Chiral Cosserat solids
University of Wisconsin
Lakes, R. S., "Elastic and viscoelastic behaviour of chiral materials", Int. J. of Mechanical Sciences, 43, 1579-1589, June (2001).
Chiral materials are not invariant to inversions: there is a distinction between right and left handed material. Material properties such as piezoelectricity and pyroelectricity, represented by tensors of odd rank, can only occur in chiral materials. Chiral effects in elasticity cannot be expressed within classical elasticity since the modulus tensor, which is fourth rank, is unchanged under an inversion. We consider effects of chirality in elastic materials described by a generalized continuum representation, specifically Cosserat elasticity. Analysis of several configurations discloses a chiral material to generate reaction moments when compressed as a slab. A chiral plate bent to hyperbolic shape is predicted to exhibit size effects from the Cosserat characteristic length, and a shear force from the chirality. This analysis can be used for the interpretation of experiments on compliant chiral materials, in particular the evaluation of the elastic constants. Viscoelastic chiral solids are examined in the context of the correspondence principle.
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Noncentrosymmetry in Micropolar Elasticity
Roderic Lakes and Robert Benedict
Adapted from
International Journal of Engineering Science, 29 (10), 1161-1167,
(1982). Download a pdf of this article.
Abstract
Consequences of noncentrosymmetry in a micropolar or Cosserat elastic solid are considered.
A solid which is isotropic with respect to coordinate rotations but not with
respect to inversions is called noncentrosymmetric, acentric, hemitropic, or chiral.
Chirality has no effect upon the classical elastic modulus tensor.
In Cosserat elasticity, chirality has an effect. A chiral Cosserat solid has three new elastic constants in addition to the six considered in the fully isotropic micropolar solid.
The chiral micropolar solid is predicted to undergo torsional deformation when
subjected to tensile load, so a chiral composite will twist when stretched. Thus chiral solids have different mechanical behavior from solids with a center of symmetry, as allowed by the more general
Cosserat elastic theory.
1. INTRODUCTION
Generalized continuum theories for mechanical behavior developed over the last
century admit degrees of freedom not considered in the classical theory of
elasticity. Common to such theories as those of the Cosserats [l], the
indeterminate couple stress theory of Mindlin and Tiersten [2] and the
micropolar theory of Eringen [3] is the assumption of couple stress and the
associated asymmetry of the force stress tensor. Generalized continuum theories
are thought to have applications in the modeling of materials with
microstructure, such as granular or fibrous materials, or materials with a
lattice structure. Micropolar theory has in recent years stimulated
considerable interest, and analytical solutions to many problems in micropolar
elasticity are available. Of particular interest to the experimentalist are the
predictions of size effects in the apparent stiffness of a cylindrical member
in torsion [4] and in bending [5]. In most published solutions, material
isotropy is assumed. Some materials, however, are not invariant to coordinate
inversions and this type of anisotropy can be expected to result in
qualitatively different behavior in comparison with isotropic solids. Some
aspects of initial stress in noncentrosymmetric Cosserat continua have been
examined [6] but geometries addressable experimentally were not considered.
Structural noncentrosymmetry is characteristic of bone, as well as synthetic
composites containing twisted fibers. In this paper the behavior of a
noncentrosymmetric micropolar elastic solid is examined.
2. NONCENTROSYMMETRIC MICROPOLAR THEORY
In classical elasticity, inversion symmetry has no effect on the elastic modulus tensor as shown in the following.
If your browser does not interpret Symbol font properly, Greek sigma, s may instead look like a bold face Latin s; Greek delta may look like a bold face Latin d.
Cijkl
= (dx
m/dxi)(dxn/dxj)(dxo/dxk)(dxp/dxl)
C
mnop
=
(-1)
dim(-1)djn(-1)dok(-1)dplCmnop
= (-1)
4
C
ijkl
= C
ijkl
.
Inversion symmetry has no effect on the elastic modulus tensor or on any tensor of even rank.
Tensor
properties of odd rank are zero if there is inversion symmetry, and can only be
nonzero if there is handedness.
Examples
include piezoelectricity, governed by a third rank tensor, and strain gradient
elastic theories which are governed by a fifth rank tensor.
Consider the constitutive eq'ns for an isotropic centrosymmetric Cosserat solid, for review. In order to account for noncentrosymmetry (chirality), the anisotropic form will be considered.
skl
=
lerr
dkl
+
(2
m
+ k
)ekl
+
eklm(rm-fm),
stress
mkl
=
a
r,r
dkl
+
b fk,l
+
g fl,k,
couple
stress
in
which
skl
is the force stress (which is a symmetric tensor in classical elasticity but is
asymmetric here), m
kl
is the couple stress (or moment per unit area), e
kl
= (u
k,l
+ u
l,k)/2
is the small strain, u is the displacement, and e
klm
is the permutation symbol. The microrotation
fk
in Cosserat elasticity is kinematically distinct from the macrorotation r
k
= (e
klmum,l)/2.
fk
refers to the rotation of points themselves, while r
k
refers to the rotation associated with movement of nearby points. The usual
Einstein summation convention for repeated indices is used and the comma
denotes differentiation with respect to spatial coordinates. In three
dimensions there are six independent elastic constants required to describe an
isotropic centrosymmetric Cosserat elastic solid,
a,
b,
g,
k,
l,
and
m.
In the linear theory of an anisotropic micropolar solid, the free energy is
given in terms of the microrotation and the micropolar strain and the A's, B's and C's are elastic constants.
The free energy density for general anisotropy is
Y
= A
0
+ A
klekl
+
(1/2)A
klmnekl
emn
+ B
klfk,l
+ (1/2)B
klmnfk,lfm,n
+ C
klmneklfm,n,
with
ekl
= e
kl
+
eklm(rm-fm).
Stress
and couple stress are given in terms of derivatives of the free energy with
respect to strain and rotation gradient.
Now we have an axial vector, therefore, the terms containing
modulus tensors in the free energy equation change sign under an inversion of spatial axes. The other
terms do not change sign, therefore, the internal energy is not invariant to
such inversions. This lack of invariance is permitted if the material does not
have a center of symmetry. The case of centrosymmetric, isotropic materials has
been treated at great length in the literature. In the present analysis, we
consider a material which is noncentrosymmetric but is isotropic with respect
to coordinate rotations.
The most general fourth order isotropic tensor may be written. The constitutive eqns (2.3) and (2.4) may be rewritten. In terms of the macrostrain and conventional notation for the elastic
constants, these may be written as in the original article.
These are the constitutive equations for a micropolar solid which is isotropic
with respect to coordinate rotations but not with respect to inversions.
Elastic constants C1, C2 and C3 are associated with noncentrosymmetry; if these
vanish, the equations of isotropic micropolar elasticity are recovered.
Consider micropolar elastic constants; if these also vanish, eqns (2.8) and
(2.9) reduce to the constitutive equations of classical isotropic, linear
elasticity theory, in which we have the Lame constants. Boundary conditions do
not depend on assumed material symmetry. One may prescribe the displacements
or the surface traction and the microrotations, or the surface couples on the
surface which has exterior normal. If tractions are specified, the boundary
conditions are given. [3]. The laws of motion also are independent of material symmetry and are given by [3].
3. RESTRICTIONS ON MICROPOLAR ELASTIC MODULI
In order that our noncentrosymmetric micropolar solid be stable, it is
necessary that the internal energy be nonnegative. From this requirement, one
may obtain restrictions on the micropolar elastic moduli. Consider the internal energy.
Using the definitions, we may rewrite the energy, as seen in the original article in the library. Observe that the quantities can be varied independently of one another. If the
first bracketed term in eqn (3.2) is the only one present, the requirement that
this term be nonnegative yields Eq. (3.3) as in classical elasticity; a quantity is identified with the Lame shear modulus. The energies represented by the second, third and fourth terms must
each be nonnegative, so for psi to be nonnegative it is necessary that Eq. (3.4) be true,
a result obtained by Eringen for the fully isotropic micropolar solid.
The product terms containing both the strain e and the micro-rotation phi cannot exist independently of the
first-fourth terms, therefore, the above approach cannot be used to restrict
the C coefficients; they can be positive or negative. However, for the total energy term to be
nonnegative, it is necessary that a negative product term not be greater in
magnitude than the sum of the corresponding positive terms containing e and A
individually.
The C coefficients associated with noncentrosymmetry are bounded by products
of combinations of classical elastic and micropolar coefficients. The
quantities K are analogous to the coupling coefficients developed in the linear
theory of piezoelectricity, and can be obtained in a similar fashion. This
correspondence is anticipated on the basis of the formal similarity between the
constitutive equations of linear piezoelectricity and those of
noncentrosymmetric micropolar elasticity, eqns (2.3) and (2.4).
4. SIMPLE TENSION
Consider a cylindrical rod of radius R, of a noncentrosymmetric micropolar
elastic solid. Let the rod be stretched by an axial force F, and let it be free
of rotational constraint, and let the lateral surface be free of force traction
and couples. Such a situation is relatively easy to realize experimentally, and
serves to illustrate the effects of the C coefficients. To solve this tension
problem, it is useful to express the constitutive equations, equilibrium
equations and strain-displacement relations in cylindrical polar coordinates.
The constitutive equations may be written as in the original article.
The equilibrium equations are given. The micropolar strains in terms of the displacements and microrotations are given. In the case of simple tension of a long cylindrical rod of radius R, the
following field of displacement and microrotation gives rise to a solution. Here I is the modified Bessel function of first order.
We may identify e with the axial strain and b0 with a twist angle per unit
length of rod, arising from the coupling produced by the C coefficients. We
observe that the Poisson-like contraction is not associated with a uniform
radial strain as is the case in classical elasticity or in centrosymmetric
micropolar elasticity [4].
The quantities are obtained by solving simultaneously the boundary condition
equations for zero force traction and zero couple on the lateral surface of the
rod, and zero net torque at the ends. The values are given in the manuscript.
In the above, the following quantities have been defined in terms of the
micropolar elastic constants.
Additional quantities are defined, in which we have modified Bessel functions
of order zero, one and two, respectively.
It is instructive to examine several special cases. If several parameters
vanish, the twist angle per unit length is given in the original manuscript.
The twist angle is proportional to the axial strain e and to the coupling
factor K0, and it tends to increase as the cylinder radius decreases. The twist
angle can be positive or negative, depending on the sign of K0. A second
special case is obtained by constraining the micro-rotation to be equal to the
macrorotation. This constraint yields a solution to the tension problem in
noncentrosymmetric indeterminate couple stress theory. The constraint is
achieved by allowing p to become infinitely large. The twist angle per unit
length becomes as given in the original manuscript.
In both special cases the twist angle is proportional to the coupling factor K0
and the axial strain e. For a thick rod of radius R much greater than the Cosserat
characteristic length, the twist angle increases as the inverse square of the radius. For a sufficiently small radius compared to the characteristic length (an unphysical situation), the twist angle per unit axial strain approaches a constant value.
5. PHYSICAL INTERPRETATION
The quantities l0 - l4 have dimensions of length and may be referred to as
characteristic lengths. l0 is the characteristic length defined in connection
with the problem of torsion in centrosymmetric micropolar theory by Gauthier
and Jahsman [4]. In generalized continuum models of structured materials, the
characteristic lengths are generally found to be related to the size of
structural elements. The quantities K1 - K5 are dimensionless. K0 and K1 are
measures of the strength of the noncentrosymmetric coupling and are analogous
to the coupling coefficients of piezoelectricity theory. K4 is equivalent to
the classical Poisson ratio, since mu + kappa /2 is the observed shear modulus. K2
and K3 are similar in nature to K4, as seen by comparing the role of lambda with
that of C1 and alpha in eqns (2.8) and (2.9). K5 represents the strength of
coupling between the macrostrain field and the microrotation field.
Micropolar elasticity and related continuum theories are thought to apply to
granular, fibrous, or composite materials. The noncentrosymmetric theory is
intended for solids containing twisted or spiraling fibers, in which one
direction of twist or spiral predominates. If the fibers are distributed
randomly in all directions, tensile specimens taken in any orientation will
appear to have the same Young's modulus, giving the impression of isotropy. Materials
may exhibit handedness on the atomic scale, as in quartz and in biological
molecules. Materials may also exhibit handedness on a larger scale, as in
composites with helical or screw shaped inclusions. Such materials can exhibit
odd rank tensor properties such as piezoelectric response. They may exhibit
torsional deformation when stretched. A material such as aluminum crystallizes
in a face-centered cubic lattice which has a center of symmetry, hence no
handedness.
6. EXPERIMENTAL
Few experiments of any kind have been performed (prior to this manuscript date
in 1982) to explore micropolar effects in real materials. Efforts to find
effects describable by indeterminate couple stress theory in metals and by
micropolar theory in a composite have been unsuccessful. Recently one of the
authors (R.L.) has found evidence of couple stress effects in human compact
bone [7,8]. There is some indication that micropolar theory is to be preferred
over indeterminate couple stress theory in describing these effects. Regarding
effects due to noncentrosymmetry, positive but very preliminary experimental
results have been found [9].
Ropes and cables containing fibers which spiral are structurally
noncentrosymmetric. It is well known that they untwist when subjected to
tensile force with no constraint on rotation, as predicted in Section 4. Use of
a continuum model for ropes is. however, questionable.
Future experiments seeking to demonstrate micropolar behavior could be
performed using the tension mode described in Section 4. This is an attractive
modality since great sensitivity is possible. It should be possible to detect
acentric micropolar effects even if the structural asymmetry is on the atomic
or molecular scale. For example, a fiber 0.07 mm dia. and 200 mm long is
typical of boron-epoxy fibers used in composites. If such a fiber were
subjected to an axial strain, the untwisting due to noncentrosymmetry could be
detected by a reflected laser beam.
Experiments based on the results in Section 4 are capable of detecting
micropolar behavior, but calculation of the nine elastic constants will be less
than straightforward. A similar complexity in the combination of elastic
constants is found in earlier work on centrosymmetric micropolar theory [4,5].
From the experimentalist's point of view, it appears that further attention to
the solution of micropolar boundary value problems is warranted.
7. DISCUSSION
Several consequences of noncentrosymmetry in micropolar elasticity have been
considered. Torsion deformation in response to tensile load, and size effects
in Poisson's ratio are predicted. Very sensitive experiments based on the
predicted behavior are possible. Macroscopic noncentrosymmetric micropolar
effects may occur in chiral composite materials with twisted or spiraling fibers.
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Acknowledgment-This research was supported in part by NIH grants 1-ROI-AM
2S863~I. brSG P507035-13 and brSG 5S07RR07035- 14.
More recently Lakhtakia at Penn State has studied propagation of waves, both acoustic and electromagnetic, in chiral materials.
Chiral structures have been observed in trees by Dr.
Cherkaev of the Mathematics Department at the University of Utah.
REFERENCES
E. and F. COSSERAT, Theorie des Corps Deformables. Hermann et Fils, Paris
(1909).
R. D. MINDLIN and H. F. TIERSTEN. Arch. Rat. Mech. Anal 11, 415 (1962).
A. C. ERINGEN, In Fracture, Edited by R. Liebowitz, 2, 621-729 (1968).
R. D. GAUTHIER and W. E. JAHSMAN. J. Appl. Mech 42, 369-374( 1975).
V. KRISHNA REDDY and N. K. VENKATAsubrAMANIAN. J. Appl. Mech 45, 429 (1978).
Y. WEITSMAN, J. Appl. Mech 34, 160 (1967).
R. S. LAKES, J Biomechanical Eng 104, 6-11 (1982).
J. F. C. YANG and R. S. LAKES, J. Biomechanical Engng 103, 275-279 (1981).
R. S. LAKES, Proc. 34th ACEMB. Houston (1981).
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Congratulations for your vigor in pursuing this esoteric branch of mechanics. As a reward, here is a brass badger. This brass badger served on the battleship USS Wisconsin; it is now in the State Capitol in Madison.
The nose of the badger is shiny because people like to squeeze it.
Here is another badger .
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