NONLINEAR TEMPERTURE DEPENDENT FAILURE ANALYSIS …...NONLINEAR TEMPERTURE DEPENDENT FAILURE...
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NONLINEAR TEMPERTURE DEPENDENT FAILURE ANALYSIS OF FINITE WIDTH COMPOSITE LAMINATES by Aniruddha P. Nagarkar Thesis submitted to the Graduate Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of APPROVED: E. G. Henneke MASTER OF SCIENCE · in Engineering Mechanics C. T. Herakovich, Chairman September, 1979 Blacksburg, Virginia M. P. Kamat
Transcript of NONLINEAR TEMPERTURE DEPENDENT FAILURE ANALYSIS …...NONLINEAR TEMPERTURE DEPENDENT FAILURE...
NONLINEAR TEMPERTURE DEPENDENT FAILURE ANALYSIS OF FINITE WIDTH COMPOSITE LAMINATES
by
Aniruddha P. Nagarkar
Thesis submitted to the Graduate Faculty of the
Virginia Polytechnic Institute and State University
in partial fulfillment of the requirements for the degree of
APPROVED:
E. G. Henneke
MASTER OF SCIENCE
· in
Engineering Mechanics
C. T. Herakovich, Chairman
September, 1979
Blacksburg, Virginia
M. P. Kamat
ACKNOWLEDGEMENT
The present work was supported under NASA Grant NGR-1250. The
author is grateful to Dr. C. T. Herakovich for his help and guidance.
Thanks are also in order to Dr. Henneke, Dr. Kamat, Mr. Marek Pindera
and Mr. Richard Boitnott. The help from in plotting
results and accurate typing is greatly appreciated.
; ;
TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS. . . . . . . • . . . . . . . . . • • • . . • • . . . . . . • . . . • . . . . . . . . . . . . . • ii
TABLE OF CONTENTS ........•...•..... : .••.................. ~...... i i.i
LIST OF TABLES . . • • • . . • . . • . . . • . . • • . • . • . . . . • . . . . . . . . . . . . . . . . . . . • . . v
LIST OF FIGURES .............•...•.........•..................... vi
CHAPTER
1. INTRODUCTION ..•.... • •.....•...•......... ·.............. 1
2. LITERATURE REVIEW . .. .. . . . . . .. . . . . .. . . . . . • . .. .. .. .. .. . . 3
3. THEORETICAL BACKGROUND . . .. . .. . . .. . .. . . • .. . .. .. .. .. . . . . 7
3.1 General Formulation . .. . .. . . .. . .. .. .. . .. .. .. . . . . . . 7
3. 1. l Finite Element Formulation . . . . . . . . . . . . . . . . 11
3.2 Mechanical Formulation .......•........... ~ ....... 12 3.3 Thermal Formualation ....•.••... ~ ..•....•..•..•... 14 3. 4 Nonlinear Analysis .. . . .. • . • . . . . .. . . . . . . • .. .. .. . . . 18
3.4.l Mechanical Load ..••....•.•..........•..... 18 3.4.2 Thermal Load .........•.•.....••.•.....•... 19
4. PRELIMINARY STUDIES . • • . • • . . . • . • • . • • • . . . . . • • • . • • • . . • . . . 23
4.1 MeshSize .....•....•.....•.•.....•••.......•...•.. 23 4.2 Averaging Finite Element Results ·•• ..••..•.••..... 25 4.3 Linear Elastic Analysis .••••.•••.•.••• ~ ••.•••..•. 30 4.4 Stress Free Temperature .....•.••.......•.•....... 30
5. STRESS AND FAILURE ANALYSIS OF LAMINATES .............. 36
5.1 Cross Ply Laminates ··········~··················· 36
5. l. 1 stress Distribution . • . . . . . . . . . . . • . . . • . . . . . 36 5.1.2 Failure Analysis .......................... 47
5.2 Angle-Ply [aminates .•...•.....•....•..•.........• 51
5 . 2 . l St res s Di str i b u t i on • . . . . . . . • . . . . . . • . . . . . . . 51 5 . 2 . 2 Fa i l u re An a 1 ys i s . . . . . . . . . . . . . . . . . •· . . . . . . . . 5 9
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Chapter
5. 5.3 Quasi ... Isotropic Laminates ....... ~ ..........•..... 63 ·
· 5.3.l Stre$s Distribution ....................•.• 63 5. 3. 2 Failure Analysis · . . . . . . . • . . . . . . . . . . . . . . . . • . 66
6. CONCLUSIONS •••••.•••••••••••••••••••••• ; •.•••••• • •••••••
BIBLIOGRAPHY· •••••. • •.• , ••••••••••••••. • •.••••••••••.•..••••••••••••••••
APPENDIX
A. CONSTITUTIVE RELATIONS ••••••••••••••••••••••• •· • • • • • • • • 81
B STIFFNESS MATRIX ••••••• ~ ••••••••••••••••••••••• ;, ••.•.• 86
C . TSAI-WU FAILURE CRITERION ••••.••••.•••.•••••••.••••••••• 88
D USERS GUIDE FOR NONCOM II • . . . • . • . • • • . • • • • . . • • • . . . . . . • . 91
E
F
G
VITA
. ··, :: ' ·. . . . . .
, MESH ES US ED : i • , • :· •• ~ ••••••• ~ ••••••••• ~ ~ ; •• ~ •• ; •••••••• 119
NONCOMil FLOW CHART ••••••• • •••••••••••••••••••••••••••. 125
1300/5208 PROPERTIES •.•.•••••••••••••••••••••••••••••• 127 .
.· . ... ·.··· . •. .· .. ' . . . .. . . . 134 . •• -• .• ·- •••• ·- ••••.•.•. • .• - ••• ·- ••• -••••• ··-· ... I! •••••••.•..•••••••••••••. _ ••••
ABSTRACT
Table
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2
3
4
5
6
LIST OF TABLES
Page
Influence of Mesh on Failure Results ....... ··········~ 24
Linear Elastic Predictions of First Failure ........... 31
Effect -Of Load Steps on the Tensor Polynomial ......... 34
Failure Mode Analysis of Cross-Ply Laminates ...•...... 52
Failure Mode Analysis of Angle-Ply Laminates ·~········ 62
Failure Mode Analysis of Quasi-Isotropic Laminates .... 74
v
LIST OF FIGURES
Figure
1 Typical Laminate Geometry 8
2 Boundary Conditions on the Quarter Section of the Laminate ....... , . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . l 0
3 Determination of Tangent Moduli with Ramberg Osgood Approximations ................................. 20
4 Typi ca 1 Percent Retention Curve . . . . . . . . . . . . . . . . • . . . . . . 21
5 Variation of az at the Interface of a [90/0]s Laminate with Mesh Size ....... ·. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 Typical Graphite/Epoxy Ply with Smallest Elements Superimposed .......................................... 27
7 896 Element 502 Mode Mesh ............................. 28
8 Averaging Scheme ...................................... 29
9 Typical a Curing Stress Distribution as a Function x . of the Number of Thermal Load Steps ...........•....... 33
10
11
12
13
ax a x cry
cry
in
in
in
in
a [0/90] . s a [90/0]s
a [0/90Js
a [90/0]s
Laminate ...... ·- ·- ....................... 38
Laminate . -........................ ' ....... 39
Laminate . . . . . . . . . .. • "' . . . . . . . . . . . . . . . ·- . . 40
·Laminate • • • • • • • e • -. •. • • • • • • • • • • • • • • • • • • • 41
14 az at the Interface in [0/90]s and [90/0]s Laminates ... 42
15 Partial Free Body Diagram of the Laminate ...........•. 43
16 az Through the Thickness for [0/90]s and [90/0]s Laminates . . . . . . . . . . . . . . • . . . . . . . . . . . . . . . . . . • . . . . . . . . . . . 45
17 ax Through the Thickness for [0/90]s and [90/0]s Laminates ............... ·· . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
18 Tensor Polynomial along the Interface in the 90° Layer for a [0/90]s Laminate ......................•... 48
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Figure . .
19 Tensor Polynomial along the lnterface in the 90° Layer for a[90/0]s Laminate ...•.•... ~................ 49
. . .
20 Tensor Polynomi ~l Through the Thickness for [0/90 ]s and [90/0]s Lam mates . , •. ·•· ... , ...••..•...•...•• ~ . . . . . . 50
21 Maximum Normalized Laminate and lnterlaminar Curing Stresses in Angle-Ply Laminates ....•.. : • . . . . . . . . . • . . . . 53
22 Maximum Norma 1 ized. tami nate and . Interlaminar .curing ·. Stresses at First Failure in Angle-Ply Laminates ;.. .. . 54
23 Lam~nate Curing Stresses in the .;.45° Layer of a [±45]s Laminate ..••................................ ·• • . . . . . . . . 56 ·
24 crz and Txy Curing Stresses in a [±45]s Laminate ........ · .57
25 Through the Thickness Distributions forthe Residual cr and T in a [±45] Laminate . . . . . . . . . . . . . . . . . . . . . . • 58 x . xz .· . s . . ..
. .
26 Tensor Polynomial along the Interfac:e for Various Angle-Ply ·Laminates>, ........•............. ~ ... ; . . . . . • . . 60
27 Through the Thickness Tensor Polynomial Distributions from Curing Stresses and Stresses at First Failure in Angle-Ply Laminates ................•...•........... · 61
. . . . 28 crx in the 90° Layer of [±45/0/90]s and [90/0/±45]s
Laminates ............ ; ............. ;.. . . . . . .. . . . . . . . • . . . 64 ·
29 cry in the go0 La'yer of [±45/0/90] 5 and [g0/0/±45]s Laminates ..... · .........•............................... ,. . . 65
30
31
32
33
34
35
oz, Txz' Tyz in a [±45/0/go]s Laminate ............... ,
oz, Txz, Tyz in a [90/0/±45]s Laminate ............... .
Through the Thickness crx Distribution in [±45/0/go]s and [90/0/±45]s Laminates ..........•..... • ............ .
· Through the Thickness o Distribution in [±45/0/go] ·. . z ..• . s and [g0/0/±45]s Laminates .............•...•.... , ..... .
Tensor Polynomial in the go0 Layer along the 01go Interface in [±45/0/g0] 5 .and [g0/0/±45] 5 Laminates .•..
Through the Thickness Tensor Polynomial Distribution in [±45/0/go]s .and [go/0/±45]s Laminates ..... ; .......... .
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69
70
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1 • · lNTRODUCTI ON
Manufacture of composites involves curing the. fiber matrix system
at elevated temperatures, followed by cooling to roomtemperature.
Cure temperatures are typically 350°F for epoxy resin systems, and 650°F
for polyimide resin systems. The thermal ·expansion cbeffic;ient mismatch ' . . .
between fibers and matrix coupled with the large temperature differentials
result in residual or curing stresses in laminates at room temperature.
These stresses may be large enough to causemicrocracking.without . ' -, '
additional applied stress, or early failure due to applied loads, in . . . . ' .
resin matrix composites. The residual thermal stresses are also.present
in metal matrix composites such as boron-'aluminum and can cause residual
plastic strains in· the metal matrix.
The purpose of this study is to analyze the response of laminates
to mechanical loads, including these residual stresses, and predict the
occurrence and mode of first failure. Researchers have. proposed various . .
ways to calculateresidual stresses, and there are a few studies of
stress-strain response to mechanical load, including residual stresses.
These studies perform the residual thermal analysis and the mechanical
load analysis separately, and assume linear elastic behavior. The
principal of superposition is used to predict the combined effect of
mechanical load and curing stresses. The present study treats the
thermal and mechanical behavior separately, but does not make the
assumption of linear elasticity. ·The residual stress field therefore
cannot be superposed on the mechanical load, but is used as. an
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initial condition. Special attention is given to the. influence of edge
effects on the stress field and occurrence of first failure. . .
The finite element analysis program NONCOM2 [l ,2,3] was modified
for this analysis .. Its efficiency and capability were.increased so as
to handle a larger number of nodes, with a choice for an in-core or an
out-of-core equation solver, and a detailed failure analysis of the
tensor polynomial failure criterion for predicting first failure.
The material system used in this study is Thorne] 300/Narmco 5208
graphite/epoxy. This system was chosen becaus.e of current NASA interests.
2. LITERATURE REVIEW
The theory of residual stress in composites does not h.ave any
counterpart in homogeneous, isotropic theories, and there are relatively
few studies reported in the literature. All the studies are based on
the assumption that the total strain is the sum of two distinct parts:
the mechanical strain, which is related to the stresses,and the 'free'
thermal strain, which .does not cause any stresses in the laminate.
Most previous studies are lamination theory solutions. They are
based on the classkal plate theory assumptions, and therefore valid
only in the interior, away from free edges. They yield only laminar
stresses. However, failure in laminates is often observed to initiate
at free edges [4], and the stress distribution there is of. interest.
Tsai presented a thermoelastic formulation for calculating thermal
stresses in 1965 [5]. This study gives the basic lamination theory
develo~ment for calculating residual stresses. A micromechani~~l
procedure for calculating residual thermal stresses was outlined by.
Hashin [6]. One of the earliest reported analytical predictions of
residual stresses, using lamination theory, is a study by Chamis [7],
in which he analyzed laminates of different material systems, stacking
sequences and fiber volume fractions. Extensive experimental studies
were conducted at the IIT Research Center by Daniel and Liber [8].
Stresses were obtained from measured strains using temperature dependent
constitutive relations.
Herakovich [9] analyzed cross ply laminates of various material
systems using finite elements, comparing stress distributions due to
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4
thermal arid mechanical loads .. The an~lysis included interlamiriar
stresses but was linear elastic, with properties independent of tempera.,.
ture. A nonlinearanalysis,which included thermal effects, wa.s - . . .
conducted byRenieri and Herakovich [l], but residual stresses formed . '· ., .- .--'.
only a part of the study, so the analysis was not very detailed. Their
formulation will be used in the present analysis.
Hahn and Pagano [10] pointed out the necessity for the inclusion of
terms corresponding to the. stress and temperature dependence of proper-. .
ties. They developed a 'total strain' theory, in which the strains and . .
stresses are calculated using temperature dependent elastic properties.
Daniel, Liber and ChanHs [ll j developed a. technique to measure
. residual strain by embedding strain gages between plies in laminates. . . .
They used this technique for measuring curing strains in Boron/Epoxy . . . . .
and S Glass/Epoxy, and calculated stresses using temperature dependent ·
constitutive relations. The thermal cycling suggested that residual . - - ' . .
stresses during the curing process were primarily due to thermal mismatch
between adjacent plies.
Chamis and Sullivan [l2l outlined a procedure for nonlinear analysis
of laminates with residual thermal stresses .. The laminate was loaded in
increments, using stresses calculated in a load step to calculate elastic
moduli for the next load step. Micromechanics.was used to predict lamina
properties, which were used iri the laminati.on theory apalysis.
Hahn [l3] concluded the· stress free temperature to be less than the
cure temperature. The method outlined in [lO] was used to calculate
residual strains, and colT)pared to experimentally determined strains.
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. . •'
Daniel and Liber [14] investigated the effect of stacking sequence on . . . .
residual stresses in Graphite/Polyimide laminates .. The strains were
determined experimentally,, .and the stresses, calculated using constitu-
tive relations, were found to be close to the transverse strength.
Wang and Crossman [15] studied edge effects due to thermal loading
on some specific laminates. They predict a peculiar behavior for a
[±45] laminate, with the existerice of 'stiff' tensile and 'soft' s . . . ·. . compressive zones in the laminate.
. .
A report by Chamis{J6] summarized work done at the NASA Lewis
Research Center on angle ply laminates over a period of 8 years. The
effect of curing stress.es on laminate warpage and fracture was studied,
experimentally and analytically using lamination theory.
Hahn and Pagano [17] used the procedure described fn [10} to calcu-
late residual thermal stresses, and studied their effect on failure in
· laminates. The curing stresses were found to influence first failure in
laminates greatly, often reducing the applied load to failure by about
half. They note that the interlaminat normal stress crz is significant
in some stacking sequences, especially at free edges, and that this
would result in failure initiating at loads much less than their calcu ...
lated values. Their analysis is based on lamination theory and cannot
predict interlaminar stresses.
Farley and Herakovich [18], .with a finite element analysis, compared
stress distributions due to mechanical; thermal, and moisture loads,
Each type of load was analyzed separately, but the study was concen~rated .
on the response of laminates to different moisture gradients.
.. ~ ...
·;.'.
" .. _ .· . :" ·: - •' . ·-. · __ :.
Kim.and Hahn [19]recently.published resiJlts of acousti.c :emissions .. ·
of laminates sµbject to mechanic~l load~; Curing stresses were in-: ..
eluded· in theJarllination thebrydevel~pment for predicting stress at · . . . - ..
which first -failure occurred. There are thus two theories for calcu-
lating residual stresses: the 'total: strain' [10] and the 1 increm~r:ital1
.·.theory that is followed bY mqst of the other studies; ·including the
·•present one; ·
'. ·;
·-· .·.;·_
·.•,
. ·;., ..
.· :--·· ..
: ·,·"
. ·~ .. •'.
. ,, ..
. ··-.· .·;
:. · .. ~· . . ,·: .. ·
. . ..
i · .·THEORETICAL BACKCiROUNO· ·. - - .
The prob1eiriu~der col1side_ratfon is the stress analysis of?yrnmetric .. · .·
lanitnates, inclyding therrn&r and-free edge' effects., ln this study· the -· .. ' .. . - ·, .:.·· . , .. . ,, . .. . . ' . . . .. ' '.
non) i~ear analysis for bofttuntaXial mechanical and th~rm~l loading-fs · ·
an . incr~mental .Pr.ocedure. · The.··.foaq is divided into nu~eroµs :steps· and ...
each loa('.I step is treated a~ a l iriear problem. -The loading, ~hethei, uniaxia·1 ·mechanical ot _thermal, is as~umed:to be -stead; -~ndunifbrm '
. . . . . ' :" .. ' . . . . ·.. .
across the laminat;e. ·
3. l General Formulation ·.·. · .. ·.· ... ·.·· ·. .• · · .. · • '· < . . - . . .
A .typical synimetric laminate is shown ·in Fig. h The behavior .of: ·.
the· l~minate ca'n be assumed to be independent •~f- the x, coordinat~s. ; As. . . .
·. . ', . · .... ·.:: . · . .:- - . . . ·.. : .·, . ... . . . . ·.· . ... . .. · </' shown by Pipes and Pagano [20] the linear strain dis pl aternent relations
can be integrated- and· manipulatedto yield the-~ol1owtng displacement ..
. field.over the ctOss-section. of the l~mtnate. ·. ·. ,·
·•· u~ -tc1z+c2)y +_(c4y+c5z+c6}x +. v(Y,z) . . . . ..
. . . .. . .. ·. x2 . .. . . · .. ·. v = (hz+C2)x -G4. 2 + v(y,z} {3.1) .· .·.
.... 2 w = · -c1 xi+ cyx - · c5 x 2 + c8 + w(y ,z)
·. · ... ·.· . . .
The displacement field has the following.~ymmetry~ W:ith r~spect to
the x~y plane:'·_-·.
u(x,y,z) ='= vCx"y,~z) · .. -·
.. v(k,y,z) · ~.-· v(x ,y,-_z) ... · · .. .
wtx.,y,z) = ~w(x,y,;..z)?
. :,'J
{3. 2a) , · ··
. : .. ·.
.• .· ..
8
. ·1;
.··~x
(0)
. ..., .
Y· ..
·(C)
( b)
FIGURE l.. TYPICAL LAMINATE GEOMETRY
9
Wtth respect to the x~z plane:
v(x,y,z) = ~v(x,-y,z)
w(x,y,z) = w(x,-y,z) ·
It has been experimentally observed [21] that at z=±H
u(x,y,±H) = ;..u{x,-y,±H)
(3.2b)
( 3. 3)
As the thickness of the laminate is small, it can be assumed that
u(x,y,z) = -u(x,-y,z)
These symmetries simplify the displacementfield to
u = c6x + U(Y~z)
v=V{y,z).
w = W(y,z)
(3 .. 4)
(3.5)
The analysis can be restricted to only a quarter section of the .
laminate (Fig. 2) with the following boundary displacement constraints:.
v(O,z) = 0
w{y,O) = a (3.6)
To complete the boundary value problem, there are the following
stress boundary conditions:
a z ( x ,y , H) = 0
Tzx(x,y,H) = a · T zy(x ,y,H) = o ( 3 .])
v=O
w=O
10
T.=O l
FIGURE 2. BOUNDARY CONDITIONS ON THE QUARTER SECTION OF THE LAMINATE
y
11
ay(b ,z) = 0
Tyx(b,z) = 0
Tyz(b,z) = 0
The material of the various laminae is orthotropic, and has a
stress strain relation with 9 independent constants. Referred to the
laminate axis, the stress strain relation transforms to
ax ell c12 c,3 0 0 c,6 EX
cry C22 c23 0 0 c26 Ey -
az C33 0 0 c36 Ez = (3,8)
Tyz C44 C45 0 Yyz
Txz Sym C55 0 Yxz
TXY c66 Yxy
The transformation equations used are given in.Appendix A.
3.1 .1 Finite Element Formulation
This boundary value problem ts cast in the finite element frame-
work. The cross section is divided into triangular elements, and the
displacement field is assumed to vary linearly within each element.
The field is represented in terms of the coordinates of the element
nodes and the nodal displacements. The total potential energy, con-
sisting of the strain energy and the potential by external forces, is
written for an element in terms of the nodal displacements and forces.
It is then minimized by partial differentiation with respect to the
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nodal displacements to get a linear system of equations relating nodal
displacements to the forces by the element 'stiffness matrix'. These
matrices are assembled, and after the imposition of boundary conditions,
the system of equations solved for displacements. The strains and
stresses in each element are calculated from the displacements of the
element nodes, the strain ~isplacement relations and the constitutive
equations.
3.2 Mechanical Formulation
Let the laminate in Fig. l be loaded with a uniform strain ~ in
the x direction. The displacement field at a cross section x=x1 becomes
u = a1 + a2y + a3z + ~x1
v = a4 + a5y + a6z
w = a7 + a8y + a9z
(3.9)
As the laminate behavior is independent of the x coordinate, x1 is arbitrary. Because the field is assumed to vary linearly over the
element, and the strain displacement relations linear, the strain over
each element is constant. This field is represented in terms of the
coordinates of the element, and the strains become:
13·
k ~Ak
'k EX
Ey av ' 1 + cv2 + CV · ' 3
Ez 1 ba1 + dw2 + 9W3 =A-
Yyz ' k bv · + dv2 +gv3 + aw1 + cw2 + ew 1 ' 3
(3.10)
Yxz bu1 + du2 + gu3
Yxy au1 + cu2 + ~U3
where Ak is the area of the.element, ui,vi,wi' (i = 1,2,3) are the u,v,
and w displacements of the nodes 1, 2 and 3 respectively, and
a,b,c,d,e,g are known constants involving nodal coordinates.
The total potential· energy is calculate.d using these strains, and
nodal forces. Minimizing it with respect to the nodal displacements
yields the foll owing set of equations:
u, k f lx
k
v, f ly w, f lz
u2 f 2x [K] v2 = f 2y (3.1n ·
w2 f 2z
U3 f 3x V3. f 3y
W3 f 3z
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. . where [Kl is the 9x9 element stiffness given in Appendix B.
The stiffrless matrices of all theelements are superposed to obtain
the global stiffness matrix, Boundary conditions are imposed as ·
foll ows ( fi g . 2) :
Displacement Boundary Conditions
v=O along y=o·and w=O along z=O
This is achieved by constra:ihing all the nodes on the 1 ine y=O against
displacement in the y direction, and those on z=O against displacement
in the z direction. Due to the assumed linear variation of the displace-
ment field, constraining two adjacent nodes also constrains the line
joining them.
Tract ion Boundary Conditions
T.=O on z=H and y=b 1 .
The traction free boundary conditions are imposed by applying
statically equivalent nodal forces. Ttie surfaces at z=H and y=b are
stress free and equivalent nodal forces are therefore zero.
The reader is refered to [l] for a detailed derivation of the
various matrices.
3.3 Thermal Formulation
The basic assumption in the thermal .formulation is that the total
strain can be written as a sum of a stress related mechanical strain
and a free thermal strain.
When a balanced symmetric laminate is subject to temperature change,
it expands or contracts uniformly. Based on this expansion the
laminate thermal expansion coefficient is defined in lamination theory.
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This definition is necessary to cqlculate strains·. in each lamina. ln a
finite element formulation however; the strai>ns are calculated from a
displacement field and the definition of the laminate expansion·
coefficient is not neces~ary.
The displacement field over the element has the same form as (3.5)
·but the uniform strain i;, instead of being known, istreated.as an
additional unknown that is corrmon to .all the elements. It is equivalent
to the laminate expansion coefficient times the temperature difference. . . .
The mechanical strain fE}a is used to calculate the strain energy
of the element.
where {d 0 is the total strain, and{dT the thern1al strain. In tenns
of the displacement field and expansion coefficient it becomes
£ z
k . 2 .· .. 2 .· t;, ~. (m a1 .+ n a2)llT
y yz ( bv1 + dv2 + gv3 + aw1 + cw2 + ew3) /Ak
Yxz ( bu1 + du2 + gu3 )/Ak
Yzy (au1 + cu2 + eu3)/Ak + 2mn(a1 - a2)llT
k
(3.13).
Minimizing the total potential energy with respect to the nodal
displacements and the unknown strain i;, results in the following set
of equations.
16
ul k
flx
vl . fly
w, flz
u2 f .· 2x
v2 f .· 2y [K]
w2 f 2z (~.14)
U3 f .. · 3x
V3 f 3y
w 3 f3z
·~ f. k
where [K] is the lOxlO element stiffness given in Appendh B.
The global sti.ffness matrix Js obtained by the superposition of the
element stiffness matrices. Boundary conditions are imposed as is the
case of the mechanical load. There i~ one more equation for determining
the uniform unknown strain ~. The force equilibrium equation for the
thermal load n E
k=l f = F = 0 k
. ( 3.15)
is used as the additional equation. The system.of equations is solved for
displacements~ and as in the case of the mechanic:al loading the strains
and stresses calculated.
The thermal response is assumed to be elastic. The stress state
17
resulting from some temperature change from T; to Tf is given by
Tf~ .· • d cr {a} = T. I[C(T )]{d~ (T}}dT .
1 . . .
(3.l6)
As exact mathematical forms for [CJ and {~~}0 are not known, continuous
integration cannot.be performed, and has to be evaluated as a summation.
nsteps {a} = E {~o;}
i=l (3.17)
Consider the ith load step in which the laminate is subject to a
temperature change.fromT1 .to T2. Bythe incremental strain theory
the incremental stress is given by
( 3 .18)
However, as pointed out by Hahn and Pagano [lOJthis equation does not
take into account the temperature dependence of elastic properties.
Their total strain theory gives the expression for the incremental
stress to be
{tiaiJIT = [C(T2)]{M: 0 (T2)} + ~T [C(T2)]{ti£ 0 (T2)} 2
The second term of the equation 3.19 is difficult to evaluate~ and the
incremental stresses are approximated as
( 3. 19)
=,
.. ·~' : . :•.·.·· ·· .. -. '·. · ..
,.· . '.::.·_. ::;·
. ·"'
,., ... ·.·-. · .. ': . ... . . ~ . : . . .
.: ·. ·'·:-"
··· 18 ..•... . .... . ,·· .... . :·.<···. ·.
· .. '.
' . '. ... :·'"
.· - .,.· .·.,·.:.
·.. . . ·.·•,.·
.~here.rm•• is .. some .~nterm~cii.~te tempe.r(lture betw~enr1 ·andTi~ chosen ..
· .. •·.· •. to he th.e mea~ in ihis ~tydy·. ·• T~e· .. te~peratur~ deper)dence;?.f pf~J~eii . -.
ties is included in the formulation. ·. ·< : ·.· · •
. ·. · .... ·. ··· .. ··•··•··· .·.· > ··•·· .·· 3.4 .·Nonlinear A~alysts->
'>. •\ . ._ , ... .........
-·. .. . .
.... · '.
. . The ·nonl i~ear ~nalys i.s .is carried out ;n an focren{~ntat f11shi'on,'
usJng da.ta obtained. in.the previous load step to. calculaie the. materi~l ··•·· .•: ... '. :: .
. '·· . '
' .. ·.,
Gonstarrts for the. cur'reht Joad step,. .. ·· · ......... _:·
··. Ramberg: Osgood paFameter~ [221 are usect to ~epresen~ .the non11n~ar · .•. . ~ ,. . . .
' . . . . . . . . ' . . "
. · . s tr~s s strain rei'a ti ons,. Typically .. . · ... : .. ,· .. ·_ ·.·· ·. ,': . . ... ·" n. · ...... ·
•. ' t :: t + ~i (J i J~'J ,2 ' . . •: ......... ·.
.. ',
":· ... ; ~.·· .,
' " .. : .· . · .. :. . .. . •',the tangentmodulus is defined. a~ '
.,··:;.
··.. : ' ·.·.·. ... , .· ...
' ..
. .- .
.·,·:·
"· '·.·:.:> r,:· .
(3,.21) .: .. ,··
.·.·. '' . ·. ·· ... ·· ....... . . Thi:! ~tress' at 1 oad ste_p: i. is ·. ·
.........
·. :.(3.22).' > . ' ., {. •"". - ' '<" A •. J'E
CJ • - L. , L1E> . , . • 1 j=l ' J . · .. , .
. ··.,.·· ... ··· ·.,.
and· the tangent n1oaulus for the i+l $_tep fs .. ;.-
. : . . ·. ; ..
· ... , ·.·:'' .. · ... ~.
' ·... : . -. . . ~ ' . :· . .. ~ . ' .
The tangent.riiod~lus for,'-each eJas_tic: const~ht is t~ltuTat~d ass,umin~t ~ ' , . ' '
... ,.·.
··. :;"··. ·.' ,_.:··_ ....
• .. ,c·: ":,
' . ,~
·' ",-''.·
.•, ... :
· .. : ·· .... :·
l9
that. there is no interaction· between the various stresses in the
nonlinear behavior.·
As seen frorn fig .. 3 there. is. some errpr in following the stress-
strain curve. This error can be minimized by iterating the. solution
in every load step, or by using smaller.load steps, as is done in this·
··study.
3.4.2 Thermal Load
. Temperature dependent elastic properties are used for the analysis
of thermal loading~ The e las tit moduli E11 , E22, G12 , etc. , Poisson 1 s
.ratios '\2, v23 , etc., and strengths X, z,s13, etc., are input at
various temperatures, in the form of percent retentions, as shown in
Fig. 4. In a given thermal load step,,the mean temperature Tm is
found and. the property linearly interpolated between the nearest higher
and lower temperatures (T1,T2) for which properties have been input.
These interpolated values are th.en used to evaluate the stiffness
·matrices. The analysis accuracy improves with a larger number of input
points, since the retention curves for elastic properties and strengths·
are approximated to greater accuracy.
3.5 Failure Criterion
The finite element analysis gives us a three dimensional state of . . . .
stress, presenting a unique opportunity t.o study. stress interaction
and failure.
Tsai and Wu [23] proposed that the failure surface be represented.
inthe form of a tensor polynomial.
r-IW
r-w <l
20
---· -.- Actua 1 Stress Strain Curve Ramberg Osgood approximation
~ 1 .~2 errors in load steps 1,2
Strain
FIGURE 3. DETERMINATION OF TANGENT MODULI WITH RAMBERG· OSGOOD. APPROXIMATIONS
~ $.... QJ Cl. 0 $....
CL
u •r-
4--0
c 0
+-' c QJ +-' QJ
0:::
+-' c QJ u s.. Cl)
CL
T2
21
0 Percent Retention Moduli at Input Temperature
T Tl m
I. ~I Load Step
Temperature
FIGURE 4. TYPICAL PERCENT RETENTION CURVE
22
F · .cr. · + F,·J·k1cr ... crkl = l . ~J lJ . . lJ . (3.24)
·. . : '
The Fij is a setond order tensor and Fijkl a foutth order tensor.
The numerical values of the terms are obtained from the material
principal strengths. The tensors simplify greatly for orthotropic
material. The transformations of the non zero terms in these tensors,
in the contracted notation, are given in Appendix C. The strength·
parameters F12 , F23 .and F13 require special biaxial loading tests
unlike all ot6er parameters which can be obtained from tenstle, com-
pressive and shear tests. Failure is predicted to occur when the value
of the polynomial is equal to .or greater than 1.0. The failure mode
can be predicted by comparing the individual contributions of the
stress components to the polynomial [24].
4. PRELIMINARY STUDIES
4.1 Mesh Size
The present analysis is conducted at the lamina level, treating the
fiber matrix sy~tem as a.continuum, and not at the micromechanical level.
The finite element method discretizes the domain b~ing analyzed.
Using finer grids, one can get a better representation of the continuum
and hence better results. The problem is deciding on the size of elements
in the mesh.
The lamination theory results are accurate in the intedor of the
laminate. The elements in that region can be much larger than those near
the free edge where large stress gr~dients exist and a finer mesh is
necessary. However, the smallest element should be large enough for the
assumption of lamina properties to be valid and not so small as to be in
the micromechanical domain.
The effect of mesh ~ize was studied by using various meshes that
were loaded with the same strain of 0.1 percent~ keeping all the material
properties constant. It was observed that not only do the stresses in
the region near the free edge change, but the maximum value of the tensor
polynomial used to predict failure changes with mesh size. A linear
elastic analysis also predicts different first failure location, for
the same laminate and the same loading, depending on the mesh used.
Table l shows the location and maximu~ value of the tensor polynomial
for various meshes {Appendix El, E2, E3, E4, E5) for a tensi 1 e load
of 0.1 percent strain. The meshes were generated using the mesh
generation code devloped by Bergner et al [25].
23
24
TABLE 1
INFLUENCE OF MESH. ON .FAILURE RESULTS
No. of Elements Failure Location Tensor in Mesh on Free Edge Po lynomi.al
124 center of top layer .238776
230 center of top layer .238876
326 •. nea,r interface in .2564352 top layer
598 near interface in .2662315 ,top lay~r
878 near interface ih .2804952 top layer
··,:'
'-:. - ·,
. . . . . ·. - .· ··' . '·· .·· ...
The stress· distributio~ 'i~ ·al$i> a functiOn of ~e·s~< .F~~ :ex.ample., .. . ...... ·. a:z for>a· [90/0]~· laminate exhibits>singuJat. behaviori'with>a .large t~n-·
sile value, at thefree ed·g~ [24:~2.6J. · Howev.er, if the grid usedis not
firie enough~· .it is.compressive· ra£her~than lar~e-t~nsile (Fig. ·5).
.. In Gr/(laminates, th.ere ~re apprbxiinately 20.:.2:~ filaments through · · ' .·, ... ... '' .. · ... - ... ' . . . . . . . . . .
the thj ckness '.in: ~~Sh· ply .. ··· ... · .. Fig. 6 ~ho~s the srnaJlest. el ernents f~ th~
useci in .·this stiJdy,. there ~r~· l, 6 el~ment_~ thY'pugh ea~h ply. • T~erefore ~
per~lernen~; there is .·Just over o~e Ji l aJnent: in.the,thicknes$ ·~iTecfi<)~. For a 1.amfnate aspect· ratio of is, the:• number o( fi. laments >calculated : .-·
:. . - ·. .. . .. . ' . . : ... - .. :. . ·.. .. . : · ..... · '. -
to be: Jn the smalie.sl ~Jeme~t is·,:approximateix ~J75,. assuming a .fiber·
. volume fraction qf'.O:~s.·,-· Thfs. mesh: Jftg.:_t5} was<modift~d s•o that it>. ·: .:·.·
.· colll cl a 1 so be used for a four 1 aye~ed J;ami nate (Fig. • 7). . . ....... · .. ·. ;•'·" ·'··. . . ·.. . .. ·.· - .· ; .·,
. . . . . :
. 'The fin.ite.element:fori!lulatiofr u:sed t.n this -invest·igation yields···
·. Const~nt· values forstresses ovey. each· element~ .• • ~WO adjacent ·~leme.nt~, in··· general, ·give di~fer~n·t value~ :·f()r the .:stress•· at. p~int$ on their .·.•··· ···
, .... ,· .' . . ..... •' : .. . ·.' ·' .· .' .: . . .· - . .. . . .. .··
common boundary~ fo,order t~. eliminate the c;liscontinuity of the
stre~ses, most. finite element aru;lyses· use an a~eraging te~chnique .•
. ; .·The foll owing' averaging sche~e ts used irl this study (Fig. 8) ...
. The illterl ami na r st res se s "z• 'yz '. : 'J\~. must ~· cont i TI 40U~ throughput . ·.
the laminate.. At a point A~ Jhese _stres.ses are averaged over the
·elements ll, i2, 13, 14.-· ·rhe:laminate stresses~ax' .ay' ;rxy inay'.be. ·
discontinuous across la~in~te interfaces, but within.ea~h ply, they ·
: '.·· ..
;, ';''
. . . ' ·: ~- :· : : . .·.·:,,"
..- . ·.
·:, · .. ·,
. ~ .·_, > .. ,·, "
. .. . ~· . . : ... ' . ~ : .
. ·.· .. ··
. .. ·, .. , · .
~·
I-<
r..n a_
2.0 - ------·~-.~-:.._J __ _
LO
26
'---- -(--r--1--I . . . I
0 0 0 0 •. ·
x
.N
CJ >-i (f)
-1. 0 -
I
-"2. 0
I . .
· -- ,~-r----c~- r - r - ~ J L--0.7
Y/B l1 DIVISICJH 0 , .c1 1 ,)
FIGURE 5. VARIATION OF crz AT THE INTERFACE OF A [90/0] 5 LAMINATE WITH MESH SIZE
I
27
(/') t-z l.J..J :::E l.J..J _J l.J..J
t-V') l.J..J _J _J c(
~ :I: t-...... 3 >-_J 0..
>-x 0 0.. l.J..J .......... l.J..J t-...... :I: 0.. 0 ex: l.J..J 0:: (/') ~o
0.. _J :::E ex: 1-4 u 0:: 1-4 l.J..J 0.. 0.. >- :::::> t- (/')
0
N
0
28
0
co 0
0
0
:::c (/) LU ::::£ LU Cl 0 z: N 0 LO
r-z: LU ::::£ LU .....I LU
~ O'\ CX)
29
FIGURE 8. AVERAGING SCHEME
30.
are continuous. At a point B they a.re averaged over elements 15
and 16.
4.3 Linear Elastic Analysis
The tensor polynomial predicts failure to occur when it attains
the value 1.0. Suppose that for an applied strain. i;, the stress state
is determined and the tensor polynomial calculated as
( 4.1)
Let failure occur at a strain of ki;, i.e.
2 ka. + k s = 1 (4.2).
This quadratic equation is- solved fo~ k, and the strain at first
failure estimated.
The stresses and tensor polynomial were evaluated for various
laminates loaded with an axial strain of O~l percent. Based on these
results, 1 k1 was calculated for the element which had the highest
value of the tensor polynomial function for each laminate.
These values are presented in Table 2, and were used to estimate
the mechanical load for first failure, and the number of load increments
in the nonlinear analysis~
4.4 Stress Free Temperature
During manufacture, ·laminates are heated to some maximum elevated
temperature. However, the bonding usually takes place ~t some lower
temperature. At .that temperature the laminate is in a stress free
3l
TABLE 2
LINEAR ELASTIC PREDICTIONS OF FIRST FAILURE
· Laminate Strain at First Fa i1 ure
· [0/90]5 0. 327956
[90/0]5 0.375117
[±10]5 0.253761
· [±15] 5 0.22304
[±30]5 0.481329
[±45]5 0.476636 ·.
[±60]5 0.456682
[±75] . s 0.423255
[90/0/±45]5 0.318849
[±45/0/90]5 0.149065 ·.
32
state. This stress free temperatureT0 is the reference temperature
for calculating the residual stresses. T0 depends on the material
system of the laminate, and to some extent, on the curing process.
Tsai [5] suggested that T0 be experimentally determined from a [±e]
unsymmetrical laminate which warps on cooling. The temperature at
which the laminate becomes flat, on reheating is T0. T300/5208 is
cured at 350°F, but the suggested values for Ta vary widely. Renieri
used a value of 27a°F [l], while Chamis always uses .the highest
temperature attained in the cure cycle as the value for Ta· A stress
free temperature of 25a°F is suggested in [17,19]. Hahn [13] reheated
warped unsymmetrical laminates, but found values of T0 varying from
250 to 3aa degrees. Ta was choosen to be 250°F for the present analysis.
4.5 Load Steps for Thermal Load
A study was conducted to evaluate the' effect of cooling the laminate
in different load steps. A [9a/aJs laminate was chosen because it ex;..
hibits the maximum mismatch in expansion coefficients. This laminate
was analysed assuming the cooling from Ta to room temperature in
1, 2, 3, 4, 5, 6, 8, and 10 load steps, and the resulting stress
distributions plotted. Typical variation of the stresses is presented
in Fig. 9. The largest value of the tensor polynomial occurred in the
same element for each load case, but had different numerical values.
These are presented in Table 3. As seen from Fig. 9, the solution for
the stresses appears to converge with increasing number of load steps.
In the finite element analysis, the stiffness matrix has to be
. 0
-I 0
ru I
0
::r I
0 . I./") I
0 . --------,---~(') (1 r ·
Uj. 0 • ~
33
0.8 1. 0
Y/B
FIGURE 9. TYPICAL ax CURING STRESS DISTRIBUTION AS A FUNCTION OF THE NUMBER OF THERMAL LOAD STEPS
34
TABLE 3
EFFECT OF LOAD STEPS ON THE TENSOR POLYNOMIJl.L.
[90/0]s LAMINATE
Maxlinum Valu~of No. of load Steps Tensor Polynomial
1 .8767522 ,·
2 .769971~.
3 ·. 7252781
4 .6949252
5 : .6705358
6 .64,97218 ..
8 .6132286 ' .;
lo .5822445
.. ·. ,,"'· . },;
··: ·. : : ~
• .! ••
.
····, ..
35
recalculated in each load step. Using a grid with 896 elements there-
fore involves an enormous amount of expensive computation. It was
decided to cool the laminate in 6 load steps of .,...JQ°F each, a compromise
between satisfactory convergence and computer cost.
5. STRESS AND FAILURE .ANALYSIS OF LAMINATES
Cross-ply, angle-ply, and some quasi-isotropic graphite/epoxy . . .
laminates were analyzed in this study. In order to obtain the stress
state in the laminate, the process of cooling the laminate to room
temperature is modelled in thermal load steps with temperature depe11-
dent properties and the nonlinear analysis of subsequent mechanical
loading is modelled as .a number of linear elastic load steps. Stress
distributions are plotted at the strain at which the first element was
predicted to fail using the tensor polynomial failure criterion. Damage
is predicted to initiate at this strain. This study does not predict
the ultimate failure strain, but does predictthe mode of first failure.
The load step for the mechanical load was decided as 0.05 percent
strain, the choice guided by the linear elastic predictions of the
strain at first failure in each laminate, and.computer cost for the
number of load steps.
5. l Cross Ply Lamfrlates
5.1.l Stress Distributions
The mismatch of the expansion coefficients between adjacent layers
is maximum in these laminates and results in very high curing stresses.
For the purpose of comparison; stress distributions are presented on.
the same plot for the following three cases.
1. Residual thermal stresses.
2. Nonlinear analysis of mechanical load at first failure
(including residual stresses).
36
'37 ., .·
3. Stresses obtained f~~m a linear elastic iinal:ysis' of pure .·.
mechanicaf.load, scaledto the firs't failure strain as' .
predicted by the nonlinear analysis. . '
The stresses crx and cry for the·. [0/90]5 and the [90/0]s are shown
. in Figs. 10, 11, 12, 13. As a result of cooling, the. laminate shrinks,
and crx is compresslve in the 0° ancftens.i.le i~ the '9ot>'1~;er, while · .
. . '·::
cry is tensile in the 0° and· compressive in the 90° layer .. ·. With the.
application· of an axial strain load, unloading occ~rs 'in.the 0°·1ayer·, .
along with ii reversal in sigh of ax, as .can be seen from Fi gs. · lO '• l l.
The stress state at first· failure consists almost entirely of the
. residual stresses and is much higher than the stress state predicted
by· the linear elasti'c analysis scaled to .the first failure strain. The
only instance of t.he .1 inear elastic prediction bei~g greater than the
nonlinear analysis is O"x in the ;p0 .• layer of the [90/Cl]s la~inate; whe.re •• > •• ' • • :
the nonlinear analysis unloads the compressive residual stress in that .. ·
layer. . . .... .
. Of the iilterlamir',tar sJresses, 1xz is zero, and the:distribution .of ' ' ' .-. .
crz versus· y/b in both. lamina~es ls shown in Fig. 14. For equilibrium
of the. free body {Fig. lSJ, Mx·= 0 and Fz = 0 ··..::
·. ··. z1 ... •· ·· ,bf·· · · ·· .. a, dZ = ·•.. ·qzdy
~'1· Y o·
·and
' ' ' Cf ' dy = 0 .· f''' ' -b' ·• z ·. ,,'' .
"·":,.
•."' ._::.
12.0
LE
T+M
T
-- -4.0 U1 ~ .... ')(
" -U1 T+M J.5
T
2. l
LE
0.1 0.2
38
0° layer
90° layer
. o. 4
Y/B
~~ t3d_ b y
o.s o.e
FIGURE 10. crx IN A {0/90] 5 LAMINATE
I .O
39
6.0 T+M
90° layer 4.0
20.0 0° layer
12.0 ~ •f--an---, .... · 9.0·· . ·· .. ·. ·• ·..•• • · .. tt=i_ b y
4.0
T
Y/8
FIGURE 11. ax IN A [90/0]'s LAMINATE
J.o
2.0
,. -f/) o.o ~ .... >-
·o o.o -(/)
-2.0
-4.0 o.o
. 4.0
T+M
T
0° 1 ayer
LE
LE ~~0° layer·
·cl=L . b y
T
T+M
0.2 0.4 o.s o.s Y/B
FIGURE 12. a IN A [0/90]5 LAMINATE .Y
l. 0
o.o
-2.0
--. (/) ."'. -4.0 .... >-(!) J.6 -(/)
o.o o.o
LE
T
T+M
T+M
T
LE
o. 2 '
41
90° 1 ayer
0° layer
0.4
Y/B
~··ho-i··· 90······ ·.· .. · t=L=1___ b y
o.s o.s
FIGURE 13. cry IN A [90/0] 5 LAMINATE
1 • 0
,.. -(I) ~ .... N
" -(I)
l. 6
o.e
Q.4
-o.o
-0.4
-o.s o.s
go 0 layer ' 0° layer
T
I I J
I
0° layer 0° layer go 0 1 ayer goo layer
Y/8
FIGURE 14. oz. AT THE INTERFACE IN [0/90] . AND [90/0J. LAMINATES . . ... s ·. s
43
z
T .J;=_
a § y
... ~ "z A
T zy
y
z
H
-b
1 ~+~ l A
z
y
FIGURE 15. PARTIAL FREE BODY DIAGRAM OF THE LAMINATE
44
az must therefore bea pure couple.
Consider the "[0/90]5 laminate. The cry in the 0 layer ts tensile . . . ,
for both, thermal and mechanical loads. For rnoment equilibrium about·
point A az must forn1 a couple, to balance the moment due to the tensile
ay. As seen from Fig. 14, az is tensile at the free edge and compressive
in the boundary layer away from the free edge producing the couple
required for equilibrium. In the [90/0]s laminate however, a for .· y
both the loads is compressive, and az has to form a couple to balance
the moment due to this cry about point A. The anaiysis predicts az to.·
reverse frorn its increasing negative value and tehd to zero or some
tensile value, exhibiting singular behavior at th'~ free edge (Fig. T4).
Such a behavior is predicted by a linear elastic analysis (Fig. 5}, and . .
reported in [24, 26]. The behavior of the [90/0]s lamln'ate is therefore ·
not the expected mirror image of the [0/90]s laminate. That the
behavior of the [90/0]s is not a mirror image of the [0/90]s laminate
was al so observed by Wang and Cross.man in [27].
Through the thickness variation of az and ax for the residual
stresses are shown in Figs. 16, 17, and compared to the distribution . .
obtained by Warig and Crossman [15], using a 1 inear elastic analysis.· . .
Though the shap~ .of the stress distributions is approximately the sa.me,.
there is significant difference in the magnitude of the stress~s. The
maximum value of az in a [0/90]s, for example, is predicted to be
2.01 ksi by this analysis compared to a value of 5.4 ksi from [l5].
This difference can be attributed to the nonl im~ar analysis using
temperature dependent elastic properties.
, I
' '
0 N
N
/ I
I /
/ , ..
0 "'.IP s.. .·s.. o·.o
·oo. O'I
::c
•" ----- .. ,. . ..... /. . \
, ,, . \
' , ' , ··', .. .,,,,' ..... ~ ... ·
. •·
. , I I
'
0
.. 45·
I l .. I I
•/~----·~·.._ ...... _ ~ -~ .;,.,;.-.-.
f I
I
'·
\ \
\
' '
t-t/z.
:o , ....... . 0
.,
0 co
0 · .. N
0
0
0 ·N·
I
--en ¥ ....,
N
c:> -en
'B .
..... ··-~······ z: ~.::C s .r-.;VJ 'o
·--~·· -~
0 z c::(
VI .r-1 0 O'I . ....... 0
··'"-' .ex: 0 4-V) V) LI.I z: ~-. u ·-.±c t-
. LLJ ::c 1-:c ~ => 0 .. ex: :c 'l'-
N b.
· .. ··, . , ..
>.
...
2.0
:r
l . 0
' N
o.o
I \. \ \ \
I I I I I I
-7~5
I I I
\ ' I I
\ \; "
/
z H
0 or
90
1 90
or
0
90/0
, I-=
' .
-F
• ..
present~.,....__-
[15]
-2.s
2.
s S
IG X
CK
Sll
b y
0/90
\ I I I I I I I I I I ...
I I I
I I I I I
7.5
FIGU
RE 1
7.
ox T
HROU
GH
THE
THIC
KNES
S FO
R [0
/90
] 5
AND
[90
/0] 5
LA
MIN
ATES
~
O'I
47
Boundary effects for the curing stre$ses can be seen in all the
stress distributions that have been plotted versus y/b an.d are a
function of the laminate aspect ratio .. The boundary layer at the free
edge due to thermal load has approximately the same thickness as that
for the mechanical load and is limited to values of y/b greater than
0.8.
5.1.2 Failure Analysis
The curing stresses in ·cross-ply laminates are very high. In a
[o;go] laminate, the stress state resulting from cooling the laminate s . . . . in 6 load steps is high enough for thetensor polynomial to predict
. .
failure. (In fact cracks are sometimes observed at the. free edge of
cross-ply laminates {1g],) For the purpose of analysis, this particular
laminate was cooled in 8 load steps. First failure was predicted to . _. ' . .
otcur at a strain of 0.05 percent in this laminate and at 0.15 perc~nt
iri the [90/0]s laminate.
The tensor.polynomial was plotted against y/b, and also through the
thickness. Failure for both laminates was predicted to initiate in the
go 0 ply at the free edge. Figs. 18 and 1g show the variation of the
tensor polynomial at the interface in the go 0 ply, as determined from
.the curing stresses and from subsequent mechanical loading. The curing
stresses are predicted to make a major contribution to initiation of
failure in regions close to the free edge in both laminates. Through
the thickness variation (Fig. 20) shows the effect of curing stresses
to be dominant in the go 0 layers in both laminates. In the 0° layers
the tensor polynomial has a negative value, which is acceptable when
_, < -r 0 % >-;.;J. 0 Q.
. tr 0 en % w ....
1 • 0
o.s
o.s
0.4
0.2
o.o o.o 0.2
48
0.4
Y/B
z ..
H r 96 •· 1 _ . b.· .·.y
o.6
. .. ~. . . .
FIGURE 18. TENSOR POLYNOMIAL ALONG THE INTERFACE.TN THE 90° LAYER. FOR A [0/90]5 .LAMINATE • . .
..J < -r 0 Z' >-..J 0 Ill..
0::-0 (/) Z' l&J .....
49 .
l . 0
T+M
o.s
0.6
T
0.4
o.o 0·0 0.4
Y/B
z
ttha, ... ···9·0··· .···.·•· .t=d_ b y
0.6 o.s I .Q
FIGURE J9. TENSOR POLYNOMIAL ALONG THE· INTERFACE IN THE 90° LAYER FOR A [90/0]s LAMINATE· .
N
0 •:
N
0 . O".I. c::i
:$.... S-o 0 0,.0
'O)
:c:
'' >50'
1-11z·
:I/) ....-i. 0 -...;.,. 0 ,0) L.,...J
·.:·' ·.
' ::E:' ··.~·.·,
'•. c-_......, ___ , . ·. ao ..
.·,~
0 .,
IX ·o. LL. J) (/)
.µJ z :ii.ii::.
,,g :c !-'.-'.' . Iii ::c ~·· ::C
.(;!)• ::> 0 a:: ' . !£._
d ('\J .LLi IX .::>· ., ·(!;I : ' ,_, Li.:
< .:· '·· ...
. . ,,
. ••' •._ :
when using the tensor polynomial failure criterion.
The tensor pol,ynornial for the element which was first to fail was
analyzed in detaJl, and t~e individual contributions from each of the
stresses are presented in Table 4. The i11dividuc:\l c:ontributions .show
the. te.nsor polynomial to be d()minated by 02 witp 7ome cC>ntrib1JtJon from
a3. Failure initiates earlier in the [0/90]s• laminate thanthe.[90/0]5 ~
This is due.tq the. differences in the a3 distriQutions for the two
1 ami nates, but the mode of failure jn both larni nates is primarily
transverse tension •. Since the. [90/0]s. is predicted to fa.il at a higher
applied strain, it is pn~fered over the f0/90]s for tensile loading .
. 5.2 Angle-Ply Laminates
5. 2 .1 Stress Distributions
The angle-ply laminates studied are the ±10, 15, 30, 45, 6Q, and ' ' - .
75. The thermal mismatch between adjacent plies ts nat·as severe as.
that in cross-ply laminates except the ±45, resulting in lower magnitudes
for the residual stresses. It is il1terest.ing to note that, in the
material principal coordinates., the residual stresses tn the cro$s-ply
and the T±45]s Jaminate are the saine.t except at the edges.
The highestabsplute value of each stress was normalized and plottecj
versus the ply angle, and figs. 2land 22 show the variation of the
laminate and the interlaminar stresses for the thermal and mechanical ,. '. -' _.· - ' . . . :.: '.-.. ' . ,
loading respettivel§. The thermal· mi.smatch in angle-ply laminates is
maximum at 45°; All the stresses attain their maximum values at 45(); . ' ' ' . . :.:· .· ' . , .. __ : - '' -.... ,
except for a which attains its maximum at 30°. This ls because the< . . . x .· .... ·. ·. •. .·· .· ·.. . .. · · ..
stress state not only depends on ttie curing strain (thermafmismatchJ,
52
TABLE 4
FAILURE MODE ANALYSIS OF CROSS-PLY LAMINATES
Laminate F2cr2 2
F22°2 F3cr3 2
F33°3 2
F44°4 2
F55°5 2
F66°6
[0/90]5 0.48977 .11862 .33672 .05607 0 .00001 0
[90/0]5 0.63599 .20003 . 15310 . 01159 0 .00003 0
Cf) Cf) L&J er ..... Cf)
0 L&J N -..J < r. er 0 :z
Cf) Cf) L&J er ..... (/)
0 L&J N -..J < r er 0 :z
I. 0
o.o
f. 0
·1 a ·I· = 2.6 ksi I Tx· Im· = J.06 ksi . x m ... · .. · .. ·· .... · Y . · · I a I ·'= 0.227. ksi Y m . .
o.o Fl~ER ORIENTATION
I a f = 0~262 ksi z m . ·1.T .. · .. 1. .. = Q. o ... 2 .·.k .... S· i yz m ····· ·.· .. ·. · I T I = 3. 3 ks i xz m ·.
TXy
o.o 20.0
FISER ORif~TATtON
FIGURE 21. MAXIMUM NORMALIZED LAMINATE AND INTERLAMlNAR CURING · STRESSES TN ANGLE".° PLY LAMINATES ·
(/) (/) lLI fr
' .... (/)
0 lLI N
..J ' < r « 0 z
(/) (/) lLI Q:" .... (/)
0 LU N -..J < r Q:" 0 z
I. 0
o. 0 '
1.0
o.o
54
o.o 20.0 40.0 60.0 so.o FIBER ORIENTAflON
.,
FIBER ORIENTATlON ..
FIGURE 22. MAXIMUM NORMALIZED. LAMINATE AND lNTERLAMiNAR CURING. STRESSES AT FTRST FAILURE IN ANGLE-PLY LAMINATES
55
but also on the elastic modulus, and E~ decreases sharply as the ply
angle increases from zero, the decrease tapering off at larger
angles [28].
The maximum values for various stres.ses occur at different fiber . .
angles, as can be seen from Fig. 22. The magnitude of "x is large at . ' . '
low angles, with its maximum at 0°, while Txz reaches its rnaximum at
15°. All other stresses attain their maxima at a fiber angle of 30°,
except T which is maximum at 45°. . xy . The distribution of curing stresses is roughly the same in all
angle-ply laminates, the difference being in the magnitudes at
different ply orientations. The curing stresses in the [±45]s are
typical and are presented, F·ig. 23 showing the lamina stresses and
Fig. 24, the interlaminar stresses. The stresses cry' Txy' and Tyz
are seen to approach zero as required by the stress free boundary
conditions. As in cross-ply laminates, the curing stresses exhibit
edge effects, with the presence of a boundary layer for y/b greater
than 0.9.
The stresses at first failure in angle~ply laminates ~re much
greater than the curing stresses. The curing stress state is of
significance ohly at the free edge~ due to the high interl~minar
stresses. Unlike cross-ply laminates, where curing stresses affect
the whole laminate, curing stresses in angle-ply laminates are at best
an edge effect, even for the [±45]s laminate.
Though the thickness variation of crx and Txz near the free edge
for a [±45]s laminate are presented {Fig. 25) and compar~d to
--en ~ .... en
·.(3 1~2 IJ( t-. en
t . xy
:.:·
.,··:
.. 56
'.,· .. ,. _.-
-._.·,_
ax .. · ·-·o. 4 · _J,;: :...._ ...... _...., __ "T"_....,....:.. __ .,..._...., ____ ,..._.,....,,_...;;...,.,......,.,...,. __ .· ....
0 . 0 0 . 2 0 . 4 0 . 6 0 ; • l . o: · .. Y/B ·"'
.. ':~.. . . ·.:· :·.
'."- . .:· . ~
· 1,.':···.
·' .. · . . "·i_ ;· .....
. ,. ··,·:
.. "/
. ,_._
. . . .: ... ·.·,
····:.·. :· ,:
40.0
,..· 20 .. 0 -· en . G. ....
N ,.. ::> :.· o.o
,.. -en A.
0 0 -
l . 0
-1.0· Y.
N
" -en
-3.0
· midplane
z
H l.>:~~ 1 ' · .. b y
o.o ·o.4 0.6 . o.a· I ~ 0
Y/B
FIGURE 24. · crz AND ··~ CURIN(l STRESSES IN A [±45]5 LAMINATE
-- ----- -
0 N
"" -------I , 'I I
'
0
H/Z
58
' '
+>
\
'
c:: Q) l/l Q) s.. 0.
\ \
\ N \ >< \
l-' \
I I I I I I
r-1 LO ,..... L.-J
' \ \
' ' ' ,,
>< b
0 . 0
If)
......
If)
N
If)
N I
....J <C :::> 0 -(/') LiJ 0::: 1.1.J :t: ' I-0::: 0 LL.
(Jf) :z: ' 0 ..... I- 1.1.J :::> I-CO<C HZ: 0::: ..... - I- :E: (/')<( - ..... ....J en 0
~ l/l - V) r-1 V'lLO
en LIJ o:::I" z +I en ~L.-J l.&.I u « -c:c .... ::c Cl) 1-Z: .....
1.1.J ::c N I- ><
I-' ::c "'Cl :::> z: O<C 0::: ::c X,' I- b
. LO C\J
1.1.J 0::: :::> C.!l -LL.
59
distributions obtained by Wang and Crossman [15]. As in the cross-ply
laminates the present solution predicts much lower stresses.
5.2.2 Failure Analysis
The tensor polynomial, as determined from the nonlinear analysis,
at first failure is shown in Fig. 26 for various fiber angles. This
figure demonstrates that the edge effects are dominant at small angles
of orientation, and the edge stress concentrations decrease with
increasing angle. At large angles, when failure is first predicted at
the free edge, elements in the interior have large values for the
tensor polynomial, hence the entire laminate is close to failure. The
tensor polynomial exhibits a small negative value for low angles. This
is acceptable in the failure criterion, and signifies that the region
with the negative values is well below failure.
Thermal stresses in angle-ply laminates are an edge effect. This
is clearly seen from Fig. 27, where the tensor polynomial has been
plotted through the thickness·at various angles, for the curing
stresses as well as for the stress state existing at first failure.
The presence of the free edge and dissimilarity of material causes
additional stress gradients at the interface~ Failure is predicted
to initiate at the interface for low angles, shift to the midplane at
45°, and shift back to the interface at 60° and stay there.
The stress state of the element where first failure was predicted
was transformed into .the material coordinate system and the individual
terms of the tensor polynomial evaluated, and presented in Table 5.
The tensor polynomi.al is completely dominated by -r13 for the 10° and
..J < -r 0 2. >-
0.9
0.1
..J o. s 0 tL
« ~ ::z "' ...
o.J
-0 •. 1 o.o
60
75
60
45
.. ~··ra· ........ +.4.5 ...•... ·· ..
. -45 .. . . . .
b y
. 30
10 15
. 0~2 o.a •. 0
Y/B FIGURE 26. TENSOR· POLYNOMIAL ALONG THE INTERFACE FOR· VARIOUS · .. ··.
ANGLE-PLY LAMINATES
75 l
Q 15
60
30 4
5 10
15
30
60
45
75
r ..
,
i .o
N
o.o
o.o
~ra
0.2
·O
. 4
0.6
o.a
TEN SO~
POL Y
NQtU
~L
FIG
UR
E.27
. TH
ROUG
H TH
E TH
ICKN
ESS·
TENS
OR P
OLYN
OMIA
L :D
ISTR
IBUT
IONS
FR
OM C
URIN
G ST
RESS
ES A
ND S
TRES
SES
AT F
IRST
FAI
LURE
IN
ANG
LE_;P
LY L
AMIN
ATES
..
1"~0
. · ..
.
°'' ~
62
TABLE 5
FAILURE MODE ANALYSTS OF ANGLE .. PLY LAMINATES
..
Laminate F2cr2 2
F22°2. F3cr3 2
F33°3 ·.. 2 F 44°4
.·
. 2 F55°5
. 2 F66°6
[±10]5 ... 07145 .00252 . 00787 . 00003 . .04935 . 90160 .02157
[±15]5 - . 10433 -.00538 -.01010 .. 00005 • 11449 .88078 .04432 ..
[±30]5 - .12406 .00761 -.07293 . .00263 .33049 .53819 .27496 .·
[±45]5 .606Q9 .18229 .03149 ~00049 0 0 .17926
:
[±60]5 . 67694 .. 22662 .03020 .00045 .03105 .00475 .03001 .
[±75]5 .71345 .25172 . 01151 .00007 . 01789 .00057 .00476 .
63
15° degree cases, and the mode of failure is therefore predicted to be
transverse shear. With increasing angl~, the contribution of -r13
decreases while that from -r23 and • 12 increase. The mode of failure
is still predicted to be transverse shear. At 45°, the polynomial• is
dominated by a2, though there is some contribution from -r12 which
decreases with increasing fiber angle. The fiber mode forangles equal
to or greater than 45° is predicted to be transverse tension.
Failure was predicted to initiate at the free edge for all
1 aminates studied. In the 10° and .15° .1 aminates first. fa.ilure occurred
· at 0.003 percent· strain, in the 45° .at 0.0045 percent strain and at
0.004 for the 30°, 60° and75° laminates. The strains at which first
failure is predicted is the same for some laminates because the strain
was applied in load steps of 0.05 percent.
5.3 Quasi-Isotropic Laminates
5.3.1 Stress Distributions
The quasi-isotropic laminates analyzed were the [±45/0/go]5 and
the [go/0/±45]s. The thermal mismatch is larger between the 0° and
go 0 degree plies than the other plies, and there is significant buildup
of curing stress in the go 0 ply in both stacking sequences.
Of the residual stresses, ax is compressive. in the 0° and tensile
in the goo ply in both laminates, and there is unloading of the 0°
plies with the application axial strain as fa the case in the cross-ply . .
laminates. Figs. 28 and 2g are plots of cr . and a in the 90° plies for . x y .· . ' the residual stresses and those at first failure. The res.idual stresses
64
5.5
go 0 layer in [±45/0/g0] 5
T+M
z
J.5 H~.·.···.·.······· .. ··.· ~
T b y
--C4 2.s ¥ -'>< 5.5
" T+M - I en
4.5 go 0 layer in [90/0/±45]5
J.5
~m_ b y
T
2.5 o.o . o. 2 Q.6 o.s I • 0
Y/B
. FIGURE 28. ax IN THE goo LAYER OF [±45/0/90]5 AND [gQ/0/±45] 5
LAMINATES
--en ¥' .... ,.. a -en
-1. 0
T -J.o
-s.o
-1.0
T+M
-9.0
-o.o
T
-4.0
-s.o
T+M
65
90° layer in the [±45/0/90]
·m· .. ·.·.-t,46"····· .• ·.·.·.·.·.··· -45'· 0
9o
-1 2. o L--r----~-,--r---,---,r-....,.----,._-.--r o.o 0.2 0 . 6 b.e 1~0
Y/B
FIGURE 29. cry IN THE 90° LAYER OF [±45/0/90]5 AND [90/0/±45]5
LAMINATES
66
make a significant contributi6n io the stress state that exists when
failure initiates.
Various interlaminar stresses are plotted at different interfaces
of both laminates in Figs. 30 and 31. The curing stresses are present
more in the form of edge effects than within the laminate. Though
•yz tends to zero as required by the boundary condition, •xz is
always predicted to exhibit singular behavior, and az also is singular
at the free edge, at all ·interfaces except the 90/0 interface where
the stress reverses from its large negative value, tending to zero
or some tensile value. Such a behavior was also predicted for a
[90/0]s laminate.
Through the thickness distributions of crx and crz are presented for
the ;·thermal and mechanical loading (Figs. 32,33). In the 0° and 90°
plies in both laminates, crz due to thermal load is comparable to the
combined stress state of curing stress and the mechanical load. The
stress distributions for crx show the unloading of the 0° layer. The
contribution of the curing stresses to the stress state at first failure
can also be seen from Figs. 32 and 33.
5.3.2 Failure Analysis
First failure in both laminates was predicted to occur in the 90°
ply. The tensor polynomial is plotted against y/b for the 90° layer
at the 0/90 interface, for both 1 ami nates (Fig. 34). The numeri ca 1
value of the polynomial is much larger than its value in other layers
in both laminates.
As the tensor polynomial in each layer is maximum at the free
67
l . 5 - ±45 interface -tn 4. o.o 0 0
'>( - ) . 5 .... T+M N -.J.o >-::> < .... -4.S
- 0/45 -tn o.o T a.
0 0
'>(
.... -1 . 0 N '>(
::> *IC .... -2.0
- l . 5 . -tn ~ .... N
c o.s midplane T -tn.
T+M -o.s
o.o 0.2 0.4 0.6 o.s Y/B
FIGURE 30. az, Txz' Tyz IN A [±45/0/90]s LAMINATE
o.o --(I) ... -N. Vo -100. :::> "" ....
o.s
.30.0
--(I) a. ...., 'o . .-o
68
': ;.·.
90/0 interface T+M
0/45 in:terface T+M
Z~·· H .
. . . .
... . . b. y
midplane
-10.o+-~r---.~--r~-r-~-T-""~.,....-~...---.~ ........ ---4-
0.0 0.2 o.·4 o.a 0•8 1 .o .. Y/B
. FIGURE 31. crz; Txz, 'yz IN A [90/0/±45]5. LAMINATE
·-.·.,_.· .... ·
4.0
J.o
~
2.0
N
l . 0
o.o
-4.0
T I
T+M
4.0
[±45
/0/9
0]5
[90/
0/±4
5]5
12.0
SIG
.I(
CK
Sll
----
·
20
.0
"""5
° CI
R '!j
O
-45'
" ...
.. 0
0 ""~
§0
dC
-+s'
FIGU
RE 3
2.
THRO
UGH
THE
THIC
KNES
S ax
DIS
TRIB
UTIO
N IN
[±
45/0
/90]
5 .
AND
[90/
0/±4
5]5
LAM
INAT
ES
2s.o
CTI
\.0
4.0
J.o
:r
2.0
.... N
) . 0
o.o
z HI
12
> -=
I b y
----
-~
-,...,
-,,..
..
-2.s
T+M
[90/
0/±4
5] s
-1 .
5 -o
.s
SIG
Z
<K
Sll
----
.. --
---
T [±
45/0
/90]
5 T+
M
o.s
I . 5
FIGUR
E 33
. TH
ROUG
H TH
E TH
ICKN
ESS
o DI
STRI
BUTI
ON I
N [±
45/0
/90]
z
. s
ANO
[90/
0/±4
5]5
LAMI
NATE
S
2.5
-.....i
0,
l . 0
o.s
o.&
0.4
..J < 0.2
g z >- o.o s a. 0: 0
l . 0 tn 'Z l&I .... o.s
Q.6
0.4
0.2
o.o o.o
T+M
T
T+M
T
0.2
71
[±45/0/90]5
I
~m 0 . ..
b y
[90/0/±45] . 5
Q.6
Y/8
~~ .. · .. ~ .. ····.·· t::::E_ b y
o.s l . 0
FIGURE 34. TENSOR POLYNOMIAL IN THE 90° LAYER ALONG THE 0/90 INTERFACE IN [±45/0/90]5. ANO [90/0/±45}5 LAMINATES
72
edge, through the thickness variation of the polynomial is plotted in
Fig. 35 for the stress state that e~ists in the laminate at the first
failure strain. The tensor polynomial for the 0° ply in both laminates
was found to be negative. This is acceptable in the failure criterion
and signifies that one of the normal stresses is compressive. The
value of the polynomial has a lower bound, and occurs for certain
compressive values of the normal stresses. The negative value implies
that the elemental stress state is well below failure.
Individual terms of the tensor polynomial in the material
coordinate system, for the element which was predicted to fail first,
are presented in Table 6. The failure for both laminates is initiated
in the 90° ply, with the dominant term in the tensor polynomial being
cr2, though there is some contribution from cr3. First failure in the
[±45/0/90]s laminate was predicted to occur at a strain of 0.001
percent, as compared to 0.0015 for the [90/0/±45]s laminate. This is
because the o3 for the [90/0/±45]s1is less than the [±45/0/90]s,
resulting in a lower contribution from o3 in the tensor polynomial. In
both laminates, however, the dominint terms correspond to cr2 and so the
mode of failure in both laminates is primarily transverse tension.
4.0
J.o
:r
2.0
... N
1 • 0
o.o
-0.2
T [9
0/0/
±45]
5
T+M
z H
------
-.. --
--. --
--
----
-.
-1-45
'" O
f't
90
-4
5 O
R
0
.:> -~
90
~-~ b
y . --
--·
---~~:
::::::
::::::
::::::
~"::::
::::= ..
........
.. ~~~~~ ....
..... -=
=:::..:
·3
......
.... .
..;,
,.--
T+M
T
[90/
0/±4
5]5
[±45
/0i9
0]5
0.2
o.&
TE
NSOR
~0LYNOf114L
1 • 0
FIGUR
E 35
. TH
ROUG
H TH
E TH
ICKN
ESS
TENS
OR P
OLYN
OMIA
L DI
STRI
BUTI
ON I
N [±
45/0
/90]
5 AN
D [9
0/0/
±45]
5 LA
MINA
TES
-..J VJ
74
TABLE 6 (
FAILURE MODE ANALYSIS OF QUASI-ISOTROPIC LAMINATES
.. ·
Laminate F2a2 2
F22cr2 F3cr3 2
F33°3 2
F44°4 2
F55°5 2
F66cr6
[±45/0/90] 5 . 52276 . 13514 . .29918 .04426 .00001 0 0
[90/0/±45] 5 .66045 . 21571 .11740 .00682 .00002 .00001 0
6. CONCLUSIONS
The present analysis has concentrated on the evaluation of curing
stresses, and the effect these stresses have on the initiation, and mode
of failure in finite width laminates under tensile load. The following
conclusions result from this study.
1. Curing stresses have to be considered in the analysis of
cross-ply laminates. The interlaminar and the laminate stresses have
significant magnitudes. Due to edge effects, the curing stress state
at the free edge is high enough for the tensor polynomial to be close
to failure.
2. In angle-ply laminates, the curing stresses are much lower
than in cross-ply and not as significant. The interlaminar curing
stresses are of significant magnitudes, but the theoretical 1 aminate
stresses are very small. Curing stresses in angle-ply laminates are
significant only near the free edge.
3. In quasi-isotropic laminates, there are significant laminate
and interlaminar curing stresses only in those plies that have fiber
orientations at 90° to the fibers in adjacent layers, i.e. in cross-
plies in the stacking sequence. At all other fiber orientations
between adjacent layers, curing stress are significant only near the
free edge.
4. The use of temperature dependent elastic' properties reduces
the magnitudes of the curing stresses significantly. Convergence of
the solution for curing stresses depends on the number of load steps
used for computing the stresses, and a single load step solution
75
76
predicts the stresses to be as much as 15 percent higher than a con-
verging 10 load step value.
5. The response of angle-ply laminates to thermal and mechanical
load is fundamentally different. The variation of a stress component with
ply orientation depends on the type of load, and the maximum value of a
stress need not occur at the same fiber orientation for thermal and
mechanical loading.
6. There are edge effects for both the thermal and the mechanical
load, with approximately the same thickness of the boundary layers.
For a laminate aspect ratio of 25, the boundary layer extends from
y/b = 0.8 to the free edge.
7. For both thermal and mechanical loading, edge effects are more
pronounced for smaller angles than for larger angles of ply orientatiori.
At large angles the tensor polynomial is almost the same across the
width of the laminate, but it is much higher at the free edge than
within the laminate for low ply angles.
8. Failure is predicted to initiate at a higher tensile strain in
the [90/0]s, than the [0/90]s laminate, and is the preferred stacking
sequence. For tensile loading, first failure occurs at a higher strain
in the [90/0/±45]s than the [±45/0/90]s laminate~ and is the preferred
stacking sequence.
9. In angle-ply laminates, failure is predicted to initiate at the
interface at low fiber angles, shift to the ~idplane at 45° and shift
back to the interface for larger angles.
This study was purely a mechanical and thermal analysis, moisture
; '
77
. effects being cpmpletely ignored. The computer code NONCOM2 however, .. ·'
is capable of ana\yzing theresponse q·f laminates to moisture .. · Som~'
areas that need investigatiOn include the following:
1. A complete hygrothermal analysis .of finite Width composite
laminates. ' . I
2 .. A detailed analysis. of curing stresses in quasi-isotropic
laminates.
3. A parametric study to determine the elastic moduli and strength
properties that ·have signifkanteffect on the· stress distribution, and
modes of failure.
4. A nonlinear analysis which takes into account the interaction
between stresses ..
'•,
··' ••• >
BIBLIOGRAPHY . . .
1. Renieri, G. o~, Herakovich, c. · T., · 11 Nonl ;n·ear Analysis ~f Laminated. · . Fibrous Composites, 11 VPI&SU Report VPI-E-76,;,10, June, 1976.
2. Humphreys, E. A., Herakovich, c~ T .. ,. 11 N9nlinear A~~lysis .of Bonded .. Joints with Thermal, Effects, 11 VPI&SU Report VPl-E"."77-19, June, 1~77.
·3. O'Brien, D. A., Herakovich, C. T., 11 Fini.te Element· Analysis of Idealized Composite.Damage Zones,11 VPI&SU Report VPI:E..;78:6, February~ 1978. · · · · · · · · .· ·
4. Reifsnider, K. L., Henneke, E. G •. II, Stinchcomb, W. W/,. noelamina .. tion in Quasi-isotropic Graphite;.·Epoxy· Lamin.ates, 11 Composite.·· · Materials: .. 'Testing· and. D:sign {Fourth .conference.}; :ASTM .sTP · 617, · American Society for Testing ana Material!!h 1977, .pp. 93-105.
5. Tsai,· s. w., 11 Stre~gth Characteristics of composite)tlaterials,i! NASA CR-224, April, 1965. ... . .
6. Hashin, z., uTheory of Fiber R~iriforced Materials·;·• NASA NASl-8818, · ·· · ·November 1970. · · · · · ·
7. Chami s, c. C. , 0 Proceedings of the. 26th Annual Co~fere(lce of the · SPI ·Reinforced Plastics/Cpmposite · Institute;'' Section·18~D, · : ·. · · Society of the Pfasti(:s Industry;. Irie~, N.Y.; 1971, p·p. l-12.
8. Daniel, I. M., Liber, T., Lamination Residual Stresses .in Fiber Composites," NASA CR-134 826,·March 1975 ... ·· · · · ·
. .
9. Herakovich, C. T., "On Thermal Edge Effects in Composite Laminates," .. Int. J. Mech. Sci., Vol. 18, pp. 129-134.,1976. · · ··
' . . . .. . . . .
10. Hahn, H. T., Pagani:>, ·N. J., 11Curin1:f Stresses in Composite Laminates, 11 J. Comp; Mat., Vol. 9~ 1975, PP• 91'.:"106~
11. Daniel, I. M.,Libe·r, T.,Chamis, C. C., i•Measurement of Residua] Strains in Boron.::Epoxy and Glass-Epoxy Laminates , 11 Composite . Reliability, ASTM STP 580-; American Soc'iety for TE!Sting. Materials; 1975, pp. 340-351. . . . . .
12 .. Chamis, C. C., Sullivan, T. L, nA Comput~tfona1 Procedure to Analyse Metal Matrix Compos.ites with Nonlinear Residual Strains, 11
Composite -Reliab,ilfty, ASTM STP 580, American Society for Testing Materials, 197~, pp. 327-339. · · ·
13. Hahn, H. T., "Residual Stresses in. Polymer Matrix Composite Laminates, 11 J. Comp~ Mat., Vol. 10, 1976, pp. 226-278.
78 '
79
14. Daniel, J. M., Liber, T., "Effect of Laminate Construction on Residual Stresses in Graphite/Polyimicle Composites," Experimental Mechanics, January, 1977, pp. 21-25. ·
15. Wang, A. S. D., Crossman, F. W., "Edge Effects on Thermally Induced Stresses in Composite Laminates, 11 J. Comp. Mat., Vol. ll, 1977, pp. 300-312.
16. ChamiS, C. C., "Residual Stresses in Angle-plied Laminates and Their Effects on Laminate Behavior, 11 NASA TM-78835.
17. Pagano, N. J., Hahn, H. T., "Eval\,lation of Composite C1,1ring Stresses," Composite MatetialS; Testing and Design,ASTM STP 617, American Society for Testing Materials, 1977, pp. 317-329.
18. Farley, G. L., Herakovich C. T., "Influence of Two-Dimensional Hygrothermal Gradients on Interlaminar Stresses Near Free Edges, Advanced Composite Materials - Environmental Effects," ASTM STP 658, American Society for Testing and Materials, 1978, pp. 143-159.
19. Kim, R. Y., Hahn, H. T., "Effect of Curing Stresses on the First Ply-failure in Composite Laminates," J. Comp. Mat., Vol. 13, 1979, pp. 2-16.
20. Pipes, R. B., Pagano, N. J., 11 Interlaminar Stresses in Composite Laminates under Uniform Axial Tension, 11 J. Comp. Mat., Vol. 4, 1970, pp. 538-548.
21. Pipes, R. B., Daniel, I.. M., "Moire Analysis of the Interl aminar Shear Edge Effect in Laminated Composites," J. Comp. Mat., Vol. 5, 1971, pp. 225-259.
22. Ramberg, W., Osgood, W. B., 11 Description of Stress-Strain Curves by Three Parameters, 11 NASA TN 902, 1943.
23. Tsai, S. W., Wu, E. M., "A General Theory of Strength for Anisotropic Materials," J.Comp. Mat., Vol. 5, 1971, pp. 58-80.
I
24. Herakovich, C. T., Nagarkar, A., O'Brien, D. A., to be presented at the ASME Winter Meeting, 1979.
25. Bergner, H. W., Davis~ J. G., Herakovich, c. T., "Analysis of Shear Test Methods for Composite Materials, 11 VPl-E-77-14, April 1977.
26. Wang, J. T. S., Dicks.on, J. N., 11lnterlaminar Stresses in Symmetric Composite Lamiriates, 11 J. Comp. Mat., Vol. 12, 1978, pp. 390-402.
80
27. Wang, A. S. D., Crossman, F. W .. , 11 Some New Results on Edge Effects in Symmetric Composite Laminates," J. Comp. Mat., Vol. 11, 1977, pp. 92-106. .
28. Jones, R. M., Mechanics of Composite Materials, Scripta Book Company, Washington, D. C., 1975.
APPENDIX A
CONSTITUTIVE RELATIONS
81
82
APPENDIX A
CONSTITUTIVE RELATIONS
The constitutive relationship1
for an orthotropic material in the I principal material directions is
fo}l = [C]({E} 1 -{a} 1 ~T)
where
c,, c12 cl 3 0 0 0
c22 c23 0 0 0
C33 0 0 0 [CJ =
C44 0 0
Symmetric C55 0
c66
a 1 El
0"2 E2
fo}l 0"3
{dl E3
= = '23 Y23
'13 Y13
'12 yl2
83
a.,
a.2
{a.}l a.3 = 0
0
0
For a e rotation about the 3 (z) axis' (Fig. l), the constitutive
relationship becomes
{cr} = [C]({E}-{a.}AT)
where
en c,2 c,3 0 0 c,6
c22 c23 0 0. c26 [CJ = C33 0 0 c36
C44 C45 0
Symmetric C55 0
c66
ax EX
cry Ey
{a} crz
{d e:z
- = Tyz Yyz
Txz Yxz
T~y . Yxy
84
ax
(J'y
{a.} = a.z
0
0
a.xy
and the various matrix and vector term~ as functions of the principal
material properties are given below (m=cose, n=sine).
4 2 2 . 4 c11 = m c11+2m n (c12+2c66 )+n c22 2 2 . 4 4 c12 = m. n ( c11 +c22 -4c66 )+(m +n' ) c12
- 2 2 cl3 = m c,3+n c23 - .• 2 2 2 2 . c16 = -mn[m c11 -n c22-(m -n )(c12+2c66 )J
- 4 2 2 . 4 c22 = n c11+2m n (c12+2c66 )+m c22 - 2 2 C23 = n C13+m C23
! - 2 1 2 22· c26 = -mn(n c11 -m c22 )+(m -n )(c12+2c66 )
I
C33 = C33
C36 = mn(C23-Cl3) 2 2 C44 = m C44+n C55
C45 = mn(C44-C55) - 2 . 2 C55 = n C44+m C55
2 2 . . 2 2 2 c66 = m n (C 11 +c22-2c12 )+(m -n ) c66
85
2 . 2 ax = m a.1+n a.2
2 2 a.y = n a.1+m a.2
a.z = a.3
a.xy = -2mn(a.1-a.2)
APPENDIX B
STIFFNESS MATRIX
86
87
.. • . ' .. .. ; i .. !'
~· .} .. :;-. .. .. ~
!' .. ... ,) J J J'
-- . !'
~ ...:· .....
~
......... ..... ..:;-J .... • .F i, J' .... ~::
.. .. "' ....... ....... .._;-
~
.~ .. .. .s ..., .. .;
.... .., .. -
~ ~ .:; :; .. ... .._; ..., "'
.. ...; ~
:: • fl !" f t .. J , ,. ...
~ ~ l { .. ~ ( { .. i .. .. -..
"";- ..; ... -- ..;· .s- .:;- ...;- ..., ...:;-• • .. • .. - '" • • .. ,.'" • • ., • • .. • • "' .... • .. • •"' • • • • • .. • • .. •• ,.., •• •• •'" • •• • • .,. • • .... • .. • • I
l.,. ~ J t ' . ..... · .... .s .. '." .J J ._,-:
~~ .. .l . ~ .. . J .
::It .. f- .} .~ s ... ...,
j J ~ .. ~ .~ ~ J ...... ... . .J '; ';i
,, 2i .. J J.., --~ .~ ........ .... ~
.. " .. "'1 .. : "-' .. ... .., ... . . ... . '~
2i .a : ) .~ Q
...,~ ... ~ ... . .., ... I> 0 ...: : . .., ,, .. ~ ., . .. .. .. . ... .., ...... .., .. .... .. ._, .. . ~ '\.
..:: ..,• ..., .. .~
'-' ..!- ..... ~ .._, ...,
.... j ... .. . .... ' ~ /~ ... .., ... ~ )
u -....... " ex
....... w ~ ,..,
-~ " >-Vl
;; J } . ~
.. ,,
~ - <('
APPENDIX C
TENSOR POLYNOMIAL FAILURE CRITERION
88
89
APPENDIX C
TENSOR POLYNOMIAL FAILURE CRITERION
The tensor polynbmial failure criterion in the contracted tensor
notation (for an orthotropic material in the principal material
directions) has the form
2 2 F1a1+F2a2+F3cr3+F11 a1+F22a2
2 2 2 2 +F33cr3+F44T23+F55T13+F66Tl2 (C.l)
+2F12a1a2+2F13a1a3+2F23a2a3 = l
where the Fi and Fij terms are as previously defined in Chapter 3.
In the xyz (laminate) coordinate. system, the tensor polynomial failure
criterion transforms (from the 1-2 to x-y by anticlockwise rotation of
+e) into
F1 a +F 1 a +F'a +F 1 a +F 1 a2 1 x 2 y 3 z 6 xy 11 x ·.FI 2 F.· I . 2 FI 2 +FI 2
+ 22°y+ 33°z+ 44tyz 55Txz
+F66T~y+2Fl6axTxy+2F26ayTxy
+2F36azTxy+2F45TyzTxz+2Fl2axay
(C.2)
where the F1 terms, as functions of the unprimed F's and e, are as
follows (m = cose, n = sine)
F' _ 2F 2F 1 - m ,+n 2
2 2 F2 = n F1+m F2
90
F3 = F3
F6 = ~2mn(F1 -F2 )
Fll = m4Fll+m2n2(F66+2Fl2)+n4F22
I _ 4 . 2 2 · · 4 F22 - n F11+m n (F66+2F12 )+m F22
F33 = F33
F• _ 2F 2F 44 - m 44+n 55 . 2 2
F~5 = n F44+m F55
F66 = 4m2n2(Fn+F22-2Fl2)+(m2-n2)2F66
2 2 . 2 2 Flfi = ~mn[2(m F11'.'"n F22 )-(m -n }(2F12+F66)]
F26 =·-mn[2(n2F11 -m2F22 )+(m2-n2)(2F12+F66 )]
F36 = -mn(Fl3-F23}
F45 = mn(F44-F55)
Fi2 = m2n2(Fll+F22-F66)+(m4+n4)Fl2
FI . - 2F + 2F 13 ..,. m l3 n 23
F23 :: n2Fl3+m2F23
These are transformations from the right handed 1-2 coordinate system
into another right hand coordinate system obtained by an anticlockwise
rotation of e0 about the 3 axis. If a ply is oriented at +e 0 from the
laminate axis, the F .. are obtained by using the above equations with 1J . the sines and cosines 6f -0°.
APPENDIX D
USERS GUIDE FOR NONCOM 2
91
Cards 1-5 (20A4)
Column
1-80
Card 6 (616)
Column
1-6 NE
7-12 NOS
13-18 NDIFM
19-24 NANG
25-30 IE LDT
31-36 I ELEM
37-42 NPLUS
Card 7 (616)
Column
1-6 NLOADS
7-12 NPSS(l)
13-18 NPSS(2)
92.
APPENDIX D
USERS GUIDE FOR NONCOM 2
Contents
Title Cards
Contents
= Number of elements
= Number of nodes
= Number of different materials
= Number of different angles
= Operating temperature indicator
0 for 70°
> O for any other temperature
= Operating moisture content indicator
0 for 0% moisture
> 0 for elevated moisture content
= 0 for in-core solution
> for out-of-core solution
Contents
= Number of load cases
= Load type number 1
= Load type number 2
etc. Repeated NLOADS times
.I I I,
NPSS(J)
Card 8 (516)
Column
l-6
7-12
NlNCRT(l)
NlNCRT(2)
93
= 1 for axial strain
= 2 for thermal
= 3 for axia 1 force
= 4 for hygroscopic
Contents
= Number of 1 oad increments for l ()ad case 1
= Number of load increme~t~ for load case 2
etc. Repeated NLOADS times
Card 9 (516)
Column --1-6 KEY(l)
7-12 KEY(2)
13-18 KEY(3)
19-24 KEY(4)
25-30. KEY(5)
Card lO (2014)
Column
l,-6 L 1 NCPR(J, I)
7-12 L 1 NCPR(K, I)
Contents
= Print indic~tor for grid
= Print indicator for strains
= Print indicator for stresses
= Print indicator for equivalent stresses
= Print indicator for displacements
KEY(l) = 0 for printing
l=l, NLOADS
Contents ;th = J increment of load case l to print
stressesi strains, displacements, etc.
= kth increment of load case l to print
etc. Repeated NLOADS times
Card 11 (2Fl2.6)
Column Contents
1-12 SMY = Scale factor for Y-coordinates
94
13-24 SMZ = Scale factor for Z-coordinates
The following card is repeated NLOADS times N=l ,NLOADS
Card 12 (2Fl2.6)
Column
1-12
13-24
ALOADS(l,N)
ALOADS(2,N)
Contents
= Load increment for first load case
= Initial load state before applying increment
Card 13 is omitted if IELET = O
Card 13 (Fl2.6)
Column Contents
1-12 DEL TOT = Constant temperature for non-thermal loading
Card 14 is omitted if IELEM = 0
Card 14 (Fl2.6)
Column
1-12 DELMOT
Contents
= Constant moisture content for non-hygro
scopic loading
The following cards are repeated NDIFM times
K=l,NDIFM
Card 15 (5El2.6)
(Cards 15-28)
Column Contents
1-12 EKll(K, l) = Ell tension modulus
13-24 EKll(K,2) = Ell ..
compression modulu~
25-36 EK22(K, 1) = E22 tension modulus
37-48 EK22(K,2) = E22 compression modulus
49-60 EK33(K,l) = E33 tension modulus
61-72 EK33( K,2) = E3j compression modulus
Card 16 (3El2.6)
Column
l-12 GK23(K) ·
13-24 GKl~(K)
25-36 GK12(K),
Card 17 (6El2.6)
-Column
1-12 SPl (K,l)
13-24 Nl(l,K,1)
Kl(l,K,l)
37:...48 SPil ( K, 1).
49-60 N1(2,K,l)
61-72
Card 18 (6El2 .. 6)
Column
1-12 SPl(K,2)
13-24 Nl ( l ,K,2)
25-36 Kl(l,K,2)
37-48 SPl( K,2)
49-:60 Nl(2~K,2)
61-72 Kl(2,K,2)
···Contents ,·., __ . . . ·. . :. .·
= G23 modulus .·· ..
· = GlJ modulus .
:;. GT2 · mq(iul us
. Contents
.l. I,
. = Elastrc· limit stressJ9r a1 _;£1 tension .·
= Ramberg_-Osgood coefficient n1 for a1 - £ 1
tension·
= Rarnberg-,Osgood coefficie,nt K1 for a1. -. £ 1
tension ' .
:: Bilinear intersect stress for a1. - £ 1
'tension • . . . .·
· . ·= :Ramberg:=O~good coefficient n2 for a1 - £ 1
fens fon .· ~ .
· = Ramberg_;Osgood coefficient K2 for. a1 - £l
tension
Contents
=
=
= Same as Card li but for al - £1 . . . ~
= compression
=
=
Card 19 (6El2.6)
Column
1-12 SP2(K,l)
13-24 N2 ( 1 ,K, l)
25-36 K2(1,K,l)···
37-48 SPI2(K, l)
49-60 N2{2,K,l)
61-72 K2 ( 2, K, 1)
Card 20 (6El2.6)
Column
1-12 SP2(K,2).
13-24 N_2 ( 1 ,K,2)
25-36 K2(1,_K,2)
37-48 SPI2(K,2)'
49-60 N2(2,K~i)
61-72 K2(2,K,2)
.· Card 21 (6El2.6)
Column
1-12 SP33(K, 1)
13-24 N33(1,K,l)
25-36 K33( 1,K,1)
37-48 SPI33( K, 1)
49-60 N33(2,k,l)
61-72 K33(2,K, l) ·.
I I _.L I -
96
Contents
= =
= ·same ~s Ca rd 17 but f:or a2 ::- £ 2
- •.· tension
=
Contents· ..
; ·.· ·.
= · Same as Card 17 but 'fo; o-2 - £ 2
= compression
. '.·.,
Contents
= = = Same as. Card 17 but fOr o 3 - £ 3
= tension
=
• • ~ • ~ •. 1 ' _ 1 l,. • .• 1_ •
97
Card 22 (6El2.6)
Column Contents
1-12 SP33(K,2) = 13-24 N33(1,K,2) = 25-36 K33(1,K,2) = Same as Card 17 but for cr3 - £ 3 37-48 SPI33(K,2) = compression
49-60 · N33(2,K,2) = 61-72 K33(2,K,2) = Card 23 (6El2.6)
Column Contents
1-12 SP23(K) = 13-24 N23(1,K) = 25-36 K23 ( 1 ,K) = Same as Card 17 but for T23 ~ Y23 37-48 SPI23(K) = shear
49-60 N23(2,K) =
61-72 K23(2,K) = Card 24 (6El2.6)
Column Contents
1-12 SP13(K) = 13-24 Nl3(1,K) = 25-36 Kl3(1,K) = Same as Card 17 but for T13 - Y13 37-48 SPI13(K) = shear
49-60 Nl3(2,K) = 61-72 Kl3{2,K) =
Gard 25 (6El2.6)
Column
1-12
13-24
25-36
37-48
49-60
61-72
SP3( K)
N3(1,K)
K3 ( 1 , K)
SPI3(K)
N3(2,K)
K3(2,K)
Card 26 (5El2.6)
Column
1-12
13-24
25-36
37-48
49-60
61-72
SL 1 (1 , K)
SL1(2,K)
Sl2(1,K)
SL2(2,K)
SL33 ( l, K)
SL33(2,K)
Card 27 (6El2.6)
Column
1-12
13-24
25-36
37-48
49-60
61-72
SL23(K)
Sll 3(K)
SL3(1,K)
XF23(K)
XF13(K)
XF12(K)
98
Contents
=
=
=
=
Same as Gard 17 but for T12 - Y12 shear
=
=
Contents.
= Ultimate stress for a1 - El tension
= Ultimate stress for a1 - El compression
= Ultimate stress for a 2 - Ez tension ' = Ultimate stress for cr2 - E2 compression
= Ultimate stress for a3 - E3 tension
= Ultimate stress for a3 - E3 compression
Contents
= Ultimate stress for T 23 - Y23 = Ultimate stress for T13 - Y13 = Ultimate stress for T12 - Y12 =. F23 interaction term
= F13 interaction term
= F12 interaction term
99
Card 28 ( 6El2. 6)
Column Contents
1-12 UK12(K,l) = Poisson's ratio v12 in tension
13-24 UK12(K,2) = Poisson's ratio vl2 in .compression
25-36 UK23(K, 1) = Poisson's ratio V23 in tension
37-48 UK23(K,2) = Poisson's ratio v23 in compression
49-60 UK13(K,l) = Poisson's ratio vl3 in tension
61-72 UK13(K,2) = Poisson's ratio V13 in compression
If no thermal or hygroscopic analysis is required, skip to card 128.
The foll owing cards are repeated NDIFM times K=l ,NDIFM.
Card 29 (16I5)
Column
1-5 NTEll (K,l)
6-10 NMEll(K,I,l)
etc. I=l,NTEll(K,l)
Contents
= Number of linear segmented temperature
points for E11 tensile modulus percent
retention curve
= Number of linear segmented moisture
points for E11 tensile modulus percent
retention curve at Ith temperature
The following two cards are repeated NTEll(K, l) time (I=l,
NTE 11 ( k , 1 ) ) .
Card 30 {lEl0.3)
Column Contents
1-10 TMPEl 1 (K, I, l) = Temperature at Ith temperature
Card 31 (8El0.3)
Column
1-10
11-20
PME 11 ( K, I , J , 1)
PRDEl 1 ( K, I , J, 1 )
100
Contents
= Moisture content at Ith temperature
= Percent retention of E11 tensile
modulus at Ith temperature and Jth
moisture content
etc. repeated J=l,NMEll(K,I,l)
Card 32 (16I5)
Column
1-5
6-10
NTEl 1 (K,2)
NM Ell (KI , 2)
Card 33 (lEl0.3)
Column
1-10 TMPEll(K,I,2)
Card 34 (8El0.3)
Column
1-10
11-20
PMEll(K,I,J,2)
PRDEll(K,I,J,2)
Contents
= =
=
=
=
101
Card 35 (16!5)
Column Contents
1-5 NTE22 ( K, 1) = 6-10 NME22(K,I,l) =
Card 36 ( lEl 0. 3)
Column
1-10 TMPE22(K,I,l) = Same as cards 29 to 31 but for E22 tensile modulus
Card 37
Column
1-10 PME22(K,I,J,l) = 11-20 PRDE22(K,I,J,l) = Card 38 (1615)
Column Contents
1-5 NTE22(K,2) =
6-10 NME22 (K, I ,2) =
Card 39 (lEl0.3)
Column
1-10 TMPE22(K,I ,2) = Same as cards 29 to 31 but for E22 Card 40 (8El0.3) compressive modulus
Column
1-10 PME22(K,I,J,2) =
ll'-20 PRDE22(K,I,J,2)
102 . '". ~·: .
Card, 41 (16!5) ' .···
Column Contents
1-5 NTE33(K,l) = 6-10 NME33(K,I ,1) .. =
Card 42 (lElo.3)
Column
1-10 TMPE33( K ,1, 1 ) ::; Same as cards 29 to 31 but for E33 .· tensile modulus
Card43 (8El0.3)
Column
1-10 PME33(K.I,J,l) ::;
11-20 PRDE33(K,I,J,l) = Card 44 (1615)
Column Contents
1-5 NTE33( K,2) = 6-10 NME33(K,I,2) =
Card 45 (lEl0.3)
Column
1-10 TMPE33(K,I,2) = Same as cards 29 to 31 but for E33 Card 46 (8El0. 3) compressive modulu$
Column
1-10 PME33 ( K, I, J ,2) ·. ·= .
11-20 PRDE33(K,I,J,2) =
103
Card 47 Cl 615)
Column Contents
1-5 NTG23{ K) = 6-10 NMG23{ K, I) = Card 48 (lEl0.3)
Column
1-10 TMPG23(K,I) = Same as cards 29 to 31 but for G23 shear modulus
Card 49 (8El0.3)
Column
1-10 PMG23(K,I,J) =
11-20 PRDG23(K,I,J) = Card 50 (1615)
Column Contents
1-5 NTG13(K) = 6-10· NMGl 2( K, I) =
Card 51 (lEl0.3)
Column
1-10 TMPGl 3 ( K, I) = Same as cards 29 to 31 but for G13 Card 52 (8El0.3) shear modulus
Column
1-10 PMG13(K,I,J) = l l-20 PRDGl3(K,I,J) =
: ,· ·.
··104
Card 53 (1615)
Column Contents
1-5 NTG12(K) = 6-10 NMG12(K,I) =
Card 54 (lEl0.3)
Column
1-10 TMPG12(K,I) = I Same as cards 29 to 31 but.for G12 \
shear modulus
Card 55 (8El0.3)
Column
1 :..10 PMG 1 2 ( K, I , J ) = ll;..20· PRDG12{K,I,J) = Card 56 (1615)
Column Contents
1-5 NTU23( K, 1) = . . ·.:
6-10 NMU23(K,I ,l) :::
Card 57 (lEl0.3)
Column
1-10 = Same as cards 29 to 31 ·but for v 23 Card 58 (8El 0. 3) · terisile Poisson's Ratio
Column
1-10 PMU23(K,I,J,l) = 11-20 PRDU23(K,I,J,l) =
105
Card 59 (1615}
Column Contents
1-5 NTU23(K,2} =
6-10 NMU23(K,I,2} =
Card 60 (1ElO.3}
Column
1-10 TMPU23( K, I ;2) = ··. Same as cards 29 to 31 but for v23
compressive Poisson's Ratio
Card 61 (8El0.3}
Colurnn
1-10 PMR23(K,I,J,2) =
11-20 PRDU23(K,I,J,2) =
Card 62 (1615}
Column Contents
l-5 NTU13(K, l) =
6-10 NMUl3(K,I, l} =
Card 63 (lEl0.3)
Column
1-10 TMPU 1 3 ( K; I, l} · = Same as cards 29 to 31 but for v13 tensil~ Poisson's Ratlo
Card 64 (8El 0. 3)
Column
' 1-10 PMU13{K,I,J,l} =
11-20 PRDU13{K,I,J,l}
·.-l ·- '· .. ._IL ••. ,. •. I. - ~
.. <."'
106
Card 65 (1615}
Column Cohtents ... ·, •'' .,,
1-5 NTU13(K,2} = 6-10 NMU13(K.I,2} •'. =
Card 66 (1ElO.3}
- ., Column
1-10 TMPU13(K,l,2} = · Same as card~·29 to 31 but for "13 ... _..:···,·.
compressiye Poisson 1·s :Ratio
Card 67 (8El0.3)
. Column
1-10 PMUl 3(K, I ,J,2) = 11-20 PRDUT3(K,J,J,2} = Card 68 (1615}
_ Column Contents
1-5 NTU12(K,l) -6-10 NMUl 2 (K, I ,l} =·
Card 69 ( l El 0. 3}
Column
1-10 TMPUl 2 ( K ~I, l} = Same as cards 29 to. 31 but for "12 Card 70 (8El0.3} . tensile Poisson 1 s Ratio
Column
1-10 PMU12(K,I ,J, l} = 11-20 PRDUl 2( K, I ,J. l} =
··. .~. I . - I I
107
Card 71 (16!5)
Column Contents
1-5 . NTUl 2 ( K ; 2) = 6-lo NMU12(K,I ,2) =
card 72 ( 1El0. 3)
Column
1-10 TMPU12(K,I,2) = :: Same as cards 29 to 31.but for v12 compressive Poisson's Ratio
Card 73 (8El 0. 3)
Column : -·
1-10 PMU12(K,l,J,2)
. 11-20 PRDUl 2(K;l,J ,2)
Card.74 (16!5) I
Column Contents
1-5 NTSl 1( K, 1) -6-10 NMS 11 ( K, I ; 1 ) =
Card 75 (lEl0.3) .·
Column ' 1-10 TMPSll(K;I,l) = . Same as cards 29 to 31 but for Xt
Card 76 (8El0.3) tens fl e strength .. ",.
Column
1-10 PMSl 1( K, I , J, 1) = 11-20 PRDSll(K,l,J,l) =
108
Card 77 (16!5)
Column Contents
1-5 NTSll(K,2) =
6-10 NMS 11 ( K, I, 2) =
Card 78 .(lEl0.3)
Column
1-10 TMPSll(K,I,2) = Same as cards 29 to 31 but for Xe
. compressive strength
Card 79 (8E10.3)
Column
1-10 PMSll (K,I ,J,2) =
11-20 PRDSll(K,I,J,2) =
Card 80 (1615)
Column Contents
1-5 NTS22(K,l) =
6-10 NMS22 ( K, I , 1 ) =
Card 81 ( 1ElO.3)
Column
1-10 TMPS22(K,I,l) = Same as cards 29 to 31 but for Vt
tensile strength
Card 82 (8El0.3)
Column
1-10 PMS22(K,I,J,1) = 11-20 PRDS22(K,I,J,l) =
109
Card 83 (16!5)
Column Contents
1-5 NTS22(K,2) =
6-10 NMS22(K,I,2) =
Card 84 (lEl0.3)
Column
1-10 TMPS22{K,I,2) = Same as cards 29 to 31 but for Y c
compressive strength
Card 85 (8El0. 3)
Column
1-10 PMS22(K,I,J,2) =
11-20 PRDS22(K,I,J,2) = Card 86 (16!5)
Column Contents
1-5 NTS33(K, 1) =
6-10 NMS33(K,I,l) =
Card 87 (1ElO.3)
Column
1-10 TMPS33(K,I,l) = Same as cards 29 to 31 but for zt
tensile strength
Card 88 (8El0.3)
1-10 PMS33 ( K, I , J, 1) =
11-20 PRDS33( K, I ,J, 1) =
110
Card 89 (1615)
Column Contents
1-5 NTS33(K,2) =
6-10 NMS33(K,l,2) =
Card 90 (1El0. 3)
Column
i~10 TMPS33(K,I,2) = Same as cards 29 to 31 but for zc ·
compr~ssive strength
Card 91 (8El0.3)
1-10 PMS33(K,l,J,2) =
11-20 PRDS33(K,I,J,2) =
Card 92 (1615)
Column Contents
1-5 NTS23(K) =
6-10 NMS23(K,I) =
Card 93 ( 1ElO.3)
Column
1-10 TMPS23 ( K, I) = Same as cards 29 to 31 but for s23
shear strength
Card 94 (8El0.3)
Column
1-10 PMS23(K,l,J) =
11-20 PRDS23(K,l,J) =
111
Card 95 (16!5)
Column Contents
1-5 NTS13(K) = 6-10 NMSl 3( K, I) =
Card 96 (lEl0.3}
Column
1-10 TMPSl 3(K, I) = Same as cards 29 to 31 but for S13
shear strength
Card 97 (HEl0.3)
Column
1-10 PMS13(K,I,J) =
11-20 PRDS13(K,I,J) =
Card 98 (1615)
Column Contents
1-5 NTS12(K) =
6-10 NMSl 2 ( K, I) =
Card 99 (lEl0.3)
Column
1-10 TMPSl 2 ( K ,r) = Same as cards 29 to 31 but for s,2 . .. shear strer;igth
Card 100 (8El0.3)
Column
1-10 PMSl 2(K, I ,J) =
11-20 PRDS12(K,I,J) =
112
Card 101 (1615)
Column Contents
1-5 NTF23(K) = 6-10 NMF23(K,I) =
Card 102 (lEl0.3)
Column
1-10 TMPF23 ( K, I) = . Same as cards 29 to 31 but for F23
interaction term
ca.rd 103 (8El0.3)
Column
1-10 PMF23( K, I ,J) = l l-20 PRDF23(K,l,J) =
Card 104 (1615)
Column Contents
1-5 NTF13(K) =
6-10 NMFl 3( K, I) =
Card 105 (lEl0.3)
Column
1-10 TMPE13(K,I) = Same as cards 29 to 31 but for F13
interaction term
Card 106 (8El0.3)
Column
1-10 PMF13(K,I ,J) =
11-20 PRDF13(K,l,J) =
113
Card 107 (16I5)
Column Contents·
1-5 NTFl 2( K)
6-10 NMFl 2(K, I) =
Card 108 (lEl0.3)
Column
1-10 TMPF12(K,I) = Same as cards 29 to 31 but for F12 interaction terms
Card 109 (8El0.3)
Column
1-10 PMF12(K,l,J) =
11-20 ·PRDF12(K,I,J) =
Card 110 ( 1615)
Column Contents
1-5 NTALl(K) =
6-10 NMALI ( K, I) = Same. as cards 29 and :30 but for a.1 ' temperature coefficielnt
Card 111 ( 1El0. 3)
Column
1-10 TMPALl (K,I) =
Card 112 (8El0.3)
Column Contents
1-10 PMALl(K,I,J) = Moisture content at Ith tempearture
ll-20 PRDAL l ( K, I ,J) = a.1 temperature coefficient at Ith
temperature and Jth motsture content
114
etc. repeated J~l. NMALl(K,I}
Card l l3 (1615)
Column Contents
1-5 . NTAL2(K). =
6-10 NMAL2(K;I) =
Card :114 (lEl0.3) ·
Column ·.>.
1-10 TMPAL2 ( K, I) - ·• .• Same as cards 110 to 112 but, for a.2 temperature coefficient
Card 115 (8El0.3)
Column
1-10 PMAL2(K,l,J}. =
11-20 PRDAL2(K,I ,J) -Card 116 (1615)
.. ·,
Column Contents
l "".5 NTAL3(Kl =
6-10 NMAL3(K,I) =
Card 117 (1ElO.3)
Column
1-10 . TMPAL3(K.tl) - .. Same as cards no to 1112 but for a.3 .•
· .. temperature coefficient
Card 118 (8El0.3)
1-10 PMAL3(K,l,J) =
11-20 PRDAL3(K,l,J) =
115
Card 119 (1615)
Column Contents
1-5 NTBTl(K) = 6-10 NMBTl (K,I) =
Card 120 (lEl0.3)
Column
1-10 TMPBTl ( K, I) = Same as cards 110 to 112 but for
s1 moisture coefficient ·
Card 121 (8El0.3)
1-10 PMBTl(K,I,J) = 11-20 PRDBTl ( K, I ,J) =
Card 122 (1615)
Column Contents
1-5 NTBT2(K) =
6-10 NMBT2(K,I) =
Card 123 ( 1ElO.3)
Column
1-10 TMPBT2(K,I) = Same as cards 110 to .112 but for
s2 moisture coefficient
Card 124 (8El0.3)
Column
1-10 PMBT2(K,I,J) =
11-20 PRDBT2(K,I,J) =
116
Card 125 ( 1615)
Column Contents
1-5 NTBT3(K) =
6-10 NMBT3( K, I) =
Ca rd 126 (lEl0.3)
Column
1-10 TMPBT3 ( K, I) = Same as cards 11 o t() 112 but for
s3 moisture coefficient
Card 127 (8El0.3)
Column
1-10 PMBT3(K,I,J) =
11-20 PRDBT3(K,I,J) =
Card 128 ( 6Fl 2. 6)
Column Contents
1-12 THE( l) = Angle number 1 in degrees
13-24 THE(2) = Angle number 2 in degrees
etc. Repeated NANG times. Angles are input according to the
stacking sequence of the laminate being analyzed.
The following card is repeated NOS times
I=l,NDS
Card 129 (4I3,2Fl2.0)
Column
1-3
4-6
7-9
INODED(I)
INODE(I, l)
INODE(I ,2)
=
=
=
Contents
I
U - displacement code
V - displacement code
10-12 INODE(I ,3)
13-24 Vv(I)
25-36 ZZ(I)
117 ·.'
= W - displacement code_
= l for force or non-zero displacement
boundary .condition
= 2 ·for prescribed zero..,. displacement ·
=
=
Y coordinate· of node I before being
staled by SMY
Z coordinate of node· I before being
sealed by. SMZ ·
The following card is repeated NE times
I=l,NE
Card 130 (6X,5I6)
Column
1-6
7-12 ND (I, l)
13-18 ND(I,2)
' 19-24 ND(I,3)
25-30 . IMAT( I)
31-36 ITHETA(I)
Card 131 {2Il 2)
Column
1-12 NDCST
13-24 NFCST .
··· Contents
= Blank·· ·. ·
= -'Node number l of el errient I
= Node number 2 of element I i
. . ~·
= Node number 3 of element.I
.. Material number of e 1 ernent I
= Angle number of element I
Contents.
= · Number of non-zero displacement
··constraints
= :Number of non-zero force constraints
If NDCST = 0 skip to card 133
· I=l ,NDCST
Card 132 (2Il2,Fl2.0)
Column
1-12
13-24
25-36
NODED( I)
MODE
DCST(I)
=
=
118
Contents
Node number of constrained node
Code fo~ constraint
= 2 for V constraint
= 3 for W constraint
= Displacement constraint increment
If NFCST = 0 no more input is required
The foll owing card is repeated NFCST times
I=l,NFCST
Card 133 (2Il2,Fl2.0)
APPENDIX E
MESHES USED
119
0 0
120
0
0
:c: 1-..... 3 :c: (,/') LJJ :::E
0
N
0
121
0
J--~~~~-1-~ ...... ~-1-~.+-~ .... ---'J---!l---i'---+~-+~-t-~'"""'~11---'1 co 0
122
0 0
C>
N
C>
123
C>
co C>
C>
C>
(/) 1--z LLI ~ LLI _J LLI
co °' l.{)
:.:r: 1--t--t 3: :.:r: (/) LLI ~
0
N
0
124
0
0
0
0
(/') t-z: l...LJ ~ l...LJ _J l...LJ
APPENDIX F
NONCOM2 FLOW CHART
125
·. -··:-
·.'".' ,126
NONCOM~ -FLOW CHART
START
CALCULATE GRID PROPERTtES .
. lNITlALIZE • VARIOUS. ARRAYS
NUMBER EQliATibrts .. IMP.oSE .. COtlSTRAlNTS
CALCULATE ELAST! C MOpu~l~~~~M· o.
CALCULATE & TRf.NSFORM LAMINA STIFFNESSES
CALCULATE & TRANSFORM STRENGTH TENSORS
EQ~~~~N~vn~" N~bAL :DISPLACEMENTS .
. CALCULATE STRAINS & STRESSES FROM .
OISP.LACEMENTS
GALCULATE TENSOR' POL YNOMlAL & CHECK
FOR FAIL.URE·
CALCULATE ELASTIC ·· MOOUL!l USING .
RETENTION CURVES
. --"".
APPENDIX G.
T3Q0/5208 PROPERTIES
·127
··/
TABL
E G.
1 ..
Ram
berg
-Osg
ood
Para
met
ers
for
grap
hite
epo
xy T
300/
5208
·
Ela
stic
Cu
rve
I Mo
d:ul u
s . (M
SI)
. Et'
. .
..
xx··
I·
··· .. 19~
2 c
· .. Ex
><
t J9
.2
t E ..
..
.11...
. l.
56
·f:C·
. »-
I l.
56
Et z·z
I
1.56
EC
..
. .
. zz
1
· 1.
56
G
. .
yz.
· .4
87
6 xz
~ .8
2 Gx
y ..
< ·
· ·
.82
'Ela
st·i
c .·
Lim
it
. (K
SI}
·go
...
87
6.3
l3.5
6.3
·. 13
.5 .
2. l
3. 5
'
3~5
"1
.2~593
2. 0
37 .
...
0 1.06
8 .
0 1.06
8.'
l ~147 .. ·.
. l.1
47
.Ll4
7.
. -n
K. ·c
p· l>·
.
l SI
.
>l4i
92xl
o:-l
6 .
.·3o
754x
n>-r
3 ·..
. . .
l
• lOOO
OxlO
.
.-~
... . .•.
._7
.. l3
324x
l0
,. .
l .1
0000
~10
•.··
. ·..
. -
--T
·.
. ] 3.
324x
l o~
· ·
. . 81
4l9x
l0 .. 7
.
. 448
82~1
0~7
·. ·..
-7
.448~2xl0
..
a*-
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Sl)
· .. 2
.82
"2
...
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.. .
K2(
PSI
2) ·
4 ·5· 6
·a I
l ·
-""2
1 · .
....
· ·
·. ·
. ·.6
842x
10 .
< 2~
82
J 4.
688
l.6
842x
l0"'
"2l
I .·
2 .• 8
2 4 •
688
. I
l.
6?4·
2xlO
.. 21
· 1 ·
_:,'
-~.
N 00
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I
·· . .--·
'·'
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..
.:~:
·.·
• I
• ..
·.'··.
,,
Prop
erty
<E:t
' .
·-xx.
. EC
···
. yy
··
•': E'.
'f~·.
'zz
.. · EC
· .
. . z
z :.G
.•. .. ·
yz.··
·
.G~z
··.G~y
··
..
. ·TA
BLE
G.2
H
ygro
ther
mal
pr
oper
ties
for
gra
phit
e ep
oxy
T300
/520
8
Perc
ent .
R~te
ntio
n of
Room
Tem
p.,
0% M
oist
ure
'Pro
pert
y:
Room
·Tem
p. ..
. .
·_ ·
. ·
. . ..
· .
. ·
·.-·-. · ·
· ·
· ·
· · ·
0% M
oistu
re.
Tem
pera
ture
-70
° F
· ·
Tem
pera
ture
~26
0°
F ··.
Tem
pera
ture
-35
0° F
. "
-Ela
stic
·.
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· . >
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8 .
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97 ~·
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1. 56
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. l·9
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3.9
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. 96.2
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· .. 85
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I;·.
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82
. ···rlo
(}io,, J
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con
tinue
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· Percen-t~~Retention o
f Ro
om .T~mp.
~ 0%
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ty·
Room
Tem
p. . .
..
. ..
0% M
oist
ure
Tem
pera
ture
;..7Q
0 -F
Te
mpe
ratu
re -
260°
F ·.
· ·. T
empe
ratu
re .-
350°
-F
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erty
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th-
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The vita has been removed from the scanned document
NONLt~EAR TE,MP~R~TURE DEl'ENOENT ~AI LURE. ANALYSlS OF fl'NITE·WIDTH COMPOSITE .LAMINATES
by
Aniruddha P. Nagarkar
(ABSTRACT}
A quasi-three dimensional, nonl,inear elastic finite element stress
analysis of finit(;! width composite laminates including curing stresses
is presented.
Cross-ply, angle-ply, a.rid some quasi"'. isotropic gt~aphite/epoxy, . .
laminates are ·studied. Curing. stresses are calculated u'sing temperature
dependent elastic properties that are input as percent retention c.u~~ies,
and stresses due to mechanical loading in the form of an axial strain ·, .
are calculated using tangent modulii, obtained by Ramberg Osgood
parameters.
It is shown that curing stresses are significant only as edge
effects in angle-ply laminates, and severe throughout the laminate
in cross-ply laminates .. The tensor polynomial failure ,criterion is
used to predict the initiation of failure, and the failure .mode ,;s pre-
dicted by examining individual contributions of the stresses to the
polynomial.
