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Point-stress analysis of continuous fiberreinforced composite materials with an elastic-plastic matrix
Compufm
& Sfru~tures
Vol.20.
No
l-3.
pp. 375-385.
1985
0045-7949185
$3.00 + .OO
0 1985 Pergamon Press Ltd
Prmted ,n the U.S.A.
COMPOSITE
MATERIALS
AND STRUCTURES
POINT-STRESS ANALYSIS OF CONTINUOUS FIBERREINFORCED COMPOSITE MATERIALS WITH AN ELASTIC-PLASTIC MATRIX ANTONIO C. RuFIN,t PETEKG. RIMBOS$ and SHERMAN D. BIGEI.OW$ Boeing Aerospace Company, Seattle, Washington. U.S.A. Abstract-The inability to accurately model the elastic-plastic behavior of certain composite materials currently under consideration by the aerospace industry. such as metal-matrix composites, has often hindered the use of these materials as efficient structural alternatives. A point-stress analysis computer code especially developed to address this issue is discussed herein, and comparisons are made between predicted and measured stress-strain behavior for various boron-reinforced aluminum laminates at room temperature and 600 “F. The computer program correlates well with most of the experimental results. but further improvements, such as the addition of a nonlinear matrix hardening model, are needed to increase accuracy.
NOMENCLATURE 3 x I vector 3 X 3 matrix load resultant M moment resultant reference time load coordinates (: = normal to plane of laminate) number of plies ply thickness local stress local strain fiber fraction (by volume)
I Ni
I-Cf
coefficient of thermal expansion Poisson’s ratio Young’s modulus shear modulus local temperature F yield surface yield stress 0, cd hardening parameter x flow rule factor P hardening law factor Superscripts m
matrix material
f fiber material -I
L, OL laminate (zero denotes midsurface) h-th ply (as in (A]!‘1 matrix inverse
Subscripts
elastic plastic load coordinates (z normal to plane of x,y,z laminate) I.&3 ply coordinates (1 parallel to fiber; 3 coincident with z) e
P
1. INTRODUCTION
Analytical methods for predicting the mechanical behavior of continuous-fiber reinforced laminated t Senior Engineer. $ Senior Specialist Engineer. § Supervisor, Structures Technology.
composite plates usually rely upon Classical Lamination Theory (CLT) as one of their key elements. CLT accounts for the orthotropic nature of individual plies and ply stacking sequence effects on overall laminate behavior. A large proportion of the computer programs designed for composite laminate analysis which utilize CLT are primarily intended for materials which consist in essence of linearly elastic fibers in linearly elastic matrices. Some of the advanced composites currently in use by the aerospace industry. however, involve markedly inelastic constituents, in particular. among matrices. Examples of such materials include metalmatrix composites (MMCs) and certain types of polymeric composites. Inelastic effects can include nonlinear elasticity, plasticity, viscoelasticity, or various combinations of these to more or less significant degrees. This study was concerned with modeling the response of linearly elastic-plastic materials reinforced with linearly elastic continuous fibers, by making use of CLT. Emphasis was placed upon continuous-fiber refinforced MMCs, but the results should equally apply to other materials with a similar linearly elastic fiber/elastic-plastic matrix arrangement. A computer program (MLAP-MMC Laminate Analysis Program) has been developed at the Boeing Aerospace Company which allows predicting the local mechanical behavior of elastic-plastic laminated composite materials under static in-plane loads, moments, and steady-state temperatures (including through-the-thickness thermal gradients). This code is partly based on an analytical model proposed by Bahei-El-Din and Dvorak [I, 21 whose main purpose was to serve as a foundation for their finite element program PAC78. The model employs the so-called “vanishing fiber diameter” approximation, according to which local phase stresses and strains in the constituent materials can be assumed to be uniform. This approximation is truly applicable only if fiber diameters are negligibly small in
375
A. RUFINet II/.
376
comparison with laminate dimensions, and it obviously neglects nonuniform fiber packing and material defects. A suitable micromechanical model can be used in conjunction with the vanishing fiber diameter approximation and CLT to arrive at incremental equations for laminate, ply matrix and fiber stresses. ply, matrix. and fiber strains. and midsurface strains and curvatures in terms of inplane loads, moments, and temperatures, as described below.
MLAP employs a reference time for internal program reference and I/O management purposes. Loads, moments, stresses, strains, curvatures and temperatures are all assumed to be piecewise lineal functions of that parameter. It is therefore convenient to express the incremental composite governing equations in terms of the derivatives of loads. moments. etc. with respect to time, even though the constitutive laws in elasticity and plasticity are not per se functions of time. A derivation of laminate, ply, fiber, and matrix governing equations is given in the Appendix. Note 2. THEORETICAL BACKGROUND AND PROGRAM that a further simplifying assumption is made in the DESCRIPTION analysis for cases where plastic deformations are As indicated in Sec. 1, the vanishing fiber di- induced in bending, namely, that ply. fiber. and maameter approximation and CLT constitute the ana- trix stresses and strains within plies subjected to such deformations can be approximated by a finite lytical basis for MLAP. Additional assumptions number of uniform steps in the through-the-thickmade in developing the current model were: (I) initially isotropic, linearly elastic matrix mate- ness direction. through a ply subdivision scheme. This necessary simplification is introduced in order rials: (2) linear kinematic work hardening matrix materials with von Mises yield surfaces: and (3) iso- to reduce the governing equations to a set of ordinary differential equattons with respect to time. tropic, linearly elastic fibers. Prior to any plastic deformations taking place in The specific plasticity model used for the matrix materials was the one postulated by Ziegler 131. the composite. MLAP utilizes generally the same after the kinematic hardening model established by methods that are used in conventional CLT codes Prager [4]. The temperature dependence of the plas- to determine midsurface strains and curvatures. i.e. via the [A]. (B], and [D] matrices and eqna (13a)ticity governing equations is based on the model (l3b) ]I()]. Ply, matrix, and fiber stresses and strains proposed by Prager, as described by Naghdi IS].
Fiber and Matrix Material (s) Data .Loads,
Ibments
and
Initialize and Determine Elastic Laminate Parameters (l4icromechanIcs)
Plastically l
Elasticity
.
Check Ply
Solution
l
Check for Plastic Load/Unload
.
Solve Incremental Plastlclty Equations (Runge-Kutta)
yield
Stresses,
Fig.
I
Ply,
Simphfied
Deforming
Fiber
MLAP
flow diagram.
Plies: All Pllcs All Tie Steps
Point-stress
Off-Axis
{All Dimnsions [1>
Sides
flat
D
Symmetrical
b
Steel
and
tabs,
0.08
In
in.
Speclmn
Tension
([151,,
1751,)
Inches)
parallel
to within
analysis
to within
+o.OOl
5.005 thick,
10’
taper.
4/Specimen
Fig. 2. Test specimen geometries.
Room Temperature Approximate Curve Used In HLAP
(Courtesy
0.5
0
1.0 STRAIN,
of
Aluminum
2.0
1.5 1O-3
Fig. 3. Stress-strain
the
Company of
2.5
3.0
in/in
curve for 6061-O.
Table 1. Material properties used m the analysis Fiber properties ((sotropic) Young’s modulus: Poisson’s ratio:
58 x IOh psi 0.25 Matrix properties
Young’s modulus: Poisson’s ratio: Proportional limit: Secondary modulus:
9.9 X IOh psi 0.33 5.0 ks) (room temperature) 1.0 ksi (600” F) 5 x IO”psi
(see Fig. 3)
America)
A. RUFINet
378
i.e. the corresponding temperature contribution [eqn (24) in the Appendix] at the beginning of each load step. When the matrix unloads, the program reverts to the elastic regime. MLAP is currently operational on a VAX-I I computer system, and it is run interactively. Figure 1 shows a simplified flow diagram of the program.
loo0 -
u-
.e.
al.
2000 -
8 s
3. VERIFICATION
1000 I
0
1.0
-_-
MEASURED HLAP
I
I
I
I
2.0
3.0
4.0
5.0
LONCITUDIMALSTRAIN,
Fig. 4. Predicted and measured B/Al (longitudinal
1D-3
load-stram direction).
in/In
curves
for [Old
then follow from eqns (2)-(Sa). Internal yield checks are performed within each load step to determine the onset of plastic deformations, using the von Mises criterion. As soon as yield is detected. program operation shifts to the solution of eqns (7). (18). (20), and (22) for ply stresses and strains. and matrix stresses and hardening parameters (see the using a variable-step fourth-order Appendix), Runge-Kutta technique. The remaining dependent variables are found from eqns (2)-(4). This method is repeated at every load step. MLAP is also designed to detect plastic unloading conditions, based on the orientation of the stress increment vector relative to the loading surface normal and .#,,“h’ --
dl’“’
#“’
dt ’
TESTS
In order to evaluate MLAP. a series of mechanical tests were conducted on a representative elastic-plastic composite. The material selected for this task was boron-reinforced 6061 aluminum (B/Al) [ 111in the as-fabricated ( - 0) condition. Tensile test specimens were machined from two four-ply thick panels (approximately 0.029 in. in thickness). one unidirectionally reinforced and the other one in a [ 2 451,Y[12] configuration. Tests were conducted on [Old, [9014, [1514 (off-axis). [75], (off-axis), [*451,Y, and [0, 901, specimens, at room temperature and at 600 “F (lo-min. soak at temperature). Off-axis tension specimen dimensions were determined using the criteria discussed by Pagan0 and Halpin [71. in an attempt to minimize the effects of end constraints on the deformations in the gage area. Figure 2 shows specimen geometries and dimensions. The specimens (one per type) were fitted with strain gage pairs ([014, [9014, and [0, 901, specimens) or strain gage rosettes [ 1514. [7514, and [ *45]., specimens) at the center, and were subjected to a singlecycle load-unload-reload program in order to generate measurable plastic strains. The load at the point of plastic unloading was established as approximately one-half of the estimated laminate failure load. The strain rate was 0.005 in./in./min. Some instrumentation problems and a premature failure limited the amount of useful data to eight specimen sets: [9014. [ t 451,. and [I 514at room temperature, and [OIJ. ]90],, [ +45],. [15]4. and 1751~at
1 .
9
s 1000
-I .4
-1.2
-1 .o TRANSVERSE
Fig. 5. Predicted
and measured
-0.8 STRAIN,10-3
load-strain
-0.6
-0.4
-0.2
0
in/in
curves
for [O]J B/AI (transverse
direction).
Point-stress analysis
dieted and measured load vs strain curves for a number of the cases indicated above. The issue of residual stresses arising from the composite fabrication process (due to the difference between fiber and matrix thermal expansions) was not considered in depth in the present study. Large residual stresses would manifest themselves in the form of unexpectedly low composite yield stresses or no elastic range at all. Neither of these phenomena were observed here, most likely due to matrix relaxation [9]. so no attempt was made to evaluate them.
300 c f!. “4 I:
379
200
100
0
LONGITUDINAL
STRAIN,
1D-3
In/in 4. DISCUSSION
Fig. 6. Predicted and measured load-strain curves for [ 15],, B/AI (off-axis) (longitudinal direction).
The results of Sec. 3, in particular the comparison between predicted and measured load vs strain curves in Figs. 4-14, should be viewed in the context of attempting to predict the behavior of a rheologically complex material with a linear elasticplastic model. As such, the model accurately predicts key features of this kind of material, for instance, the onset of yield and the Bauschinger effect upon plastic unloading, and to a lesser extent, per-
600 “F. MLAP runs were made for all these cases. The material properties used in the analysis are listed in Table 1. Room temperature matrix data follows from Fig. 3. The level of yield stress degradation with temperature for the aluminum matrix was taken from 6061-T6 data [S] due to the lack of similar data for 6061-o. Figures 4-14 show pre-
-
400
Room Temperature MEASURED -
--
HLAP
TRANSVERSE
STMI
13, lO’3
In/in
Fig. 7. Predicted and measured load-strain curves for [IQ B/Al (off-axls) (transverse
---
-1.4
direction).
IlAP
-1.2
--1.0
45’
-0.0 STRAIN,
-0.6 10m3
-0.4
in/in
Fig. 8. Predicted and measured load-strain curves for [ 1514B/Al (off-axis) (45” direction).
380
0
1.0
3.0
4.0
STRAIN,
lO’3
2.0
LONGITUDINAL
5.0
6.0
In/in.
Fig. 9. Predicted and measured load-strain curves for [ 24.51, B/Al (longitudinal direction).
Room Temperature ---
MEASURED MAP 7
I
-3.5
300
I
I -3.0
-2.5
-2.0
TRANSVERSE
STRAIN,
-1.5
-1 .o lo-’
-0.5
0
in/in.
Fig. IO. Predicted and measured load-stram curves for [ 2451, B/Al (transverse
direction).
400 Room Tompcrature
300
-
MEASURED
---
t!LAP
45.
STRAIN,
10m3
In/in
Fig. 11. Predicted and measured load-strain curves for [ 2 451, B/Al (45” direction).
Point-stress
381
analysis
ZOO.
f
Strain Gage Failure at c=1.4*10e2
150 -
MEASURED HLAP
-
_-_
0
1.0
2.0
I.0
LONGITUDINAL
5.0
4.0 STRAIN,
In/in.
7.0
6.0
8.0
10m3 in/in.
Fig. 12. Predicted and measured load-strain curves for [ 2451, B/Al (longitudinal direction).
600°F _-_
-5.0
-6.0
-4.0 TRANSVERSE
y
MEASURED HLAP
-3.0
-2.D
STRAIN,
1O’3
-1 .o
200
0
in/in.
Fig. 13. Predicted and measured load-strain curves for [ 2451, B/Al (transverse
200__-
direction).
6000~ REASURED KAP .
150a
u..
45O STRAIN,
tO’3
lnfln
Fig. 14. Predicted and measured load-strain curves for [ rt45], B/AI (45” direction),
382
A.
RIJFIN ef ul.
manent strains induced by plastic deformation. None of these parameters could be estimated with a purely elastic CLT model. Figures 4-14 also show that, in general, MLAP tends to underestimate the elastic moduli of the composite and overestimate the magnitudes of plastic strains. While these errors are not significant along the fiber direction. they are more pronounced in the direction normal to the fibers. These effects are directly attributable to two factors: ( I ) limitations of the micromechanical model, and (3) the nonlinear nature of matrix hardening (MLAP assumes linear matrix hardening). The micromechanical model combines an upper bound model for the stiffness in the fiber direction with a lower bound model in the transverse direction. as shown in the Appendix. Obviously. the lower bound model is too conservative. No changes in the definition of the model are possible without major changes in the analysis, though. On the other hand, modifying the hardening law to account for nonlinear matrix plasticity should be a relatively simpler task. All it requires is replacing the constant flk’ in eqn (5b) by a suitable function of the hardening parameters or the dissipation work, as suggested by Ziegler 131. This change would not require any major modifications in MLAP’s structure. However, no welldocumented satisfactory nonlinear hardening formulations have been found in the literature for the plasticity model used in this study. As a result. future efforts for improving MLAP will have to be directed toward setting up and incorporating in the program such a formulation.
5.
4. W. Prager, The theory of plasticity: A survey of recent achievements. Proc. Inst. Mech. Eng. 169, 41-57 (1955). 5. P. M. Naghdi, Stress-strain relations in plasticity and thermoplasticity. Pkuticicy. Proceedings of the Second Symposium on Naval Structural Mechanics (Edited bv E. H. Lee and P. S. Svmonds). DD. 121-169. Pergamon Press, Elmsford, N:Y. (1960): 1 6. R. M. Jones, Mechanics sf Composite Materiuls. Chap. 4. p. 147. McGraw-Hill, New York (1975). influence of end con7. N. J. Paean0 and J. C. Haloin. . straint in the testing of anisotropic bodies. J. Compos. Mater. 2, 18-31 (1968). Military Standurdizutron Handbook 8. Anonymous, Met&c Muterials and Elements for Aerospace Vehicle Structures. MIL-HDBK-SC. Vol. I. pp. 3215 (1978). 9. J. E. Schoutens. Introduction to Metul Matrix ComDosites. MMCIAC Tutorial Series. MMC. No. 272. iune, 1982, pp. 7-6 and 7-7. 10. All equation numbers listed in the text correspond to equations in the Appendix. 11. Flat panels, 47% fiber by volume. Produced by Amercom Inc.. Chatsworth, CA. 12. s stands for “symmetric” relative to midplane of laminate.
APPENDIX Theo? The basic laminate, ply, fiber, and matrrx governing equations are given below: (1) Equilibrium and compatibility
(1)
CONCLUSIONS
There is a need for analytical methods for modeling the stress-strain behavior of continuous-fiber reinforced elastic-plastic composites. Boeing’s elastic-plastic point-stress analysis code, MLAP. can be used for that purpose, although further improvement in the matrix plastic hardening model is needed in order to enhance the program’s capabilities. Ac~noM,/ed~menr.c-The authors would like to express their gratitude to the engineering personnel and staff of the Structural Development, Manufacturing Technology. Materials and Structures Test Laboratories, and Parts. Materials, and Processes organizations of the Boeing Aerospace Company who collaborated in this program. for their valuable help. This study was funded as an independent research and development effort by the Boeing Aerospace Company. REFERENCES
Y. A. Bahei-El-Din and G. J. Dvorak, Plasticity analysis of laminated composite plates. Trans. ASME .I. Appl. Mech. 49, 740-746 (1982). Y. A. Bahei-El-Din and G. J. Dvorak, Plasticity analysis of fibrous composites. Trans. ASME J. Appl. Mech. 49, 327-335 (1982). H. Ziegler, A modification of Prager’s hardening rule. Q. Appl. Mach. XVII, 55-65 (1959).
(2) Micromechanics a@ at
_ a@’
a(Tf?,(h) v for
_
at
at
+ &,
ij#
f,~ = 1.2,
!!$!Y + !!S$.E)
for i,
11.
j
=
rj
#
I,
2,
(
a& _=-=_ at (3) Constitutive
aEf{“’
at
a#] at
II.
(3)
relations
(i) Fibers
1
a#) -=_ at
Ef”’ (
a&&’ _=at
a@’ 1 -_ Ef’“’ c at
aEf?’
-- 1
aa(i”)
cf(h’
at
-= ilf
-a&“’ _ at
vf,n, a(TS!“’
at +
+ ,*fcll >
-ark)
at ’
aaflk’ + a*flr) _aFh) at ’ at > (4)
Point-stress (ii) Matrices m(k) 1 a&k) -_ at = jjiZi at ( m(h) aazz
1
A=--_ ar
Em’k’
m(h) ae12 -=_=at
383
Combining eqns (2) and (7),
aelI
a$PJ
analysis
(
1
at m(h) ael2
A
,,m(k)
m(l) aull _
+
at
+
at
aP
a*mlA)
_
a*mCh)
aP _ at ’
(Elastic plies).
at
G”(h)
m(k) aal
,,m(k)
at ’ (8)
t5a) So,for equilibrium,
(iii) Yield surface Frn’h’
(up
-
cxy9
=
m(A) (a,, +
_
(up
&/k))2
m(h)
-
(all
-
c&k),
_
+
_ &‘)
a$yh’)’
af’h’)
(c&h
In(h) 3(u,z
Ub)
+
_ oJ$G): (PA’)
= 0. -
(iv) Flow rule e
I [R”‘]
’ {O”‘} s
dz
,jFm(k)&,m(hl
at =-a@'
(5)
at ’
(c!cJ~+(!?!$)?g
= $ [FL’]-’ (l:^I
aAm -=
flh’nz x
at
(g$)($)
= 7 [Fk’]-’
j-:^, ; {F]
z[R”‘]-’
dz
[7”“] (z}
dz
(SC) +
5 (E”W - ENA)) For linear hardening, fi,k’ = 4 Em”” -V””
[ 7”)] { $}
dz - j-:1, z[R”“] - ’ {O@‘} $
dz) ,
(9)
(v) Hardening law ap(hl II
at
where the superscript (p) in the load and moment resultants now denoted the contribution to {dNldr} and {dMl dt}, respectively, from plies deforming plastically. Assembling the results of eqns (6) and (9),
=
al*.m,k’
- at =
(t.g y
+ (S) $
m(hl (uZk’ - wnn ) E!L&
(5d)
f?t”
Making use of classical lamination theory (CLT). for plies deforming elastically,
(6) where the superscript (e) in the load and moment resultants denotes the contribution to {dN/dt} and {dMldt}, respectively, from plies deforming elastically. These equations follow from equilibrium and eqn (2). with matrices [Al, [Bl, and [DI defined in the usual sense [6], containing terms from all plies in the elastic regime. The last term in both eqns (6) represents the temperature contributions to the load and moment resultants. On the other hand, for plies deforming plastically, ply strains can be expressed as follows:
= ([B]+~L~“‘]~‘(~:“,z[R’*‘]-‘dz) x ,P];{g
+ $,T”‘l-‘(
+
([Dl
~:~,r’[R”‘]-‘dz)
[F”])
384
RUFIN et ul.
A.
For plies deforming
plasttcally.
Therefore. Next.
rewrite
(I 1) as
eqns (IO) and
follows:
(I?) It can then be readtly
shown [6] that
+ :17’“‘llLl{i,I
+ :lr”llMl~S~~
from eqns (31 and (13). Rearrangmg,
tl3b)
+ [Q’“‘,
‘[7’“‘[
x (IL,
+ :,M,J
where
I4
= IF1
’ + [Fl_ ‘IGI [[HI
-
[K] = [Fl-
‘lG][[G]lF[
‘[G]
[Ml
~ [Gl[F]
‘[G]] ~’
= [[HI
-
(not to be confused with moment plies deforming elastically.
lG][Fl lH[]-’
‘lGl]p’. = [L[‘.
resultants).
Now.
for
(II) Al50. ply stresses can be expressed stresses as follows:
in term,
An additional set of equattons tained by noticing that
Then.
definmg
dtr’“‘/d/
can be ob-
from eqn (70).
of matrix
+
[
,Ph'][Q'h']-'
+
Then.
({p}
-
7’h’] ([K] +
[pq[Q’h’]-
‘{#“‘})
F
+ IP”“I[Q”‘l
‘[7”‘1 (14 + ,7LLl){<,)
+ [P”‘][Q”‘]
’
IF/“1 ([ICI + ;IMl){izJ (22)
(16)
+ ~lFA’11L115,/ + :lr”‘ll~l{521. by vtrtue of eqns 12) and
( 13).
Tht> equation
yields
[F”][.T]
- [ r”],y*‘i’]$+
[Q*“‘][ r”‘]m’([7’A’][J]
+~[~k’][L]{~,})+[Q*““][r~‘l
-‘([~~‘I[Kl+z[~“‘1[~1{~2}). (17)
Equattons (16). (17). and (22) constitute a set of simultaneous partial dtfferential equattons with respect to : and f. These equations cannot be readily solved, however. if one or more plies have been stresses beyond yield under bending or twisting, due to the fact that stress and strain gradients in the ; direction withrn those plies are not necessarily linear (as they would be if the plies were deforming elastically). This difficulty can be at least partially avotded if each of such plies IS treated, for analysis purposes. as a stack of a limited number of thinner plies wherem stresses and strains are uniform. Then, for the individual. subdivtded plies.
and the factor ; m eqns (16). (17). (20). and (22) can be replaced by i, i.e. the position of the subdivided ply centroid along the z axis. Finally. plastic loading/unloading conditions can be determined by making use of the followmg crrteria 151:
Point-stress &Cl”“”
w
#“I”” +--=()
aF”’
a(~;‘~’ dr
dt
F”“‘” = 0 (neutral loading). #““hl dFh1 #,?I”” d,$z’h’ +--co, 8~:;‘~
F”“”
dt
aF”
= 0 (unloading).
aF”“h’
#A’
dt
’
385
analysis
au$“’ F”““’
w
+
dt
dF’“‘h’ d7”‘” > o
@’
dl
’
= 0 (loading).
(24) (indicial notation)
& Sfru~tures
Vol.20.
No
l-3.
pp. 375-385.
1985
0045-7949185
$3.00 + .OO
0 1985 Pergamon Press Ltd
Prmted ,n the U.S.A.
COMPOSITE
MATERIALS
AND STRUCTURES
POINT-STRESS ANALYSIS OF CONTINUOUS FIBERREINFORCED COMPOSITE MATERIALS WITH AN ELASTIC-PLASTIC MATRIX ANTONIO C. RuFIN,t PETEKG. RIMBOS$ and SHERMAN D. BIGEI.OW$ Boeing Aerospace Company, Seattle, Washington. U.S.A. Abstract-The inability to accurately model the elastic-plastic behavior of certain composite materials currently under consideration by the aerospace industry. such as metal-matrix composites, has often hindered the use of these materials as efficient structural alternatives. A point-stress analysis computer code especially developed to address this issue is discussed herein, and comparisons are made between predicted and measured stress-strain behavior for various boron-reinforced aluminum laminates at room temperature and 600 “F. The computer program correlates well with most of the experimental results. but further improvements, such as the addition of a nonlinear matrix hardening model, are needed to increase accuracy.
NOMENCLATURE 3 x I vector 3 X 3 matrix load resultant M moment resultant reference time load coordinates (: = normal to plane of laminate) number of plies ply thickness local stress local strain fiber fraction (by volume)
I Ni
I-Cf
coefficient of thermal expansion Poisson’s ratio Young’s modulus shear modulus local temperature F yield surface yield stress 0, cd hardening parameter x flow rule factor P hardening law factor Superscripts m
matrix material
f fiber material -I
L, OL laminate (zero denotes midsurface) h-th ply (as in (A]!‘1 matrix inverse
Subscripts
elastic plastic load coordinates (z normal to plane of x,y,z laminate) I.&3 ply coordinates (1 parallel to fiber; 3 coincident with z) e
P
1. INTRODUCTION
Analytical methods for predicting the mechanical behavior of continuous-fiber reinforced laminated t Senior Engineer. $ Senior Specialist Engineer. § Supervisor, Structures Technology.
composite plates usually rely upon Classical Lamination Theory (CLT) as one of their key elements. CLT accounts for the orthotropic nature of individual plies and ply stacking sequence effects on overall laminate behavior. A large proportion of the computer programs designed for composite laminate analysis which utilize CLT are primarily intended for materials which consist in essence of linearly elastic fibers in linearly elastic matrices. Some of the advanced composites currently in use by the aerospace industry. however, involve markedly inelastic constituents, in particular. among matrices. Examples of such materials include metalmatrix composites (MMCs) and certain types of polymeric composites. Inelastic effects can include nonlinear elasticity, plasticity, viscoelasticity, or various combinations of these to more or less significant degrees. This study was concerned with modeling the response of linearly elastic-plastic materials reinforced with linearly elastic continuous fibers, by making use of CLT. Emphasis was placed upon continuous-fiber refinforced MMCs, but the results should equally apply to other materials with a similar linearly elastic fiber/elastic-plastic matrix arrangement. A computer program (MLAP-MMC Laminate Analysis Program) has been developed at the Boeing Aerospace Company which allows predicting the local mechanical behavior of elastic-plastic laminated composite materials under static in-plane loads, moments, and steady-state temperatures (including through-the-thickness thermal gradients). This code is partly based on an analytical model proposed by Bahei-El-Din and Dvorak [I, 21 whose main purpose was to serve as a foundation for their finite element program PAC78. The model employs the so-called “vanishing fiber diameter” approximation, according to which local phase stresses and strains in the constituent materials can be assumed to be uniform. This approximation is truly applicable only if fiber diameters are negligibly small in
375
A. RUFINet II/.
376
comparison with laminate dimensions, and it obviously neglects nonuniform fiber packing and material defects. A suitable micromechanical model can be used in conjunction with the vanishing fiber diameter approximation and CLT to arrive at incremental equations for laminate, ply matrix and fiber stresses. ply, matrix. and fiber strains. and midsurface strains and curvatures in terms of inplane loads, moments, and temperatures, as described below.
MLAP employs a reference time for internal program reference and I/O management purposes. Loads, moments, stresses, strains, curvatures and temperatures are all assumed to be piecewise lineal functions of that parameter. It is therefore convenient to express the incremental composite governing equations in terms of the derivatives of loads. moments. etc. with respect to time, even though the constitutive laws in elasticity and plasticity are not per se functions of time. A derivation of laminate, ply, fiber, and matrix governing equations is given in the Appendix. Note 2. THEORETICAL BACKGROUND AND PROGRAM that a further simplifying assumption is made in the DESCRIPTION analysis for cases where plastic deformations are As indicated in Sec. 1, the vanishing fiber di- induced in bending, namely, that ply. fiber. and maameter approximation and CLT constitute the ana- trix stresses and strains within plies subjected to such deformations can be approximated by a finite lytical basis for MLAP. Additional assumptions number of uniform steps in the through-the-thickmade in developing the current model were: (I) initially isotropic, linearly elastic matrix mate- ness direction. through a ply subdivision scheme. This necessary simplification is introduced in order rials: (2) linear kinematic work hardening matrix materials with von Mises yield surfaces: and (3) iso- to reduce the governing equations to a set of ordinary differential equattons with respect to time. tropic, linearly elastic fibers. Prior to any plastic deformations taking place in The specific plasticity model used for the matrix materials was the one postulated by Ziegler 131. the composite. MLAP utilizes generally the same after the kinematic hardening model established by methods that are used in conventional CLT codes Prager [4]. The temperature dependence of the plas- to determine midsurface strains and curvatures. i.e. via the [A]. (B], and [D] matrices and eqna (13a)ticity governing equations is based on the model (l3b) ]I()]. Ply, matrix, and fiber stresses and strains proposed by Prager, as described by Naghdi IS].
Fiber and Matrix Material (s) Data .Loads,
Ibments
and
Initialize and Determine Elastic Laminate Parameters (l4icromechanIcs)
Plastically l
Elasticity
.
Check Ply
Solution
l
Check for Plastic Load/Unload
.
Solve Incremental Plastlclty Equations (Runge-Kutta)
yield
Stresses,
Fig.
I
Ply,
Simphfied
Deforming
Fiber
MLAP
flow diagram.
Plies: All Pllcs All Tie Steps
Point-stress
Off-Axis
{All Dimnsions [1>
Sides
flat
D
Symmetrical
b
Steel
and
tabs,
0.08
In
in.
Speclmn
Tension
([151,,
1751,)
Inches)
parallel
to within
analysis
to within
+o.OOl
5.005 thick,
10’
taper.
4/Specimen
Fig. 2. Test specimen geometries.
Room Temperature Approximate Curve Used In HLAP
(Courtesy
0.5
0
1.0 STRAIN,
of
Aluminum
2.0
1.5 1O-3
Fig. 3. Stress-strain
the
Company of
2.5
3.0
in/in
curve for 6061-O.
Table 1. Material properties used m the analysis Fiber properties ((sotropic) Young’s modulus: Poisson’s ratio:
58 x IOh psi 0.25 Matrix properties
Young’s modulus: Poisson’s ratio: Proportional limit: Secondary modulus:
9.9 X IOh psi 0.33 5.0 ks) (room temperature) 1.0 ksi (600” F) 5 x IO”psi
(see Fig. 3)
America)
A. RUFINet
378
i.e. the corresponding temperature contribution [eqn (24) in the Appendix] at the beginning of each load step. When the matrix unloads, the program reverts to the elastic regime. MLAP is currently operational on a VAX-I I computer system, and it is run interactively. Figure 1 shows a simplified flow diagram of the program.
loo0 -
u-
.e.
al.
2000 -
8 s
3. VERIFICATION
1000 I
0
1.0
-_-
MEASURED HLAP
I
I
I
I
2.0
3.0
4.0
5.0
LONCITUDIMALSTRAIN,
Fig. 4. Predicted and measured B/Al (longitudinal
1D-3
load-stram direction).
in/In
curves
for [Old
then follow from eqns (2)-(Sa). Internal yield checks are performed within each load step to determine the onset of plastic deformations, using the von Mises criterion. As soon as yield is detected. program operation shifts to the solution of eqns (7). (18). (20), and (22) for ply stresses and strains. and matrix stresses and hardening parameters (see the using a variable-step fourth-order Appendix), Runge-Kutta technique. The remaining dependent variables are found from eqns (2)-(4). This method is repeated at every load step. MLAP is also designed to detect plastic unloading conditions, based on the orientation of the stress increment vector relative to the loading surface normal and .#,,“h’ --
dl’“’
#“’
dt ’
TESTS
In order to evaluate MLAP. a series of mechanical tests were conducted on a representative elastic-plastic composite. The material selected for this task was boron-reinforced 6061 aluminum (B/Al) [ 111in the as-fabricated ( - 0) condition. Tensile test specimens were machined from two four-ply thick panels (approximately 0.029 in. in thickness). one unidirectionally reinforced and the other one in a [ 2 451,Y[12] configuration. Tests were conducted on [Old, [9014, [1514 (off-axis). [75], (off-axis), [*451,Y, and [0, 901, specimens, at room temperature and at 600 “F (lo-min. soak at temperature). Off-axis tension specimen dimensions were determined using the criteria discussed by Pagan0 and Halpin [71. in an attempt to minimize the effects of end constraints on the deformations in the gage area. Figure 2 shows specimen geometries and dimensions. The specimens (one per type) were fitted with strain gage pairs ([014, [9014, and [0, 901, specimens) or strain gage rosettes [ 1514. [7514, and [ *45]., specimens) at the center, and were subjected to a singlecycle load-unload-reload program in order to generate measurable plastic strains. The load at the point of plastic unloading was established as approximately one-half of the estimated laminate failure load. The strain rate was 0.005 in./in./min. Some instrumentation problems and a premature failure limited the amount of useful data to eight specimen sets: [9014. [ t 451,. and [I 514at room temperature, and [OIJ. ]90],, [ +45],. [15]4. and 1751~at
1 .
9
s 1000
-I .4
-1.2
-1 .o TRANSVERSE
Fig. 5. Predicted
and measured
-0.8 STRAIN,10-3
load-strain
-0.6
-0.4
-0.2
0
in/in
curves
for [O]J B/AI (transverse
direction).
Point-stress analysis
dieted and measured load vs strain curves for a number of the cases indicated above. The issue of residual stresses arising from the composite fabrication process (due to the difference between fiber and matrix thermal expansions) was not considered in depth in the present study. Large residual stresses would manifest themselves in the form of unexpectedly low composite yield stresses or no elastic range at all. Neither of these phenomena were observed here, most likely due to matrix relaxation [9]. so no attempt was made to evaluate them.
300 c f!. “4 I:
379
200
100
0
LONGITUDINAL
STRAIN,
1D-3
In/in 4. DISCUSSION
Fig. 6. Predicted and measured load-strain curves for [ 15],, B/AI (off-axis) (longitudinal direction).
The results of Sec. 3, in particular the comparison between predicted and measured load vs strain curves in Figs. 4-14, should be viewed in the context of attempting to predict the behavior of a rheologically complex material with a linear elasticplastic model. As such, the model accurately predicts key features of this kind of material, for instance, the onset of yield and the Bauschinger effect upon plastic unloading, and to a lesser extent, per-
600 “F. MLAP runs were made for all these cases. The material properties used in the analysis are listed in Table 1. Room temperature matrix data follows from Fig. 3. The level of yield stress degradation with temperature for the aluminum matrix was taken from 6061-T6 data [S] due to the lack of similar data for 6061-o. Figures 4-14 show pre-
-
400
Room Temperature MEASURED -
--
HLAP
TRANSVERSE
STMI
13, lO’3
In/in
Fig. 7. Predicted and measured load-strain curves for [IQ B/Al (off-axls) (transverse
---
-1.4
direction).
IlAP
-1.2
--1.0
45’
-0.0 STRAIN,
-0.6 10m3
-0.4
in/in
Fig. 8. Predicted and measured load-strain curves for [ 1514B/Al (off-axis) (45” direction).
380
0
1.0
3.0
4.0
STRAIN,
lO’3
2.0
LONGITUDINAL
5.0
6.0
In/in.
Fig. 9. Predicted and measured load-strain curves for [ 24.51, B/Al (longitudinal direction).
Room Temperature ---
MEASURED MAP 7
I
-3.5
300
I
I -3.0
-2.5
-2.0
TRANSVERSE
STRAIN,
-1.5
-1 .o lo-’
-0.5
0
in/in.
Fig. IO. Predicted and measured load-stram curves for [ 2451, B/Al (transverse
direction).
400 Room Tompcrature
300
-
MEASURED
---
t!LAP
45.
STRAIN,
10m3
In/in
Fig. 11. Predicted and measured load-strain curves for [ 2 451, B/Al (45” direction).
Point-stress
381
analysis
ZOO.
f
Strain Gage Failure at c=1.4*10e2
150 -
MEASURED HLAP
-
_-_
0
1.0
2.0
I.0
LONGITUDINAL
5.0
4.0 STRAIN,
In/in.
7.0
6.0
8.0
10m3 in/in.
Fig. 12. Predicted and measured load-strain curves for [ 2451, B/Al (longitudinal direction).
600°F _-_
-5.0
-6.0
-4.0 TRANSVERSE
y
MEASURED HLAP
-3.0
-2.D
STRAIN,
1O’3
-1 .o
200
0
in/in.
Fig. 13. Predicted and measured load-strain curves for [ 2451, B/Al (transverse
200__-
direction).
6000~ REASURED KAP .
150a
u..
45O STRAIN,
tO’3
lnfln
Fig. 14. Predicted and measured load-strain curves for [ rt45], B/AI (45” direction),
382
A.
RIJFIN ef ul.
manent strains induced by plastic deformation. None of these parameters could be estimated with a purely elastic CLT model. Figures 4-14 also show that, in general, MLAP tends to underestimate the elastic moduli of the composite and overestimate the magnitudes of plastic strains. While these errors are not significant along the fiber direction. they are more pronounced in the direction normal to the fibers. These effects are directly attributable to two factors: ( I ) limitations of the micromechanical model, and (3) the nonlinear nature of matrix hardening (MLAP assumes linear matrix hardening). The micromechanical model combines an upper bound model for the stiffness in the fiber direction with a lower bound model in the transverse direction. as shown in the Appendix. Obviously. the lower bound model is too conservative. No changes in the definition of the model are possible without major changes in the analysis, though. On the other hand, modifying the hardening law to account for nonlinear matrix plasticity should be a relatively simpler task. All it requires is replacing the constant flk’ in eqn (5b) by a suitable function of the hardening parameters or the dissipation work, as suggested by Ziegler 131. This change would not require any major modifications in MLAP’s structure. However, no welldocumented satisfactory nonlinear hardening formulations have been found in the literature for the plasticity model used in this study. As a result. future efforts for improving MLAP will have to be directed toward setting up and incorporating in the program such a formulation.
5.
4. W. Prager, The theory of plasticity: A survey of recent achievements. Proc. Inst. Mech. Eng. 169, 41-57 (1955). 5. P. M. Naghdi, Stress-strain relations in plasticity and thermoplasticity. Pkuticicy. Proceedings of the Second Symposium on Naval Structural Mechanics (Edited bv E. H. Lee and P. S. Svmonds). DD. 121-169. Pergamon Press, Elmsford, N:Y. (1960): 1 6. R. M. Jones, Mechanics sf Composite Materiuls. Chap. 4. p. 147. McGraw-Hill, New York (1975). influence of end con7. N. J. Paean0 and J. C. Haloin. . straint in the testing of anisotropic bodies. J. Compos. Mater. 2, 18-31 (1968). Military Standurdizutron Handbook 8. Anonymous, Met&c Muterials and Elements for Aerospace Vehicle Structures. MIL-HDBK-SC. Vol. I. pp. 3215 (1978). 9. J. E. Schoutens. Introduction to Metul Matrix ComDosites. MMCIAC Tutorial Series. MMC. No. 272. iune, 1982, pp. 7-6 and 7-7. 10. All equation numbers listed in the text correspond to equations in the Appendix. 11. Flat panels, 47% fiber by volume. Produced by Amercom Inc.. Chatsworth, CA. 12. s stands for “symmetric” relative to midplane of laminate.
APPENDIX Theo? The basic laminate, ply, fiber, and matrrx governing equations are given below: (1) Equilibrium and compatibility
(1)
CONCLUSIONS
There is a need for analytical methods for modeling the stress-strain behavior of continuous-fiber reinforced elastic-plastic composites. Boeing’s elastic-plastic point-stress analysis code, MLAP. can be used for that purpose, although further improvement in the matrix plastic hardening model is needed in order to enhance the program’s capabilities. Ac~noM,/ed~menr.c-The authors would like to express their gratitude to the engineering personnel and staff of the Structural Development, Manufacturing Technology. Materials and Structures Test Laboratories, and Parts. Materials, and Processes organizations of the Boeing Aerospace Company who collaborated in this program. for their valuable help. This study was funded as an independent research and development effort by the Boeing Aerospace Company. REFERENCES
Y. A. Bahei-El-Din and G. J. Dvorak, Plasticity analysis of laminated composite plates. Trans. ASME .I. Appl. Mech. 49, 740-746 (1982). Y. A. Bahei-El-Din and G. J. Dvorak, Plasticity analysis of fibrous composites. Trans. ASME J. Appl. Mech. 49, 327-335 (1982). H. Ziegler, A modification of Prager’s hardening rule. Q. Appl. Mach. XVII, 55-65 (1959).
(2) Micromechanics a@ at
_ a@’
a(Tf?,(h) v for
_
at
at
+ &,
ij#
f,~ = 1.2,
!!$!Y + !!S$.E)
for i,
11.
j
=
rj
#
I,
2,
(
a& _=-=_ at (3) Constitutive
aEf{“’
at
a#] at
II.
(3)
relations
(i) Fibers
1
a#) -=_ at
Ef”’ (
a&&’ _=at
a@’ 1 -_ Ef’“’ c at
aEf?’
-- 1
aa(i”)
cf(h’
at
-= ilf
-a&“’ _ at
vf,n, a(TS!“’
at +
+ ,*fcll >
-ark)
at ’
aaflk’ + a*flr) _aFh) at ’ at > (4)
Point-stress (ii) Matrices m(k) 1 a&k) -_ at = jjiZi at ( m(h) aazz
1
A=--_ ar
Em’k’
m(h) ae12 -=_=at
383
Combining eqns (2) and (7),
aelI
a$PJ
analysis
(
1
at m(h) ael2
A
,,m(k)
m(l) aull _
+
at
+
at
aP
a*mlA)
_
a*mCh)
aP _ at ’
(Elastic plies).
at
G”(h)
m(k) aal
,,m(k)
at ’ (8)
t5a) So,for equilibrium,
(iii) Yield surface Frn’h’
(up
-
cxy9
=
m(A) (a,, +
_
(up
&/k))2
m(h)
-
(all
-
c&k),
_
+
_ &‘)
a$yh’)’
af’h’)
(c&h
In(h) 3(u,z
Ub)
+
_ oJ$G): (PA’)
= 0. -
(iv) Flow rule e
I [R”‘]
’ {O”‘} s
dz
,jFm(k)&,m(hl
at =-a@'
(5)
at ’
(c!cJ~+(!?!$)?g
= $ [FL’]-’ (l:^I
aAm -=
flh’nz x
at
(g$)($)
= 7 [Fk’]-’
j-:^, ; {F]
z[R”‘]-’
dz
[7”“] (z}
dz
(SC) +
5 (E”W - ENA)) For linear hardening, fi,k’ = 4 Em”” -V””
[ 7”)] { $}
dz - j-:1, z[R”“] - ’ {O@‘} $
dz) ,
(9)
(v) Hardening law ap(hl II
at
where the superscript (p) in the load and moment resultants now denoted the contribution to {dNldr} and {dMl dt}, respectively, from plies deforming plastically. Assembling the results of eqns (6) and (9),
=
al*.m,k’
- at =
(t.g y
+ (S) $
m(hl (uZk’ - wnn ) E!L&
(5d)
f?t”
Making use of classical lamination theory (CLT). for plies deforming elastically,
(6) where the superscript (e) in the load and moment resultants denotes the contribution to {dN/dt} and {dMldt}, respectively, from plies deforming elastically. These equations follow from equilibrium and eqn (2). with matrices [Al, [Bl, and [DI defined in the usual sense [6], containing terms from all plies in the elastic regime. The last term in both eqns (6) represents the temperature contributions to the load and moment resultants. On the other hand, for plies deforming plastically, ply strains can be expressed as follows:
= ([B]+~L~“‘]~‘(~:“,z[R’*‘]-‘dz) x ,P];{g
+ $,T”‘l-‘(
+
([Dl
~:~,r’[R”‘]-‘dz)
[F”])
384
RUFIN et ul.
A.
For plies deforming
plasttcally.
Therefore. Next.
rewrite
(I 1) as
eqns (IO) and
follows:
(I?) It can then be readtly
shown [6] that
+ :17’“‘llLl{i,I
+ :lr”llMl~S~~
from eqns (31 and (13). Rearrangmg,
tl3b)
+ [Q’“‘,
‘[7’“‘[
x (IL,
+ :,M,J
where
I4
= IF1
’ + [Fl_ ‘IGI [[HI
-
[K] = [Fl-
‘lG][[G]lF[
‘[G]
[Ml
~ [Gl[F]
‘[G]] ~’
= [[HI
-
(not to be confused with moment plies deforming elastically.
lG][Fl lH[]-’
‘lGl]p’. = [L[‘.
resultants).
Now.
for
(II) Al50. ply stresses can be expressed stresses as follows:
in term,
An additional set of equattons tained by noticing that
Then.
definmg
dtr’“‘/d/
can be ob-
from eqn (70).
of matrix
+
[
,Ph'][Q'h']-'
+
Then.
({p}
-
7’h’] ([K] +
[pq[Q’h’]-
‘{#“‘})
F
+ IP”“I[Q”‘l
‘[7”‘1 (14 + ,7LLl){<,)
+ [P”‘][Q”‘]
’
IF/“1 ([ICI + ;IMl){izJ (22)
(16)
+ ~lFA’11L115,/ + :lr”‘ll~l{521. by vtrtue of eqns 12) and
( 13).
Tht> equation
yields
[F”][.T]
- [ r”],y*‘i’]$+
[Q*“‘][ r”‘]m’([7’A’][J]
+~[~k’][L]{~,})+[Q*““][r~‘l
-‘([~~‘I[Kl+z[~“‘1[~1{~2}). (17)
Equattons (16). (17). and (22) constitute a set of simultaneous partial dtfferential equattons with respect to : and f. These equations cannot be readily solved, however. if one or more plies have been stresses beyond yield under bending or twisting, due to the fact that stress and strain gradients in the ; direction withrn those plies are not necessarily linear (as they would be if the plies were deforming elastically). This difficulty can be at least partially avotded if each of such plies IS treated, for analysis purposes. as a stack of a limited number of thinner plies wherem stresses and strains are uniform. Then, for the individual. subdivtded plies.
and the factor ; m eqns (16). (17). (20). and (22) can be replaced by i, i.e. the position of the subdivided ply centroid along the z axis. Finally. plastic loading/unloading conditions can be determined by making use of the followmg crrteria 151:
Point-stress &Cl”“”
w
#“I”” +--=()
aF”’
a(~;‘~’ dr
dt
F”“‘” = 0 (neutral loading). #““hl dFh1 #,?I”” d,$z’h’ +--co, 8~:;‘~
F”“”
dt
aF”
= 0 (unloading).
aF”“h’
#A’
dt
’
385
analysis
au$“’ F”““’
w
+
dt
dF’“‘h’ d7”‘” > o
@’
dl
’
= 0 (loading).
(24) (indicial notation)