ON SOME ASPECTS IN THE SPECIAL THEORY OF
GRADIENT ELASTICITY
B. S. Altan and E. C. Aifantis
Center of Mechanics, Materials and Instabilities
Michigan Technological University, ME-EM Dept.
Houghton, MI 49931-1295
ABSTRACT
In this paper a special form of gradient-dependent elasticity is considered.
The motivation for considering higher-order gradients of strains in elasticity is
discussed. Equilibrium equations and boundary conditions are discussed. The
relationship between the special form of gradient elasticity adopted in this
study and mixture or nonlocal theories is considered. Solutions to certain prob-
lems including the propagation of harmonic waves, the longitudinal vibrations
of a beam, and the displacement field in an infinite medium weakened by a line
crack are given.
1. INTRODUCTION
Although the basic idea of taking into account not only the first but also
the higher gradients of the displacement field in the expression for the strain
energy function can be traced all the way back to Bernoulli and Euler a corre-
sponding formulation did not attract the attention of scientists for a long time.
After Voigt [1] briefly indicated the role of the gradients of rotation in elasticity,
E. and F. Cosserat [2] gave the first systematic treatment of the rotation gradi-
ents and the associatedutsrpolec couple-stresses. The Cosserats were drawn to the gen-
eral concept of a continuous medium each point of which has six degrees of
freedom (three displacements and three rotations) similar to rigid bodies. This
concept was already known in various theories of rods and shells, and they
extended this notion in a rigorous way to three-dimensional continuous media.
The novel feature in their theory was the appearance of couple-stresses in the
equations of motion. As a consequence of the Cosserat theory, the stress tensor
is not symmetric as in the classical theory of elasticity
For almost fifty years, not much attention was given to such generaliza-
tions in continuum mechanics. Hellinger [3] and Von Heun [4] drew attention to
231
Vol. 8, No. 3, 1997
On Some Aspects in the Special Theory of Gradient Elasticity
the problem of asymmetric stress of the Cosserat medium. Jaramillo [5] con-
structed a generalization of the classical theory of infinitesimal elastic deforma-
tions based on the assumption that the strain energy density was a quadratic
function of the second- order spatial derivatives of the displacement field, as
well as the first- order spatial derivatives and velocity components. Since
Jaramillo kept the stresses in their classical form, thus disregarding couple
stresses, he went on to impose some unnatural restrictions on the dependence of
the strain energy density upon the second- order spatial derivatives of the dis-
placement field. Truesdell [6] elaborated upon the balance equations for the
Cosserat continuum. Ericksen & Truesdell [7] developed a purely kinematical
description of Cosserat continua emphasizing the cases of rods and shells. They
also suggested a natural generalization of the Cosserat continuum. The orienta-
tion of a volume element can be represented mathematically by three mutually
perpendicular unit vectors. The Cosserats formulated their theory by assuming
that this triad (directors) is rigid. In Ericksen & Truesdell's generalization of
Cosserat continuum, the orientation vectors were stretchable and did not
remain mutually orthogonal. An interesting connection between the kinematics
of a Cosserat continuum and the theory of continuous distribution of disloca-
tions was pointed out by Guenther [8]. A modern treatment of a continuum of
grade 2 (i.e. a material whose strain energy density is a function of the second-
order spatial derivatives of the displacement field, in addition to the first- order
spatial derivatives) was given by Truesdell and Toupin [9a]. They also discussed
the indeterminacy of the couple-stress tensor in the Cosserat theory. Grioli [10]
gave the first general and correct treatment of elastic materials of grade 2,
whose strain energy function was of the same form as the Cosserats' strain
energy function. Toupin [11] has derived the associated constitutive equations
for finite deformation of perfectly elastic materials. Upon linearization, Toupin's
results are identical with those which were obtained, for example, by Aero and
Kuvshinskii [12]. In his study, Toupin [11] also reviewed the foundations of the
theory of grade 2 elastic materials, corrected the formula for the couple-stresses
given by Truesdell and Toupin [9a], pointed out that the Cosserat continuum
was a peculiar subclass of the grade 2 elastic materials, and studied the propa-
gation of plane sound waves. Schaeffer [13] solved some explicit boundary value
problems for a two-dimensional Cosserat medium so as to illustrate some of the
novel features of the theory. Mindlin & Tiersten [14] gave an extensive analysis
on the derivations of the finite and linearized equations for the Cosserat contin-
uum and also discussed previous derivations in detail. Moreover, they extended
many of the classical results on uniqueness theorems, stress functions, funda-
mental solutions, propagation of plane waves (they showed that the propagat-
ing waves were accompanied by non-propagating waves in Cosserat
continuum), thickness-shear vibrations of an infinite plate, stress concentra-
tions and singularities, stresses around spherical and cylindrical cavities in an
infinite body under tension, nuclei of strains, etc. They also provided explicit
232
B.S. Altan and E.C. Aifantis
Journal of the Mechanical Behavior of Materials
solutions to certain boundary value problems illustrating the novel features of
the theory. Mindlin [15] derived a linear generalized Cosserat theory for a
three-dimensional elastic continuum, in which the constitutive equations were
identical to those obtained by Toupin [11], and studied the propagation of plane
waves. Green & Rivlin [16] developed a more general theory by considering
higher-order surface and body force multipoles. They studied the kinematics
and the nature of higher-order force multipoles extensively, and gave the consti-
tutive equations of a generalized elasticity, by also employing an appropriate
energy equation and an entropy production inequality. In a subsequent paper,
Green & Rivlin [17] developed a general theory for multipolar displacement and
velocity fields with corresponding multipolar body and surface forces, as well as
multipolar stresses. They accomplished this by using an energy principle, an
entropy production inequality, and invariance conditions under superposed
rigid body motion. They also showed that their previous work (Green & Rivlin
[16]) is special case of that developed in [17]. Toupin [18] reviewed the models
developed for continuous media with couple-stresses, identified the concepts
and principles of continuum mechanics common to all models and devised a
mathematical machinery for easy and precise expression of the basic ideas and
assumption pertaining to each model. In addition, Toupin [18] pursued quite
another direction which also leads to a modification of the familiar concept of
stress. Instead of introducing rigid or deformable material points, he expressed
the relative position vector of a material point x' in the neighborhood of the
material point χ in terms of the successively higher-order gradients of the dis-
placement vector at the point x. Furthermore, he argued that it is a quite natu-
ral generalization to assume that the strain energy density depends on not only
the first but also on the higher gradients of the displacement field. Toupin [18]
also showed that a stress-free configuration (natural state) for materials of
grade 2 is an exception which is a rule in the classical theory of elasticity.
The boundary layer effect in crystals was known for a long time and
observed by low-energy electron diffraction experiments (see, for example,
Germer, MacRae & Hartman [19]). Toupin & Gazis [20] illustrated the relation
between the strain-gradient elasticity and atomic lattice (with nearest neighbor
and next nearest neighbor interaction) theories and explored the consequences
of an initial, homogeneous, self-equilibrating stress field. Later, Gazis & Wallis
[21] modeled the free-surface of a crystal by considering a semi-infinite, one-
dimensional, monatomic lattice with nearest and next nearest neighbor interac-
tions including a harmonic interaction between nearest neighbors at the end.
They showed that the particles near the free-surface must move to a new equi-
librium position, and that the force constants characterizing small oscillations
of these particles will be different from those of the infinite crystals. Mindlin
[22] formulated a linearized theory for an elastic solid in which the strain
energy density is a function of the strain and its first and second gradients. He
233
Vol. 8, No. 3, 1997
On Some Aspects in the Special Theory of Gradient Elasticity
showed that cohesive force and surface-tension were intrinsically included in
this theory. Also, an explicit solution for the strain and surface-tension, result-
ing from the separation of a solid along a plane was given; and a comparison
was made with an analogous lattice model.
Higher-order gradients of constitutive variables have also been employed
in other branches of continuum mechanics. In 1901, Korteweg formulated a con-
stitutive equation for the Cauchy stress that included density gradients, in
order to model the fluid capilarity effect. Theories of Korteweg's type have also
been employed to analyze the structure of liquid-vapor interfaces by Aifantis &
Serrin [24,25]. Motivated by the success of this approach for fluid interface
problems, Triantaiyllidis & Aifantis [26] formulated a nonlinear theory for
hyperelastic materials by adding the second deformation gradient into the
strain energy function to analyze the pre-and post-localization behavior of
deformation. It was shown that the width and direction of the localized defor-
mation zone could be described (without the occurrence loss of ellipticity in the
governing equations), in contrast to the classical results, higher-order gradients
of strain or other constitutive variables had been already considered for analyz-
ing dislocation patterns, microvoids and other material microstructures in sol-
ids by Aifantis and his co-workers [27-33]. These theories provided a means to
account for internal length scale and size effects in inelastic material behavior
in contrast to standard theories which could not capture these and other pat-
tern-forming instabilities effects.
In this paper, a special form of gradient-elasticity, which is based on the
linear version of the constitutive equations obtained by Triantaiyllidis & Aifan-
tis [24], is employed. Our purpose is to discuss non-classical implications of the
simplest possible gradient elasticity theory and, thus, our motivation is com-
pletely different than the fundamental continuum mechanics works reviewed
above. In the following section, the derivation of the field equations and the
boundary conditions of the special form of gradient-elasticity considered in this
study is briefly introduced. In the subsequent sections the relationship between
the special form of gradient-elasticity, the nonlocal elasticity, and the mixture
theory are pointed out. The three basic modes of cracks are formulated and
solved. Then, the propagation of plane waves in an infinite medium is consid-
ered. In the last section, natural frequencies and modes during the longitudinal
vibration of a bar are discussed. Some of the results reviewed in the present
paper have discussed by the authors separately in previous publications. It was
decided, however, to include a summary of these results here for completeness
and for the convenience of the reader who is not very familiar with the field.
234
B.S. Altan and Ε. C. Aifantis Journal of the Mechanical Behavior of Materials
2. FIELD EQUATIONS AND BOUNDARY CONDITIONS
A customary approach in obtaining a constitutive equation in elasticity is
to assume the existence of a strain energy density, which is taken as a function
of the symmetric part of the first gradient of the displacement field
w = νν(ε
0
·) (1)
where ε^ is the symmetric part of the displacement field
e
ij = ;
+
";,«)
(2)
where
Uj
is the Cartesian component of the displacement vector and indices fol-
lowing a comma, as usual, denote partial derivatives with respect to the space
coordinates. In gradient elasticity, the strain energy density function is
assumed to depend not only on the first gradients but also on the second gradi-
ents of the displacement field
w = w(E
ijt
e
ijik
) (3)
Since we are dealing with single valued displacement fields one can easily
establish a one-to-one correspondence between ε^ and ujjk (see Mindlin &
Eshel [23]). The most general form of the strain energy density function for a
linear, isotropic, gradient-dependent elastic material is
C
l
£
ij, j
£
ik, k
+ C
2
ε
ιϊ, k
E
kj, j
+ C
3
E
ii, k
£
jj, k
+ C
A
e
ij, k
e
ij, k
+ C
5
e
ij, k
Z
kj, i ^
For the special form of gradient elasticity that we consider here we assume that
c
3
and
C4
are the only non-vanishing gradient coefficients. More specifically, we
take the strain energy density function as
w = ^λε,,ε.. + μ ε
ί;
·ε
ί7
+ ^
k
+ μ ε
ί;
· ^
k
) (5)
where c denotes a newly introduced strain gradient parameter which is the only
non-standard coefficient of the theory.
235