- Email: [email protected]
Functionally graded metal matrix composite tubes
Composites Engineering, Vol. 5, No. 7, pp. 891-900, 1995 Copyright Q 1995ELsevierScienceLtd Printed in Great Britain. All rights reserved 0961-9526/95 $9.50+ .00
Pergamon
0961-9526(95) 00023-2 FUNCTIONALLY GRADED METAL MATRIX COMPOSITE TUBES Robert S. Salzar Department of Civil Engineering & Applied Mechanics, University of Virginia, Charlottesville, VA 22903, U.S.A.
(Received 15 December 1994; final version accepted 17 January 1995) A b s t r a c t - - C u r r e n t advanced manufacturing techniques allow the continuous variation of fiber or inclusion volume fraction in metal matrix composites. With this technology, it is now possible to tailor a composite to the expected loads by using the constituent materials to redistribute the stress and strain states through the material. Lighter and more structurally efficient components will be obtained through this grading process. The focus of this paper is the evaluation of the effects of material property and fiber volume grading on the overall mechanical response of metal matrix composite tubes subjected to mechanical loadings. This is accomplished through the development of a fully elastic-plastic axisymmetric generalized plane strain tube model. This analytical model incorporates a micromechanics algorithm in order to determine the elastic-plastic response of a heterogeneous fiber-reinforced composite cylinder. An arbitrary number of heterogeneous concentric cylinders can be included in the model, each with independent material properties. The inelastic analysis is performed through the method of successive elastic solutions. The optimization algorithm used in conjunction with this solution procedure utilizes the method of feasible directions and accepts any combination of design variables, constraints, and objective functions. As an example of the effectiveness of this grading, it is possible to vary the fiber volume fraction in an SiC/Ti-24Al-11Nb tube in such a way that the effective stress at the critical inner surface of an internally pressurized tube is reduced. For a 50.8 m m (2in) thick tube with an internal radius of 25.4 m m (1 in) and an internal pressure of 206.8 M P a (30 ksi), a uniform 40°7o fiber volume fraction distribution results in a tube that begins to plastically yield at the inner radius. By grading the fiber volume fraction, the tube now behaves elastically under the same pressure loading, allowing the tube to have a wall thickness of 25.4 m m (1 in) before plastic yielding begins. This grading results in a 6007o weight saving.
INTRODUCTION
Composites, whether organic, metallic, or ceramic based, have only recently seen use in a wide range of applications. These uses range from everyday items such as sporting goods and automobiles to advanced aerospace structures. With composites being used successfully in an increasing number of applications, advances in the manufacture of composite structures have enabled material designers to examine additional applications for these materials. Since most composite materials are comprised of either continuous fibers or inclusions embedded in a matrix, they can be tailored to a specific load by varying the volume fraction of the inclusion material. These "functionally graded materials" can be applied to many problems currently facing the composite design community. These problems include the stress concentrations resulting from free edge effects as well as from the discontinuity at the fiber/matrix interface. Significant work in the area of functionally graded materials was performed by Japanese researchers (see, for example, Yamanouchi et al., 1990) who emphasized the design of interfaces transitioning from pure ceramic materials to metals for durable thermal barrier coatings. This type of composite architecture is useful in high temperature applications where ceramic thermal barrier coatings are needed to protect metal matrix composites from the effects of prolonged high temperature exposure. To analyze these graded ceramic-metal interfaces, Williamson et al. (1993) utilized the finite element method and discussed the optimum design of microstructure in order to control plasticity and potentially damaging stresses. Also using finite elements, Suresh et al. (1994) analyzed the elastic-plastic response of functionally graded composites subject to thermal cycling. Aboudi et al. (1994) developed an efficient micromechanical theory that accurately predicts the response of graded composites subject to thermal gradients. 891
892
R.S. Salzar
With a previously developed multiple concentric cylinder model that accurately predicts the elastic-plastic residual stress state in metal matrix composites, Pindera et al. (1993, 1994) analyzed the effects of interface microstructure on the overall stress state in the composite by the incorporation of the three-dimensional version of the "method of cells" micromechanics solution (Aboudi, 1991). By using this multiple concentric cylinder model, Salzar and Barton (1994) developed a design program that determines the optimum interface layer properties in a composite in order to achieve a prescribed residual stress state at the end of processing. Salzar (1994) and Salzar et al. (1995a, b) extended the solution methodology used for the concentric cylinder model to include the elastic-plastic response of a fiber reinforced metal matrix composite tube. This was accomplished by incorporating the two-dimensional "method of cells" into the solutions to the governing differential equations for transversely isotropic and monoclinic axisymmetric tubes. The resulting analytical model allows for the efficient analysis of a composite tube comprised of any combination of ply thickness, fiber volume fraction, and fiber angle. This study examines the potential for strength and weight savings in metal matrix composite pressure vessels through the functional grading of properties through the thickness. Because the examples considered here involve axisymmetrically loaded structures with radially dependent deformation fields, this functional grading of properties involves varying the fiber volume fraction through the thickness. This grading is necessary to reduce certain key stresses in the composite pressure vessels and to control the amount of plasticity, while at the same time maintaining the overall stiffness properties of the cylinder. By radially redistributing the stresses occurring in the composite tube, reductions in the thickness of the pressure vessel walls are possible. If significant savings in weight can be made through functional grading of fiber volume fraction, the advantage of using composites in many additional applications will become much more apparent. ANALYTICALMODEL The analytical model consists of an arbitrary number of isotropic homogeneous and fiber reinforced heterogeneous elastic-plastic cylindrical shells perfectly bonded together (Fig. 1). In Fig. 1, a denotes the inner radius, b the outer radius, and r is the radial distance from the center of the cylinder. The model requires that the constituent materials be characterized by a bilinear stress-strain curve defined by the elastic modulus, yield stress, and the slope of the strain hardening line. Possible loading conditions include axisymmetric axial and thermal loadings, axial torque, and both internal and external pressure. Using this model, it is possible to efficiently examine a wide variety of competing tube designs through the variation of the individual ply's fiber volume fraction and fiber orientation. A displacement formulation is used to obtain the solution for the concentric tube assemblage subject to the specified ~boundary conditions. Using the assumption of generalized plane strain, and limiting loading to only axisymmetric loads, the following displacement field in the individual layers of the assemblage is obtained: IIk = CoX ,
V k -~. ~ o X r ,
w k =
wk(r)
(1)
where eo is the uniform longitudinal strain for all layers, Yo is the angle of twist per unit length of the tube, and w(r) is the radial displacement in each layer. The above displacement field yields the strains within each layer in the form: du exx
.
d x.
w(r) . to,
.
Coo
r
dw(r) ,
err
=
~
, dr
dv Y~O =
~d x =
Yor.
(2)
Substituting the inelastic constitutive equations for a transversely isotropic and monoclinic (including orthotropic) homogeneous material into the equilibrium equation, Oarr - +
Or
(Trr - - (~00
r
=
0
(3)
893
Functionally graded metal matrix composite tubes X
I
0
~',,
Fig. 1. Geometric representation of a typical composite tube assemblage.
yields the governing differential equations for the radial displacement of a transversely isotropic inelastic tube,
w 1 (Ci3 -- Ci2) ep Ci3 de p + -- --, r2 r C33 C33 dr
dZw 1 dw dr 2 + r dr
i = 1,2, 3
(4)
as well as for a monoclinic (including orthotropic) inelastic tube, d2w 1 dw dr 2 + r dr
C22 W (C21 - C13) •0 fC26 Z -2~36' -r- + \ C33 r2 ~'33 C33 /]"~Yo +-
1 (Ci3 - Ci2)(~(r) + O~i A T ) r
C33
t~3i deP(r) + C33 dr '
i=
1,2,3,6 (5)
where
eP°~ = ~eP~
~'Pr~ YPoJ
and
I~ ~EPJ
~00}
~(~rr~ \OLxO'/
=
c~2
(6)
Og3 ~6
for notational purposes. Solving the transversely isotropic equation gives
w(r) = A i r + - - +A 2
r
r Ir
q- 5 ~rk_1
1 ~r 2rr Jrk_l
(C3i + Czi) e~(r')r' dr' C33
(C3i - C2i) e~(r') dr' C33
r' '
i = 1,2, 3
(7)
894
R.S. Salzar
and for the monoclinic (including orthotropic) equation,
(012 -- 013)
A2
w(r) = AI rx + -~
+
(026 - 2036) (0i3 - 0i2) (033 _ 022) ~0 r + (-~33 --"~22) y0r2 q- (~33 ~22) ~ i A T
1 I r (Ci2 + ~.0i3) tXep(t) dt -1- ~
rk_ 1
rX f r 22 rk-i
033
(Oi2-- AOi3)eP(t)~x,
i=
1,2,3,6
(8)
033
where 2 = (C22/C33)w2 and r~_l < r _< rk. By reformulating the above equations in terms of the interfacial radial displacements, and imposing continuity of tractions and displacements as well as the appropriate boundary conditions within the concept of the "local/global stiffness matrix" formulation (Pindera, 1991), the global system of equations for an arbitrarily laminated tube takes the form of [Al{w} = [P} - [ f l ( T - To) - {g]
(9)
where A is comprised of the effective elastic properties of each layer, w is comprised of the interfacial radial displacements, axial strain, and axial twist of the tube, P involves the mechanical forcing terms, f involves the thermal terms, and g is a function of the integrals of plastic strain. These terms are defined explicitly in Salzar (1994). This system of equations is solved iteratively using Mendelson's method of successive elastic solutions (Mendelson, 1983). This involves dividing the load history into several small load increments and solving the global system of equations in an iterative fashion at each of these load steps. In addition, to incorporate the effects of the presence of fibers in the tube, the micromechanics "method of cells" is integrated into the solution methodology. The method of cells is used to compute the effective elastic properties of a fiber/matrix composite layer, as well as the inelastic strains in the composite by first calculating the inelastic strains in the individual phases, and then combining these inelastic strains into an "effective inelastic strain". These "subcell" strains, as well as the effective strains are then updated at the end of each load step in the following manner: ~P.(r)current
= t i jP( r ) p r e v i o u s +
deP(r).
(10)
Through the calculation of the effective composite strains, as well as the effective elastic and thermal properties for the composite, the effective elastic-plastic behavior of the composite can be defined as, (T = C*(~ - gP - o/* AT) (11) where # is the effective stress, C* is the effective elastic stiffness tensor, g is the total effective strain, EP is the effective plastic strain, and or* A T is the effective thermal strain. OPTIMIZATIONMODEL The optimization algorithm used in this model incorporates the "method of feasible directions" based PC software package DOT (DOT, 1993). This method, briefly outlined here, searches the n-dimensional design space along the constraint boundaries for the "global optimum". Let the purpose of the optimization be to
Minimize (or Maximize)F(X q) where X q = X q-1 + o~*Sq
(12)
where X q is the vector of design variables, Sq is the search direction in n-dimensional space, and or* is the distance traveled along the one-dimensional search direction Sq. In this method, the "usable sector" is defined as any direction (S) in the design space that improves the objective function. This can be defined mathematically as VF(Xo) • S _< 0
(13)
Functionally graded metal matrix composite tubes
895
where VF(Xo) is the gradient of the objective function at the given design point. The "feasible sector" is defined as any direction (S) in the design space which does not violate the design constraints. This can mathematically be defined as Vg~(X0)" S < 0
(14)
where Vgm(X0) is the gradient of the constraint function at the present design point. The intersection of these two sectors is called the "usable-feasible" sector. Any search direction in this sector will improve the design, and, at the same time, not violate any of the design constraints. Once a suitable search direction is found, a one-dimensional search is performed to locate the minimum (or maximum) in that direction. The newly located design point is then defined as the starting place for the next directional search. This process is repeated at each new design point until the convergence criteria is reached (in this case, the Kuhn-Tucker conditions) (Vanderplaats, 1984). RESULTS
For the first example, consider a 50.8 mm (2 in) thick Ti-24AI-11Nb tube with an internal radius of 25.4mm (1 in) pressurized internally with 206.8 MPa (30ksi). The material properties used in the analysis of this and other examples in this paper are located in Table 1. The effective stress through the thickness of this tube is shown in Fig. 2, with the elastic modulus through the tube being the constant value for Ti-24AI-11Nb (Fig. 3). For this tube, a small amount of plastic deformation is noticed along the inner radius. Subsequently, the tube is divided into 20 equally thick cylinders, and each cylinder's elastic modulus is allowed to vary between 13 790.0 MPa (2000 ksi) and 172400.0 MPa (25 000 ksi). The objective function for this problem is to minimize the difference between each layer's effective stress and the average effective stress for the tube, and takes the form: 20
Min ~
[O"eff -
_eft Oave.
(15)
i=1
This objective function causes the effective stress in the tube to become evenly distributed through the thickness. By performing this unconstrained optimization problem, an idealized elastic modulus distribution is obtained that allows for a more even distribution of effective stress through the tube (Figs 2 and 3). With this modulus distribution, the effective stress at the critical inner surface has been reduced approximately 54°70 by redistributing the stress toward the outside of the tube. This graded tube remains elastic under the same internal pressure that yielded the homogeneous tube. Of course, being able to develop materials with such specific elastic moduli is not possible with the current state of technology. However, with the use of fiber reinforced composites becoming increasingly common, manufacturers are currently able to vary the fiber volume fraction in a fairly controlled manner. So, to examine a more realistic problem, consider a SiC fiber-reinforced Ti-24Al-11Nb matrix composite tube with the same dimensions and internal pressure loading as in the previous example. The fibers are aligned along the long axis of the tube (0°). The tube with a uniform 40°70 fiber volume fraction distribution experiences initial yielding at this (i.e. 206.8 MPa (30 ksi)) pressure (Fig. 4). The transverse modulus for the uniform tube is shown in Fig. 5. Utilizing the Table 1. Material properties of fiber and matrix constituents at 75°F (Arnold et al., 1990; Aboudi, 1991)
Elastic modulus ( × 103 MPa) CTE ( × 10 -6 i n / i n / ° C ) Poisson's ratio Yield stress (MPa) Hardening slope ( × 103 MPa)
SiC
Boron
Ti-24AI- 11Nb
Aluminum
399.9 3.53 0.25 ---
413.7 4.50 0.20 ---
110.3 9.00 0.26 371.6 23.0
68.9 11.07 0.31 55.2 0.0179
R. S. Salzar
896 400
i
I-- o Ti-24AI-11Nb Tube 1
(2. 300
ej
200
>
1 O0 I.d
0
.
1.0
,
1.2
.
i
1.4
,
i
t .6
.
i
1.8
A
i
2.0
•
.
2.2
•
'
2.4
.
!
2.6
,
i
2.8
.
i
3,0
r/o Fig. 2. Effective stress distributions for an internally pressurized (206.SMPa (30ksi)) homogeneous Ti-24AI-11Nb tube and a functionally graded elastic modulus tube with all other properties equivalent to the Ti-24Al-1 lNb. The homogeneous tube plastically yields at the inner surface while the graded tube behaves elastically.
2.00E5 ~
1.0
1.2
1 .4
t .6
1.8
2.0 r/o
2,2
2.4
2.6
2,8
3.0
Fig. 3. Elastic modulus distributions for a homogeneous Ti-24Al-11Nb tube and a functionally graded elastic modulus tube with all other properties equivalent to the Ti-24AI-11Nb.
same objective function as before in an unconstrained optimization analysis, and allowing the fiber volume fraction in each subdivided layer to vary between 0070 (pure matrix) and 75°/'0, the idealized transverse modulus distribution (Fig. 5) yields the effective stress distribution shown in Fig. 4. The average fiber volume fraction for this graded tube is approximately 56°70. Note that no plasticity is occurring at the inner radius of the tube. Grading the fiber distribution has the effect of lowering the effective stress at the inner radius by redistributing the stress outward through the tube. Now that the tube is not yielding plastically for the internal pressure load, a thinner tube can now be used to accomplish the same purpose. By reducing each of the subdivided layers equally until the tube begins to plastically yield along the inner radius, a new wall thickness of 25.4 mm (1 in) is obtained. This new design for the pressure vessel demonstrates a weight saving of 60070 over the original uniform fiber volume design. Taking a 50.8 mm (2 in) thick 0 ° boron/aluminum tube with a uniform fiber volume fraction of 40°7o and an internal radius of 25.4 mm (1 in), an internal pressure of 29.0 MPa (4.2 ksi) is found to initiate plastic yielding (Fig. 6). The uniform transverse modulus for this tube is shown in Fig. 7. By utilizing the objective function used in the previous examples, a distribution of fiber volume is found (Fig. 7) that reduces the effective stress at the inner radius. Now, this graded tube can withstand an internal pressure of 34.5 MPa (5 ksi) before plastic yielding initiates (Fig. 6). The average fiber volume fraction for this graded tube is approximately 54°7o.
Functionally graded metal matrix composite tubes
400
897
o
407.FVF, Pin=206.8 MPo
o
Graded
FVF, P i n = 2 0 6 . 8
MPo
0
(3300
200 > (3
I O0
I
.0
i
i
1 .2
1 .4
i
i
i
1.6
i
i
i
i
i
2.0
1 .8
,
i
2.2
i
2.4
i
i
2.6
i
J
2.8
i
3.0
r/o Fig. 4. Effective stress distributions for a 50.8mm (2in) thick, 4007o volume fraction SiC/Ti-24AI-I 1Nb tube, a 50.8 mm (2 in) thick volume fraction graded tube, and a 25.4 mm (1 in) thick volume fraction graded tube, all internally pressurized with 206.8 MPa (30 ksi). Note that the uniform tube and the 25.4 mm (1 in) thick graded tube have begun plastic yielding while the 50.8 mm (2 in) thick graded tube remains elastic.
3.00E5 o (32.50E5 O3 ~3
2.00E5 0
::LY
1.50E5 I1)
.............
. ^ 1.00E5 . . . . . . .
o u o
407. FVF Graded FVF Graded FVF
F---
5.00E4
,
1 .0
, , , , n , i 1.2 1.4 1.6 1.8
i
i
i
2.0
i
i
2.2
i
2.4
i
i
2.6
i
2.8
i
3.0
r/a Fig. 5. Transverse modulus distributions for a 50.8 mm (2 in) thick, 4007o fiber volume fraction SiC/Ti-24AI-11Nb tube and fiber volume fraction graded tube, and a 25.4 mm (1 in) thick fiber volume fraction graded tube. For the graded tubes, there is pure matrix at the inner surface and a fiber volume fraction of 7507o at the outer surface.
I
60
I "~o 0_
50
\
o u
I
4-07. F-'-VF, P i n = 2 9 . 0 MPa Graded FVF, Pin=29.0 MPa
,._j
o3
40
oo
30
CO
> o
20
2 Ld
10
0
i
.0
i
1.2
i
i
1.4
i
i
1.6
i
i
1.8
i
i
2.0
i
I
2.2
i
i
i
2.4
2.6
i
i
2.8
i
i
5.0
r/e Fig. 6. Effective stress distribution for a 40070 uniform and functionally graded volume fraction boron/aluminum tube under 29.0MPa (4.2ksi) internal pressure, and a functionally graded volume fraction tube under 34.5MPa (5.0ksi) internal pressure. The uniform tube under 29.0 MPa (4.2 ksi) pressure and the graded tube under 34.5 MPa (5.0 ksi) pressure have begun plastic yielding while the graded tube under 29.0MPa (4.2 ksi) internal pressure is elastic throughout. COE 5/7-K
898
R . S . Salzar
o a
13.
2.00E5
0
407. F'VF
~
:
2.2
2.4
Groded~F~_ ___
= a = = = c
1.50E5
S
1.00E5 c
F5.00E4 1.0
1.2
1.4
1.6
1.8
2.0
2.6
2.8
3.0
r/o Fig. 7. Transverse modulus distribution for the 4007ouniform and functionally graded boron/ aluminum tubes. The graded tube has pure matrix along the inner radius and 75070fiber volume fraction along the outer radius.
Now that the mechanics of grading heterogeneous tubes is better understood, consider a hybrid cylindrical presssure vessel with an inner radius of 25.4 mm (1 in), made up o f two concentric 2 5 . 4 m m ( l i n ) thick layers. The innermost layer is a boron/aluminum composite and the outer layer is SiC/Ti-24Al-11Nb. First, consider a tube where the fiber volume fraction is a uniform 40°70 through the thickness. Under an internal pressure load of 68.9 M P a (10 ksi), this tube plastically yields approximately 20.3 mm (0.8in) into the boron/aluminum layer (Fig. 8). The effective transverse modulus for this tube design is shown in Fig. 9. By applying the previously used objective function that evenly distributes the effective stress throughout the tube, the transverse modulus distribution shown in Fig. 9 produces the redistributed effective stress shown in Fig. 8. The grading results in a reduction of the effective stress at the inner radius from 73.SMPa (10.7ksi) to 53.3 M P A (7.7 ksi). However, instead of selecting a fiber distribution that allows the tube to remain totally elastic, the satisfaction of the objective function required that the amount of plasticity throughout the tube be maximized. This differs from the previous examples due to the presence of a significant amount of plasticity in the initially uniform tube. This results in an increase in the effective plastic strain at the tube's inner radius from 0.001 174 to 0.002 112. In the case of the graded tube under the internal pressure load, the entire layer o f boron/aluminum has now plastically yielded, while the SiC/Ti-24Al-11Nb layer remains elastic in both cases.
80 i '~
Boron/Aluminum
SiC/Ti-24AI-11
Nb
60
oi
,50 GO 4-0 •~
30 20
o o
lO 0
, .0
i 1.2
,
407. FVF, P i n = 6 8 . 9 MPa G r o d e d FVF, P i n = 6 8 . 9 MPa I 1.4-
,
i 1.6
,
J I .8
,
i 2.0
,
i 2.2
,
i 2.4
,
i 2.6
,
i 2,8
.
i 5.0
r/a Fig. 8. Effectivestress distribution for a 40% uniform and functionallygraded volume fraction hybrid boron/aluminum : SiC/Ti-24AI-I 1Nb tube. Note the plastic yield line at 45.7 mm (1.8 in) in the uniform tube.
Functionally graded metal matrix composite tubes
o_ 2.00E5
o o
899
407. F V F Groded FVF
5
]
,
~:
iC/Ti-24AI'-I 1 Nb
p500E4
i 1.0
, 1.2
1 .¢
1.6
1.8
2.0
J 2.2
,
i 2.4
,
i 2.6
,
i 2.8
,
, 3.0
r/a Fig. 9. Transversemodulus distribution of a 40070uniform and functionallygraded fiber volume fraction hybrid boron/aluminum : SiC/Ti-24Al-1lNb tube.
DISCUSSION As demonstrated, a more efficient distribution of stress can be achieved in an internally pressurized tube through the tailoring of the composite microstructure through the thickness of the member. By varying the fiber volume fraction, it is possible to redistribute some of the stress from the critical regions of the tube to regions experiencing a lower stress state. This redistribution of stress allows for structural members that can withstand higher loads at the present size, or be resized, taking advantage of the new stress state to decrease the size of the member, thus realizing significant weight savings. For an internally pressurized tube, some observations on the ideal material property distribution are noted. Where the effective stress is highest at the inner surface of the tube, the optimization results in the lowest fiber volume fraction allowed. This means that in order to transfer some of the stress away from the inner surface to the remainder of the tube, a compliant material is required at this location. As long as the applied load is such that a significant amount of plasticity does not occur and the constituent materials strain harden, the objective function used herein will result in a lower effective stress along the inner surface of the tube and the amount of plastic strain will be reduced or eliminated. If the applied load is large enough to cause significant plastic strains in the tube, and the constituent materials do not strain harden significantly, the objective function may result in higher levels of plasticity. The hybrid tube experiences this phenomenon because of the aluminum matrix's lack of strain hardening and because there was significant plastic deformation at the beginning of the optimization procedure. The only way that the objective function can be satisfied (i.e. evenly distribute the effective stress) was to fully plasticize the boron/aluminum layer. The SiC/Ti-24Al-11Nb tubes did not behave in this way because of the high plastic hardening of the matrix and the magnitude of the loads being applied. Of course, the effectiveness of grading is governed by the properties of the constituent phases. In order to obtain the " o p t i m u m " effective stress distribution in a given tube, the properties required may be different from those obtainable from a heterogeneous mix of the phases or even pure fiber or matrix. In addition, the amount of fiber volume fraction must remain between 0°7o (pure matrix) and about 75O7o for a hexagonal packing array. Though the introduction of fibers into the matrix may not produce the exact properties needed to completely redistribute the stresses, this grading of fiber volume always appears to have some beneficial effect on the overall stress state, though the effect on plastic deformations may be detrimental. To further increase the efficiency of composite tubes, the fiber orientation must be varied in addition to the fiber volume fraction. Though this optimization problem will involve twice the number of design variables and require considerably more computer time, the resulting design will provide the ultimate efficiency and weight savings.
900
R.S. Salzar CONCLUSIONS
By functionally grading the fiber volume fraction t h r o u g h the thickness o f an internally pressurized tube, tube configurations with lower effective stress along the inner radius can be achieved, leading to lighter or stronger structures. This type o f grading can also reduce or eliminate the a m o u n t o f plastic deformations occurring in the tube, reducing the possibility o f rupture, a l t h o u g h care must be taken when dealing with materials with a low plastic yield stress. While present technology can partially accomplish this a m o u n t o f grading, improvements in m a n u f a c t u r i n g must be m a d e in order to create fully functional grading and to improve cost efficiency. Composites were developed for their high strength-to-weight ratio. Fiber-reinforced composites allow tailoring the material to a specific load and geometry by varying fiber volume fraction and fiber orientation. By using design algorithms like the one employed in this investigation, a c o m p o s i t e ' s specifications can be designed in such a way as to improve the already efficient strength-to-weight ratio. A l t h o u g h the calculations for the elastic-plastic response o f fiber-reinforced composites are complex, the marriage o f such analytical solutions with optimization algorithms results in some o f the most efficient design tools available. REFERENCES Aboudi, J. (1991). Mechanics of Composite Materials: A Unified Micromechanical Approach. Elsevier, Amsterdam. Aboudi, J., Pindera, M.-J. and Arnold, S. M. (1994). Elastic response of metal matrix composites with tailored microstructures to thermal gradients. Int. J. Solids Struct. 31, 1393-1428. DOT User's Manual, Version 4.00. VMA Engineering, Vanderplaats, Miura & Asssociates, Inc., Goleta, CA, 1993. Mendelson, A. (1983). Plasticity: Theory and Applications. Robert E. Krieger Publishing Company, Malabar, FL. Pindera, M.-J. (1991). Local/global stiffness matrix formulation for composite materials and structures. Comp. Engng 1, 69-80. Pindera, M.-J., Freed, A. D. and Arnold, S. M. (1993). Effects of fiber and interfacial layer morphologies on the thermoplastic response of metal matrix composites. Int. J. Solids Struct. 30, 1213-1238. Pindera, M.-J., Williams, T. O. and Arnold, S. M. (1994). Thermoplastic response of metal-matrix composites with homogenized and functionally graded interfaces. Comp. Engng 4, 129-145. Salzar, R. S. (1994). Optimization of layered metal matrix composite cylinders. Ph.D. Dissertation, University of Virginia, Charlottesville, VA. Salzar, R. S. and Barton, F. W. (1994). Residual stress optimization in metal-matrix composites using discretely graded interfaces. Comp. Engng 4, 115-128. Salzar, R. S., Pindera, M.-J. and Barton, F. W. (1995a). Elastoplastic analysis of layered metal matrix composite cylinders, Part I: Theory, submitted. Salzar, R. S., Pindera, M.-J. and Barton, F. W. (1995b). Elastoplastic analysis of layered metal matrix composite cylinders, Part II: Applications, submitted. Suresh, S., Giannakopoulos, A. E. and Olsson, M. (1994). Elastoploastic analysis of thermal cycling: layered materials with sharp interfaces. J. Mech. Phys. Solids 42, 979-1018. Vanderplaats, G. N. (1984). Numerical Optimization Techniquesfor Engineering Design: With Applications. McGraw-Hill, New York. Williamson, R. L., Rabin, B. H. and Drake, J. T. (1993). Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part I. Model description and geometrical effects. J. Appl. Phys. 74, 1310-1320. Yamanouchi, M., Koizumi, M., Hirai, T. and Shiota, I. (1990). Proc. 1st Int. Syrup. Functionally Gradient Materials, Sendal, Japan.
Pergamon
0961-9526(95) 00023-2 FUNCTIONALLY GRADED METAL MATRIX COMPOSITE TUBES Robert S. Salzar Department of Civil Engineering & Applied Mechanics, University of Virginia, Charlottesville, VA 22903, U.S.A.
(Received 15 December 1994; final version accepted 17 January 1995) A b s t r a c t - - C u r r e n t advanced manufacturing techniques allow the continuous variation of fiber or inclusion volume fraction in metal matrix composites. With this technology, it is now possible to tailor a composite to the expected loads by using the constituent materials to redistribute the stress and strain states through the material. Lighter and more structurally efficient components will be obtained through this grading process. The focus of this paper is the evaluation of the effects of material property and fiber volume grading on the overall mechanical response of metal matrix composite tubes subjected to mechanical loadings. This is accomplished through the development of a fully elastic-plastic axisymmetric generalized plane strain tube model. This analytical model incorporates a micromechanics algorithm in order to determine the elastic-plastic response of a heterogeneous fiber-reinforced composite cylinder. An arbitrary number of heterogeneous concentric cylinders can be included in the model, each with independent material properties. The inelastic analysis is performed through the method of successive elastic solutions. The optimization algorithm used in conjunction with this solution procedure utilizes the method of feasible directions and accepts any combination of design variables, constraints, and objective functions. As an example of the effectiveness of this grading, it is possible to vary the fiber volume fraction in an SiC/Ti-24Al-11Nb tube in such a way that the effective stress at the critical inner surface of an internally pressurized tube is reduced. For a 50.8 m m (2in) thick tube with an internal radius of 25.4 m m (1 in) and an internal pressure of 206.8 M P a (30 ksi), a uniform 40°7o fiber volume fraction distribution results in a tube that begins to plastically yield at the inner radius. By grading the fiber volume fraction, the tube now behaves elastically under the same pressure loading, allowing the tube to have a wall thickness of 25.4 m m (1 in) before plastic yielding begins. This grading results in a 6007o weight saving.
INTRODUCTION
Composites, whether organic, metallic, or ceramic based, have only recently seen use in a wide range of applications. These uses range from everyday items such as sporting goods and automobiles to advanced aerospace structures. With composites being used successfully in an increasing number of applications, advances in the manufacture of composite structures have enabled material designers to examine additional applications for these materials. Since most composite materials are comprised of either continuous fibers or inclusions embedded in a matrix, they can be tailored to a specific load by varying the volume fraction of the inclusion material. These "functionally graded materials" can be applied to many problems currently facing the composite design community. These problems include the stress concentrations resulting from free edge effects as well as from the discontinuity at the fiber/matrix interface. Significant work in the area of functionally graded materials was performed by Japanese researchers (see, for example, Yamanouchi et al., 1990) who emphasized the design of interfaces transitioning from pure ceramic materials to metals for durable thermal barrier coatings. This type of composite architecture is useful in high temperature applications where ceramic thermal barrier coatings are needed to protect metal matrix composites from the effects of prolonged high temperature exposure. To analyze these graded ceramic-metal interfaces, Williamson et al. (1993) utilized the finite element method and discussed the optimum design of microstructure in order to control plasticity and potentially damaging stresses. Also using finite elements, Suresh et al. (1994) analyzed the elastic-plastic response of functionally graded composites subject to thermal cycling. Aboudi et al. (1994) developed an efficient micromechanical theory that accurately predicts the response of graded composites subject to thermal gradients. 891
892
R.S. Salzar
With a previously developed multiple concentric cylinder model that accurately predicts the elastic-plastic residual stress state in metal matrix composites, Pindera et al. (1993, 1994) analyzed the effects of interface microstructure on the overall stress state in the composite by the incorporation of the three-dimensional version of the "method of cells" micromechanics solution (Aboudi, 1991). By using this multiple concentric cylinder model, Salzar and Barton (1994) developed a design program that determines the optimum interface layer properties in a composite in order to achieve a prescribed residual stress state at the end of processing. Salzar (1994) and Salzar et al. (1995a, b) extended the solution methodology used for the concentric cylinder model to include the elastic-plastic response of a fiber reinforced metal matrix composite tube. This was accomplished by incorporating the two-dimensional "method of cells" into the solutions to the governing differential equations for transversely isotropic and monoclinic axisymmetric tubes. The resulting analytical model allows for the efficient analysis of a composite tube comprised of any combination of ply thickness, fiber volume fraction, and fiber angle. This study examines the potential for strength and weight savings in metal matrix composite pressure vessels through the functional grading of properties through the thickness. Because the examples considered here involve axisymmetrically loaded structures with radially dependent deformation fields, this functional grading of properties involves varying the fiber volume fraction through the thickness. This grading is necessary to reduce certain key stresses in the composite pressure vessels and to control the amount of plasticity, while at the same time maintaining the overall stiffness properties of the cylinder. By radially redistributing the stresses occurring in the composite tube, reductions in the thickness of the pressure vessel walls are possible. If significant savings in weight can be made through functional grading of fiber volume fraction, the advantage of using composites in many additional applications will become much more apparent. ANALYTICALMODEL The analytical model consists of an arbitrary number of isotropic homogeneous and fiber reinforced heterogeneous elastic-plastic cylindrical shells perfectly bonded together (Fig. 1). In Fig. 1, a denotes the inner radius, b the outer radius, and r is the radial distance from the center of the cylinder. The model requires that the constituent materials be characterized by a bilinear stress-strain curve defined by the elastic modulus, yield stress, and the slope of the strain hardening line. Possible loading conditions include axisymmetric axial and thermal loadings, axial torque, and both internal and external pressure. Using this model, it is possible to efficiently examine a wide variety of competing tube designs through the variation of the individual ply's fiber volume fraction and fiber orientation. A displacement formulation is used to obtain the solution for the concentric tube assemblage subject to the specified ~boundary conditions. Using the assumption of generalized plane strain, and limiting loading to only axisymmetric loads, the following displacement field in the individual layers of the assemblage is obtained: IIk = CoX ,
V k -~. ~ o X r ,
w k =
wk(r)
(1)
where eo is the uniform longitudinal strain for all layers, Yo is the angle of twist per unit length of the tube, and w(r) is the radial displacement in each layer. The above displacement field yields the strains within each layer in the form: du exx
.
d x.
w(r) . to,
.
Coo
r
dw(r) ,
err
=
~
, dr
dv Y~O =
~d x =
Yor.
(2)
Substituting the inelastic constitutive equations for a transversely isotropic and monoclinic (including orthotropic) homogeneous material into the equilibrium equation, Oarr - +
Or
(Trr - - (~00
r
=
0
(3)
893
Functionally graded metal matrix composite tubes X
I
0
~',,
Fig. 1. Geometric representation of a typical composite tube assemblage.
yields the governing differential equations for the radial displacement of a transversely isotropic inelastic tube,
w 1 (Ci3 -- Ci2) ep Ci3 de p + -- --, r2 r C33 C33 dr
dZw 1 dw dr 2 + r dr
i = 1,2, 3
(4)
as well as for a monoclinic (including orthotropic) inelastic tube, d2w 1 dw dr 2 + r dr
C22 W (C21 - C13) •0 fC26 Z -2~36' -r- + \ C33 r2 ~'33 C33 /]"~Yo +-
1 (Ci3 - Ci2)(~(r) + O~i A T ) r
C33
t~3i deP(r) + C33 dr '
i=
1,2,3,6 (5)
where
eP°~ = ~eP~
~'Pr~ YPoJ
and
I~ ~EPJ
~00}
~(~rr~ \OLxO'/
=
c~2
(6)
Og3 ~6
for notational purposes. Solving the transversely isotropic equation gives
w(r) = A i r + - - +A 2
r
r Ir
q- 5 ~rk_1
1 ~r 2rr Jrk_l
(C3i + Czi) e~(r')r' dr' C33
(C3i - C2i) e~(r') dr' C33
r' '
i = 1,2, 3
(7)
894
R.S. Salzar
and for the monoclinic (including orthotropic) equation,
(012 -- 013)
A2
w(r) = AI rx + -~
+
(026 - 2036) (0i3 - 0i2) (033 _ 022) ~0 r + (-~33 --"~22) y0r2 q- (~33 ~22) ~ i A T
1 I r (Ci2 + ~.0i3) tXep(t) dt -1- ~
rk_ 1
rX f r 22 rk-i
033
(Oi2-- AOi3)eP(t)~x,
i=
1,2,3,6
(8)
033
where 2 = (C22/C33)w2 and r~_l < r _< rk. By reformulating the above equations in terms of the interfacial radial displacements, and imposing continuity of tractions and displacements as well as the appropriate boundary conditions within the concept of the "local/global stiffness matrix" formulation (Pindera, 1991), the global system of equations for an arbitrarily laminated tube takes the form of [Al{w} = [P} - [ f l ( T - To) - {g]
(9)
where A is comprised of the effective elastic properties of each layer, w is comprised of the interfacial radial displacements, axial strain, and axial twist of the tube, P involves the mechanical forcing terms, f involves the thermal terms, and g is a function of the integrals of plastic strain. These terms are defined explicitly in Salzar (1994). This system of equations is solved iteratively using Mendelson's method of successive elastic solutions (Mendelson, 1983). This involves dividing the load history into several small load increments and solving the global system of equations in an iterative fashion at each of these load steps. In addition, to incorporate the effects of the presence of fibers in the tube, the micromechanics "method of cells" is integrated into the solution methodology. The method of cells is used to compute the effective elastic properties of a fiber/matrix composite layer, as well as the inelastic strains in the composite by first calculating the inelastic strains in the individual phases, and then combining these inelastic strains into an "effective inelastic strain". These "subcell" strains, as well as the effective strains are then updated at the end of each load step in the following manner: ~P.(r)current
= t i jP( r ) p r e v i o u s +
deP(r).
(10)
Through the calculation of the effective composite strains, as well as the effective elastic and thermal properties for the composite, the effective elastic-plastic behavior of the composite can be defined as, (T = C*(~ - gP - o/* AT) (11) where # is the effective stress, C* is the effective elastic stiffness tensor, g is the total effective strain, EP is the effective plastic strain, and or* A T is the effective thermal strain. OPTIMIZATIONMODEL The optimization algorithm used in this model incorporates the "method of feasible directions" based PC software package DOT (DOT, 1993). This method, briefly outlined here, searches the n-dimensional design space along the constraint boundaries for the "global optimum". Let the purpose of the optimization be to
Minimize (or Maximize)F(X q) where X q = X q-1 + o~*Sq
(12)
where X q is the vector of design variables, Sq is the search direction in n-dimensional space, and or* is the distance traveled along the one-dimensional search direction Sq. In this method, the "usable sector" is defined as any direction (S) in the design space that improves the objective function. This can be defined mathematically as VF(Xo) • S _< 0
(13)
Functionally graded metal matrix composite tubes
895
where VF(Xo) is the gradient of the objective function at the given design point. The "feasible sector" is defined as any direction (S) in the design space which does not violate the design constraints. This can mathematically be defined as Vg~(X0)" S < 0
(14)
where Vgm(X0) is the gradient of the constraint function at the present design point. The intersection of these two sectors is called the "usable-feasible" sector. Any search direction in this sector will improve the design, and, at the same time, not violate any of the design constraints. Once a suitable search direction is found, a one-dimensional search is performed to locate the minimum (or maximum) in that direction. The newly located design point is then defined as the starting place for the next directional search. This process is repeated at each new design point until the convergence criteria is reached (in this case, the Kuhn-Tucker conditions) (Vanderplaats, 1984). RESULTS
For the first example, consider a 50.8 mm (2 in) thick Ti-24AI-11Nb tube with an internal radius of 25.4mm (1 in) pressurized internally with 206.8 MPa (30ksi). The material properties used in the analysis of this and other examples in this paper are located in Table 1. The effective stress through the thickness of this tube is shown in Fig. 2, with the elastic modulus through the tube being the constant value for Ti-24AI-11Nb (Fig. 3). For this tube, a small amount of plastic deformation is noticed along the inner radius. Subsequently, the tube is divided into 20 equally thick cylinders, and each cylinder's elastic modulus is allowed to vary between 13 790.0 MPa (2000 ksi) and 172400.0 MPa (25 000 ksi). The objective function for this problem is to minimize the difference between each layer's effective stress and the average effective stress for the tube, and takes the form: 20
Min ~
[O"eff -
_eft Oave.
(15)
i=1
This objective function causes the effective stress in the tube to become evenly distributed through the thickness. By performing this unconstrained optimization problem, an idealized elastic modulus distribution is obtained that allows for a more even distribution of effective stress through the tube (Figs 2 and 3). With this modulus distribution, the effective stress at the critical inner surface has been reduced approximately 54°70 by redistributing the stress toward the outside of the tube. This graded tube remains elastic under the same internal pressure that yielded the homogeneous tube. Of course, being able to develop materials with such specific elastic moduli is not possible with the current state of technology. However, with the use of fiber reinforced composites becoming increasingly common, manufacturers are currently able to vary the fiber volume fraction in a fairly controlled manner. So, to examine a more realistic problem, consider a SiC fiber-reinforced Ti-24Al-11Nb matrix composite tube with the same dimensions and internal pressure loading as in the previous example. The fibers are aligned along the long axis of the tube (0°). The tube with a uniform 40°70 fiber volume fraction distribution experiences initial yielding at this (i.e. 206.8 MPa (30 ksi)) pressure (Fig. 4). The transverse modulus for the uniform tube is shown in Fig. 5. Utilizing the Table 1. Material properties of fiber and matrix constituents at 75°F (Arnold et al., 1990; Aboudi, 1991)
Elastic modulus ( × 103 MPa) CTE ( × 10 -6 i n / i n / ° C ) Poisson's ratio Yield stress (MPa) Hardening slope ( × 103 MPa)
SiC
Boron
Ti-24AI- 11Nb
Aluminum
399.9 3.53 0.25 ---
413.7 4.50 0.20 ---
110.3 9.00 0.26 371.6 23.0
68.9 11.07 0.31 55.2 0.0179
R. S. Salzar
896 400
i
I-- o Ti-24AI-11Nb Tube 1
(2. 300
ej
200
>
1 O0 I.d
0
.
1.0
,
1.2
.
i
1.4
,
i
t .6
.
i
1.8
A
i
2.0
•
.
2.2
•
'
2.4
.
!
2.6
,
i
2.8
.
i
3,0
r/o Fig. 2. Effective stress distributions for an internally pressurized (206.SMPa (30ksi)) homogeneous Ti-24AI-11Nb tube and a functionally graded elastic modulus tube with all other properties equivalent to the Ti-24Al-1 lNb. The homogeneous tube plastically yields at the inner surface while the graded tube behaves elastically.
2.00E5 ~
1.0
1.2
1 .4
t .6
1.8
2.0 r/o
2,2
2.4
2.6
2,8
3.0
Fig. 3. Elastic modulus distributions for a homogeneous Ti-24Al-11Nb tube and a functionally graded elastic modulus tube with all other properties equivalent to the Ti-24AI-11Nb.
same objective function as before in an unconstrained optimization analysis, and allowing the fiber volume fraction in each subdivided layer to vary between 0070 (pure matrix) and 75°/'0, the idealized transverse modulus distribution (Fig. 5) yields the effective stress distribution shown in Fig. 4. The average fiber volume fraction for this graded tube is approximately 56°70. Note that no plasticity is occurring at the inner radius of the tube. Grading the fiber distribution has the effect of lowering the effective stress at the inner radius by redistributing the stress outward through the tube. Now that the tube is not yielding plastically for the internal pressure load, a thinner tube can now be used to accomplish the same purpose. By reducing each of the subdivided layers equally until the tube begins to plastically yield along the inner radius, a new wall thickness of 25.4 mm (1 in) is obtained. This new design for the pressure vessel demonstrates a weight saving of 60070 over the original uniform fiber volume design. Taking a 50.8 mm (2 in) thick 0 ° boron/aluminum tube with a uniform fiber volume fraction of 40°7o and an internal radius of 25.4 mm (1 in), an internal pressure of 29.0 MPa (4.2 ksi) is found to initiate plastic yielding (Fig. 6). The uniform transverse modulus for this tube is shown in Fig. 7. By utilizing the objective function used in the previous examples, a distribution of fiber volume is found (Fig. 7) that reduces the effective stress at the inner radius. Now, this graded tube can withstand an internal pressure of 34.5 MPa (5 ksi) before plastic yielding initiates (Fig. 6). The average fiber volume fraction for this graded tube is approximately 54°7o.
Functionally graded metal matrix composite tubes
400
897
o
407.FVF, Pin=206.8 MPo
o
Graded
FVF, P i n = 2 0 6 . 8
MPo
0
(3300
200 > (3
I O0
I
.0
i
i
1 .2
1 .4
i
i
i
1.6
i
i
i
i
i
2.0
1 .8
,
i
2.2
i
2.4
i
i
2.6
i
J
2.8
i
3.0
r/o Fig. 4. Effective stress distributions for a 50.8mm (2in) thick, 4007o volume fraction SiC/Ti-24AI-I 1Nb tube, a 50.8 mm (2 in) thick volume fraction graded tube, and a 25.4 mm (1 in) thick volume fraction graded tube, all internally pressurized with 206.8 MPa (30 ksi). Note that the uniform tube and the 25.4 mm (1 in) thick graded tube have begun plastic yielding while the 50.8 mm (2 in) thick graded tube remains elastic.
3.00E5 o (32.50E5 O3 ~3
2.00E5 0
::LY
1.50E5 I1)
.............
. ^ 1.00E5 . . . . . . .
o u o
407. FVF Graded FVF Graded FVF
F---
5.00E4
,
1 .0
, , , , n , i 1.2 1.4 1.6 1.8
i
i
i
2.0
i
i
2.2
i
2.4
i
i
2.6
i
2.8
i
3.0
r/a Fig. 5. Transverse modulus distributions for a 50.8 mm (2 in) thick, 4007o fiber volume fraction SiC/Ti-24AI-11Nb tube and fiber volume fraction graded tube, and a 25.4 mm (1 in) thick fiber volume fraction graded tube. For the graded tubes, there is pure matrix at the inner surface and a fiber volume fraction of 7507o at the outer surface.
I
60
I "~o 0_
50
\
o u
I
4-07. F-'-VF, P i n = 2 9 . 0 MPa Graded FVF, Pin=29.0 MPa
,._j
o3
40
oo
30
CO
> o
20
2 Ld
10
0
i
.0
i
1.2
i
i
1.4
i
i
1.6
i
i
1.8
i
i
2.0
i
I
2.2
i
i
i
2.4
2.6
i
i
2.8
i
i
5.0
r/e Fig. 6. Effective stress distribution for a 40070 uniform and functionally graded volume fraction boron/aluminum tube under 29.0MPa (4.2ksi) internal pressure, and a functionally graded volume fraction tube under 34.5MPa (5.0ksi) internal pressure. The uniform tube under 29.0 MPa (4.2 ksi) pressure and the graded tube under 34.5 MPa (5.0 ksi) pressure have begun plastic yielding while the graded tube under 29.0MPa (4.2 ksi) internal pressure is elastic throughout. COE 5/7-K
898
R . S . Salzar
o a
13.
2.00E5
0
407. F'VF
~
:
2.2
2.4
Groded~F~_ ___
= a = = = c
1.50E5
S
1.00E5 c
F5.00E4 1.0
1.2
1.4
1.6
1.8
2.0
2.6
2.8
3.0
r/o Fig. 7. Transverse modulus distribution for the 4007ouniform and functionally graded boron/ aluminum tubes. The graded tube has pure matrix along the inner radius and 75070fiber volume fraction along the outer radius.
Now that the mechanics of grading heterogeneous tubes is better understood, consider a hybrid cylindrical presssure vessel with an inner radius of 25.4 mm (1 in), made up o f two concentric 2 5 . 4 m m ( l i n ) thick layers. The innermost layer is a boron/aluminum composite and the outer layer is SiC/Ti-24Al-11Nb. First, consider a tube where the fiber volume fraction is a uniform 40°70 through the thickness. Under an internal pressure load of 68.9 M P a (10 ksi), this tube plastically yields approximately 20.3 mm (0.8in) into the boron/aluminum layer (Fig. 8). The effective transverse modulus for this tube design is shown in Fig. 9. By applying the previously used objective function that evenly distributes the effective stress throughout the tube, the transverse modulus distribution shown in Fig. 9 produces the redistributed effective stress shown in Fig. 8. The grading results in a reduction of the effective stress at the inner radius from 73.SMPa (10.7ksi) to 53.3 M P A (7.7 ksi). However, instead of selecting a fiber distribution that allows the tube to remain totally elastic, the satisfaction of the objective function required that the amount of plasticity throughout the tube be maximized. This differs from the previous examples due to the presence of a significant amount of plasticity in the initially uniform tube. This results in an increase in the effective plastic strain at the tube's inner radius from 0.001 174 to 0.002 112. In the case of the graded tube under the internal pressure load, the entire layer o f boron/aluminum has now plastically yielded, while the SiC/Ti-24Al-11Nb layer remains elastic in both cases.
80 i '~
Boron/Aluminum
SiC/Ti-24AI-11
Nb
60
oi
,50 GO 4-0 •~
30 20
o o
lO 0
, .0
i 1.2
,
407. FVF, P i n = 6 8 . 9 MPa G r o d e d FVF, P i n = 6 8 . 9 MPa I 1.4-
,
i 1.6
,
J I .8
,
i 2.0
,
i 2.2
,
i 2.4
,
i 2.6
,
i 2,8
.
i 5.0
r/a Fig. 8. Effectivestress distribution for a 40% uniform and functionallygraded volume fraction hybrid boron/aluminum : SiC/Ti-24AI-I 1Nb tube. Note the plastic yield line at 45.7 mm (1.8 in) in the uniform tube.
Functionally graded metal matrix composite tubes
o_ 2.00E5
o o
899
407. F V F Groded FVF
5
]
,
~:
iC/Ti-24AI'-I 1 Nb
p500E4
i 1.0
, 1.2
1 .¢
1.6
1.8
2.0
J 2.2
,
i 2.4
,
i 2.6
,
i 2.8
,
, 3.0
r/a Fig. 9. Transversemodulus distribution of a 40070uniform and functionallygraded fiber volume fraction hybrid boron/aluminum : SiC/Ti-24Al-1lNb tube.
DISCUSSION As demonstrated, a more efficient distribution of stress can be achieved in an internally pressurized tube through the tailoring of the composite microstructure through the thickness of the member. By varying the fiber volume fraction, it is possible to redistribute some of the stress from the critical regions of the tube to regions experiencing a lower stress state. This redistribution of stress allows for structural members that can withstand higher loads at the present size, or be resized, taking advantage of the new stress state to decrease the size of the member, thus realizing significant weight savings. For an internally pressurized tube, some observations on the ideal material property distribution are noted. Where the effective stress is highest at the inner surface of the tube, the optimization results in the lowest fiber volume fraction allowed. This means that in order to transfer some of the stress away from the inner surface to the remainder of the tube, a compliant material is required at this location. As long as the applied load is such that a significant amount of plasticity does not occur and the constituent materials strain harden, the objective function used herein will result in a lower effective stress along the inner surface of the tube and the amount of plastic strain will be reduced or eliminated. If the applied load is large enough to cause significant plastic strains in the tube, and the constituent materials do not strain harden significantly, the objective function may result in higher levels of plasticity. The hybrid tube experiences this phenomenon because of the aluminum matrix's lack of strain hardening and because there was significant plastic deformation at the beginning of the optimization procedure. The only way that the objective function can be satisfied (i.e. evenly distribute the effective stress) was to fully plasticize the boron/aluminum layer. The SiC/Ti-24Al-11Nb tubes did not behave in this way because of the high plastic hardening of the matrix and the magnitude of the loads being applied. Of course, the effectiveness of grading is governed by the properties of the constituent phases. In order to obtain the " o p t i m u m " effective stress distribution in a given tube, the properties required may be different from those obtainable from a heterogeneous mix of the phases or even pure fiber or matrix. In addition, the amount of fiber volume fraction must remain between 0°7o (pure matrix) and about 75O7o for a hexagonal packing array. Though the introduction of fibers into the matrix may not produce the exact properties needed to completely redistribute the stresses, this grading of fiber volume always appears to have some beneficial effect on the overall stress state, though the effect on plastic deformations may be detrimental. To further increase the efficiency of composite tubes, the fiber orientation must be varied in addition to the fiber volume fraction. Though this optimization problem will involve twice the number of design variables and require considerably more computer time, the resulting design will provide the ultimate efficiency and weight savings.
900
R.S. Salzar CONCLUSIONS
By functionally grading the fiber volume fraction t h r o u g h the thickness o f an internally pressurized tube, tube configurations with lower effective stress along the inner radius can be achieved, leading to lighter or stronger structures. This type o f grading can also reduce or eliminate the a m o u n t o f plastic deformations occurring in the tube, reducing the possibility o f rupture, a l t h o u g h care must be taken when dealing with materials with a low plastic yield stress. While present technology can partially accomplish this a m o u n t o f grading, improvements in m a n u f a c t u r i n g must be m a d e in order to create fully functional grading and to improve cost efficiency. Composites were developed for their high strength-to-weight ratio. Fiber-reinforced composites allow tailoring the material to a specific load and geometry by varying fiber volume fraction and fiber orientation. By using design algorithms like the one employed in this investigation, a c o m p o s i t e ' s specifications can be designed in such a way as to improve the already efficient strength-to-weight ratio. A l t h o u g h the calculations for the elastic-plastic response o f fiber-reinforced composites are complex, the marriage o f such analytical solutions with optimization algorithms results in some o f the most efficient design tools available. REFERENCES Aboudi, J. (1991). Mechanics of Composite Materials: A Unified Micromechanical Approach. Elsevier, Amsterdam. Aboudi, J., Pindera, M.-J. and Arnold, S. M. (1994). Elastic response of metal matrix composites with tailored microstructures to thermal gradients. Int. J. Solids Struct. 31, 1393-1428. DOT User's Manual, Version 4.00. VMA Engineering, Vanderplaats, Miura & Asssociates, Inc., Goleta, CA, 1993. Mendelson, A. (1983). Plasticity: Theory and Applications. Robert E. Krieger Publishing Company, Malabar, FL. Pindera, M.-J. (1991). Local/global stiffness matrix formulation for composite materials and structures. Comp. Engng 1, 69-80. Pindera, M.-J., Freed, A. D. and Arnold, S. M. (1993). Effects of fiber and interfacial layer morphologies on the thermoplastic response of metal matrix composites. Int. J. Solids Struct. 30, 1213-1238. Pindera, M.-J., Williams, T. O. and Arnold, S. M. (1994). Thermoplastic response of metal-matrix composites with homogenized and functionally graded interfaces. Comp. Engng 4, 129-145. Salzar, R. S. (1994). Optimization of layered metal matrix composite cylinders. Ph.D. Dissertation, University of Virginia, Charlottesville, VA. Salzar, R. S. and Barton, F. W. (1994). Residual stress optimization in metal-matrix composites using discretely graded interfaces. Comp. Engng 4, 115-128. Salzar, R. S., Pindera, M.-J. and Barton, F. W. (1995a). Elastoplastic analysis of layered metal matrix composite cylinders, Part I: Theory, submitted. Salzar, R. S., Pindera, M.-J. and Barton, F. W. (1995b). Elastoplastic analysis of layered metal matrix composite cylinders, Part II: Applications, submitted. Suresh, S., Giannakopoulos, A. E. and Olsson, M. (1994). Elastoploastic analysis of thermal cycling: layered materials with sharp interfaces. J. Mech. Phys. Solids 42, 979-1018. Vanderplaats, G. N. (1984). Numerical Optimization Techniquesfor Engineering Design: With Applications. McGraw-Hill, New York. Williamson, R. L., Rabin, B. H. and Drake, J. T. (1993). Finite element analysis of thermal residual stresses at graded ceramic-metal interfaces. Part I. Model description and geometrical effects. J. Appl. Phys. 74, 1310-1320. Yamanouchi, M., Koizumi, M., Hirai, T. and Shiota, I. (1990). Proc. 1st Int. Syrup. Functionally Gradient Materials, Sendal, Japan.