S. Schmauder and M. Meyer: Dundurs' Parameters and Elastic Constants
© Carl Hanser Verlag, München
1992
Siegfried Schmauder and Markus Meyer
(Max-Planck-Institutfür
Metallforschung,
Institutfür
Werkstoffwissenschaft,
Seestraße 92, D-7000 Stuttgart 1)
Correlation Between Dundurs'
Parameters and Elastic Constants
Dedicated to Professor Dr. phil., tekn. dr. h. c., Dr. techno E. h. Hellmut F. Fischmeister
on the occasion of his 65th birthday
The correlation between the four elastic constants of two
adjoint isotropic materials and Dundurs ' parameters a and ß is
elucidated. Whereas ais a function of elastic moduli alone, ß
depends on Poisson's ratios in addition to the elastic moduli.
The linear dependency of ß on ais explicitly derived in this
work. From these a- ß relationships it can be determined whether or not the assumption of negligible ß values is applicable
für a given material combination. These figures are powerful
means for the design of composites and provide information
about which elastic properties of the components are required
für obtaining certain Dundurs ' parameters. The method is
demonstrated for two metal/ceramic systems.
Korrelation zwischen Dundurs-Parametern
elastischen Konstanten
son's ratio, and the index Ci = 1,2) refers to material "I" and
"2", respectively. Although the Dundurs' parameters are
applicable for a variety of bonded materials and joints, we
shall refer to these materials as simply a "bimaterial" for convenience.
Bogy [2] has shown that the upper and lower limits of the
constant a of any bimaterial are
- 1.0::;;; a::;;; 1.0
and that the constant ß is bounded for the plane strain and
plane stress cases, respectively
% - 0.25 ::;;;ß ::;;;%
und
In dieser Arbeit wird der Zusammenhang zwischen den vier
elastischen Konstanten zweier aneinandergrenzender Materialien und den dazugehörigen Dundurs-Parametem a und ß
beleuchtet. Während a nur eine Funktion der elastischen
Moduln ist, war schon immer bekannt, daß ß zusätzlich von
den Poissonzahlen abhängt. In dieser Arbeit wird zum ersten
Mal die lineare Abhängigkeit zwischen a und ß explizit
gezeigt. Graphische Darstellungen zeigen, ob für eine gegebene Materialkombination die Annahme vemachlässigbarer
ß-Werte zutrifft oder nicht. Die Bilder stellen ein wichtiges
Hilfsmittel zum Design von Verbundwerkstoffen dar und verschaffen Informationen darüber, welche elastischen Eigenschaften der Komponenten zur Erzielung bestimmter Dundurs-Parameter erforderlich sind. Die Methode wird an zwei
Metall/Keramik-Systemen
demonstriert.
(3)
~a-
+ 0.25
0.125 ::;;;ß ::;;;~
(plane strain)
a + 0.125
(plane stress)
(4)
(5)
One useful property of the Dundurs ' parameters is that only
the sign of a and ß is reversed when materials 1 and 2 are
exchanged; thus, it is sufficient to examine a single case (e.g.
a;;;; 0) to fully understand the elastic properties of the bimaterial.
Schmauder [3] has shown that the expression for parameter
a can be simplified in terms of the Young's moduli of the
phases
E!-Et
a=W
2
+ E+I
(6)
In Eq. (6), Et is given by
1.0
1 Introduction
0.8
Since their discovery, Dundurs ' parameters a and ß have been
proven useful in characterizing the elastic properties of bimaterials and other material joints [1 to 13]. Using the original
notation ofDundurs [1] andBogy [2], one can write a and ß as
a = k(1\:1
k(1\:1 + 1) - (1\:2 +
+ 1) + (1\:2 +
ß=
k(1\:1 k(1\:J
+
1)
1)
1)- (1\:2 - 1)
+ (1\:2 + 1)
1)
0.6
CI. 0.4
(1)
0.2
(2)
0.2
In Eqs. (1) and (2), k is the ratio ofthe shear moduli ,uz/,ul, 1\:i is
Muskhelishwili's constant [14] and is equal to (3-4vi) für
plane strain and (3-v ;)/(1+v i) forplane stress, where Viis Pois524
0.8
1.0
Fig. 1. Dependence of a on ratio of Young's moduli.
Z.Metallkd.83
(1992) 7
S. Schmauder and M. Meyer: Dundurs' Parameters and Elastic Constants
ß
0.3
1_
0.2
••
...
0"
0.1
.",.
0.0
""..1~:99.11l1....
•
1.0
>Oe
Q
0.8
- 0.1
0.6
o
o
6
•
•
Ceramic ,Mo,W. B,C
Cu/ Ceramic ,Mo.W.B,C
Ni / Ceramic . Mo.W,B.C
Ti,Zr,Hf,V,Nb /Ceramic.Mo.W
Ta.Cr.Mo.W /Ceramic
• '-44 / Plastics
/Others
-0.2
0.2
Fig. 2.
a-
ß diagram for bimaterials
of isotropie eomponents
(plane strain)
(plane stress)
Al /
(plane strain) [9].
to reduce the number of parameters [10]. It is interesting to
note that when VI = V2 = 1/3, Eq. (11) is exact. Equation (11) is
also valid for many composites with components which have
andEjare Young's moduli ofthe components (i = 1,2). Thus, a unequal Poisson's ratios.
Hutchinson [11] has argued that Dundurs ' second parameter
is easily obtained using the ratio of the plane moduli Et and
may be ignored in many practical cases since ß is small
E2 (Fig. I).
Because of the importance of Young's moduli, ais of pri- according to Eqs. (9) and (11). Nevertheless, except for the
case of identical Poisson's ratios, the correlation between ß
mary interest in studies on the e1astic response of composites.
and elastic constants of involved phases remained unc1ear.
For typical material combinations, the elastic properties are
The relationship between ß and the elastic constants (Eq. (2))
often within a range given by
provides the basis for resolving these uncertainties and for
correlating
aand ßin a predictive sense for arbitrary Poisson's
025 < Et < 4
(8)
. = E:fj=
ratios. The role of ß is discussed in detail in the following
section.
Thus, the value of Dundurs' parameter ais limited to
EI
Er = {EJ(1-
vt)
lai;;;; 0.6
(7)
(9)
This statement is supported
Suga [4, 9]. A few examples
shown in Fig. 2. In addition,
noted that many technically
follow
by a systematic investigation by
of different material joints are
Meyer and Schmauder [7] have
relevant material combinations
2 Dundurs' Second Parameter ß
The relationship in Eqs. (1) and (2) between the four elastic
constants and ß can be rewritten as (e. g. in [12] for plane strain
deformation)
I
~-OI<ß<~
4
. =
=4+
01.
(plane strain)
(10)
in agreement with Eq. (4). Equation (10) is sometimes simplified to
a
ß=-4
(plane strain)
Z. Metallkd. 83 (1992) 7
(11)
ß=
ß
2:
E:fj
I-I -
2vI
VI
_
Et
I1-
2V2
V2
=!2 E:fj (1- VI)
- Et (1Et+E2
..
(plane stram)
r.+ . r.+
V2)
(plane stress)
(12)
(13)
We discuss the cases of identical and unequal Poisson' s ratios
in sections 2.1 and 2.2, respectively.
525
S. Schmauder and M. Meyer: Dundurs' Parameters and Elastic Constants
0.5
0.5
0.4
0.4
0.3
0.3
i3
fYex
0.2
0.2
0.1
0.1
0.2
0.1
0.3
0.2
0.4
0.5
0.4
0.6
0.8
1.0
ex
v
b)
a)
Fig. 3. Variation of ß/a für identical Poisson's ratios v = v I = V2.
Fig. 4. Linear a - ß dependence für identical Poisson's ratios v = v I = V2 under (a) plane
strain, and (b) plane stress deformation.
0.0
2.1 Identical Poisson's Ratios
For identical Poisson's ratios
(6), (12) and (13) that
1 (1 - 2v)
ß =:2
/1
••\
(v
ß=!!.3
= v 1= V2), it follows from Eqs.
.
a
(plane stram)
(14)
(plane stress)
It is also interesting to note that for the case of identical
Poisson's ratios v = VI = V2 ;;;; 0.2 and for positive a values
which fulfill Eq. (9), Eq. (14) leads to a narrow range of
possible ß values (Fig. 4) given by
0;:;:;ß;:;:; 0.25
(plane stress)
(16)
(17)
(15)
2.2 Unequal Poisson's Ratios
Equations (14) and (15) are evaluated in Fig. 3 and Eq. (11) is
derived from Eq. (14) by setting Poisson's ratio to v = 1/3.
However, one mayaIso
observe that many other linear
relationships between a and ß may be derived by changing v.
Under plane stress assumptions a relationship similar to Eq.
(11) holds for v = 1/3
526
For the more realistic case of unequal Poisson's ratios, the
following expressions can be derived from Eqs. (6), (12) and
(13)
Z. Metallkd. 83 (1992) 7
S. Schmauder and M. Meyer: Dundurs' Parameters and Elastic Constants
0.4 [-;ane
st;~~
0.4
ß
steelJceramic
0.0
plane strain
0.3
0.3
0.1 -'
I
I: AlJceramic
n
~
0.2
0.1
WC/Co!
.~
:~{.C/Co
0.0
-0.1
-0.1
0.025
o.ooo~
0.1
0.2
0.3
0.4
O.S
-0.2
-0.2
0.0
0.2
0.8
0.6
0.4
1.0
0.0
0.2
a.
0.6
0.4
0.8
1.0
a.
~
~
Fig. 7. Distribution
of typical metal/ceramic
![(
ß=4
VI
1- 2V2)
V2
1 - 2vI
VI
11- 2vI
+ 1a + ( 1-
systems with different metals in the
_
1 - 2V2)]
V2
1-
(plane strain)
(18)
(plane stress)
(19)
When both materials possess the same Poisson's ratio Eqs.
(18) and (19) can be simplified to Eqs. (14) and (15). The existence of a linear relationship between a and ß was first noted by
Bogy [13]. However, its form was not given explicitly until
now. It can be seen that the range of possible ßvalues is limited
to a narrow regime when a and Poisson' s ratio of one material
are fixed. The form of Eqs. (18) and (19) provides one with a
simple method of predicting Dundurs ' second parameter ß for
technically important composites with Poisson's ratios limited as follows: 0.2 ;;;; V], V2 ;;;; 0.4; this is shown in Fig. 5. In
combination with Eq. (6), ß can be graphically presented as a
function of elastic constants of involved materials (Fig. 6).
3 Application to Metal/Ceramic Systems
Steel and Al are frequently used as the metal part in metall
ceramic systems (Figs. 7a and 7b). From the discussion ofthe
previous sections resulting Dundurs' parameters and the scattering of the data in the a - ß-diagram are now weIl understood.
As shown in Figs. 7a and 7b, a large scatter is present for
smaIl ß values in systems with small mismatch of Young's
moduli; this is basically due to the mismatch in Poisson's
ratios. Conversely, one may expect larger ß values and less
scatter in systems where I al ;;;;0.5 (Fig. 7b).
Generally, small ß values can be achieved in typical metall
ceramic systems with I a I ;;;; 0.2 when ceramics with smaIl
Poisson's ratios are used.
4 Concluding Remarks
For any given material combination a is obtained from Eq.
(6). The simple graphical presentations of the linear interrelation between a and ß provide a powerful tool to determine ß in
a simple manner. These graphs mayaiso be applied to design
Z. Metallkd. 83 (1992) 7
a-
ß diagram (plane strain): (a) steel, (b) A!.
new composites with pre-defined elastic material characteristics. Thus, only certain Dundurs' parameters a and ß can be
achieved for composites where the elastic constants of one of
the two adjoining materials are already fixed. Figure 5
together with Eqs. (18) and (19) provide the means to determine the range of admissible Young's moduli and Poisson's
ratios of materials to be joined. The trends discussed in this
paper for a and ß are typical for many metallceramic systems.
This work was supported by the German Research Foundation
(DFG project EI 53/13-1); the support is gratefully acknowledged.
Thanks are due to Dr. Steve Rozeveld for valuable suggestions in
improving the manuscript.
Literature
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3. Schmauder, S.: CF! (Ceramic Forum International) 2 (1987)
101.
4. Suga, T.: Ph. D. Thesis, University of Stuttgart (1983).
5. Schmauder, S.: in "Metal-Ceramic Interfaces", Acta Scripta
Metallurgica Proceedings, Se ries 4, M. Rühle, A. G. Evans, M.
F. Ashby, J. P. Hirth (eds.), Pergamon Press, Oxford, England
(1990) 413.
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Eng. Mater. Struct. (1992) in press.
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917.
10. Schmauder, S.: CFI (Ceramic Forum International) 3 (1988) 33.
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Metallurgica Proceedings" Series 4, M. Rühle, A. G. Evans, M.
F. Ashby, J. P. Hirth (eds.), Pergamon Press, Oxford, England
(1990) 295.
12. Jensen, H. M.: Eng. Fract. Mech. (1992) submitted.
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(Reeeived Oetober 22, 1991)
527
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