3.1.1. Constitutive Model and Progressive Damage Analysis for Composite Panels
In
Figure 7, (1, 2, 3) and (
n,
s,
z) are the main axis and the off-axis coordinate systems of the orthotropic composite material, respectively, whereas
θ is the angle between 1 and the
z-direction.
When
θ = 0°, the three-dimensional constitutive equation of the composite is as follows:
Equation (2) can be written as
where
C is the stiffness matrix of the main axis coordinate system, θ ≠ 0°, and the stiffness matrix of the off-axis coordinate system can be obtained as follows:
where
T is the transformation matrix, calculated as follows:
Substituting Equation (5) into Equation (4), the three-dimensional constitutive equation of the composite under the off-axis coordinate system is obtained by
where
is the three-dimensional stiffness coefficient under the off-axis coordinate system. A more detailed expressions can be found in Reference [
7].
The Hashin criterion was used as the damage initiation criterion of the composite. The Hashin criterion is a model correlation criterion, which can distinguish the four failure modes of fiber tensile, fiber crush, matrix tensile, and matrix crush, defined below.
(1) Fiber tensile failure
(3) Matrix tensile failure
(4) Matrix crush failure
where
Fi (
i =
ft,
fc,
mt,
mc) are the damage parameters corresponding to the four failure modes. When
Fi ≥ 1, the composite material is destroyed;
XT and
XC are the longitudinal tensile strength and compressive strength, respectively;
YT and
YC are the transverse tensile strength and compressive strength;
SL and
ST are the shear strength. The engineering elastic constants and ultimate strength values of the composite materials used in the panels are shown in
Table 3.
When a single layer of the composite material is damaged in strength, its mechanical properties are attenuated to some extent. In this paper, the stiffness degradation was based on the linear degradation mode of fracture toughness, as shown in
Figure 8, where σ
0,i (
i =
ft,
fc,
mt,
mc) is the initial damage equivalent stress,
δ0,i is the initial damage displacement,
δf,i is the complete damage displacement, and
di is the damage state variable. It can be seen from
Figure 8 that the equivalent strain energy of the composite material under complete failure can be calculated as follows:
Table 4 shows the fracture energy parameters of the carbon-fiber composites used in the specimens [
8]. When the composite material fails completely, the equivalent strain energy
Ws is equal to its fracture energy
. At this time, the complete damage displacement can be obtained as
The degree of deterioration of the mechanical properties of the material is characterized by the damage state variable
di, and their expression is as follows:
The constitutive equation of composite monolayers in the process of damage evolution is as follows:
where
Cd is the stiffness matrix considering the damage, calculated as
where
;
df,
dm, and
ds are state variables of fiber damage, matrix damage, and in-plane shear damage, respectively, which can be expressed as follows
The relationship between the effective stress matrix and the real stress matrix is as follows:
M is the damage coefficient matrix and is calculated by
3.1.3. Adhesive Interface Element Damage Analysis Model
The interface performance was simulated by placing interface elements of COH3D8. COH3D8 is a three-dimensional eight-node interface element with thickness as shown in
Figure 11 [
11].
The constitutive equation of COH3D8 in the linear elastic range is as follows [
11]:
where
tn,
ts, and
tt are the normal stress and the two shear stresses of the adhesive interface element respectively,
εi =
δi/
T0 (
i =
n,
s,
t),
T0 is the thickness actually calculated for the adhesive element,
δi is the relative displacement of the top surface and the bottom surface of the cohesive element in the corresponding direction, and
Kii is a stiffness factor. The secondary stress criterion was used as the starting criterion, shown in Equation (23).
The damage evolution of COH3D8 elements was based on the mixed mode Benzeggagh–Kenane energy criterion [
12], shown below.
where
Gi (
i =
n,
s,
t) represents the strain energy release rates corresponding to open, slip, and tear cracks, respectively,
represents the critical strain energy release rates of the three kinds of cracks,
is the damage evolution variable (the layers are fully stratified when
), and
η is the damage factor. For carbon-fiber epoxy materials,
η is generally between 1 and 2. The material parameters of the adhesive element layer are shown in
Table 5 [
13].