Research Papers

Analysis of Cone Wave Reflection in Finite-Size Elastic Membranes and Extension of the Ballistic Impact Problem From Elastic to Viscoelastic Membranes

[+] Author and Article Information
Amit Singh

Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: sing0335@umn.edu

Sinan Keten

Mechanical Engineering,
Civil and Environmental Engineering,
Northwestern University,
Evanston, IL 60208

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received March 11, 2018; final manuscript received April 24, 2018; published online May 21, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(8), 081004 (May 21, 2018) (7 pages) Paper No: JAM-18-1143; doi: 10.1115/1.4040119 History: Received March 11, 2018; Revised April 24, 2018

The transverse ballistic impact on a two-dimensional (2D) membrane causes a truncated deformation cone to develop in the wake of tensile implosion waves. Here, the cone wave reflected from the finite boundaries of the elastic membrane has been studied analytically. A first-order linear nonhomogeneous differential equation for the ratio of the reflected cone wave front velocity to the speed of tensile waves is derived, which is further used to calculate the traveling time taken by the reflected cone wave to reach to the projectile surface. Since the reflected wave starts when the membrane is already in a deformed configuration, the speed of the reflected cone wave is a function of radius r in the cylindrical coordinates as opposed to almost constant speed of the incoming cone wave studied in the literature. The analytical results are validated with molecular dynamics (MD) simulations of the ballistic impact of projectiles onto a single layer of coarse-grained (CG) graphene. In the second part of the paper, we analyze the membrane impact problem for linear isotropic viscoelastic materials and find that the tensile wave speed for stresses and displacements is the same as that obtained in the case of a linear isotropic elastic material. We also show that only under special conditions, self-similar solutions for the cone wave are possible in viscoelastic materials modeled by Maxwell, Kelvin–Voigt, or a combination of similar models. Our findings lay some grounds on which further studies on the ballistic response of viscoelastic materials can be performed.

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Grahic Jump Location
Fig. 1

Axisymmetric membrane shape during an MD simulation of the ballistic impact on graphene membrane at time: (a) t = −tp when the projectile hits the membrane, (b) t = 0 when the cone wave reaches the membrane edge, (c) t = t1 > 0 when reflected cone wave is traveling toward the projectile surface, (e) t = trp when the reflected cone wave hits the projectile surface, and (f) immediately after trp when failure occurs. (d) Illustration of the cylindrical coordinate system (r, ϕ, y) and displacements (u, 0, v) of the material element with center coordinates (r, ϕ, 0). The solid black curve corresponds to the incoming cone wave at t = 0 and the solid blue curve with cross corresponds to the reflected cone wave for a time t > 0. The blue dashed line is just a sketch to reflect a smooth transition of the solid blue curve with cross to the remainder of the solid black curve. The colors are visible in web version of the paper.

Grahic Jump Location
Fig. 2

The local in-plane strain εt at time t = 0 as a function of radial distance r when the incoming cone wave hits the edge of the membrane. The fluctuating solid red curve and the smooth solid blue curve correspond to MD and analytical results, respectively. The colors are visible in web version of the paper.

Grahic Jump Location
Fig. 3

Time ratio (trp/tp) versus ζ = rp/a expressed in Eq. (17)



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