Composites Engineering, Vol. 4, No. 1, pp. 107-114. 1994.
Printed in Great Britain.
0961-9526194 S6.00+ .W
Q 1993 Pergamon Press Ltd
FUNCTIONALLY GRADED MATERIALS FOR
YIELD SUPPRESSION AT STRESSCONCENTRATORS
‘Department
J. M. DUVA,+ B. R. OLSON+ and H. N. G. WADLEY~
of Applied Mathematics and *Department of Materials Science and Engineering,
University of Virginia, Charlottesville, VA 22903, U.S.A.
(Received 16 June 1993; final version accepted 6 July 1993)
Abstract-Vapor-deposition
techniques are emerging as an economical and flexible way of
fabricating multiphase materials with spatially varying architectures. The aim of this work is to
design architectures that exploit the concept of functional grading. We consider two basic
problems; the first is the computation of the response of an infinite plate with a circular hole to
a remote symmetric radial load, and the second is the computation of the response of an infinite
matrix to a uniform temperature change when the matrix is perfectly bonded to an infinitely long,
rigid, circularly cylindrical fiber with a zero coefficient of thermal expansion. For each configuration the variation of the material properties with position is determined so that the body yields
everywhere at the same load, and we compute how this variation can be effected through variation
of the distribution of two phases. We show that for each configuration the load required to initiate
yielding can be doubled if the material is engineered properly.
1. INTRODUCTION
Economical and flexible vapor deposition processes are emerging for fabricating multiphase materials (Hsiung, 1993). The potential flexibility of this and other related processes
allows the material designer to specify both the volume fractions of the constituent phases
and (to some extent) the microstructure of the material as a function of position within a
component. The practical consequence of this flexibility is that the material properties can
be made to vary within limits imposed by the properties of the constituent phases and the
relationships between the properties and the volume fractions of the phases. Our aim here
is to begin exploring how this emerging capacity to fabricate engineered inhomogeneous
materials can be exploited.
Because yielding under monotonic loads and fatigue damage and failure under cyclic
loading are ubiquitous engineering problems, we focus on the computation of the elastic
limit-the
load causing the onset of plastic deformation-for
a pair of commonly
encountered configurations. In each case we seek to determine a variation in the volume
fraction of the phases in a two-phase material that results in yielding everywhere in the
body at the same load (“simultaneous yielding”). It will be shown that for both cases (but
not for all material combinations) the loading required to initiate plastic deformation in
a functionally graded material can be doubled through the proper design of the material.
Figure 1 is a schematic representation of a jet vapor deposition process. An inert gas,
such as helium, is forced through a nozzle by a pressure difference created with mechanical
vacuum pumps. A vapor source within the nozzle injects atoms of the deposit into the
flowing inert gas stream, where they are transported to the deposition site or target.
Although the atoms in the flow may be highly reactive, the flow velocity is sufficiently
large and the vapor concentration sufficiently low so that three-body collisions (and thus
reactions) do not occur. The target can be translated or rotated under multiple nozzles
emitting different vapors, and it can be heated or cooled. By controlling these process
features the volume fractions (and spacings) of the different materials can be spatially
varied. Laminated structures with a length scale of lo-500nm have been fabricated
(Hsiung, 1993). Particulate structures can also be fabricated and the size of the particulate
phase(s) controlled by heat treatment. It is our intent here to show how this flexibility in
processing might be exploited to improve the performance of components containing
stress concentrators.
Qxr:t-n
107
J. M. DUVA et al.
108
JET VAPOR DEPOSITION
Ref.
Jet PromsaCorporatbn.
New
Haven,CT,
Patent No. 4.7W.O@2 Nov. 29, IS@
Fig. 1. In the jet vapor deposition process represented schematically above, material from the
source is carried to the target or substrate by a high-velocity stream of inert gas. The deposition
site and the deposition rate are easily controlled, and multiple sources and nozzles can be used
(after Hsiung et al., 1993).
Given the shape of a body, the properties of the material of which the body is made,
and the system of loads and constraints acting on the body, a traditional mechanical
analysis requires the equations of equilibrium and compatibility and the constitutive
relations and delivers, in general, a system of partial differential equations that, when
solved, give the displacement, strain and stress distributions in the body. In what follows
we will specify the shape of the body and the loads (except for the magnitude) along with
a simultaneous yielding constraint, and use the governing equations to compute the
displacement, strain and stress distributions in the body as well as the spatial distribution
of phases, and thus the spatial distribution of material properties. The simultaneous
yielding constraint is enforced by requiring that the von Mises yield criterion
(a1 - a2)2 + (a2 - a#
+ (63 - a1)2 = 2Y2
(1)
be satisfied at every point in the body. In (1) the ok, k = 1,2,3, are the principle stresses,
and Y is the yield strength. Note that all of these quantities will be, in general, functions
of position. We illustrate the approach by analyzing the following two problems.
2. CONFIGURATION
1
Consider an infinite, thin plate with a circular hole of unit radius, subjected to a
remote, symmetric, in-plane tension as shown in Fig. 2. As the hoop stress concentration
Fabrication of multiphase materials
109
Fig. 2. Configuration 1 is a thin, layered two-phase composite plate with a circular hold of unit
radius subjected to remote in-plane symmetric tension. A schematic of the plate cross-section is
shown to illustrate a radial variation in volume fractions of the constituents.
for a homogeneous plate with a circular hole is 2, yielding occurs at the edge of the hole
at r = 1 when the remote load reaches one half the yield strength of the material. We
assume that plane-stress conditions hold everywhere, even in the vicinity of the hole. The
plate is composed of layers of two materials distributed symmetrically about the midplane
whose thicknesses, and thus the volume fraction c of material 1, vary only with radial
position in a manner to be determined. Figure 2 also shows a section of the plate
illustrating schematically these variations. Budiansky et al. (1992) have examined the
concept of “modulus tailoring” to reduce the magnitude of the stress concentrations in
the individual layers of a laminated plate with a hole. Here we seek the distribution of
phases that will result in simultaneous yielding, that is, we are interested in “strength
tailoring”.
We note as well that Mansfield (1950) first investigated neutralizing the
stress concentration in the vicinity of a hole by tailoring the thickness of a homogeneous
plate.
The governing equations are the yield condition (l), the equilibrium equation,
a; + $, - a& = 0,
the compatibility
(2)
equation,
&, = (WJ,
and the isotropic linear elastic constitutive
EE, = ur - vue
(3)
equations
and
In the above, the prime denotes differentiation
EC@= ug - va,.
(4)
with respect to r. Note that the in-plane
110
J. M.
DUVA
et al.
u
c = c(r)
Fig. 3. Configuration 2 is an infinite isotropic two-phase composite body perfectly bonded to a
single infinitely long rigid circularly cylindrical fiber. Plain-strain conditions hold. The assembly
is subjected to a uniform temperature change.
elastic modulus E, the yield strength Y and Poisson’s ratio v are all functions of position
in general.
We assume for simplicity that both the in-plane elastic modulus E and the yield stress
Y follow a simple rule of mixtures (see the Appendix). The rule of mixtures approximation for the in-plane elastic modulus is reasonably accurate (see Christensen, 1979), but
such an approximation for the yield stress is very crude; note that for the present we make
no use of the observed square-root relation between yield strength and layer thickness for
microlaminates. Poisson’s ratio is taken to be constant, again for convenience.
The governing equations can be nondimensionalized by introducing the dimensionless
variables
a=a,
r,
and
e,Ez
El
’
where the subscripts 1 and 2 refer to the two phases present in the plate. Eliminating the
strains and the circumferential stress from the five governing equations results in a
coupled system of two ordinary nonlinear differential equations
0’ = f&,
0, c, Y, 4
(6)
and
c’ = fc(r, 0,6’, c, y, e)
(7)
which are presented explicitly in the Appendix. We solve this system with the initial
conditions o( 1) = 0, as the hole is traction free, and c( 1) = c,, , where c,, must be between
0 and 1. Given c,,, whether a physically reasonable solution exists will depend on the
values of y and e. The magnitude of the load is determined by the remote value of the yield
strength and the simultaneous yield condition.
We take co = 0 and calculate as follows. Given a value of e, a value of y is chosen.
A fourth-order Runge-Kutta algorithm is used to integrate the coupled system out to a
large radius. If the value of y is too small, c will grow to exceed 1 for some r. If this
occurs, a larger value of y is used and the calculation is repeated. In this way the boundary
Fabrication of multiphase materials
Modulus
111
Ratio e
Fig. 4. The simultaneous yielding map on the ey-plane shows, at a glance, the material combinations that can be used to create a plate with a hole that will satisfy the simultaneous yield constraint.
A value of 0.25 for Poisson’s ratio was used in constructing the map.
in the ey-plane is located which separates material combinations for which simultaneous
yielding is possible from those combinations for which it is not. For e and y values on the
boundary, c approached 1 from below as r grew unbounded. Such a “map” is shown in
Fig. 4, along with regions representing several different material combinations. The table
below shows the material properties used.
Material
E
(GPa)
Y @Pa)
a (lo-6 oc-1)
Al
AW%
cu
Nb
Ni
0.03-0.6
2.0-5.0
110
105
206
70-80
300-400
0.07-0.36
0.17
0.10-0.20
23.6
7.1-9.6
16.5
7.2
13.0
Figure 5 shows the radial variation of c, the volume fraction of material 1 (aluminum),
Q, dimensionless radial stress, and E = E/El and Y = T/Y, defined in the Appendix. The
radial stress far from the hole approaches the yield stress, consistent with the simultaneous yielding requirement. As mentioned above, the initiation of yielding for the
nonhomogeneous plate at this load represents a doubling of the load required to initiate
yielding in a homogeneous plate of material 1. The nonhomogeneous plate is virtually
homogeneous outside of one hole radius from the hole boundary. The hoop strain at the
surface of the hole will be Y,/E, for the nonhomogeneous plate and Y,/E, for the
homogeneous plate; for the aluminum-aluminum
oxide combination used to construct
Fig. 5, the hoop strain is 8/5 times larger in the nonhomogeneous plate than in the
homogeneous plate.
3. CONFIGURATION
2
Consider an infinite matrix perfectly bonded to a single infinitely long rigid circular
cylinder, as shown in Fig. 3. The matrix consists of two phases. To simplify the analysis
we assume the matrix is deposited on the fiber in such a way that it is macroscopically
isotropic (not layered) and its properties vary only as a function of radial position, and we
assume the coefficient of thermal expansion of the fiber is zero. The assembly is subjected
J. M. DWA et al.
112
Radius r
Fig. 5. For configuration 1 with an aluminum-aluminum
oxide material combination, the yield
strength, modulus, radial stress and volume fraction of ahuninurn are plotted against the radial
position r.
to a uniform temperature change. For a homogeneous matrix, yielding will occur at
the fiber/matrix interface when the product of temperature change, the modulus and the
coefficient of thermal expansion reaches one half the yield strength. We calculate the
variation in the distribution of the phases in the matrix which results in simultaneous
yielding throughout the matrix.
The governing equations are the yield condition (l), the equilibrium equation (2), the
compatibility equation (3), and the isotropic linear thermo-elastic constitutive relations
EC, = a, + (r TE
and
EEL = Og + CXTE
(8)
where CYis the coefficient of thermal expansion and Poisson’s ratio has been taken to be
zero for convenience. Note that in general E, Y and Q are all functions of the radius r.
We assume that the elastic modulus E, the yield strength Y, and the coefficient of thermal
expansion Q! all obey the rule of mixtures. The governing equations can be nondimensionalized by introducing the dimensionless variables of (5) along with
where again the subscripts 1 and 2 refer to the two phases present in the matrix. Eliminating
the strains and the circumferential stress from the five governing equations again results
in a coupled system of two ordinary nonlinear differential equations
0’ = go@, Q, c, a, Y, 4
(10)
c’ = g,@, 0, Q’, c, a, Y, 4
(11)
and
which are presented explicitly in the Appendix.
We solve this system with the initial conditions se(l) = 0, consistent with the perfect
bonding of the matrix to the rigid fiber, and c(l) = co. As with configuration 1, we
integrate using a fourth-order Runge-Kutta algorithm. The magnitude of the load is
determined by the remote material properties and the simultaneous yielding condition.
One subtlety that arises in this context that does not arise for problem 1 involves (10). This
expression comes from application of the quadratic formula to the yield condition. It is
easy to check that if the yield condition is to be satisfied everywhere, the square-root
function must switch branches for some r > 1. Because the radial stress and its derivatives
are expected to be continuous, the switch must occur where the root vanishes.
Figure 6 shows the variation with radial position r of the volume fraction of material
1, the yield strength, the modulus, the coefficient of thermal expansion, and the radial
stress for configuration 2 with an aluminum-aluminum
oxide matrix. The yield strength
Fabrication of multiphase materials
113
Fig. 6. For configuration 2 with an aluminum-aluminum
oxide material combination, the yield
strength, modulus, coefficient of thermal expansion, radial stress and volume fraction of
aluminum are plotted against the radial position r.
F, the modulus ,??and the coefficient of thermal expansion d have been normalized with
respect to the material 1 (aluminum) values. In this case the full flexibility of the material
system is not required to satisfy the condition of simultaneous yielding; the volume
fraction of aluminum at r = 1 is about 0.69. As in the case of mechanical loading, the
body is virtually homogeneous beyond a fiber radius of the interface. The temperature
change required to initiate yielding for the inhomogeneous material is twice that for
homogeneous aluminum.
4. SUMMARY
We have shown, in the context of two simple model calculations, that the flexibility
inherent in some vapor deposition processes can be used to double the load (both thermal
and mechanical) required to initiate plastic deformation. Our aim was not to recommend
a particular material design for either configuration, nor to advocate the use of our
algorithm in general. Rather, we aimed at demonstrating, through analysis and as directly
as possible, the potential of this emerging new technology for functionally graded
materials synthesis.
Ackrzowledgements-This
work was supported by the Advanced Research Projects Agency (Program Manager,
W. Barker) and the National Aeronautics and Space Administration
Headquarters (Program Manager,
R. Hayduk) through NASA grant NAGW 1692.
REFERENCES
Budiansky, B. J., Hutchinson, J. W. and Evans, A. G. (1992). On neutral holes in tailored, layered sheets.
Harvard University Report MECH-198, Harvard University, Cambridge, MA.
Christensen, R. M. (1979). Mechanics
of Composites.
John Wiley, New York.
Hsiung, L. M., Zang, J. Z., McIntyre, D. C., Golz, J. W., Halpem, B. C., Schmidt, J. J. and Wadley, H. N. G.
(1993). Structure and properties of jet vapour deposited Al-A&O, nano-scale laminates. Scripta Metall.
in press.
Mansfield, E. H. (1950). Neutral holes in a plane sheet: reinforced holes which are elastically equivalent to the
uncut sheet. R. A. E. Report Structures 90, Reports and Memoranda No. 2815.
APPENDIX
The rule of mixtures gives
Y = CY, + (1 - c)y, = Y,(c + (1 - c)y) = r, F,
E = cE,
+
(1 - c)& = E&c + (1 - c)y) = E,E,
(A.1)
64.2)
and
a = Cal + (1 - c)cQ = a,(c + (1 - c)y) = cr,fi
64.3)
for the yield strength Y, the modulus E and the coefficient of thermal expansion CY.For compactness these
formulae will be used in the equations below.
114
J. M. DWA
et
al.
For configuration 1, the equilibrium equation can be used to solve for u) in terms of u, and 0;. The result
can be substituted into the von Mises yield criterion and the quadratic formula can be used to solve for u:. On
nondimensionalizing, one obtains
Q’ = -u + d7-22
2r
= f&,
0, c, Y, 6.
(A.4)
The positive sign is selected for the root because we expect the radial stress to increase monotonically from its
value of zero at r = 1.
Using the stress-strain relations to write the compatibility equation in terms of stresses, and using the
equilibrium equation to eliminate ug, one can obtain an expression for c’ that involves u;; this term can be
eliminated by differentiation of the yield condition and substitution of the resulting expression for a;. The
equation for c’ is
The system of differential equations is derived in a similar way for configuration
u, = -u - GE f d4P2 - 3(u + GE)2
= g,(r, 0, c, a, Y, 4
2r
2:
(‘4.6)
and
3rd2
c’ = 2
[
_
Y(1 - v) - (G(l - e) - E(1 - a))@ + til? + ru’)
- (1 - e)(ru’ + u + &f)(ru’
B
= g,(r, u, u’, c, a, Y, e).
+ &7 + $C!?) -I
I
(A.7)
In (A.6) the negative root must be used from r = 1 out to the radius at which the root vanishes, then the positive
root must be used, as discussed in Section 3. Note that the dimensionless load parameter Q, E, T/ Y, must be unity
to satisfy the condition of simultaneous yielding far from the fiber. Thus it does not appear explicitly in (A.6)
and (A.7).