Proceedings of ICTACEM 2014
International Conference on Theoretical, Applied, Computational and Experimental Mechanics
December 29-31, 2014, IIT Kharagpur, India
ICTACEM-2014/109
Progressive failure analysis of lamianted composite under
biaxial loading
Appaso M Gadadea* Achchhe Lalb and B. N. Singhc
a*
Research Scholar, MED,S. V. National Institute of Technology, Ichchhanath, Surat,Gujrat-395 007,India.
Asst. Professor, MED,S. V. National Institute of Technology, Ichchhanath, Surat,Gujrat-395 007,India.
c
Professor and Head, AED, Indian Institute of Technology, Kharagpur, West Bengal-721302, India.
b
ABSTRACT
This paper presents progressive failure analysis of laminated composite plate under biaxial loading using
advanced Puck failure criterion. A highly efficient Reddy‟ higher order shear deformation theory (HSDT) with
seven degrees of freedom and C0 continuity used for computing accurate ply by ply stresses. A progressive
degradation model used for degradation of material properties. The results obtained are compared with Tsai-Wu
theory and literatures. The first-ply failure and last ply failure envelopes are obtained for various lamination
scheme, thickness ratio and aspect ratio.
Keywords:
laminate.
Progressive failure analysis, Reddy‟s higher order theory, Puck failure criterion, Composite
1. INTRODUCTION
The increasing demand of laminated composite materials in structural applications
necessitates a thorough understanding of their behaviour under various loading conditions. In
particular the trend towards higher design levels has led to the need to understand failure
initiation and to predict strength. Composite materials are key element to improve energy
efficiency of future planes, trains, ships and automobiles. In stiffness critical applications
composites have delivered their promised weight savings, but this is not a case in strength
critical applications and needs more attention in this area. The development of more accurate
models and accurate quantification of material parameters used for failure prediction required
for attaining higher efficiency in strength critical applications. Due to orthotropic nature of
composite laminate failure prediction of composite laminate is a complex phenomenon. The
detailed literature review of failure prediction models show that the model formulated by
Puck et al [1] to be the most accurate, but Tsai-Wu [2] and Hashin [3] model developed in
1960 and 70s become popular and used in industrial applications widely. The Tsai-Wu and
Hashin model used widely because only six material parameters are required for failure
prediction, whereas Pucks model requires at least eleven material parameters. Over the last
three decades numerous failure theories or models have been proposed by many researchers
to cater this need. A considerable amount of strength remains unutilised even after first ply
failure of composite. Therefore the prime objective of the present study is to predict last ply
1
failure load by using more accurate stiffness degradation model and Puck failure criterion so
as to utilize complete available strength of structure.
2. HIGHER ORDER SHEAR DEFORMATION THEORY(HSDT)
Higher-order theories can represent the kinematics better, may not require shear correction
factors, and can yield more accurate interlaminar stress distributions. However, they involve
higher-order stress resultants that are difficult to interpret physically and require considerably
more computational effort. In the C0-HSDT type, two additional variables have been included
in the displacement field, and hence only the first derivative of transverse displacement is
needed.
2.1 Displacement Field
The origin of the material coordinates is at the middle of the laminate as shown in the figure
1. The present theory uses a displacement approach, much like in the Reissner-Mindlin type
theories. However, the displacement field chosen is of a special form. The form is directed by
the satisfaction of the conditions that the transverse shear stresses vanish on the plate surfaces
and be nonzero elsewhere. This requires the use of a displacement field in which the in-plane
displacements are expanded as cubic functions of the thickness coordinate and the transverse
deflection is constant through the plate thickness.
1
h 1= - h/2
h2
x
k
hk
h k+1
N
hN
h N+1 = h/2
z
Figure 1. Geometry of the laminated composite plate.
We begin with the displacement field [5],
u u f1 ( z )1 f 2 ( z )1
v v f 2 ( z )2 f 2 ( z ) 2
ww
(1)
2
Where u , v and w denote the displacements of a point along the
x
y
z coordinate
axis u, v, and w are the corresponding displacements of a point on the mid plane, 1 and 2
are the rotations at z = 0 of normal to the mid-surface with respect to y and x, axes,
respectively. 1 ( w, x ) and 2 ( w, y ) is the slope along x, y respectively. The function
f1 ( z ) and f 2 ( z ) given in eq. (1) can be written as
f1 ( z ) C1 z C2 z 3 and
f 2 ( z ) C4 z 3
(2)
Where C1 1; C2 C4 4 3h2
The displacement vector for the modified C0 continuous model can be written as
u
v
w 2
1 2 1
(3)
where, (,) denotes the partial differential.
2.2 Strain Displacement Relations
The strain displacement relations are obtained by using Von-Karman theory. The strain
vectors corresponding to displacement fields are
x
u
v
u v
u w
v w
, xz
, y , xy
, yz
,
x
y
x y
z x
z y
The linear strain vectors corresponding to displacement fields are,
l 1, 2, 3, 4, 5, 6
T
with
,
1 1o z k1o z 2 k12 , 2 2o z k2o z 2 k22 , 3 0,
4 4o z 2 k42 , 5 5o z 2 k52 , 6 6o z k6o z 2 k62 ,
(4)
(5)
where 10 = u / x , k10 = x / x , k12 = - 4/3h2( x / x + 2 w / x 2 ), 20 = u / y ,
k20 = y / y , k22 =-4/3h2( y / y + 2 w / y 2 ), 60 u / y v / x , k60 x / y y / x ,
k62 4 / 3h2 ( x / y y / x 2 2 w / xy) , 40 y w / y,
k42 4 / h2 ( y w / y) , 50 x w / x, k52 4 / h2 ( x w / x) .
2.3 Constitutive Equation (stress-strain relation)
Stress is related to strain by following relation
Q
(6)
3
x 1 Q11
Q
12
2
y
xy 6 Q16
yz 4 0
xz
5
0
Q12
Q16
0
Q 22
Q 26
0
Q 26
Q 66
0
0
0
Q 44
0
0
Q 45
0
x
0 y
0 xy ,
Q 45 yz
xz
Q 55
(7)
where Qij are elastic material constants.
2.4 Potential Energy of the Laminate
The potential energy of a laminated composite plate undergoing deformation is given as [4]
U
1
T
dV ,
2V
(8)
where is stress vector, 1 2 6 4 5
T
(8)
Further the linear potential energy of the laminate is given by using equation (7) and (8) as
U
1
l T Q l dV .
2v
1
0
here l l , and 0
0
0
and
l
(9)
0 0 z 0 0 z3
1 0 0 z 0
0 1 0 0 z
0 0 0 0 0
0 0 0 0 0
0
0
0
0
0
0
0 0
0
3
0
0 0
0
0
3
0 0
0
z
0
0
z
0
0
1 0 z
0 1 0
20 60 k10 k20 k60 k12 k22 k62 40 50 k42 k52 ,
0
1
2
0
0
0 ,
0
z 2
(10)
(11)
Linear potential energy becomes,
Ul
T
1
l T T QT l dV ,
2V
(12)
This can further be written as,
UL
Where,
T
1
L D L dA,
2A
D
T
NL zk
k 1 zk 1
T
A1
B
Q T dz E
0
0
(13)
B E
C1 F1
F1 H
0
0
0
0
0
0
0
A2
C2
4
0
0
0
C2
F2
(14)
with
A
2ij ,
A1ij ,
Bij ,
C1ij ,
F2ij
C2ij ,
Eij ,
Q 1
NL zk
(k )
k 1 zk 1
H ij
F1ij ,
(k )
k 1 zk 1
z 4 dz,
z,2
,
ij
Q 1
NL zk
ij
,
z,
z,2
z,3
z,4
z 6 dz , For i,j=1,2,6,
For i,j=4,5,
L ,
(15)
The equation (11) can be rewritten as
(16)
L
where L is differential operator.
UL
The linear potential energy becomes
1
T LT DL dA.
2A
(17)
3. FINIE ELEMENT MOEL
For the finite element analysis the (17) equation can be written as
1
( e )T LT DL ( e ) dA.
e 1 2 A( e )
Ul
NE
(18)
here NE is number of elements used for messing the plate.
Displacement vector in equation (3) can be written in terms of shape functions as
Ni i ,
NN
(19)
i 1
here i represent node number. And Ni is shape function at ith node.
N
q
For an element, it can be written as
(e)
(e)
(e)
,
putting above values of in equation (18) we get
(20)
(e)
1
q ( e )T N ( e )T LT DLN ( e ) q ( e ) dA
2
e 1
A( e )
Ul
NE
(21)
1
q ( e)T B( e)T DB( e) q ( e) dA,
2 A( e )
Elemental potential energy can be written as
Ul (e)
where
B
(e)
here B
(e)
(22)
L N ,
(23)
B2
(24)
B1
(e)
B3 BNN ,
with [Bi] = [L]Ni.
i=1, 2, 3,……, NN
5
(25)
Thus finally elemental potential energy can be written as
1
U l ( e ) q ( e )T K ( e ) q ( e )
2
(26)
Where Element bending stiffness matrix
K (e)
B ( e )T DB( e ) dA,
(27)
(e)
A
Here K(e) are computed numerically by transforming existing coordinate system to natural
coordinate system and , the equations (27) can be represented as
BiT DB j det Jd d ,
1 1
K
(e)
ij
(28)
1 1
x / x2 /
Here J 1
,
x1 / x2 /
is Jacobian.
When numerical integration is adopted, the elemental matrices in equations (28) becomes
Kij( e ) 1 2 WpWq BiT DB j det J ,
N
N
p 1 p 1
(29)
Here Wp and Wq are weights used in Gaussian quadrature. Thus total linear potential
energy of system becomes,
1
U L q ( e )T K ( e ) q ( e ) ,
e 1 2
NE
(30)
The governing equation for computing deflection of a composite laminated plate under
biaxial loading is given by
Kij qi Fi
(31)
Where Kij Kije is global stiffness matrix and Fi Fx Fy 0 0 0 0 is a force
NE
e 1
vector and qi deflection vector of a composite laminated plate.
4. PUCK FAILURE CRITERION
The new criterin developed by Puck et al [1], is based on physical foundations as proposed by
Hashin et al [6]. The criterion is essentially based on the hypothesis of Coulomb [7] and
Mohr [8] and modified by Paul [14]. It is therefore formulated for a rotatable coordinate
system, which is referred to as the failure plane, the plane where the brittle failure occurs.
The summarized Puck failure criterion is shown in Table 1 below.
6
Table 1. Summarized Puck failure criterion
Type of Failure
Failure Mode
Fibre Failure
Tension
Fibre
f 12
1
2
m f 2 10 6 1
1
1C
Ef1
6
YT 2
2
1
1 PP
PP
S12 YT
S12
S12
Mode A
2
2
2
p P 2 1
2
2 Y
6
2 C 1
2 1 p S YC 2
21
Mode C
R
2
1
2
6 pP 2
S12
Mode B
Parameter
Relationships
Condition of validity
f 12
1
m f 2 1
1
1T
Ef1
Compression
Inter
Failure
.... 0
Failure Condition
2 1 P
YC
S12
2 P
.... 0
2 0
2 0 0
2
R
6 6c
2 0 0
6 6c
2
R
R
YC
1 ; P
PP ; 6c S12 1 2P
1 2 P
S12
S12
5. DEGRADATION MODEL
Puck and Schürmann [1] account for material property degradation for IFF in Modes A, B
and C in plane stress by introducing a dimensionless degradation factor denoted by η. For no
degradation η= 1, and values of η less than one correspond to degraded material property
values. The Mode A degradation factor is denoted by ηa and for Mode B and C degradation
factor is denoted by η(-). Hence, the range of the matrix degradation factors is
0 a 1 and 0 1
(32)
The onset of a matrix crack in Mode A (transverse tension) occurs when the failure index
FIM = 1, and at the crack location the moduli drop to zero. The gradual decrease in Mode A
is represented by reduced moduli ηaE2 and ηaG12 and a reduced Poisson‟s ratio ηaν12. The
average stresses in the failed ply are functions of η, that is σ22 (ηa) and σ12 (ηa) , and they are
computed from the laminate strains using the reduced moduli ηaE2 and ηaG12, and a reduced
Poissons ratio ηaν12. The value of ηa is computed by enforcing failure index FIM= 1 after the
onset of the Mode A failure. Modes B and C are shear failures impeded by a compressive
normal stress and the cracks do not open. So there is no degradation to the modulus E2 and
the Poisson‟s ratio ν12, but the modulus G12 is degraded by the factor η(-). After computing the
shear stress σ12(η(-)) from the laminate strains for the reduced shear modulus η(-)G12, the
failure index for Mode B is kept equal to one to find the value of η(-). For increasing load the
7
failure mode can change from B to C, and at the point of transition the magnitude of the shear
stress is a maximum. For further increasing load in Mode C the failure index for Mode C is
kept equal to one to find the value of η(-). The value modulus G12 is then reduced till the value
of σ22 becomes equal to YC.
5. PROGRESSIVE FAILURE ANALYSIS
Define Initial State
Degrade properties using η1 putting
in the laminate elastic stiffness in
matrix fiber failure
Establish linear static equation
[Kij]{qi}={Fi}
Compute stresses in terms of η1
for unit load
Obtain Ply by Ply stresses and strains for unit
load varying action plane from 0 to 360
Assuming FI as 1 find out value of η1
Obtain FI and LF by Puck Failure Criterion
Again putting the value of η1 in the
laminate elastic stiffness matrix
Minimum of load factor will be FPF
Compute Load Factor by
using Failure criterion
Degrade properties of materials via
degradation model
Minimum of Load factor will be LPF
Figure 1. Progressive failure analysis floe chart.
6. RESULTS AND DISCUSSION
A finite element code is developed in MATLAB for solving displacement field, by using a
nine noded isoparametric element with seven degrees of freedom.
6.1 Problem Description
The laminates considered for failure analysis are made of AS-4 graphite/epoxy material.
Properties of this material are listed in Table2. The dimensions of plate considered are unit
8
dimensions. Stresses and strains are determined by HSDT and validated against literatures. A
different in plane loading conditions is considered and response of plate is obtained.
Table 2. Material properties of AS-4 graphite/epoxy pre-peg [1]
Properties
Value
Properties
Value
E1
126 Gpa
Xt
1950 MPa
E2
11 Gpa
Xc
1480 MPa
E3
10.8 Gpa
Yt =Zt
48 MPa
G12=G13
6.6 Gpa
Yc=Zc
200 MPa
G23
3.4 Gpa
R
79 MPa
Table 3. Mechanical Properties of composite laminate [9]
Property
Longitudinal tensile failure strain,
T
Value
1.38
(%)
Longitudinal compressive failure strain C (%)
1.175
Major Poisson's ratio v12
0.28
Through thickness Poisson's ratio v23
0.4
Transverse tensile failure strain
0.436
2T (%)
Transverse compressive failure strain
In-plane shear failure strain
12u (%)
2C (%)
2
2
6.2 Convergence Study
The convergence study shows that 4x4 mesh will be sufficient for further analysis. Also
present results matches closely with Tolson et al [10], Reddy et al [12, 4], Ocha et al [11] and
3D elasticity solution [15].
Table 4. Convergence study and comparison of central deflection of square laminated [0/90] s cross ply plate
subjected to non dimensionalised sinusoidal load with simply supported boundary conditions with different
aspect ratios(aspect ratio s=a/h=10, E1/E2=25.0, G12/G23=2.5, υ12= υ23=0.25)
Aspect
Ratio
(a/h)
Mesh
density
10
Central deflection
Present
Tolson et al[10]
Ocha et al [11]
Reddy et
al [12]
Reddy et
al [4]
3D
Elasticity
Solution [16]
2×2
1.6919
1.693
--------
1.534
1.643
1.709
3×3
1.6816
--------
--------
--------
4×4
1.6733
1.671
--------
--------
5×5
1.6733
--------
--------
--------
6×6
1.6733
--------
1.790
--------
9
20
2×2
1.1911
1.188
--------
1.136
3×3
1.1826
--------
--------
--------
4×4
1.1798
1.177
--------
--------
5×5
1.1734
--------
--------
--------
6×6
1.1730
--------
1.216
--------
1.163
1.189
6.3 Validation Study
The results for in plane loading conditions are validated with Reddy and Reddy [14] and
deflections for transverse loading conditions are validated with Kam et al.[10].
Table 5.Validation of non-dimensionalised first-ply failure load for different symmetric and antisymmetric
laminates with clamped end boundary conditions under in plane tensile loading
Lamination scheme
First ply failure load
Difference
(I) Present
(II) Reddy and Reddy [14]
(I) –(II)/ (I) %
[45/-45/90/0/45/90/-45/0]s
1.1896e+006
1.167367e+006
1.87
[45/-45/0/90/45/0/-45/90]s
1.1874e+006
1.167367e+006
1.68
[45/0/-45/0/-45/90/0/45]s
1.4734e+006
1.455385e+006
1.22
[45/0/-45/0/-45/0/45/0]s
2.1555e+006
2.123379e+006
1.49
[45/-45/45]t
9.6114e+007
9.2514644e+007
3.74
[-45/45/-45/45]t
4.4309e+007
4.3535484+007
1.75
Table 6. Validation of deflection of laminated plate subjected to uniform load with simply supported (SS1) and
clamped (CC1) boundary conditions.
BC
SS1
CC1
Lamination Scheme
Normalized Load
(P/E2)(a/h)^4
Normalized central deflections (w/h)
Present
Kam et al [10]
Reddy & Reddy[14]
[45/-45/90/0/45/90/-45/0]s
9689.1
1.9603
1.950
1.956
[45/-45/0/90/45/0/-45/90]s
9030.1
2.1282
1.913
1.919
[45/0/-45/0/-45/90/0/45]s
6177.7
1.8150
1.842
1.845
[45/0/-45/0/-45/0/45/0]s
4517.8
1.4510
1.838
1.840
[45/-45/90/0/45/90/-45/0]s
5144.9
1.0408
1.282
1.283
[45/-45/0/90/45/0/-45/90]s
3833.8
0.9034
1.163
1.164
[45/0/-45/0/-45/90/0/45]s
2442.6
0.7175
1.055
1.053
[45/0/-45/0/-45/0/45/0]s
2083.6
0.6603
1.066
1.064
A failure envelope of the composite laminate plate subjected to unit biaxial load is shown in
figure 2.
10
LPF Present
1500
LPF Puck et al [5]
FPF Present
1000
FPF Puck et al [5]
MPa
500
y
0
-500
-1000
-1500
-1500
-1000
-500
.
x
0
MPa
500
1000
1500
Figure 2. Validation of present first ply and last ply failure envelope for [0/45/-45/90]s made up of CFRP material subjected
to bi-axial loading with Puck et al [5].
6.4 Biaxial failure envelopes for various loading conditions
2
x 10
7
Puck Failure Envelopes-Load Space a/h=50.Nx/Ny=1/2,Ny
1
FPF
LPF
1.5
x 10
7
Puck Failure Envelopes-Load Space,a/h=100,Nx/Ny=1/2, Ny
0.8
0.6
1
0.4
y
0.2
0
0
y
0.5
-0.2
-0.5
-0.4
-1
-0.6
-1.5
FPF
LPF
-0.8
-2
-1
-0.5
0
0.5
x
-1
-5
1
x 10
0
5
Nx
7
(a)
x 10
6
(b)
Figure 3. Biaxial failure envelope for [0/+45/-45/90]s laminate under Nx/Ny=1/2 with (a) a/h=50 and (b)
a/h=100
1
x 10
7
Puck Failure Envelopes-Load Space,a/h=100,Nx/Ny=1/-1, Nx
2
FPF
LPF
0.8
x 10
7
Puck Failure Envelopes-Load Space,a/h=50,Nx/Ny=1/-1, Ny
FPF
LPF
1.5
0.6
1
0.4
0.5
y
0
y
0.2
-0.2
0
-0.5
-0.4
-1
-0.6
-1.5
-0.8
-1
-1
-0.5
0
x
0.5
-2
-2
1
x 10
-1.5
-1
-0.5
0
7
(a)
x
0.5
1
1.5
2
x 10
7
(b)
Figure 4. Biaxial failure envelope for [0/+45/-45/90]s laminate under Nx/Ny=1/-1 with (a) a/h=50 and (b)
a/h=100
11
1
x 10
7
Puck Failure Envelopes-Load Space[0/15]2s a/h=40
0.5
0
x 10
7
Puck Failure Envelopes-Load Space[0/30]2s, Nx/Ny=1/1, Nx
0
-1
-0.5
-2
y
y
-1
-3
-1.5
-4
-2
-5
LPF
FPF
-6
-7
-4
-3
-2
-1
0
1
2
3
FPF
LPF
-3
-3
4
x 10
x
-2.5
-2
-1
0
7
(a)
0.5
x 10
7
1
2
Puck Failure Envelopes-Load Space[0/45]2s,Nx/Ny=1/1,Nx
x 10
7
Puck Failure Envelopes-Load Space[0/90]2s, Nx/Ny=1/1, Nx
2
0
FPF
LPF
1.5
-0.5
1
0.5
FPF
LPF
y
y
7
(b)
2.5
-1
3
x 10
x
0
-0.5
-1.5
-1
-2
-1.5
-2.5
-3
-2.5
-3
-2
-2
-1
0
x
1
2
3
x 10
-2
-1
0
7
(c)
1
2
x
3
x 10
7
(d)
Figure 5. Biaxial failure envelope for (a) [0/15]2s ,(b) [0/15]2s ,(c) [0/15]2s (d) [0/15]2s laminate with a/h=40 under
Nx/Ny=1/1
A figure 3 shows biaxial failure envelope for [0/+45/-45/90]s laminate under Nx/Ny=1/2 with
(a) a/h=50 and (b) a/h=100. A figure 4 shows biaxial failure envelope for [0/+45/-45/90]s
laminate under Nx/Ny=1/-1 with (a) a/h=50 and (b) a/h=100. A figure 5 Biaxial failure
envelope for (a) [0/15]2s ,(b) [0/15]2s ,(c) [0/15]2s (d) [0/15]2s (a/h=40 and Nx/Ny=1/1).
7. CONCLUSIONS
The response of composite laminated plate for under different biaxial loading is obtained.
The result shows considerable difference between first-ply and last-ply failure load. The
failure load increases with increase in thickness of plate and decrease in ply angle.
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12
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Dr Appaso M Gadade