Rasmus ÿstergaard forside 100%.indd - Solid Mechanics

Interface crack in sandwich specimen 303
P1
P2
M1
M2
Neutral axis
δ
a
x2
x1
L
face sheet
H
core h #1
face sheet
Fig. 1 Interface cracking of a sandwich specimen with equal thickness face sheets is analyzed. Material numbers are shown
properties of the materials be given by the shear modulus
μm and the Poisson’s ratio νm, where subscript m
takes the value 1 for the core and 2 for the face sheets. As
shown by Dundurs (1967, 1969) under certain restrictionsthestressfieldinatwo-dimensionalbimaterialbody
depends only on two non-dimensional combinations of
the elastic moduli. Defining κm = 3 − 4νm for plane
strain and κm = (3 − νm)/(1 + νm) for plane stress, the
Dundurs parameters can be expressed as
α = μ1(κ2 + 1) − μ2(κ1 + 1)
μ1(κ2 + 1) + μ2(κ1 + 1) and
β = μ1(κ2 − 1) − μ2(κ1 − 1)
. (1)
μ1(κ2 + 1) + μ2(κ1 + 1)
Alternatively, α can also be expressed as
− 1
α = , (2)
+ 1
where is the ratio between the Young’s modulus for
the core and the face sheets, respectively
= Ē1/Ē2, (3)
where Ēm = Em for plane stress and Ēm = Em/(1 −
ν2 m ) for plane strain. Here, Em denotes the Young’s
modulus of material number m. In some of the results
in the present work α will be used and in other results
will be used.
Just ahead of the crack tip (θ = 0◦ , see Fig. 2) the
stresses are dominated by a singular stress field of the
form (Rice 1988):
σ22 + iσ12 = 1
√ 2π Kr iɛ−1/2 , (4)
where K = K1 + iK2 is the complex stress intensity
factor, i = √ −1, r is the radial distance along the x1-
axis and ɛ is the oscillatory parameter given as
ɛ = 1
2π ln
1 − β
1 + β
.
H
#2
#2
M3
The crack opening components un are defined from
the displacements un of two material points coinciding
in the un-deformed state
P3
un = un(r,θ = 90 ◦ ) − un(r,θ =−90 ◦ ),
where n takes the values 1 and 2, that refers to the
direction in the coordinate system in Fig. 2. The crack
opening components are related to the complex stress
intensity factor, K , through
u2 + iu1 =
where
c1 + c2
2 √ 2π(1 + 2iɛ)cosh(πɛ) Kriɛ+1/2 ,
(5)
cm = κm + 1
.
μm
Note that (5) predicts interpenetration of crack faces
close to the crack tip. However, as discussed elsewhere
(Rice 1988), the contact zone is usually small and can
be neglected. The energy release rate can be related to
the complex stress intensity factor through (Rice 1988):
c1 + c2
G =
16 cosh2 |K |2
(6)
(πɛ)
and here |K | is the modulus of K .
For an interface problem the mode mixity is defined
as the phase angle of Kliɛ ψ = tan −1
ℑ(Kliɛ )
ℜ(Kliɛ
, (7)
)
where l is an arbitrary chosen length parameter and ℑ,
ℜ denote the imaginary and real parts, respectively. In
the forthcoming analysis the parameter l is set equal to
the thickness of the core, h.
123