Table Of ContentINTERNATIONAL CENTRE FOR MECHANICAL SCIENCES
C 0 U R SE S AN D L E C T U R E S - 'lio. ::;r;
MAREK SOKOLOWSKI
POLISH ACADEMY OF SCIENCES
THEORY OF COUPLE- STRESSES
IN BODIES WITH CONSTRAINED ROTATIONS
COURSE HELD AT THE DEPARTMENT
FOR MECHANICS OF DEFORMABLE BODIES
JULY 1970
UDINE 1970
SPRINGER-VERLAG WIEN GMBH
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© 1972 by Springer-Verlag Wien
Originally published by Springer-Verlag Wien New York in 1972
ISBN 978-3-211-81143-6 ISBN 978-3-7091-2943-2 (eBook)
DOI 10.1007/978-3-7091-2943-2
PREFACE
These notes represent the material of the
author's leatures given at the CISM summer aourses
in Udine 1970.
The author is indebted to Prof. Luigi
Sobrero, Searetary General of CISM for his kind
invitation to deliver these leatures. As always,
the staff of CISM should be thanked for the trouble
they take in printing, proofreading and publishing
these leatures.
Udine, July 1970
M. Sokolowsky.
1. Introduction
In the last ten, twelve years a rapid development of
the theory of Gosserat-type media, bodies with additional inter
nal degrees of freedom, the couple-stress theory and the theo
ry of bodies with internal microstructure can be observed. It
is not the aim of this series of lectures to characterize general
ly the above mentioned theories, to outline their fundamental
physical and mathematical assumptions and to discuss the pos
sibilities of their practical applications in numerous fields of
technology. Some of these problems are subject to detailed dis
cussion in the lecture to be held in the same period of time at
Udine by Professor R. Stojanovic and Professor W. Nowacki;
some of these problems arenot completely cleared in the world
literature so far- like, for instance, the extent of possible tech
nological applications of the couple-stress theory to real poly
crystalline bodies.
Generally speaking, two trends can be observed in the
current development of the theories mentioned above; one of
them consists in a constant broadening of their mathematical
and physical basis; increasing the number of internal degrees
of freedom, introducing a number of new elastic and material
constants; trying to embrace the largest possible number of
physical phenomena and to generalize most of the results known
6 Problems of Couple-Stress Theory. ..
from the classical theories of deformable bodies.
The other tendency can be characterized by the efforts
to obtain certain new solutions in this domain based, however,
on possibly simple models of media, and on the lowest possible
nurober of additional assumptions, material constants, etc.,
in order to obtain solutions which could successf•1.1ly be com
pared with certain experimental results which cannot be ex
plained on the basis of classical theories of elastic media. The
so-called theory of bodies with constrained rotations offers a
good opportunity forthistype of investigations. Under the as
sumption of isotropy, centrosymmetry and homogeneity, the
nurober of additional material constants is reduced to two (or
even one, depending on certain additional assumptions); a ser
ies of problems of well-known classical problems of elasticity
can be solved, within the framework of this generalized con
cept of elastic medium, in a more or less simple manner; even
certain closed-form solutions can be derived what, naturally,
makes the discussion of results particularly simple. In spite
of the comparative simplicity of the model, the results are
found to differ considerably from the classical ones, andin
addition to quantitative, certain qualitative differences with
respect to the classical solutions are established.
The aim of this series of lectures is to demonstrate
how certain classical solutions of the elasticity theory can be
"translated" intG the slightly more generallanguage of the
Intro-duction 7
couple-stress theory and to stress the fundamental differences
between the classical and new results, thus increasing the pos
sibility of determining the class of physical phenomena which
can be explained only on the basis of the generalized model of
elastic bodies, and which could not be explained as long as the
simple Hookean elastic solid was considered.
2. Fundamental Equations
The derivation of the fundamental equations of the
couple-stress theory presented in this section is mainly based
on paper [1] by W. T. Koiter with certain modifications. The
indicial notation (Cartesian tensors) is used throughout the pa
oer. The summation convention holds for repeated indices (ex
cept for the index n denoting the direction normal to the sur
face). The permutation symbol is denoted .e.i.~it and assumes
the values of +1, -1, 0, depending on whether the permutation
of indices is even, odd or some indices are repeated. Vectors,
denoted in print by bold-face letters, are underlined in this
text.
Introducnon Most of the classical text-books on elasticity
and mechanics of deformable bodies do not introduce the notion
of couple-stresses; it is tacitly assumed that the interaction of
individual particles in the body can be reduced to simple forces
or force vectors holding the entire body in equilibrium.
8 Problems of Couple-Stress Theory. ..
The simple elementary reasoning shows how the con..:
cept of couple-stresses can be lntroduced into the analysis of
the problern of transmission of forces through continuous media.
Envisage an elementary parallelopiped frequently used
in classical elasticity to derive the equilibrium conditions. The
dimensions of the parallelopiped are usually assumed to be
small enough to expect that the loads acting on a face, say,
:x: = con5t.,are uniformly distributed over the entire reetangle
1
d:x:2d:x3 ; thus, the resultant force can be applied in the center
of the reetangle which simplifies the further derivations (Fig.l}1i.
d X?,
0'1-
2
.,
)(
"0 ----
0'31
dx
Fig. 1
The procedure is justified in the following manner.
A ssume that the load acting on face dX2 dx3 consists ofuniform
ly distributed shearing stresses o-1, ,o-31 and of normal stresses
<r11 • 0" + O"X3 being a linear function of 'X3 and independent of X2 .
0 1
Fundamental Equations 9
Writing down the conditions of equilibrium of moments of all
fo rces acting on the element with respect to the Xl-axis, a co!!_
tribution of the non-uniform <r11- stres s distribution (stres ses
acting on the oppositeface are disregarded forsakeofsimplicity)
is added to the usual sum
Adding up these results one obtains
Now, the expression in brackets contains terms of different or
der of magnitude: with d:t~--+ 0 the last term involving ·the finite
value 0"• tancx of the stress gradient and the first power of the
'1
infinitesimal magnitude dxi. can be neglected when compared
with the first term in parenthesis and the usual result O"n• O"a1
is obtained.
The above procedure cannot be, however, applied in
the three following cases.
(a) The dimensions d.x~ of the element cannot tend to
zero owing to a finite size of the individual particles (grains)
of the body; the body cannot be replaced by a continuous medium
model. This case leads to the equations governing the behavior
of bodies with microstructure where the ·stt·ess tensorwas
10 Problems of Couple-Stress Theory. ..
found to be non-symmetric.
b) The coefficient o- is not finite; if
1
o- = tan a - oo ,
1
the stress gradient becomes infinite, and such situation is en
countered at singular points of the stress field, at points of in
finite stress concentration. The possibility of application of the
couple-stress theory to the stress concentration problems was
suggested by several authors : W. T. Koiter [ 1] E. Sternberg
[2] and others.
c} The forces of interaction of the element with the sur
rounding body cannot be reduced to force vectors (represented
by "arrows" in Fig. 1). If the forces of interaction are of a
Teally polaT nature (which, for instance, holds true in magnet
ic materials characterized by magnetic dipols-single magnetic
poles do not appear in nature), the whole reasoning ceases to
be of any physical meaning.
These remarks (which do not pretend tobe of any rig
orous mathematical value) and other, much more extensive
studies of the problern of force transmission indicate that a
detailed analysis of the so-called non-symmetric elasticity may
prove interesting and maylead to results important both from
the purely theoretical and the applied, technological point of
view.
Equation of Continuity It is obvious that the derivation of the
