PSI - Issue 35

Deniz ÇelikbaŞ et al. / Procedia Structural Integrity 35 (2022) 269 – 278 D. C¸ elikbas¸ / Structural Integrity Procedia 00 (2021) 000–000

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Finally, in Eq. (1), σ ∗ f represents the fractured material strength given in Eq. (6), where B represents damaged strength coe ffi cient, M represents damaged strength exponent, and σ f max represents normalized maximum fracture strength. σ ∗ f = B ( P ∗ ) M (1 + C ln ˙ ε ∗ ) ≤ σ f max ) (6) In the material model, the hydrostatic pressure of undamaged material can be calculated from Eq. (7). P = K 1 µ + K 2 µ 2 + K 3 µ 3 ( Compression ) K 1 µ ( Tension ) (7) where K 1 , K 2 and K 3 are the equation of state constants. The term µ is computed from µ = ρ/ ( ρ 0 − 1) , where ρ is the current density and ρ 0 is the initial density. The pressure increases with the initiation of damage. The fraction, the value between 0 ≤ β ≤ 1 , of the elastic energy loss is converted into hydrostatic potential energy / pressure. The pressure increases e ff ects the bulking pressure. If the material is undamaged, then the bulking pressure is 0. Hugoniot elastic limit given in Eq. (8). where G represents the shear modulus. The pressure strength components for normalization at HEL are given by Eq. (9). P HEL = K 1 µ HEL + K 2 µ 2 HEL + K 3 µ 3 HEL σ HEL = 1 . 5( HEL − P HEL ) (9) The mechanical properties of the JH2 material model are given in Table 1. HEL = K 1 µ HEL + K 2 µ 2 HEL + K 3 µ 3 HEL + G (4 / 3) µ HEL 1 + µ HEL (8)

Table 1. Mechanical parameters for alumina ceramic (MAT 110 – Johnson Holmquist Ceramics) Toussaint and Polysois (2019). Parameter Symbol Value

Unit

kg / m 3

Density

3860 90 . 16 2 . 139

ρ

Shear Modulus

G A B C m

GPa

Intact normalized strength parameter Fractured normalized strength parameter

− − − −

0 . 31

s − 1

Strain rate dependence

0

Fracture strength parameter Intact strength parameter

0 . 6 0 . 6

n

1

s −

1

Reference strain rate

˙ ε 0

Maximum tensile strength

T

0 . 2

GPa

σ f

20

1 x 10 2 . 79 1 . 46

Maximum normalized fracture strength

−

max

Hugonoist elastic limit

HEL P HEL

GPa MPa

Pressure component at the Hugonoit elastic limit

Fraction of elastic energy loss converted to hydro-static energy

1

− − −

β

0 . 025

Parameter for plastic strain to fracture

D 1 D 2 K 1 K 2 K 3

Parameter for plastic strain to fracture (exponent)

0 . 5

Equation of state constants Bulk modulus Second pressure coe ffi cient

130 . 95

GPa GPa GPa

0 0

Elastic constant

2.3. Projectile Material Model

Steel projectile is modeled with the Simplified Johnson-Cook material model. By using this material model, the simulation time decreases by 50% due to neglected thermal e ff ects and damage Livermore Software Technology Corporation (2018). In this model, the yield strength can be expressed as: σ y = ( A + B ¯ ε p n )(1 + C ln ˙ ε ∗ ) (10)