4 Finite Element Method Study of the Protection …
59
p is an area of external pressure action, and the dot above the symbol means a partial
derivative of time t. Over repeated index is summation. Strain rates are defined in
the current state metric:
˙
ε i j =
˙
U i, j + ˙
U j,i
/2,
i, j = 1, 3
,
˙
U i, j = ∂ ˙
U i /∂ X j , X j = X j | t=0 +
t
0
˙
U j dt.
(4.2)
Elastic–plastic deformation of metals and alloys is described by the relations of
flow theory with kinematic and isotropic hardening (Volkov and Korotkikh 2008).
4.2.1 MHS Filler Modeling
MHS filler is modeled by a continually homogeneous, orthotropic, physically
nonlinear medium (Hallquist 1998; Demareva et al. 2014). Strains and stresses are
defined in the local coordinate system x = [x 1 , x 2 , x 3 ]. The relationship of elastic
deformations and stresses is established on the basis of the generalized Hooke’s law:
˙
σ i j = C i jkl ˙
ε kl , i, j = 1, 3,
C i jkl =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
C 1111 C 1122 C 1133
C 1122 C 2222 C 2233 0
C 1133 C 2233 C 3333
C 1212
0
C 2323
C 3131
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(4.3)
For highly porous metals, it can be assumed that in the initial state, the Poisson’s
ratio is negligible. With this in mind, the ratio (4.3) takes the form:
˙
σ i j = A i j ˙
ε i j , A i j = C i ji j , i, j = 1, 3.
(4.4)
As the porous material is compressed, its elastic characteristics are changed. The
following ratio is accepted:
A i j = A
0
i j + β
A
s
i j − A
0
i j
(4.5)
where indices “0” and “s” mark the elastic constants of the filler in the initial state
and when the pores are completely closed. The parameter β is determined by the
current value of the relative volume v = V /V 0
59
p is an area of external pressure action, and the dot above the symbol means a partial
derivative of time t. Over repeated index is summation. Strain rates are defined in
the current state metric:
˙
ε i j =
˙
U i, j + ˙
U j,i
/2,
i, j = 1, 3
,
˙
U i, j = ∂ ˙
U i /∂ X j , X j = X j | t=0 +
t
0
˙
U j dt.
(4.2)
Elastic–plastic deformation of metals and alloys is described by the relations of
flow theory with kinematic and isotropic hardening (Volkov and Korotkikh 2008).
4.2.1 MHS Filler Modeling
MHS filler is modeled by a continually homogeneous, orthotropic, physically
nonlinear medium (Hallquist 1998; Demareva et al. 2014). Strains and stresses are
defined in the local coordinate system x = [x 1 , x 2 , x 3 ]. The relationship of elastic
deformations and stresses is established on the basis of the generalized Hooke’s law:
˙
σ i j = C i jkl ˙
ε kl , i, j = 1, 3,
C i jkl =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
C 1111 C 1122 C 1133
C 1122 C 2222 C 2233 0
C 1133 C 2233 C 3333
C 1212
0
C 2323
C 3131
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(4.3)
For highly porous metals, it can be assumed that in the initial state, the Poisson’s
ratio is negligible. With this in mind, the ratio (4.3) takes the form:
˙
σ i j = A i j ˙
ε i j , A i j = C i ji j , i, j = 1, 3.
(4.4)
As the porous material is compressed, its elastic characteristics are changed. The
following ratio is accepted:
A i j = A
0
i j + β
A
s
i j − A
0
i j
(4.5)
where indices “0” and “s” mark the elastic constants of the filler in the initial state
and when the pores are completely closed. The parameter β is determined by the
current value of the relative volume v = V /V 0