r - The Hong Kong Polytechnic University

In the CMR, the interface during the failure process is symmetrical about the middle of the bonded joint so only
half of the interface (from x = 0 to 0.5L) needs to be considered in this analysis. Assuming that the bond length
is sufficiently large so that L > 2a d , (if L≤2a d no debonding state exists), the interface progressively experiences
the following states during the loading process:
1) softening–rigid (S–R) state (Figure 5a);
2) debonding–softening–rigid (D–S–R) state (Figure 5b,c); and
3) debonding–softening (D–S) state (Figure 5d,e).
Governing Equations and Solutions for Different States of Interfaces
The equilibrium, constitutive relations and the interface compatibility requirements for a differential segment of
the plated beam between two flexural cracks are used to derive the governing differential equation to analyse the
initiation and propagation of plate end debonding in the plated beam. Considering the local deformation model
shown in Figure 4 for the case of 0 < δ(x) ≤ δ f , the governing equation for the interfacial slip for the softening
interface is obtained as:
2
d δ ( x)
2
+ λ δ ( x)
= λ
2 δ
2
f
(2)
dx
and the axial force in the plate is given by
2
τ f b2
⎛ dδ
( x)
2 ⎞
N2(
x)
= ⎜ + mλ
M ( x)
⎟
(3a)
2
T
2G
⎝ dx
f λ
⎠
where
2
2
τ f b2
⎡( y + )( + + )
⎤
=
1 y2
y1
ta
y2
1 1
1 ⎡ y + ⎤
λ ⎢
+ + ⎥ ; =
1 y
m
2
⎢
⎥
2G
f ⎣ E1I1
+ E2I2
E1
A1
E2A2
⎦
2 ⎣(
E1I1
+ E2
I (3b,c)
λ 2)
⎦
in which y 1 and y 2 represent the distances from the bottom of adherend-1 (the original beam) and the top of
adherend-2 (the plate) to their respective centroids; N 2 (x) and M T (x) refer respectively to the axial force in the
plate and total bending moment due to applied external loading; t, b, E, A and I represent the thickness,
breadth, elastic modulus, cross-sectional area and second moment of area of the adherends and adhesive
respectively; and subscripts 1, 2 and a respectively refer to beam, plate and adhesive.
The interfacial slip, the interfacial shear stress and the axial force in the plate for the softening region in
different states of interface can be found by solving Eq. 2 for the appropriate boundary and continuity
conditions.
Softening–rigid (S–R) interface
The interface remains in the S–R state as shown in Figure 5a until the debonding bending moment M Td is
reached at the left plate end (x = 0) as the loads are increased gradually from zero. The general solution for the
softening region of the interface (0 ≤ x ≤ a) with 0 < δ(x) ≤ δ f is given by
δ ( x)
= A1 sin[ λ(
x − a)]
+ B1
cos[ λ(
x − a)]
+ δ f
(4a)
τ f
τ ( x)
= − ( A1 sin[ λ(
x − a)]
+ B1
cos[ λ(
x − a)]
)
(4b)
δ
where
f
2
( A λ cos[ λ(
x − a)]
− B λ sin[ λ(
x − a)]
+ mλ
M ( ))
N2(
x)
= m1
1
1
x . (4c)
2
f b2
2
f
τ
m 1 = (4d)
2G
λ
T
The boundary and continuity conditions at the softening region of the interface are:
N 2 (x) = 0 at x = 0
N 2 (x) is continuous at x = a
δ(x) = 0 at x = a
(5a)
(5b)
(5c)
The constants of integration A 1 and B 1 and the softening length a are obtained by applying Eqs 5a-c to Eqs 4a-c
as given below.
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