The Yield Criterion of
Laminated Media
P. C. CHOU, B. M. McNAMEE
AND
D. K. CHOU
Drexel
University
. 19104
Philadelphia Pa
,
(Received December 27, 1972)
ABSTRACT
This study examines the yield criteria for anisotropic laminated media.
It will be shown that for laminated media with isotropic layers, the critenon of Tsai and Wu is a direct extension of Von Mises’. Also presented
here is a set of equations governing the relative positions of the yield
ellipses. Furthermore, a general expression for the yield condition of a
laminated medium composed of generally anisotropic layers is obtained.
INTRODUCTION
criterion for a laminated medium with anisothe independent yield strength constants of the
in terms of the strength constants and elastic properties of the individual
In formulating the yield criterion, the laminated medium is transformed into
yield
Tcomposite
tropic layers by characterizing
HIS STUDY PROPOSES a
layers.
an equivalent medium which is homogeneous but anisotropic.
In studying the yielding of homogeneous, anisotropic media, Hill [ 1] proposed a
criterion that is quadratic in form. In his criterion there are only six independent
constants, the principal strength constants. His interaction terms are not independent ; they are in terms of three of the six principal strength constants. Furthermore, his criterion is limited to specially orthotropic materials with plastic incompressibility, and with equal strength in tension and compression. It is defined only
in the coordinate of material symmetry. Thus, Hill’s criterion can not be used for
materials with general anisotropy.
More recently, Tsai and Wu [2] proposed a general strength theory for anisotropic materials based on the concept of characterizing the independent strength
constants as tensor quantities as suggested by Gol’denblat and Kopnov [3]. These
strength components, as they are called, satisfy the invariant requirements of
J. COMPOSITE MATERIALS, Vol. 7 (January 1973),
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22
coordinate transformation similar to the compliance matrix. The number of independent strength tensors is reduced for materials exhibiting symmetry. It is a very
convenient form which is easily applicable to practical problems. Nevertheless, the
general form of Tsai and Wu’s criterion is an assumption made arbitrarily.
In this paper, we will show that a yield criterion for a laminated medium with
isotropic layers will have the same form as the one proposed by Tsai and Wu when
the Von Mises’ yield criterion is applied to the individual layers. The formulation of
this yield criterion is based on the assumptions made according to a smearing
technique proposed by Chou, Carleone and Hsu [4]. The Von Mises’ yield criterion
is applied to the individual layers to determine the incipient yielding conditions of
the composite.
Further, we will show that when the laminated composite consists of anisotropic
layers and each layer yields according to the criterions of Tsai and Wu, the yield
locus of the composite is an envelope formed by many intersecting ellipses. Each
ellipse represents the yield locus of one layer under a combined state of equivalent
stresses.
All tensors are represented in contracted notation and superscripts designate the
quantities associated with the corresponding layer. Quantities without superscripts
are associated with the equivalent medium.
YIELD CRITERION FOR A LAMINATED MEDIUM
WITH ISOTROPIC LAYERS
Consider a laminate made up of n layers of different isotropic materials rigidly
bonded together and subjected to a general state of loads. The materials in the
layers are assumed to be homogeneous and have equal strength in tension and
compression. The laminate is then smeared into an equivalent medium by the
lamination theory proposed by Chou, Carleone, and Hsu [4]. In this case, the
equivalent medium is transversely isotropic with the 1-2 plane as the isotropic plane.
According to the lamination theory of Reference [4], the stress-strain relation
of the equivalent medium can be expressed as
where Sq are functions of the
the lamina stress components
following form [5] :
properties of the layers. Further, it can be shown that
related by the equivalent stress components in the
are
where
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and for isotropic
layers
determine which layer will first reach the
limiting state of stress established by the yield criteria for each layer. Here, the Von
Mises yield criterion will be used to govern the yielding of each layer,
Thus, Equation (2) enables
When
us to
expressed in tensor form, for the kth layer, Equation (3) becomes
where
and
Xk
is the
yield stress in uniaxial
tension.
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Substituting Equation (2) into Equation (4) and simplifying yields,
where
Equation (5) is the yield criterion for the equivalent medium caused by yielding
layer k. Therefore, the yield condition of a laminate is, in general, represented by n equations like Equation (5) and each equation is a hyper-ellipsoid in
in the
the six-dimensional equivalent stress space. These ellipsoids can intersect one
another or can enclose one another. For the former case, a yield envelope is
produced. For the latter case, the smallest ellipsoid governs the yielding of the
composite under the combined equivalent stress states.
in terms of the
Due to the lengthy expression, no explicit formula for
strength and elastic properties of the individual layers will be presented. However,
m Equation (5) becomes:
by permutating the indices,
F~
F§
It can be immediately seen that Equation (6) has the same form as the one
derived by Tsai and Wu [2] for a transversely isotropic material and both have five
independent constants.
An explicit yield equation in terms of the yield strength and elastic properties of
the individual layers will be given for a repeated bilaminar medium composed of
two isotropic materials subjected to a state of plane stress in the 1-2 plane as
shown m Figure 1. Thus, the previous three-dimensional formulation is reduced to a
two-dimensional stress problem. If the loads are applied in the 1-2 plane, and a3
0, Ft in Equation (5) can be expressed as follows:
a4
Qs
=
=
=
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+7gure
l. 7 1
where Ek and ph
respectively. The
pit al olement of a repeated bllammar medium
under a
plane load
modulus and the Poisson’s ratio of the kth layer
Vk
symbol
represents the volume fraction of layer k. Also, it can
are
the
Young’s
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which agrees with the stability condition
be shown that
surface
failure
as discussed in [2].
closed
a
for
required
Therefore, we arrive at the conclusion that the yield criterion presented here for
the laminated medium is a direct extension of the Von Mises criterion; it has the
same requirements as the criterion proposed by Tsai and Wu. In other words, Tsai
and Wu’s criterion can be viewed as a direct extension of the Von Mises criterion
for a laminated medium.
An example will now be presented in which we will study the yield surfaces of
three layered plates with isotropic layers subjected to loads applied in the 1-2
plane. Each plate is composed of two materials that are arranged in a repeated
fashion (Figure 1 ). The first plate consists of structural steel and annealed
aluminum. The second is made up of structural steel and aluminum, but the
aluminum has been heat treated. The third is layered with high strength steel and
malleable cast iron. All the physical properties of the laminas of the three bilaminar
media are listed in Table 1. In all three cases, the laminas are of equal thickness.
Equations of this plane stress problem have been coded on a computer, and the
results are plotted in the two-dimensional equivalent stress space in Figures 2, 3,
and 4.
(F11 )2 - (Fkl 2 )2 > 0
Table 1.
Physical Properties of Laminas in Example
1
For Case 1, the ellipse for yielding in the aluminum layer is enclosed by the
ellipse for the steel as shown in Figure 2. Thus, aluminum always yields first in this
case. However, if the aluminum is first heat treated and attains a higher yield
strength, the ellipse for the steel is enclosed by the ellipse for the aluminum, see
Figure 3. Thus, the steel always yields first for Case 2. For Case 3, where the cast
iron has both higher Young’s modulus and yield strength than aluminum Figure 4
shows that the ellipses intersect one another to form a failure envelope for the
composite. Thus, the materials with a lower yield stress in a laminated composite
do not necessarily reach the established yield criteria first under combined stress
states.
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Figure 2. Yield loci plotted on
Figure 3. Yield loci plotted on
a, - a2
plane, steel and aluminum.
a, - az plane, steel and aluminum.
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Figure 4. Yield loci plotted on a, - (Tplane, steel and cast iron.
Equation (5) which is the equation of an ellipse, when written in terms
principal state of stress and for a plane stress loading condition reduces to
where k
these
=
1, 2. It
ellipses have
can
the
be shown that the
of the
major axis, ak, and the minor axis bk, of
following form
where
Therefore, the criteria for the ellipses to intersect one another is either a’ > a2
and bl < b2, or a’ < a2 and b’ > b2. From Equation (9) the factors Ak and Bk
determine the ellipses will enclose one another or intersect one another, and the
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following conclusions
can
be drawn:
the
ellipses will
the
ellipses will intersect
enclose
one
another.
one
another.
Both Ak and Bk are independent of the thickness of the lamina. Thus,
the thickness of the laminae does not determine the relative position of ellipses.
3)
LAMINATES WITH ANISOTROPIC LAYERS
The formulation for the laminates with isotropic layers discussed above can be
generalized to laminates with anisotropic layers. The only change to be made is that
the Von Mises yield criterion for each layer can be replaced by an anisotropic yield
criterion, e.g., Tsai and Wu’s strength criterion. Similar to Equation (2), the lamina
stresses and equivalent stresses are related by the following equation, [5, 6] .
where
and
Tsai and Wu’s criterion for the individual
layers are of the following form,
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where
strength tensors of the second and fourth rank respectively.
By substituting Equation (10) mto Equation (11), the yield equation for
laminate with anisotropic layers is obtained as follows:
Fkand F~,.
i
are
’I
the
where
and
Therefore, Equation (12) is the yield condition for a laminate with anisotropic
layers caused by yielding in the kth layer. In addition, Equation (12) takes into
consideration materials in which the magnitude of the tensile yield stress is
different from the compression yield stress.
In the following example, the material under study is S-Glass/xP-251 Resin [7].
Its properties are shown in Table 2. We will first investigate a laminate with a
cross-ply of [0°/90°] [ [8] and then one with an angle-ply [45°/-45°] s. Also, we
will study each lammate for two assumed possible cases. One case is that the yield
strengths of the laminas are equal in tension and compression; the other case is that
they are not equal. The results of these four examples are plotted in the equivalent
stress space, as shown m Figure 5 to Figure 9. Since experimental data for the
interaction term
are not available for the S-Glass/xP-251Resin, we choose the
theoretical upper bound value [2] for the present analysis. Furthermore, we will
consider the loads are applied in the 1-2 plane.
Because the mteraction term
is the theoretical upper bound value, the
results of this example show that the yield loci become open curves (Figures 5, 6,
and 8), which have been indicated by Collins and Crane [9] . Moreover, the angle of
inclination of the elhpses, 6 is defined by the following form (Figure 8)
Fi
Fi
It is interesting to note that the direction of 0 depends not only on the sign
but also on the sign of the difference of
Thus this explains why the
and
of P121
Fl
F22.
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Table 2. Physical Properties of the Fibrous
Composite in Example 2
Figure 5. Yield loci plotted on
a, - a plane,
S-Glass/xP-251 resin, [00/900Js
0° yield locus rotate in a clockwise (negative) direction and the 90° rotates in
counter-clockwise (positive) direction in Figures 5, 6 and 8 for the chosen Fi 2 .
a
CONCLUDING REMARKS
By applying the Von Mises Yield Criterion to each layer of a laminated medium
isotropic layers, we are able to arrive at a yield equation for the composite
with
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Figure
6. Yield loci plotted
Figure
7. Yield loci
on
plotted on
a, - a2 plane,
°1 - °2
S-Glass/xP-251 resin, {O° /900s.
plane, S Glass/xR251 resin,
/45°%45°Js .
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Figure 8. Yield loci plotted on
°1 - °2
plane, S Glass/xR251 resin,
[0° /900J S.
Figure 9. Yield loci plotted on
a, - a,
plane, S-Glass/xP-251 resin,
/45&dquo;/45’ / s’
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I
same form as the one proposed by Tsai and Wu for anisotropic elastic
it
bodies.Thus,
may be considered as direct extension of the Von Mises criterion.
Furthermore, we have obtained a set of equations governing the relative position of
which has the
yield
layers.
the
loci. These may be used
as
design guidelines for a laminate with isotropic
For the laminates with anisotropic layers, we have shown that for the examples
studied all the ellipses intersect one another, but we are unable to conclude whether
there could be other cases where the ellipses enclose one another.
The yield surface of a laminated composite is the inscribed envelope of many
ellipses. If this envelope can be approximated by another ellipse, the yield problem
of laminates can be greatly simplified.
It should be noted that the presented approach can be applied to a threedimensional state of stress. It is not limited to the conventional &dquo;plate bending and
stretching&dquo; type of loading. In applying our approach to plane stress problems, we
have neglected the effect of bending-stretching coupling. For certain arrangement
of the layers in a laminate, such as symmetric stacking or thin repeated layers, the
coupling effect disappears or becomes negligible.
ACKNOWLEDGMENT
This research was partially supported by Army Ballistic Research Laboratory
DAADOS-70-C-0175 and a NASA grant for the University Development Program at
Drexel University.
REFERENCES
1. R. Hill, The Mathematical Theory ,
of Plasticity Oxford, 1956.
2. S. W. Tsai and E. M. Wu, "A General Theory of Strength for Anisotropic Materials," J
.
, Vol. 5 (1971), p. 58.
Composite Materials
3. I. I. Gol’denblat and V. A. Kopnov, "Strength of Glass Reinforced Plastics in the Complex
Stress State," Mekhanika Polimerov
, Vol. 1, No. 2 (1965), p. 70.
4. P. C. Chou, J. Carleone and C. M. Hsu, "Elastic Constants of Layered Media," J
. Composite
, Vol. 6 (1972), p. 80.
Materials
5. P. C. Chou, B. M. McNamee and D. K. Chou, "The Yield Criterion of Laminated Media,"
will appear in the Proceedings of the Sixth St. Louis Symposium of Composite Materials in
Engineering Design, May 1972. Also DU Mechanics and Structures Report 72-5, 1972.
6. N. J. Pagano, "Exact Moduli of Anisotropic Laminates" will appear in Composite Materials
edited by G. P. Sendeckyj, Academic Press.
7. "Plastic for Aerospace Vehicles, Part I Reinforced Plastics," Military Handbook, MILHDBK-17A, January 1971, p. 3-72, Department of Defense.
8. "Structural Design Guide for Advanced Composite Application," Advanced Composite
Division, Air Force Material Laboratory, Air Force Systems Command, Wright-Patterson Air
Force Base, Ohio, August 1969.
9. B. R. Collins and R. L. Crane "A Graphical Representation of the Failure Surface of a
. ,
Composite," J
Composite Materials Vol. 5 (1971), p. 408.
35
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