A Kumar

E T Onat

G R Cowper

S T Ariaratnam

W Prager

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BRIEF NOTES 
References 
1 Prager, W., "Three-Dimensional Plastic Flow Under Uniform Stress," 
Rev. Fac. Sci. Instanbul, Vol. 19,1954, pp. 23-26. 
2 Cowper, G. R., and Onat, E. T., "The Initiation of Necking and Buckling 
in Plane Plastic Plow," Proceedings of the 4th U.S. National Congress on 
Applied Mechanics, Vol. 2,1962, pp. 1023-1030. 
3 Kumar, A., and Ariaratnam, S. T., "Uniqueness of Deformation of 
Thin-Walled Rigid-Plastic Cylinders Under Internal Pressure, Tension, and 
Torque," International Journal of Solids and Structures, Vol. 11, 1975, pp. 
1211-1217. 
4 Kumar, A., and Ariaratnam, S. T., "Uniqueness and Stability of Rigid-
Plastic Cylinders Under Internal Pressure and Axial Tension, "International 
Journal of Solids and Structures Vol. 12,1976, pp. 525-535. 
Yield Surface Characteristics 
Arising From Orthorhombic 
Symmetry 
B. StorSkers1 
Introduction 
One of the fundamental problems of the mathematical theory of 
plasticity consists of a description of the change of the yield surface 
induced by plastic flow, accurately enough to reproduce the behaviour 
of real materials. Theoretical predictions of plastic flow in members 
subject to yield is commonly based on isotropic yield functions, 
notably the quadratic one proposed by von Mises. In certain situa-
tions, however, metals exhibit pronounced directional properties as 
regards yield. An obvious situation is when the current state has been 
attained by processes like rolling. If such an operation has commenced 
from an isotropic state it will induce three mutually preferred direc-
tions in the material resulting in orthorhombic symmetry. 
When modeling the behavior of real materials by aid of experi-
mental data, the existence of preferred (or symmetry) directions at 
any generic instant is frequently assumed, knowingly or not. It seems 
though that the explicit consequences of such an imposition have not 
always been fully appreciated and it is the present purpose to eluci-
date some implications in particular for the case of plane stress. 
Orthorhombic Yield Surface Models 
Suppose that a yield state is defined by a scalar condition 
/(<7,A)=0 (1) 
at any generic instant where, referred to a fixed Cartesian coordinate 
system xi, a denotes Cauchy stress and A any set of scalar or tensor-
valued constitutive variables defining the current material state. 
Several popular models fall within this formulation. 
In an early proposal by Hill [1], exhibiting orthorhombic symmetry, 
/ is quadratic in the stress components and A stands for six indepen-
dent scalars being functionals of the deformation history. Recently, 
[2], this yield condition has been generalized to allow for noninteger 
•powers of the stress components. 
When attempting to account for the (uniaxial) Bauschinger effect 
in more general circumstances, Prager [3] proposed the concept of 
kinematic hardening by introducing a symmetric second rank tensor 
determining a translation of the yield locus induced by plastic flow. 
Subsequently second rank tensors have been introduced in a more 
elaborate way implying rotation and distortion of the yield surface 
1 Department of Strength of Materials and Solid Mechanics, The Royal In-
stitute of Technology, S-10044 Stockholm 70, Sweden. 
Manuscript received by ASME Applied Mechanics Division April, 1979; final 
revision, July, 1979. 
in stress space. For a recent proposal and an account cf. [4]. Still 
though whatever intricate way a symmetric second rank tensor is 
introduced, it possesses three orthogonal principal directions and an 
immediate consequence is that orthorhombic symmetry prevails for 
the material state at any generic state of deformation. 
A more general approach to the use of state variables in order to 
represent the internal structure of materials has for some time been 
advocated by Onat. Several ingenious methods to determine the 
number and rank of internal variables necessary to obtain realistic 
representations have been proposed by Onat and Pardshisheh [5]. 
Subsequently in a constitutive theory for elastic-plastic solids based 
on a set of even rank irreducible tensors as state variables, symmetry 
requirements arising in particular from a single symmetric second 
rank tensor have been discussed by these writers [6]. 
Explicit Symmetry Properties 
As in the present circumstances the material symmetry operations 
consist of 180° rotations around the preferred directions, x\ (say), then 
on introducing an appropriate integrity basis, the yield condition is 
expressible as 
/ ( f f l l , ?22> ^ 3 3 , 5 l 2 2 , 0 2 3 2 , <T312, O ^ O ^ S S l . A) = 0 (2) 
in obvious notation. 
It is immediately clear then that, when referred to the preferred 
directions, the yield function is invariant with respect to pair-wise 
changes of sign of shear stresses and eventually single changes when 
the triple product in (2) is absent. Experimental evidence of the sec-
ond event indicates that a quadratic yield function might suffice to 
model material properties. 
In the degenerate case of transverse isotropy, with respect to X3, 
(2) reduces to 
g(bzn + <X22, ffno^ - oi2
2, (T33, (T232 + (T312, I o\, A) = 0 (3) 
by aid of an appropriate basis, [7]. 
In the particular case when A stands for a symmetric second rank 
tensor, a, the arguments of the yield function reduce to the single and 
mutual invariants of a and a and the yield condition (1) is expressible 
as 
h (tr a, tr <rz, tr a3, tr a, tr a2, tr a3, 
tr {aa), tr (aa2), tr (a2a), tr (<r2a2)) = 0 (4) 
with respect to any coordinate system. 
In such models it is common that the yield function also depends 
on a number of scalars transforming in an invariant manner and 
representing, in a geometric formalism, for instance expansion or 
shrinkage of the corresponding yield surface in stress space. In the 
present context, however, there is no loss of generality in suppressing 
the presence of these. 
In the fully degenerate case when a is spherical, naturally the yield 
function is isotropic. Note though that a common assumption is that 
a is deviatoric and then this case must be excluded unless a van-
ishes. 
When investigating the applicability of different hardening models 
to real material behavior, it is customary to subject material specimens 
to various stress histories and from measurements determine the 
associated yield surfaces in stress space. Due to obvious practical 
difficulties, almost invariably subsequent yield surfaces are deter-
mined by experiments in the subspace of plane stress and the explicit 
consequences of (2), or the particular form (4), are then perhaps best 
visualized in such a space. It must be remembered, however, that in 
a general situation the preferred material directions are unknown and 
only the components of the stress tensor are conceptually measur-
able. 
Suppose then that a state of plane stress is prevailing in a material 
plane perpendicular to a preferred direction as might be the case for 
instance when dealing with rolled sheet metal. Accordingly the yield 
condition (2) reduces to 
f(cfn ^22, &i2
2, A) = 0 (5) 
Journal of Applied Mechanics DECEMBER 1979, VOL. 46 / 961 Copyright © 1979 by ASMEDownloaded From: https://appliedmechanics.asmedigitalcollection.asme.org on 07/02/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use
BRIEF NOTES 
As a consequence 
a 12 = 0 (6) 
represents a symmetry plane in the (three-dimensional) stress 
space. 
In order to see the implication of (6) when the preferred directions 
are unknown and the plane stress tensor a„ff is referred to an arbitrary 
coordinate system x„, a = 1, 2, it proves advantageous to introduce 
the transformation 
ffl2* = V/2<T12 (7 ) 
As explained by Hahn [8], a rotation by an angle 6 of a Cartesian 
coordinate frame X1X2 in a physical plane corresponds, as regards an 
associated plane second rank symmetric tensor T (say), to a pure rigid 
rotation by an angle 20 around the line T n — T22 = 0, T\i = 0, in a 
coordinate frame of a reduced space spanned by Tu, T22, V2T12. 
Thus, utilizing (7) in the present case, for any basic coordinate system 
chosen, the yield surface will retain its shape; only the coordinate 
frame in the reduced stress space will be rotated. 
When referred to a frame x„, rotated an (unknown) angle 6 with 
respect to the preferred material in-plane directions, the symmetry 
plane, SP (say), given by (6) is in the associated transformed stress 
space represented by 
-7= (022 - tfn) sin 20 + \/2o-i2 cos 20 = 0 (8) 
as illustrated in Fig. 1. 
It is evident then that, for the particular case of anisotropy being 
discussed, points of the intersection between this plane and the yield 
surface must represent stress states having common principal axes 
and, in particular, coinciding with those of the preferred material 
directions. . . " ' 
Thus, when inspecting the shapes in stress space of experimentally 
determined yield surfaces, under the restrictions previously intro-
duced, it might immediately be established whether subsequent 
yielding may, with some accuracy, be theoretically predicted on the 
assumption of orthorhombic symmetry and as a special case by means' 
of a single symmetric second rank tensor apart from scalars. If such 
circumstances prevail, the rotation of the preferred material directions 
may be determined at subsequent plastic states. Plausibly such in-
formation as regards the evolution of the directions of principal axes 
is of value when judging the relevance of different growth laws for 
state variable tensors as for instance those discussed by Ziegler [9] for 
kinematic hardening. 
Results from a two-dimensional normal stress-shear stress sub-
space, as commonly obtained from combined tension and torsion tests, 
will in general not suffice for the proposed type of investigation as has 
been indicated in a preliminary investigation of the tensor approach 
by Fardshisheh [10]. In the present framework this is immediately 
obvious from Fig. 1 as the mirror images of points in the vertical 
coordinate planes with respect to the symmetry plane SP, will in 
general be located in the same planes only for the fortuitous case 
0 = 0 . ' . 
An extensive and careful investigation of the hardening properties 
of pure aluminum under complex stress paths has been carried out 
by Phillips and associates in the 3-space of plane stress. A visual in-
spection of some subsequent yield surfaces depicted by Phillips and 
Kasper [11] is, however, discouraging as regards desired symmetries 
except for very simple stress histories. It must be underlined though 
that the shapes of yield surfaces are indeed sensitive to the definition 
of yield, [12, 13]. Phillips and Rasper's results are based on a pro-
portional limit definition of yield (3jtt plastic probing strain). The 
outcome might have been quite different had any other common yield 
definition been adopted. Furthermore it must be remembered that 
the present argument rests upon the prerequisite condition that one 
preferred direction is perpendicular to the material plane. Such cir-
cumstances, however, seem plausible in the experiments reported by 
Fig. 1 Yield surface and symmetry plane (SP) in reduced space of plane 
stress 
Phillips and coworkers, as the specimens used had enjoyed a stress-
free isotropic state. In this context it is interesting to note that, when 
interpreting experimental results for aluminum, copper, and brass, 
with apparent success, Phillips and Weng, [14], have been led to in-
troduce a yield surface representation based on two second rank 
tensors and a scalar. 
. It is beyond the present scope to scan the literature for further 
experimental results or to explore the restrictions imposed on other 
types of constitutive functions. The foregoing arguments may, how-
ever, for instance be applied to potential surfaces of constant dissi-
pation rate for creeping materials. 
Acknowledgments 
This line of inquiry was proposed by Prof. E. T. Onat, Yale Uni-
versity. A pertinent comment by Dr. B. Raniecki, Polish Academy of 
Sciences, is gratefully acknowledged. 
References 
1 Hill, R., The Mathematical Theory of Plasticity, Clarendon Press, 
Oxford, 1950. 
2 Hill, R, "Theoretical Plasticity of Textured Aggregates," Mathematical 
Proceedings of Cambridge Philosophical Society, Vol. 85, 1979, pp. 179-
191. 
3 Prager, W., "The Theory of Plasticity—A Survey of Recent. Achieve-
ments," Proceedings of the Institution of Mechanical Engineers, Vol. 169,1955, 
pp. 41-57. 
4 Tanaka, M., and Miyagawa, Y., "On Generalized Kinematic Hardening 
Theory of Plasticity," Ingenieur-Archiv, Vol. 44,1975, pp. 255-268. 
5 Onat, E. T., and Fardshisheh, P., "Representation of Creep of Metals," 
Oak Ridge National Laboratory Report No. ORNL-4738,1972. 
6 Fardshisheh, F., and Onat, E. T., "Representation of Elastoplastic Be-
havior by Means of State Variables," Foundations of Plasticity, Vol. II, ed., 
Sawzuk, A., Noordhoff, Leyden, 1974, pp. 89-114. 
7 Green, A. E., and Adkins, J. E., Large Elastic Deformations, Clarendon 
Press, Oxford, 1960. 
8 Hahn, H. T., "A Derivation of Invariants of Fourth Rank Tensors," 
Journal of Composite Materials, Vol. 8, 1974, pp. 2-14. 
9 Ziegler, H., "A Modification of Prager's Hardening Rule," Quarterly 
of Applied Mathematics, Vol. XVII, 1959, pp. 55-65. 
10 Fardshisheh, F., Dissertation, Yale University, 1973. 
11 Phillips, A., and Kasper, R., "On the Foundations of Thermoplasti-
city—An Experimental Investigation," ASME JOURNAL OP APPLIED ME-
CHANICS, Vol. 40,1973, pp. 891-896. 
12 Phillips, A., "Experimental Plasticity. Some Thoughts on Its Present 
Status and Possible Future Trends," Foundations of Plasticity, Vol. II, ed., 
Sawzuk, A., Noordhoff, Leyden, 1974, pp. 193-233. 
13 Michno, M. J., Jr., and Findley, W. N., "An Historical Perspective of 
Yield Surface Investigations for Metals," International Journal of Nonlinear 
Mechanics, Vol. 11, 1976, pp. 59-82. 
14 Phillips, A., and Weng, G., "An Analytical Study of an Experimentally 
Vertilied Hardening Law," ASME JOURNAL OP APPLIED MECHANICS, Vol. 
42,1975, pp. 375-378. 
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Uniqueness of Deformation of Thin-Walled Rigid-Plastic Cylinders Under Internal Pressure, Tension, and Torque

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Uniqueness of Deformation of Thin-Walled Rigid-Plastic Cylinders Under Internal Pressure, Tension, and Torque

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