- Email: [email protected]
A modeling analysis for splitting failure of orthotropic laminates with a surface notch
Composites
Science and Technology 56 (1996) 1201-1207 0 1996 Elsevier Science Limited
Printed PII:
in Northern
Ireland.
SO266-3538(96)00083-8
All rights reserved 0266-3538/96/$15.00
A MODELING ANALYSIS FOR SPLITTING FAILURE OF ORTHOTROPIC LAMINATES WITH A SURFACE NOTCH J. H. Zhao Department of Modern Mechanics,
& J. Wei
University of Science and Technology of China, Hefei, Anhui 230027, People’s Republic of China
(Received 20 September 1995; revised 17 May 1996; accepted 30 May 1996) Abstract A simple model is presented for the failure behavior of surface notched laminated composites subjected to tensile loading. The model is capable of simulating split growth and predicting the ultimate tensile load for glass/epoxy cross-ply laminates with a surface notch. In simulating, the classical shear-lag theory including the resistance effect against split growth appearing between the adjacent split surfaces, and the strain energy release rate criterion for fracture mechanics are adopted. The program of simulating split growth to final failure was developed. Numerical results were compared with experimental data for glass/epoxy orthotropic laminates with a surface notch. It is indicated that the proposed model and simulating program in the work are available. 0 1996 Elsevier Science Limited Keywords:
splitting failure, numerical simulation
laminate,
tions,“4’5 a simple mechanical model is presented for predicting split growth and the ultimate load in surface notched glass/epoxy cross-ply laminates subjected to tensile loading. By means of the shear-lag effect existing analysis, including the resistance between the adjacent split surfaces, the control equations for the displacements of the model elements were derived, and the strain energy release rate for split growth could be calculated from the displacements obtained. The procedures of numerically simulating the process of split growth to final failure was developed according to the strain energy release rate criterion in fracture mechanics. Numerical results were compared with the experimental data for glass/epoxy orthotropic laminates containing a surface notch. Satisfactory agreement was found between the numerical results and experimental data.
surface notch, 2 SIMPLIFIED
MODEL
AND
GOVERNING
EQUATIONS
1 INTRODUCTION
2.1 General
The mechanical behavior of fiber-reinforced laminated composites with surface notches has received considerable attention from a number of researchers.lm9 Test studies have showed that there are two typical patterns of failure for these laminates. The first is that little damage appears near the surface notch until final failure occurs, as happens for most surface notched graphite/epoxy laminates.2.3 The second pattern is that splitting (delamination and debonding) initiates near the surface notch and then increases in extent leading to final failure, as for most glass/epoxy laminates with a surface notch.1,4’5 For the first type of failure, Lo et al.* and Sendeckyj3 have presented elementary properties of materials analyses to predict the failure loads of laminates. As for the second case of failure, Im et al.’ have proposed a relatively simple model which is capable of predicting some features of the splitting (delamination) process, but cannot be used for the initial and final stages of the splitting process and the ultimate load. On the basis of careful experimental observa-
description
of the model
As has been pointed out’,4,5 for glass/epoxy laminates with a surface notch, the damage caused by loading is mainly delamination between the reference layer (notched layers) and the constraint layer (unnotched layers) and debonding near notch tips in the reference layer. The delamination and debonding grows with tensile load in the tensile direction. It is also known that the propagating fronts of the delaminated region are arc-shaped, but far from being semi-circular, and the central length of the delaminated region is somewhat larger than the debonded region. From these experimental observations, a simplified model is presented as in Figs 1 and 2. Because of symmetry, only one-half of the orthotropic laminate with a surface notch is shown in Fig. l(a), where y is the axial distance measured from the location of surface notch, and L, W and T are the half length, width and thickness of the laminate, respectively. PO is the total load applied at the ends of laminate. The details of the model are as follows. 1201
1202
J. H. Zhao, J. Wei
regions is limited to the model elements 2 and 3. So that the axial stresses in elements A and B are independent of y, i.e. the axial displacements of elements A and B depend linearly on y only. (4) Delamination occurs between elements 1 and 3, as well as 2 and 3, debonding occurs between elements 1 and 2. (5) In the splitting region, a resistance against split growth exists between adjacent split surfaces. This effect may be a result of fiber bridging or friction between the adjacent split surfaces, and matrix yielding at the front of the propagating split, and is defined as the apparent shear-lag stress So (ASLS), as shown in Fig. l(b). 3. It is assumed that sr = ~(1 - 1,/1,)(0 I so 5 s,), where 1, is half the critical split length and so the ASLS to initiate splitting. f, and so are dependent on the character of the surface notched laminate and will be determined by the tests. s, is the shear strength of matrix.
(b) Fig. 1. Schematic
illustration of the mechanics material analysis.
model for
1. For
simplicity of calculation, the difference between the central lengths of the delamination and the debonding is neglected. In what follows, delamination and debonding will both be referred to as splitting, so that the split region is box-shaped, as shown in Fig. l(a), where ld stands for the half split length. 2. It is assumed that: (1) the glass/epoxy orthotropic laminate with a surface notch consists of seven model elements. Their cross-sections are shown in Fig. 2, in which an element with a shaded cross-section represents the notched part which has m layers. Therefore the notch depth, H = mt + (m - l)h, where t is the thickness of each layer and h is the thickness of the matrix between two neighboring plies or between two model elements. (2) Each model element carries only an axial stress which is a function of y and the matrix between two neighboring elements sustains only shear stress. (3) The influence of the surface notch on the stresses of neighboring
2.2 Governing equations Let El, E2, E3, EA and Es denote the Young’s moduli of the materials corresponding to elements 1, 2, 3, A and B along the loading direction, F,, F,, F3, FA and FB the cross-sectional area of corresponding elements, respectively. G, stands for the shear modulus of the IEdriX. UA, (TB and UA, uB are the stresses and the displacements in elements A and B along the loading direction. According to the assumptions in Section 2.1, the relationship UA = CT, x
y/E,
= UB X y/E,
(1)
should exist. Considering split damage with length Id along the loading direction y, let ul, u2, u3 and ul, u2, u3 denote the displacements of elements 1, 2, 3 along the direction y in the split and the non-split regions, respectively. Introducing the following dimensionless variables: l&=
&rzy
J
E,F,G,,,H
G,H
EIFrh ’
h
Q p,’
77 = ki
E,F,G,H
vi= GH m E1eh
J
h
J
’
SF=
ui PO
EIFIHh s, ~J Gm PO
(2) The dimensionless axial load, e, on the cross-section of the ith (i = 1, 2, 3, A, B) element is related to the dimensionless displacement by: p. _ pi _
Eifi dG(or K)
’ PO E,F, I
Fig. 2. Cross-section
i;
view showing surface notched plane.
d5
where pi is the true axial load applied on the cross-section of the ith element. According to the shear-lag theory and taking into account the resistance effect, we obtain the following
1203
Modeling analysis of orthotropic laminates
governing equations for the equilibrium of any section in the split and non-split regions:
3.2 Solutions (1) For n I 5 eqns (7)-(9) following form:
(14)
where V = [V,,V,,V,]’ is a column vector of unknown dimensionless displacements, and [D] = [Djj] is a coefficient matrix of [V] in eqns (7)-(9), respectively; Q = [q1,q2,q31T is a column vector which consists of the coefficients of 5 in eqns (7)-(9). Equation (14) is a system of inhomogeneous, second-order, linear ordinary differential equations. When t+ ~0,dV/dt should be finite. Accordingly, the general solution to eqn (14) is given by
and
Vi = ic,e-*Pe,,
d2V, -dt2
to the
$=[D]V+Q(
(4)
(5)
can be reduced
+ Sit
i = 1, 2, 3
(15)
j=l
(8)
where hi (j = 1,2,3) are the three eigenvalues of [D], ej, is the eigenvector corresponding to 5 and Cj is a vector containing the three unknown constants. The S, is also an unknown vector and will be calculated as follows. Considering eqn (3) and substituting eqn (15) into eqn (12), we obtain u-
WB )
B - EAFA + 2EBFs (
(16)
Substituting eqn (16) into Q of eqn (14), we obtain for n 5 5, in which, FI = Ha, F2 = Ht, F3 = (2t + c + 2h)t, and
s12=;; -, s,x=$, 1
1
2
2
3 BOUNDARY
3
Q = Q[a,(P,)]
(17)
As load PO is given, Si in the general solution eqn (15) can be solved from the following equation:
SIB=
3
CONDITIONS
= D;& + q/
D;Fs, + q; = 0
(18)
where Djj = Dir + Dp.
AND
(2) For 0 5 5 5 n: from eqn (4) and the condition eqn (11) we have
SOLUTIONS 3.1 Boundary conditions Symmetry across the xz plane is satisfied by requiring that U*(O)= U,(O) = 0
(when 5 = 0)
(10)
~~=~(2+x)s’(l-~~~2+cI where C4 is a constant to be determined. (5), we obtain
(19) From eqn
while U, = Cge-+J t C 6e’45+ SiBuB 5
d&(O) -=O d5
(whent=O)
(11) +(1-1,)+?,
stands for the free stress on the notched cross-section. Force equilibrium condition on the end face is PI + 2P, + l’? + PA + 2P, = 1 (when t-+ m)
Continuous such that
(12)
(20)
where Ai = S,, is an eigenvalue, while Cs and Cs are constants to be determined. The solution of eqn (6) is given by:
conditions should be satisfied when [ = n
Wri) = V(T),
W(q)
7
=
W(v)
7
(i=1,2,3)
(13)
2t + a +p H
(21)
J. H. Zhao, J. Wei
1204
in which A25= (F,IHh) + 2(tlH), while C7 and C8 are constants to be determined. Making use of the boundary conditions eqns (10) and (13) yields a system of linear algebraic equations for determining the unknown constants which stand for a vector C = [C, C2 C3 C4 C5 C6 C, CXIT as follows: AC=f
(22)
where A is a 8 X 8 matrix, f is a vector of length 8. The elements of A and f can easily be determined from the general solutions in eqns (15), (19), (20) and (21). Obviously A and f, and hence C, are all dependent on the dimensionless split length 7.
4 THE ENERGY
Based on mechanics, propagation
RELEASE
RATE
the theory of linear elastic fracture the energy release rate, G, for the of a longitudinal split can be expressed as
G2$
(23)
P
where U is a half the strain energy of the split laminate, A = (2H + a& is the area of split region, or G= ‘_“I 2H+a&
P
=1 2H+a
&$:I,.
(24)
The total strain energy contains the extension strain energies of elements (note: the strain energies of the elements A,B are independent of 7) and the shear strain energies of the matrices between the adjacent elements. The extension strain energies are expressed as
‘)( Ui - ~ )2d~, i-j=2-B,3-A BzA = a + 2h + 2t
(26)
for
=
1 I
GmBij 2h2
J
El&h a (VJ - y)2d& G,,,H _,,
i-j=l-2,1-3,2-3,2-B,3-A
where A, f are the known functions of 7. Accordingly, as PO, r,, n are given, once the vector C is determined by eqn (22), then Xi/al7 will be found from eqn (28). 5 TESTS AND NUMERICAL OF SPLITTING FAILURE
SIMULATIONS
5.1 Tests for splitting failure The surface notched specimens were made from glass/epoxy cross-ply laminates [O”/Oo/900/Oo]s with fiber volume fraction of 0.53. Details of the fabrication of the laminates and specimens have been given elsewhere.4,s*8 Table 1 gives specimen dimensions for this work. For each type of specimen listed in Table 1, three to five samples were tested. For a unidirectional ply in these specimens, the longitudinal and transverse Young’s moduli are EL = 2.37 X lo4 MPa, ET = 3.60 X lo3 MPa, and Poisson’s ratio is v12 = 0.28. The shear modulus of the matrix is G,,, = 1.38 X lo3 MPa. The Young’s moduli E1,E2,E3,EB and EA for the model elements shown in Fig. 2 can be calculated from EL and ET. The tensile tests on surface-notched specimens were performed in a universal testing machine at a cross-head speed of 1 mm min-‘. The technique of measuring split length is similar to that described previously.’ 5.2 The numerical simulations for splitting failure process By examining the experimental data for the fourth type of specimen listed in Table 1, we chose a half critical split length Z, = 14 mm as the calculation parameter in the following simulations.
For each group of data for (coo, so), where go is the average stress applied at the end of the notched laminate (obviously, PO= go (F, + I$ + 4 + 2F, + FA)), G versus ld/lc were calculated by the procedures in
split
! w,_,
(28)
5.2.1 The relationship between energy release rate G and split length ld
The shear strain energies are expressed as
where B12 = Bzs = H, regions, and
to find G by eqns (24)-(27), the calculation of aC,/an should be performed. Differentiating eqn (22) with respect to 7, we have
(27)
where B12 = Bzs = H, B13 = a, Bz3 = t, B3A = a + 2h + 2h, for non-split regions. As pointed out earlier, Cj is the function of 77,hence
Table 1. The size of specimens Type
L
W
T
t
h
HIT
a/W
1 2 3 4 5 6
170
20
2
0.20
0.05
0.50
0.50 0.25 0.50 o-25 0.50 0.25
0.75 0.25
1205
Modeling analysis of orthotropic laminates
0 Fig.
IL1
&-““..
as
3. The relationship between the energy release rate and
the split length.
Sections 3 and 4. Such results are plotted in Fig. 3. The plots show that for s0 = 0 the split process will rapidly tend to become unstable. This is obviously not in agreement with the experimental observations, but by increasing so, a stable process of split growth will be achieved. This means that consideration of the resistance effect between the adjacent split surfaces has a significant effect in simulating the process of split growth until failure. 5.2.2 The determination of the critical energy release rate and the ASLS to initiate splitting.
(1) By using the measured load fl which initiates the splitting of a specific type of specimen in Table 1, for example, the first specimen, the stress gi to initiate splitting can be calculated. For the calculated aj and taking so = 0, a series of values G@) which corresponds to a series of values @) (ranging from ld = O.Ol0.1 mm with step Al, = O-01 mm) can be determined. Then the curve of G(“) versus @) is obtained by regression by using the formula G = a0 + aI& + a& According to the definition of critical energy release rate, and letting id + 0, it is seen that G = G,* = a, is the initial splitting energy release rate corresponding to so= 0. (2) After obtaining G,* in step (l), for a suitable value of so(O< so < s,), starting with (TV= Ed= 0, and taking steps AId = O-1 mm, Ace = 2.5 MPa, a series of values G’“’ may similarly be determined for the first type of specimen in Table 1, but in the calculation, if G(“) < G,*, increasing Acre, the calculation must be repeated; otherwise holding a0 unchanged and increasing Al, is necessary. We proceed in this way until the moment- when, without an increase in go, increasing only Aid will result in G’“’ 2 G,*. At that time the u. is the critical splitting failure stress, u,, and the corresponding ld is the critical split length, 1,. It is the beginning of the unstable split growth when r. = uc or ld = 1,. Such a procedure of calculation is the process of numerical simulation of split growth. In the process, a series of data for a0 and ld may be obtained, then a curve of u. versus ld will be fitted. (3) Using G$, still for the first type of specimen, choosing several different values of so and repeating step (2), several curves of a0 versus fd are constructed. Comparing the curves with the test data for the first
0
2
4
14
16
18
Fig. 4. The predicted curves and test data for the first type
of specimens.
type of specimen by the least squares method, the optimum value of so is selected (denoted by $). (4) Letting so = s 8, repeating step (l), a new G,* will be obtained. (5) Using the new G,*, repeating step (3), again a new value of so*will be obtained.The calculation shows that stable values of G,* = G, =274Jm-’ and so*= so = 30 MPa can be obtained by repeating the operation only two to three times. Figure 4 shows the test data for the first type of specimen and its optimum fitted curve corresponding to the values of G, and so. In the following simulations, G, and so will be considered as a true laminate property characterizing longitudinal splitting failure. 5.2.3 The numerical simulations of the splitting process and comparison
with tested data
Using parameters G, = 274 J m-* and so = 30 MPa determined in Section 5.2.2, the process of splitting for the other five types of specimen in Table 1 were
~
Prediied fa HTT=0.5afVW0.25 Measured & H/T=0.5,aVV0.25
5011 0 Fig.
2
4
8 6 SPLIT LENGTH
10
12
I
I
14
16
[mm)
5. The predicted curve and test data for the second type
of specimens.
J. H. Zhao, J. Wei
1206
simulated numerically by the method described in step 2 of Section 5.2. The curves of co versus ld which present the process of split growth are shown in Figs 5-7. In the figures the test data corresponding to each type of specimen are pointed in order to make a comparison. During the numerical simulations, the predicted values of the initial splitting stress (T,, the splitting failure stress V~ and the half critical split length 1, are calculated at the same time. These predicted values and the corresponding tested values are all listed in Table 2. 3501
I 2
0
14
4 SLTL&T~"
16
1
(tzl)
Fig. 6. The predicted curves and test data for the fifth and sixth types of specimens.
REMARKS
A simplified mechanical model is proposed to simulate splitting failure in glass/epoxy laminates with a surface notch in this paper. On the basis of the model, and by using the shear-lag analysis including the effect of resistance between the adjacent split surfaces, a procedure for simulating the whole course of splitting failure including split growth until final failure has occurred. The good agreement by comparing the results of numerical simulation with experimental data shows the potential of the present model and the simulating procedure. The results also show that it is very important and necessary to include the resistance effect between the adjacent split surfaces in the simulation. The proposed approach can be used to determine the critical energy release rate and ASLS to initiate splitting which are the characteristics of a laminate. The stresses to initiate splitting, splitting failure and the critical split length were predicted. The predicted values agree with the measured ones and the relative errors are within 10%. It is indicated that the initial split and splitting failure stresses will decrease with increasing depth and width of the surface notch, while there is no obvious dependence of the critical split length on the depth and width of the surface notch; therefore, it can be said that the critical split length may also be a characteristic parameter for split extension. ACKNOWLEDGEMENTS
250
The authors wish to acknowledge the National Natural Science Foundation of China, for financial support of this study (19072060) and postgraduate Z. Hu for his help with laboratory tests.
lz
200
P
!
6 CONCLUDING
150
REFERENCES
( - -
Predktedbot
twr=0.75,am=05
~
L
50
-
0
I
I
2
4
I SPL&&i
I
lb ----~ 12 (mm)
I4
--’ 1
Fig. 7. The predicted curves and test data for the third and fourth types of specimens
1. Im, J. et al., Surface crack growth in fiber composites, MIT Report, NASA CR-135094 (1976). 2. Lo, K. H. & Wu, E. M., Serviceability of composites surface damages. Fibrous Composites in Structural Design, ed. E. M. Lenoe, et al. Plenum Press, New York, 1978, pp. 459-466.
Table 2. The values of prediction No. ISS CSGS CSGL
Prediction Test Prediction Test Prediction Test
and test for specimens
2
3
4
5
6
90.5 96.0 279.1 278.0 12.9 1 I.3
83.0 70.0 231.3 1X8.0 12.0 13.4
95.6 128.0 275.1 256.0 12.2 14.6
72.9 80.0 269.1 246.0 13.0 10.8
82.9 80.0 301.7 294.0 12.4 16.1
ISS-initial splitting stress (MPa); CSFS-critical CSGL--critical splitting length (MPa).
splitting
failure
stress
(MPa);
Modeling analysis of orthotropic laminates 3. Sendeckyj, G. P., Surface notches in composites. Fracture of Composite Materials, ed. G. C. Sih & V. P. Tamuzs. Martinus Nijhoff, The Hague, 1982, pp. 115-137. 4. Zheng, S. & Zhao, J., Investigation on damage of laminates with a surface crack by improved deply technique. .I. Exp. Mech., 3 (1988) 371-377. 5. Zhao, J., Characteristics on damage morphology of laminates with a surface notch. Acta Mater. Compos. Sin., 10 (1993) 1-5. 6. Wei, J. & Zhao, J., The Felicity effect of the acoustic emission in composites. Acta Mater. Compos. Sin., 9 (1992) 6.5-69.
1207
7. Zhao, J. & Ji, F., A variational formulation and analysis of a undirectional composite with broken fibers over a finite region. Theor. Appl. Fract. Mech., 13 (1990) 217-224. 8. Zhao, J. & Yang, Y., Fatigue behaviour of glass fiber/epoxy cross-ply laminates with a surface notch. J. Exp. Mech., 8 (1993) 303-310. 9. Zhao, J. et al., The numerical simulation of splitting failure for unidirectional composites with a surface notch. Progress in Composites, Aeronautical Industry, 1994, pp. 851-854 (in Chinese).
Science and Technology 56 (1996) 1201-1207 0 1996 Elsevier Science Limited
Printed PII:
in Northern
Ireland.
SO266-3538(96)00083-8
All rights reserved 0266-3538/96/$15.00
A MODELING ANALYSIS FOR SPLITTING FAILURE OF ORTHOTROPIC LAMINATES WITH A SURFACE NOTCH J. H. Zhao Department of Modern Mechanics,
& J. Wei
University of Science and Technology of China, Hefei, Anhui 230027, People’s Republic of China
(Received 20 September 1995; revised 17 May 1996; accepted 30 May 1996) Abstract A simple model is presented for the failure behavior of surface notched laminated composites subjected to tensile loading. The model is capable of simulating split growth and predicting the ultimate tensile load for glass/epoxy cross-ply laminates with a surface notch. In simulating, the classical shear-lag theory including the resistance effect against split growth appearing between the adjacent split surfaces, and the strain energy release rate criterion for fracture mechanics are adopted. The program of simulating split growth to final failure was developed. Numerical results were compared with experimental data for glass/epoxy orthotropic laminates with a surface notch. It is indicated that the proposed model and simulating program in the work are available. 0 1996 Elsevier Science Limited Keywords:
splitting failure, numerical simulation
laminate,
tions,“4’5 a simple mechanical model is presented for predicting split growth and the ultimate load in surface notched glass/epoxy cross-ply laminates subjected to tensile loading. By means of the shear-lag effect existing analysis, including the resistance between the adjacent split surfaces, the control equations for the displacements of the model elements were derived, and the strain energy release rate for split growth could be calculated from the displacements obtained. The procedures of numerically simulating the process of split growth to final failure was developed according to the strain energy release rate criterion in fracture mechanics. Numerical results were compared with the experimental data for glass/epoxy orthotropic laminates containing a surface notch. Satisfactory agreement was found between the numerical results and experimental data.
surface notch, 2 SIMPLIFIED
MODEL
AND
GOVERNING
EQUATIONS
1 INTRODUCTION
2.1 General
The mechanical behavior of fiber-reinforced laminated composites with surface notches has received considerable attention from a number of researchers.lm9 Test studies have showed that there are two typical patterns of failure for these laminates. The first is that little damage appears near the surface notch until final failure occurs, as happens for most surface notched graphite/epoxy laminates.2.3 The second pattern is that splitting (delamination and debonding) initiates near the surface notch and then increases in extent leading to final failure, as for most glass/epoxy laminates with a surface notch.1,4’5 For the first type of failure, Lo et al.* and Sendeckyj3 have presented elementary properties of materials analyses to predict the failure loads of laminates. As for the second case of failure, Im et al.’ have proposed a relatively simple model which is capable of predicting some features of the splitting (delamination) process, but cannot be used for the initial and final stages of the splitting process and the ultimate load. On the basis of careful experimental observa-
description
of the model
As has been pointed out’,4,5 for glass/epoxy laminates with a surface notch, the damage caused by loading is mainly delamination between the reference layer (notched layers) and the constraint layer (unnotched layers) and debonding near notch tips in the reference layer. The delamination and debonding grows with tensile load in the tensile direction. It is also known that the propagating fronts of the delaminated region are arc-shaped, but far from being semi-circular, and the central length of the delaminated region is somewhat larger than the debonded region. From these experimental observations, a simplified model is presented as in Figs 1 and 2. Because of symmetry, only one-half of the orthotropic laminate with a surface notch is shown in Fig. l(a), where y is the axial distance measured from the location of surface notch, and L, W and T are the half length, width and thickness of the laminate, respectively. PO is the total load applied at the ends of laminate. The details of the model are as follows. 1201
1202
J. H. Zhao, J. Wei
regions is limited to the model elements 2 and 3. So that the axial stresses in elements A and B are independent of y, i.e. the axial displacements of elements A and B depend linearly on y only. (4) Delamination occurs between elements 1 and 3, as well as 2 and 3, debonding occurs between elements 1 and 2. (5) In the splitting region, a resistance against split growth exists between adjacent split surfaces. This effect may be a result of fiber bridging or friction between the adjacent split surfaces, and matrix yielding at the front of the propagating split, and is defined as the apparent shear-lag stress So (ASLS), as shown in Fig. l(b). 3. It is assumed that sr = ~(1 - 1,/1,)(0 I so 5 s,), where 1, is half the critical split length and so the ASLS to initiate splitting. f, and so are dependent on the character of the surface notched laminate and will be determined by the tests. s, is the shear strength of matrix.
(b) Fig. 1. Schematic
illustration of the mechanics material analysis.
model for
1. For
simplicity of calculation, the difference between the central lengths of the delamination and the debonding is neglected. In what follows, delamination and debonding will both be referred to as splitting, so that the split region is box-shaped, as shown in Fig. l(a), where ld stands for the half split length. 2. It is assumed that: (1) the glass/epoxy orthotropic laminate with a surface notch consists of seven model elements. Their cross-sections are shown in Fig. 2, in which an element with a shaded cross-section represents the notched part which has m layers. Therefore the notch depth, H = mt + (m - l)h, where t is the thickness of each layer and h is the thickness of the matrix between two neighboring plies or between two model elements. (2) Each model element carries only an axial stress which is a function of y and the matrix between two neighboring elements sustains only shear stress. (3) The influence of the surface notch on the stresses of neighboring
2.2 Governing equations Let El, E2, E3, EA and Es denote the Young’s moduli of the materials corresponding to elements 1, 2, 3, A and B along the loading direction, F,, F,, F3, FA and FB the cross-sectional area of corresponding elements, respectively. G, stands for the shear modulus of the IEdriX. UA, (TB and UA, uB are the stresses and the displacements in elements A and B along the loading direction. According to the assumptions in Section 2.1, the relationship UA = CT, x
y/E,
= UB X y/E,
(1)
should exist. Considering split damage with length Id along the loading direction y, let ul, u2, u3 and ul, u2, u3 denote the displacements of elements 1, 2, 3 along the direction y in the split and the non-split regions, respectively. Introducing the following dimensionless variables: l&=
&rzy
J
E,F,G,,,H
G,H
EIFrh ’
h
Q p,’
77 = ki
E,F,G,H
vi= GH m E1eh
J
h
J
’
SF=
ui PO
EIFIHh s, ~J Gm PO
(2) The dimensionless axial load, e, on the cross-section of the ith (i = 1, 2, 3, A, B) element is related to the dimensionless displacement by: p. _ pi _
Eifi dG(or K)
’ PO E,F, I
Fig. 2. Cross-section
i;
view showing surface notched plane.
d5
where pi is the true axial load applied on the cross-section of the ith element. According to the shear-lag theory and taking into account the resistance effect, we obtain the following
1203
Modeling analysis of orthotropic laminates
governing equations for the equilibrium of any section in the split and non-split regions:
3.2 Solutions (1) For n I 5 eqns (7)-(9) following form:
(14)
where V = [V,,V,,V,]’ is a column vector of unknown dimensionless displacements, and [D] = [Djj] is a coefficient matrix of [V] in eqns (7)-(9), respectively; Q = [q1,q2,q31T is a column vector which consists of the coefficients of 5 in eqns (7)-(9). Equation (14) is a system of inhomogeneous, second-order, linear ordinary differential equations. When t+ ~0,dV/dt should be finite. Accordingly, the general solution to eqn (14) is given by
and
Vi = ic,e-*Pe,,
d2V, -dt2
to the
$=[D]V+Q(
(4)
(5)
can be reduced
+ Sit
i = 1, 2, 3
(15)
j=l
(8)
where hi (j = 1,2,3) are the three eigenvalues of [D], ej, is the eigenvector corresponding to 5 and Cj is a vector containing the three unknown constants. The S, is also an unknown vector and will be calculated as follows. Considering eqn (3) and substituting eqn (15) into eqn (12), we obtain u-
WB )
B - EAFA + 2EBFs (
(16)
Substituting eqn (16) into Q of eqn (14), we obtain for n 5 5, in which, FI = Ha, F2 = Ht, F3 = (2t + c + 2h)t, and
s12=;; -, s,x=$, 1
1
2
2
3 BOUNDARY
3
Q = Q[a,(P,)]
(17)
As load PO is given, Si in the general solution eqn (15) can be solved from the following equation:
SIB=
3
CONDITIONS
= D;& + q/
D;Fs, + q; = 0
(18)
where Djj = Dir + Dp.
AND
(2) For 0 5 5 5 n: from eqn (4) and the condition eqn (11) we have
SOLUTIONS 3.1 Boundary conditions Symmetry across the xz plane is satisfied by requiring that U*(O)= U,(O) = 0
(when 5 = 0)
(10)
~~=~(2+x)s’(l-~~~2+cI where C4 is a constant to be determined. (5), we obtain
(19) From eqn
while U, = Cge-+J t C 6e’45+ SiBuB 5
d&(O) -=O d5
(whent=O)
(11) +(1-1,)+?,
stands for the free stress on the notched cross-section. Force equilibrium condition on the end face is PI + 2P, + l’? + PA + 2P, = 1 (when t-+ m)
Continuous such that
(12)
(20)
where Ai = S,, is an eigenvalue, while Cs and Cs are constants to be determined. The solution of eqn (6) is given by:
conditions should be satisfied when [ = n
Wri) = V(T),
W(q)
7
=
W(v)
7
(i=1,2,3)
(13)
2t + a +p H
(21)
J. H. Zhao, J. Wei
1204
in which A25= (F,IHh) + 2(tlH), while C7 and C8 are constants to be determined. Making use of the boundary conditions eqns (10) and (13) yields a system of linear algebraic equations for determining the unknown constants which stand for a vector C = [C, C2 C3 C4 C5 C6 C, CXIT as follows: AC=f
(22)
where A is a 8 X 8 matrix, f is a vector of length 8. The elements of A and f can easily be determined from the general solutions in eqns (15), (19), (20) and (21). Obviously A and f, and hence C, are all dependent on the dimensionless split length 7.
4 THE ENERGY
Based on mechanics, propagation
RELEASE
RATE
the theory of linear elastic fracture the energy release rate, G, for the of a longitudinal split can be expressed as
G2$
(23)
P
where U is a half the strain energy of the split laminate, A = (2H + a& is the area of split region, or G= ‘_“I 2H+a&
P
=1 2H+a
&$:I,.
(24)
The total strain energy contains the extension strain energies of elements (note: the strain energies of the elements A,B are independent of 7) and the shear strain energies of the matrices between the adjacent elements. The extension strain energies are expressed as
‘)( Ui - ~ )2d~, i-j=2-B,3-A BzA = a + 2h + 2t
(26)
for
=
1 I
GmBij 2h2
J
El&h a (VJ - y)2d& G,,,H _,,
i-j=l-2,1-3,2-3,2-B,3-A
where A, f are the known functions of 7. Accordingly, as PO, r,, n are given, once the vector C is determined by eqn (22), then Xi/al7 will be found from eqn (28). 5 TESTS AND NUMERICAL OF SPLITTING FAILURE
SIMULATIONS
5.1 Tests for splitting failure The surface notched specimens were made from glass/epoxy cross-ply laminates [O”/Oo/900/Oo]s with fiber volume fraction of 0.53. Details of the fabrication of the laminates and specimens have been given elsewhere.4,s*8 Table 1 gives specimen dimensions for this work. For each type of specimen listed in Table 1, three to five samples were tested. For a unidirectional ply in these specimens, the longitudinal and transverse Young’s moduli are EL = 2.37 X lo4 MPa, ET = 3.60 X lo3 MPa, and Poisson’s ratio is v12 = 0.28. The shear modulus of the matrix is G,,, = 1.38 X lo3 MPa. The Young’s moduli E1,E2,E3,EB and EA for the model elements shown in Fig. 2 can be calculated from EL and ET. The tensile tests on surface-notched specimens were performed in a universal testing machine at a cross-head speed of 1 mm min-‘. The technique of measuring split length is similar to that described previously.’ 5.2 The numerical simulations for splitting failure process By examining the experimental data for the fourth type of specimen listed in Table 1, we chose a half critical split length Z, = 14 mm as the calculation parameter in the following simulations.
For each group of data for (coo, so), where go is the average stress applied at the end of the notched laminate (obviously, PO= go (F, + I$ + 4 + 2F, + FA)), G versus ld/lc were calculated by the procedures in
split
! w,_,
(28)
5.2.1 The relationship between energy release rate G and split length ld
The shear strain energies are expressed as
where B12 = Bzs = H, regions, and
to find G by eqns (24)-(27), the calculation of aC,/an should be performed. Differentiating eqn (22) with respect to 7, we have
(27)
where B12 = Bzs = H, B13 = a, Bz3 = t, B3A = a + 2h + 2h, for non-split regions. As pointed out earlier, Cj is the function of 77,hence
Table 1. The size of specimens Type
L
W
T
t
h
HIT
a/W
1 2 3 4 5 6
170
20
2
0.20
0.05
0.50
0.50 0.25 0.50 o-25 0.50 0.25
0.75 0.25
1205
Modeling analysis of orthotropic laminates
0 Fig.
IL1
&-““..
as
3. The relationship between the energy release rate and
the split length.
Sections 3 and 4. Such results are plotted in Fig. 3. The plots show that for s0 = 0 the split process will rapidly tend to become unstable. This is obviously not in agreement with the experimental observations, but by increasing so, a stable process of split growth will be achieved. This means that consideration of the resistance effect between the adjacent split surfaces has a significant effect in simulating the process of split growth until failure. 5.2.2 The determination of the critical energy release rate and the ASLS to initiate splitting.
(1) By using the measured load fl which initiates the splitting of a specific type of specimen in Table 1, for example, the first specimen, the stress gi to initiate splitting can be calculated. For the calculated aj and taking so = 0, a series of values G@) which corresponds to a series of values @) (ranging from ld = O.Ol0.1 mm with step Al, = O-01 mm) can be determined. Then the curve of G(“) versus @) is obtained by regression by using the formula G = a0 + aI& + a& According to the definition of critical energy release rate, and letting id + 0, it is seen that G = G,* = a, is the initial splitting energy release rate corresponding to so= 0. (2) After obtaining G,* in step (l), for a suitable value of so(O< so < s,), starting with (TV= Ed= 0, and taking steps AId = O-1 mm, Ace = 2.5 MPa, a series of values G’“’ may similarly be determined for the first type of specimen in Table 1, but in the calculation, if G(“) < G,*, increasing Acre, the calculation must be repeated; otherwise holding a0 unchanged and increasing Al, is necessary. We proceed in this way until the moment- when, without an increase in go, increasing only Aid will result in G’“’ 2 G,*. At that time the u. is the critical splitting failure stress, u,, and the corresponding ld is the critical split length, 1,. It is the beginning of the unstable split growth when r. = uc or ld = 1,. Such a procedure of calculation is the process of numerical simulation of split growth. In the process, a series of data for a0 and ld may be obtained, then a curve of u. versus ld will be fitted. (3) Using G$, still for the first type of specimen, choosing several different values of so and repeating step (2), several curves of a0 versus fd are constructed. Comparing the curves with the test data for the first
0
2
4
14
16
18
Fig. 4. The predicted curves and test data for the first type
of specimens.
type of specimen by the least squares method, the optimum value of so is selected (denoted by $). (4) Letting so = s 8, repeating step (l), a new G,* will be obtained. (5) Using the new G,*, repeating step (3), again a new value of so*will be obtained.The calculation shows that stable values of G,* = G, =274Jm-’ and so*= so = 30 MPa can be obtained by repeating the operation only two to three times. Figure 4 shows the test data for the first type of specimen and its optimum fitted curve corresponding to the values of G, and so. In the following simulations, G, and so will be considered as a true laminate property characterizing longitudinal splitting failure. 5.2.3 The numerical simulations of the splitting process and comparison
with tested data
Using parameters G, = 274 J m-* and so = 30 MPa determined in Section 5.2.2, the process of splitting for the other five types of specimen in Table 1 were
~
Prediied fa HTT=0.5afVW0.25 Measured & H/T=0.5,aVV0.25
5011 0 Fig.
2
4
8 6 SPLIT LENGTH
10
12
I
I
14
16
[mm)
5. The predicted curve and test data for the second type
of specimens.
J. H. Zhao, J. Wei
1206
simulated numerically by the method described in step 2 of Section 5.2. The curves of co versus ld which present the process of split growth are shown in Figs 5-7. In the figures the test data corresponding to each type of specimen are pointed in order to make a comparison. During the numerical simulations, the predicted values of the initial splitting stress (T,, the splitting failure stress V~ and the half critical split length 1, are calculated at the same time. These predicted values and the corresponding tested values are all listed in Table 2. 3501
I 2
0
14
4 SLTL&T~"
16
1
(tzl)
Fig. 6. The predicted curves and test data for the fifth and sixth types of specimens.
REMARKS
A simplified mechanical model is proposed to simulate splitting failure in glass/epoxy laminates with a surface notch in this paper. On the basis of the model, and by using the shear-lag analysis including the effect of resistance between the adjacent split surfaces, a procedure for simulating the whole course of splitting failure including split growth until final failure has occurred. The good agreement by comparing the results of numerical simulation with experimental data shows the potential of the present model and the simulating procedure. The results also show that it is very important and necessary to include the resistance effect between the adjacent split surfaces in the simulation. The proposed approach can be used to determine the critical energy release rate and ASLS to initiate splitting which are the characteristics of a laminate. The stresses to initiate splitting, splitting failure and the critical split length were predicted. The predicted values agree with the measured ones and the relative errors are within 10%. It is indicated that the initial split and splitting failure stresses will decrease with increasing depth and width of the surface notch, while there is no obvious dependence of the critical split length on the depth and width of the surface notch; therefore, it can be said that the critical split length may also be a characteristic parameter for split extension. ACKNOWLEDGEMENTS
250
The authors wish to acknowledge the National Natural Science Foundation of China, for financial support of this study (19072060) and postgraduate Z. Hu for his help with laboratory tests.
lz
200
P
!
6 CONCLUDING
150
REFERENCES
( - -
Predktedbot
twr=0.75,am=05
~
L
50
-
0
I
I
2
4
I SPL&&i
I
lb ----~ 12 (mm)
I4
--’ 1
Fig. 7. The predicted curves and test data for the third and fourth types of specimens
1. Im, J. et al., Surface crack growth in fiber composites, MIT Report, NASA CR-135094 (1976). 2. Lo, K. H. & Wu, E. M., Serviceability of composites surface damages. Fibrous Composites in Structural Design, ed. E. M. Lenoe, et al. Plenum Press, New York, 1978, pp. 459-466.
Table 2. The values of prediction No. ISS CSGS CSGL
Prediction Test Prediction Test Prediction Test
and test for specimens
2
3
4
5
6
90.5 96.0 279.1 278.0 12.9 1 I.3
83.0 70.0 231.3 1X8.0 12.0 13.4
95.6 128.0 275.1 256.0 12.2 14.6
72.9 80.0 269.1 246.0 13.0 10.8
82.9 80.0 301.7 294.0 12.4 16.1
ISS-initial splitting stress (MPa); CSFS-critical CSGL--critical splitting length (MPa).
splitting
failure
stress
(MPa);
Modeling analysis of orthotropic laminates 3. Sendeckyj, G. P., Surface notches in composites. Fracture of Composite Materials, ed. G. C. Sih & V. P. Tamuzs. Martinus Nijhoff, The Hague, 1982, pp. 115-137. 4. Zheng, S. & Zhao, J., Investigation on damage of laminates with a surface crack by improved deply technique. .I. Exp. Mech., 3 (1988) 371-377. 5. Zhao, J., Characteristics on damage morphology of laminates with a surface notch. Acta Mater. Compos. Sin., 10 (1993) 1-5. 6. Wei, J. & Zhao, J., The Felicity effect of the acoustic emission in composites. Acta Mater. Compos. Sin., 9 (1992) 6.5-69.
1207
7. Zhao, J. & Ji, F., A variational formulation and analysis of a undirectional composite with broken fibers over a finite region. Theor. Appl. Fract. Mech., 13 (1990) 217-224. 8. Zhao, J. & Yang, Y., Fatigue behaviour of glass fiber/epoxy cross-ply laminates with a surface notch. J. Exp. Mech., 8 (1993) 303-310. 9. Zhao, J. et al., The numerical simulation of splitting failure for unidirectional composites with a surface notch. Progress in Composites, Aeronautical Industry, 1994, pp. 851-854 (in Chinese).