahedral cells and based on joint application of the Wilkins method for calculating interior points of the body and the Johnson method for calculating the contact interactions [2, 5, 6]. Three-dimensional domain was successively partitioned into tetrahedrals with subroutines of automatic meshing. The ideology and methodology of applying a probabilistic approach to solids fracture are completely described in [7]. TEST CALCULATIONS AND EXPERIMENT Figure 1 presents the results [8] of numerical simulation of ricocheting a steel ball-projectile with a diameter of 8 mm at interaction with a 9.5 mm-in-thick titanium barrier with a diameter of 80 mm for a time moment t = 13.23 µs. The projectile initial velocity was 3600 m/s, the angle of impact from the normal to the barrier was 75 deg. Figure 1 presents the computational domain: (a) top view; (b) side view; (c) end view. Figure 1d presents the photos of the steel ball in the initial state and the trace in the titanium plate after their interaction.

(b)

(a)

(c) Fig. 1.

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In numerical study we obtained the following values of crater parameters, namely, the major crater axis was 30.5 mm, the minor axis was 15 mm, and the crater depth was 8.2 mm. The experimental data were characterized by the following values, namely, the major crater axis was 28 mm, the minor axis was 16 mm, and the crater depth was 7 mm. COMPUTATIONAL RESULTS Previously, the interaction of cylindrical elements with barriers at lower angles of impact and erosion fracture mechanism making it impossible to consider the interaction of the fragments with each other and with the barrier was considered in [9, 10]. In [11], the authors numerically investigated the interaction of ice projectiles with the barriers made of aluminum alloy and asbotextolite. The problem was solved in a two-dimensional formulation for the case of axial symmetry. In this paper, the interaction of a group of three steel balls with an aluminum plate is considered in the three-dimensional formulation. A ball radius was 2.8 mm. The thickness of each plate is 3.5 mm sized 50×50 mm. The impact velocity is 1500 m/s. The balls were arranged in a circle of radius 7.5 mm. Figure 2 shows the results of interaction of an aluminum plate with a cluster of steel balls. Figure 2a presents the configuration of balls, that is the barrier system at instant of time t = 15 μs for an impact angle of 0 deg. Figure 2b presents the plane section of the computational domain. It is thus evident that after the collision the area of undestroyed material between the holes was quite large and the velocity of projectiles after the barrier breaking fell to 1130 m/s (Fig. 2b). Figure 3 presents the configuration of balls, that is the barrier system at instant of time t = 38.4 μs for an impact angle of 45 deg. from the normal. Here 3a and 3b are the general views and 3c presents the plane section. The increase in the impact angle to 45 deg. (Fig. 3) caused the decrease in the area of undestroyed material between the punched holes and the formation of a significant fragmentation flow behind the barrier. Figures 3a and 3b show different views of the punched holes and the fragmentation field. The plane section of balls, namely the barrier system and velocity distribution in the section is shown in Fig. 3c. Such presentation of the results obtained is used for the subsequent figures. In the case under consideration, there are two leading balls followed by the third one. The velocity of the balls after breaking fell to 980 m/s.

(a)

(b) Fig. 2.

Changing the direction of the impact by 180 deg. (Fig. 4), when there is one leading ball, one can observe cracks in the bridges between the holes and the velocity of the projectiles also falls to 980 m/s. Figure 4 presents the configuration of balls, namely, the barrier system at instant of time t = 40.4 μs for an impact angle of 45 deg. from the normal. Here Figs. 4a and 4b are the general views and Fig. 4c presents the plane section. RUSSIAN AERONAUTICS

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(c)

Fig. 3.

(a)

(b)

(c)

Fig. 4.

A similar change in the number of the leading balls was used in the two following calculations at a 60 deg. deviation of the impact direction from the normal to the barrier surface (Figs. 5 and 6). In these cases, one observes complete destruction of the bridges between the holes in the aluminum plate and the formation of a powerful fragmentation flow behind the back surface of the barrier. Figures 5a–5b and Figs. 6a–6b present the configuration of balls–barrier system at instant of time t = 34 μs for the angle of impact 60 deg. Figures 5c and 6c show the plane sections of the computational domain. The increase in the angle of impact led to the increase in the punctured thickness of the plate material and, as a consequence, to lower residual velocities of the projectiles behind the barrier.

(a)

(b) Fig. 5.

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(a)

(b)

(c)

Fig. 6.

It should be noted that velocity distribution was determined for central sections of the barrier and the ball lying on this line (Fig. 5c and Fig. 6c). The next group of calculations was related to the study of projectile ricocheting at the following angles of impact, namely, 70 deg. (Fig. 7), 75 deg. (Fig. 8), and 78 deg. (Fig. 9).

(a)

(b) Fig. 7.

(a)

(b)

(c)

(d) Fig. 8.

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Figures 7a and 7b present the general view and plane section of the ball–barrier system at instant of time t = 14.3 µs at the angle of impact 70 deg. With the increase in the angle of impact the area of the deformed material and the values of aluminum plate deflections decreased. If at the angle of impact 70 deg. (Fig. 7b) we see the formation of cracks on the barrier face at the contact with the projectile, then at the angle of impact 75 deg. (Fig. 8d) the number and size of cracks decrease. Figure 8 presents the configuration of the ball–barrier system at instant of time t = 10 µs (Figs. 8a and 8b) and t = 20 µs at the angle of impact 75 deg. Figures 8a and 8b show the general views, and Figs. 8c and 8d present the plane sections of the system under consideration at the instants of time mentioned. At an angle of impact of 78 deg. (Fig. 9d) one observes no cracks and reduced area of dents caused by balls-projectiles interactions with the plate. The ball–barrier interaction at the instants of time t = 10 µs (Figs. 9a and 9c), t = 44.5 µs (Figs. 9b and 9d) the angle of impact 78° is presented in the general form in Figs. 9a and 9b and as the plane sections in Figs. 9c and 9d.

(a)

(b)

(c)

(d) Fig. 9.

CONCLUSIONS The probabilistic approach proposed and numerical technique developed on its basis enable us to simulate the processes of barrier breaking in a wide range of angles of impact. The decrease in the angle of collision causes projectile ricocheting and change in the nature of the aluminum plate deformation. It is possible to explore projectile and barrier fragmentations as well as the nature of the forming fields of fragmentation behind the barrier. ACKNOWLEDGMENTS The work was supported by the Ministry of Education and Science of the Russian Federation in the framework of state task no. 2014/223, project no. 1567.

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