p8-gradient elasticity in comparison with classical continuum elasticity
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GRADIENT ELASTICITY IN COMPARISON
WITH CLASSICAL CONTINUUM ELASTICITY
Vasilios Nikitas1, Alexandros Nikitas2, Nikolaos Nikitas3*
1 Head of the Department of Development, Energy and Natural Resources
Region of Eastern Macedonia and Thrace, Regional Unity of Drama,
Drama 66100, Greece
E-mail: nikitasv@pamth.gov.gr
2 ................................................................
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E-mail: ................................
3 ................................................................
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E-mail: ................................
* Corresponding author
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Abstract
Gradient elasticity theory is founded on the gradient equation of motion deduced for a
discrete system by means of the discrete (force-displacement) formulation of Hookes
law. This equation of motion appears to include as a special case the equation of
motion derived from the continuum (stress-strain) formulation of Hookes law. Thus,
gradient elasticity appears to include classical continuum elasticity as a special case.
The paper focusing on one-dimensional systems, derives the fundamental formulas of
gradient elasticity for short and long-range interactions, compares them with the
corresponding formulas of classical elasticity and concludes that classical continuum
elasticity cannot be a special case of gradient elasticity, but instead, gradient elasticity
must be a special case of classical continuum elasticity.
Keywords: Hookes law, gradient elasticity, classical continuum elasticity.
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Contents
Abstract
Nomenclature
1. Hookes law and equation of motion: discrete and continuum formulation
2. Derivation of the general gradient equation of motion
3. Gradient equation of motion of an infinite uniform discrete system
4. Gradient equation of motion for only short-range interactions
5. Derivation of the basic equations of classical continuum elasticity
6. The continuum formulation of Hookes law as the generalized form of the law
7. Comparison between gradient and classical continuum elasticity
8. Conclusions
References
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Nomenclature
jia, a distance, distance between the i and j mass points
( ) x;t strain at the position x of a one-dimensional continuum
E elasticity modulus (Youngs modulus)
ijE elasticity modulus of an ideal massless spring connecting the i and j
mass points
iF resultant internal force acting on the i mass point
ijF internal force on the i mass point due to the action of the j mass point
k , spring stiffness
ijk stiffness of an ideal massless spring connecting the i and j mass points
im, m mass, mass of the i mass point
N number of mass points of a discrete system
i, mass density, mass density around point ix of a continuum
( ) x, t stress at the position x of a one-dimensional continuum
( )ij i x ,t stress at the position ix of an ideal massless spring connecting the i and
j mass points
t time
iu displacement of the i mass point
( )u x,t x strain at point x of a one-dimensional continuum
x coordinate of a point of a continuum at its at-rest position
ix coordinate of the i mass point at its at-rest position
1. Hookes law and equation of motion: discrete and continuum formulation
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Hookes law as applied to an ideal massless spring connecting the i and j mass points
of a discrete system and representing their interaction relates the spring force ( )ijF t to
the displacements ( )iu t and ( )ju t of the mass points as follows [1 p.166]
( ) ( ) ( ) with constant stiffness of the massless springij ij j i ijF t k u t u t k = = . (1.1)
In the case of a free vibrating one-dimensional discrete system with N mass points
lying on the coordinate axis x , the discrete formulation (1.1) of Hookes law and
Taylors expansion of the displacement differences ( )j iu u for = 1, 2, ,j NK lead to
the gradient equation of motion of the i discrete mass point
2 22
2 21 1
1 1
2
N Ni i i
ij ji ij jii ij j
u u uk a k a
m x mt x= =
= + +
) )
3 43 4
3 41 1
1 1
6 24
N Ni i
ij ji ij jii ij j
u uk a k a
m mx x= =
+ + +
) )
L . (1.2)
where iu , im and ix orx in n niu x stand for the displacement at time t, the mass,
and the natural(i.e. for 1 2 0Nu u u= = = =K ) position, respectively, of the i mass point,
jia stands for thefinite distance magnitude ( )j ix x , i.e. ji j ia x x= ,
ijk stands for the stiffness of the ideal massless spring connecting the i and j
mass points and representing their interaction.
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On the other hand, Hookes law as applied to a point of a one-dimensional continuum
may be written in the formulation [2 p.492 eq.()]
( ) ( ) x;t E x;t = , (1.3)
where ( ) x;t , ( ) x;t and E stand for the stress, the strain and a constant elasticity
coefficient (namely, Youngs modulus or elasticity modulus), respectively, at the point
x of the elastic continuum.
In the case of a free vibrating one-dimensional continuum for sufficiently small strains,
the continuum formulation (1.3) of Hookes law combined with the Newtons 2nd
axiom leads to the partial differential equation of motion [3 pp.406-409]
( ) ( )2 222 2
u x,t u x,t c
t x
=
, (1.4)
where c denotes a constant velocity magnitude (namely, wave propagation velocity).
Taking the gradient equation of motion (1.2) for representative of a partial differential
equation and comparing with the partial differential equation of motion (1.4) on the
arbitrary assumption that
2 2
1
1
2
N
ij jii j
k a cm =
=
)
, (1.5)
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gradient elasticity theory supports [4 p.61] that the inclusion of the series of spatial
derivatives 3 3 4 4i i iu x , u x , u x , L in the gradient equation of motion (1.2)
makes the partial differential equation (1.4) be a special case of the gradient equation
(1.2). Equation (1.2) is reduced to equation (1.4), when the series of the spatial
derivatives 3 3 4 4i i iu x , u x , u x , L becomes negligible. Moreover, for gradient
elasticity theory, the partial differential equation (1.4) represents only short-range
(local) interactions around the position x , while the gradient equation of motion (1.2)
represents both shortand long-range (nonlocal) interactions [4 pp.60-61]. The long-
range interactions are accounted for in the right-hand member of the gradient equation
of motion (1.2) by means of the series of the higher-order spatial derivatives
3 3 4 4i iu x , u x , L .
Within this theoretical frame, the partial differential equation of motion (1.4) is
conventionally deemed to be a special case of the gradient equation of motion (1.2).
And since the discrete formulation (1.1) of Hookes law results in the gradient equation
of motion (1.2), it seems to follow thatthe continuum formulation (1.3) of Hookes law
resulting in the partial differential equation of motion (1.4) should be a special case of
the discrete formulation (1.1) of Hookes law. However, this conventional view cannot
hold true; for in par. 6, the discrete formulation (1.1) of Hookes law is shown to be
directly derived from the continuum formulation (1.3) of Hookes law as a special case
of it.
Further, the gradient equation of motion (1.2) for a discrete system requires the
application of Taylors expansion to an ideal continuous displacement distribution
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( )u x , t enclosing all discrete displacements of the system, thereby requiring the
existence and continuity of the spatial derivatives
2 2 3 3 4 4u x, u x , u x , u x , L of ( )u x,t at every point [5 p.92-94]. In contrast,
the partial differential equation of motion (1.4) requires only the existence of the spatial
derivative 2 2u / x , and hence, this derivative may not be differentiable or even
continuous, which means that the higher-order spatial derivatives 3 3 4 4u x , u x , L
may not exist. Consequently, the gradient equation of motion (1.2) requires conditions
of differentiability for the distribution ( )u x , t that form a special case of the
corresponding conditions required by the partial differential equation of motion (1.4).
This actually avoids considering that the partial differential equation of motion (1.4) is a
special case of the gradient equation of motion (1.2).
2. Derivation of the general gradient equation of motion
We consider a free vibrating one-dimensional discrete system lying along the coordinate
axis x with both short-range (ornearest neighbour) and long-range interactions. It is
worth noting that the magnitudes of the short-range interactions are far superior to those
of the long-range interactions [4 p.25]. Any mass point of the discrete system is
connected through ideal massless springs to all the other mass points of the discrete
system, with the forces developed in the ideal massless springs representing the
interactions of the mass points. Hence, the interaction force ijF imposed by the j mass
point on the i mass point may be evaluated by applying the discrete formulation (1.1)
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of Hookes law to the ideal spring joining the i and j mass points, provided that the
stiffness ijk is known and the displacement difference ( )j iu u are given as data.
These latter may be derived from the spatial derivatives
2 2 3 3 4 4i i i iu x , u x , u x , u x , L at the i mass point with reference to an ideal
continuous displacement distribution ( )u x,t compatible with all displacements of the
discrete system, using Taylors expansion, viz.
2 3 42 3 41 1 1
1 1 2 3 4
2 3 42 3 42 2 2
2 2 2 3 4
2 2 3 3 4 4
2 3 4
2 6 24
2 6 24
2 6 24
i i ii i i ii i
i i ii i i ii i
i N i i Ni i Ni i
N i Ni
a a au u u uu u a
x x x x
a a au u u uu u a
x x x x
. . . . . . . . . . . . . . . . . . . . .
u a u a u a u
u u a x x x x
= + + + +
= + + + +
= + + + +
L
L
L
(2.1)
The system of the ( )1N linear equations (2.1) discloses that the ( )1N spatial
derivatives 2 2 3 3 4 4i i i iu x , u x , u x , u x , L at the i mass point can uniquely
be determined by the ( )1N displacement differences ( ) ( ) ( )1 2i i N iu u , u u , , u u K ,
and vice versa, on the assumption that all terms of higher than ( )1N order are
negligible. Even if some displacement differences ( )j iu u result in zero interaction
forces ijF due to zero stiffness coefficients ijk , the ( )1N equations (2.1) will still
determine uniquely the ( )1N spatial derivatives
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2 2 3 3 4 4i i i iu x , u x , u x , u x , L in terms of the ( )1N displacement
differences ( ) ( ) ( )1 2i i N iu u , u u , , u u K , and vice versa.
The determination of the spatial derivatives 2 2 3 3 4 4i i i iu x , u x , u x , u x , L
by means of equations (2.1) suffices for the determination of the ideal continuous
displacement distribution ( )u x , t . Indeed, ( )u x,t can be derived for given spatial
derivatives 2 2 3 3 4 4i i i iu x , u x , u x , u x , L from Taylors expansion
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )2 32 3
2 32 6
i ii i ii i
x x x xu t u t u t u x,t u t x x
x x x
= + + + +
L . (2.2)
In line with Hookes law in the discrete formulation (1.1), the interaction forces ijF
imposed by every j mass point of the discrete system on the i mass point may be
expressed by means of Taylors expansions (2.1) as below
( )
( )
2 2 3 3 4 4
1 1 1
1 1 1 1 1 2 3 4
2 2 3 3 4 4
2 2 2
2 2 2 2 2 2 3 4
2 6 24
2 6 24
i i i i i i ii i i i i
i i i i i i ii i i i i
iN iN
u a u a u a uF k u u k a
x x x x
u a u a u a uF k u u k a
x x x x
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
F k
= = + + + +
= = + + + +
=
) )L
) )L
)( )
2 2 3 3 4 4
2 3 42 6 24
i Ni i Ni i Ni iN i iN Ni
u a u a u a uu u k a
x x x x
= + + + +
)L
(2.3)
After summing up all interactions ijF as described by equations (2.3), the resultant iF of
all the short-range and long-range interactions imposed on the i mass point may be put
into the formulation
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for any ,
i , i n i , i n
i n ,i i n , i
i
k k
a a n a i n
m m
+
+
=
= = =
, (3.1)
then, it is deduced that
2 1 2 1 2 1
1 1
2 2 2 2
1 1 1
0
for 1 2 3
2
ij ji ij ji ij ji
j j i j i
ij ji ij ji ij ji ij ji
j j i j i j i
k a k a k a
, , ,
k a k a k a k a
= = + =
= = + = = +
= + = =
= + =
) ) )
K
) ) ) ), (3.2)
and inserting in equation (2.4) with j , ,= K implies
22
21
ii ij ij ji
j j i
uF F k a
x
= = +
= = +
)
4 64 6
4 61 1
1 1
12 360
i iij ji ij ji
j i j i
u uk a k a
x x
= + = +
+ + +
) )
L , (3.3)
which by Newtons 2nd axiom (2.5) for im m= gives the gradient equation of motion
2 22
2 21
1i iij ji
j i
u uk a
mt x
= +
= +
)
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4 64 6
4 61 1
1 1
12 360
i iij ji ij ji
j i j i
u uk a k a
m mx x
= + = +
+ + +
) )
L , (3.4)
with all odd-order spatial derivatives 3 3 5 5i i iu x , u x , u x , K exluded.
4. Gradient equation of motion for only short-range interactions
By its very definition the gradient equation of motion (1.2) refers to both the short-range
and the long-range interactions developed within the discrete system.
In the case of only short-range (or nearest neighbour) interactions, the interactions on
the i mass point are due to only the nearest neighbour 1i + and 1i mass points, viz.
1 1i i ,i i ,iF F F+ = + . (4.1)
If we further simplify the case by adopting
1 1
1 1 for any ,
i , i i , i
i , i i , i
i
k k k
a a a i n
m m
+
+
= == ==
(4.2)
then, it is deduced that
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1 1 1 13 5 7
1 1 1 1
0i i i i
ij ji ij ji ij ji ij ji
j i j i j i j i
k a k a k a k a+ + + +
= = = =
= = = = = ) ) ) )
L , (4.3)
and inserting all the above equations in equation (2.4) for 1 1j i , i= + implies
2 4 64 62
1 1 2 4 612 360
i i ii i ,i i ,i
u u uk a k aF F F k a
x x x+
= + = + + +
L . (4.4)
Equation (4.4) and Newtons 2nd axiom (2.5) for im m= result in the gradient equation
of motion
2 2 4 62 4 6
2 2 4 612 360
i i i iu u u uk a k a a
m mt x x x
= + + +
L , (4.5)
which excludes all odd-order spatial derivatives 3 3 5 5i i iu x , u x , u x , K .
By Taylors expansion of the ideal displacement distribution ( )u x,t of a discrete system
around the position of the i mass point, it is deduced that
2 3 4 5 62 3 4 5 6
1 2 3 4 5 62 6 24 120 720
i i i i i ii i
u u u u u ua a a a au u ax x x x x x
= + + + + L , (4.6)
and hence,
( )2 4 64 6
21 1 2 4 6
2
12 360
i i ii i i
u u ua au u u a
x x x
+
+ = + + +
L . (4.7)
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For sufficiently small distance a , the spatial derivative 2 2iu x approximates
( )2 1 12 2
2, for sufficiently small
i i iiu u uu
ax a
+ +
, (4.8)
which combined with equation (4.7) implies
4 64 6
4 60, for sufficiently small
12 360
i iu ua a ax x
+ +
L , (4.9)
and inserting in equation (4.5) yields
2 22
2 2, for sufficiently smalli i
u uk aa
mt x
. (4.10)
5. Derivation of the basic equations of classical continuum elasticity
From the continuous displacement distribution ( )u x, t of a one-dimensional continuum
undergoing a longitudinal free vibration we can directly derive the strain at any position
x in the continuum as the derivative ( )u x,t x at the position x . Thus, for an
infinitesimal longitudinal element of length dx and cross-sectional area dA located at
the position x in the continuum, Taylors expansion of the difference ( )du x,t between
the displacements at the ends of the element along axis x is reduced to the exact
relation
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( )( )
d du x,t
u x,t xx
=
. (5.1)
The resultant of interactions imposed on the infinitesimal element by the left-hand part
of the continuum may be considered as an internal force ( ) d x,t A produced by the
action of an internal stress ( ) x, t on the cross-sectional area dA of the left-hand
boundary of the element. Similarly, the resultant of interactions imposed on the
infinitesimal element by the right-hand part of the continuum may be considered as an
internal force ( ) ( ) d x,t x,t A + produced by the action of an internal stress
( ) ( ) x,t x,t + on the cross-sectional area dA of the right-hand boundary of the
element, with ( ) x, t standing for the difference of the two internal stresses. By
definition the internal forces ( ) d x,t A and ( ) ( ) d x,t x,t A + represent the
resultant interactions between the infinitesimal element and the left-hand and right-
hand parts of the continuum, thereby representing both of the short-range and long-
range interactions.
It is worth noting that the difference( ) x, t
must actually represent a differential
( )d x, t . Indeed, the equation of motion of the infinitesimal element equals
( ) ( ) d d d x,t A x A u x,t = && , (5.2)
which for the real case of afinite acceleration ( )u x,t && implies that the difference ( ) x, t
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must be infinitesimal, thereby implying that the difference operator must equal the
differential operator d , that is, d . Then, the internal stresses imposed by two parts
of a continuum on their common boundary must equal to each other. In conclusion, the
continuity of the internal stress ( ) x, t as to the position x seems to be quite reasonable.
Considering that the infinitesimal longitudinal element behaves as an infinitesimal
longitudinal spring with constant stiffness k , Hookes law in the discrete formulation
(1.1) as applied to the ends of the infinitesimal element yields
( ) ( )d d x,t A k u x,t = . (5.3)
Equation (5.3) after inserting equation (5.1) can equally be rewritten in the following
form relating the internal stress ( ) x,t to the strain ( )u x,t x
( )( )d
d
u x,tk x x, t
A x
=
. (5.4)
Since by definition the dimensions dx and dA and the stiffness k remain unchanged at
any instant in time, and also, the magnitudes of the stress ( ) x,t and strain ( )u x,t x
are finite, it follows that the coefficient d dk x A in equation (5.4) must be a finite
constant E for the considered position x in the continuum, that is,
dfinite constant of the continuum
d
k xE
A
= . (5.5)
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Then, substituting in equation (5.4) gives the continuum formulation of Hookes law
( )( )u x,t
x,t E x
=
. (5.6)
Newtons 2nd axiom as applied to an infinitesimal longitudinal element of the free-
vibrating one-dimensional continuum can be put into the formulation
( ) ( )2
2
x,t u x,t
x t
=
, (5.7)
with ( ) x,t x acting as an internal body force per unit volume responsible for only
the accelerated motion of the infinitesimal element. It is reasonable to assume that the
internal stress ( ) x,t is a differentiable function of the position x . Otherwise, instead
of the unique differential ratio ( ) x,t x there could be an indeterminate or infinite
ratio ( ) x ,t dx in the left-hand member of equation (5.7) contradicting the existence
of a given finite acceleration ( )2 2u x,t t .
The continuum formulation (5.6) of Hookes law after differentiation becomes
( ) ( )2
2
x,t u x,t E
x x
=
, (5.8)
and combining with Newtons 2nd axiom (5.7) gives
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( ) ( )2 2
2 2
u x,t u x,t E
t x
=
, (5.9)
which coincides with the partial differential equation (1.4) after inserting the symbolism
2Ec
= . (5.10)
6. The continuum formulation of Hookes law as the generalized form of the law
The basic difference between the discrete formulation (1.1) and the continuum
formulation (1.3) or (5.6) of Hookes law is that the former can only apply to an ideal
massless spring of finite length, while the latter can apply to any massless or
nonmassles spring of infinitesimal length. From this viewpoint, it seems that the
continuum formulation (5.6) of Hookes law is more general than the discrete one. And
besides, as shown in this paragraph, the discrete formulation of Hookes law can be
derived from the continuum one, which proves that the latter is the generalized form of
Hookes law.
By applying Newtons 2nd axiom (5.7) to the ideal massless (i.e. with zero mass density
0 = ) spring connecting the i and j mass points of a discrete system, it is deduced that
( ) 0 x,t x = , which means that ( ) x,t the same all over the spring= , for any finite
acceleration ( )2 2u x,t t . Putting ( ) 0 x,t x = in the differential form (5.8) of the
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continuum formulation (5.6) of Hookes law yields ( )2 2 0u x,t x = , which
necessitates
( ) ( ) ( )for a massless spring connecting and mass points
j i
ji
u t u t u x,t i j
x a
=
. (6.1)
Equation (6.1) and the continuum formulation (5.6) of Hookes law result in
( ) ( ) ( ) for a massless spring connecting and mass pointsj iji
E x,t u t u t i j
a = , (6.2)
with E denoting the elasticity modulus in the massless spring. After substituting
( ) ( ) with total cross - sectional area of the springij
ijji
F t x,t A A
E Ak
a
= =
=
)(6.3)
equation (6.2) can be given the discrete formulation (1.1) of Hookes law, and hence,
the discrete formulation of Hookes law (1.1) must be considered as a special case of
the continuum formulation (5.6) of Hookes law. Consequently, the gradient equation of
motion (1.2) resulting from the discrete formulation of Hookes law (1.1) must be
considered as a special case of the equation of motion resulting from the continuum
formulation (5.6) of Hookes law, i.e. the partial differential equation of motion (1.4).
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7. Comparison between gradient and classical continuum elasticity
Using the notion of the stress derivative ( ) x,t x for the continuum representation of
a discrete system undergoing a free vibration, we can apply the classical continuum
formulation of Newtons 2nd axiom (5.7) and define the stress derivative ( ) x,t x at
all mass points of the discrete system in terms of the acceleration ( )2 2u x,t t and the
mass density of the considered continuum representation of the discrete system.
Then, applying the gradient formula of motion (1.2) to the acceleration ( )2 2u x,t t ,
we can express the stress derivative ( ) x,t x in terms of the spatial derivatives
2 2 3 3 4 4i i i iu x , u x , u x , u x , L and conclude a gradient stress-strain
relation that allows the foundation of the gradient elasticity theory.
Indeed, combining the gradient equation of motion (1.2) with the classical continuum
formulation of Newtons 2nd axiom (5.7) yields
( ) 222
1 12
N Ni i i
ij ji ij jii ij j
x ,t u u k a k a
x m x m x= =
= + +
) )
3 43 4
3 41 16 24
N Ni i
ij ji ij jii ij j
u u k a k a
m mx x= =
+ + +
) )L
, (7.1)
which for the case of an infinite uniform discrete system is reduced by equation (3.4) to
( ) 222
1
i iij ji
j i
x ,t uk a
x m x
= +
= +
)
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4 64 6
4 61 1
12 360
i iij ji ij ji
j i j i
u u k a k a
m mx x
= + = +
+ + +
) )
L , (7.2)
while for the case of only short-range interactions is reduced by equation (4.5) to
( ) 2 4 62 4 6
2 4 612 360
i i i i x ,t u u u k a k a a
x m mx x x
= + + +
L . (7.3)
It is obvious that equation (7.1) for the stress derivative ( ) x,t x can consistently be
derived from the differentiation of the general gradient stress-strain relation
( ) 2
1 12
N Ni
i ij ji i ij jii ij j
u x ,t k a u k a
m m x= =
= + +
) )
2 33 4
2 31 1
6 24
N Ni i
ij ji ij jii ij j
u u k a k a
m mx x= =
+ + +
) )
L , (7.4)
whose main characteristic is not only the inclusion of the higher-order derivatives of the
strain ( )u x,t x in addition to the strain ( )u x,t x , but also the inclusion of the
displacement iu . The displacement iu and other terms may be eliminated in a few
cases. To specify, for an infinite uniform discrete system, equation (7.4) is reduced to
( ) 2
1
ii ij ji
j i
u x ,t k a
m x
= +
= +
)
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3 54 6
3 51 1
12 360
i iij ji ij ji
j i j i
u u k a k a
m mx x
= + = +
+ + +
) )
L , (7.5)
and for a discrete system with only short-range interactions, equation (7.4) is reduced to
( )3 52 4 6
3 512 360
i i ii
u u u k a k a a x ,t
m x m x x
= + + +
L . (7.6)
The gradient elasticity theory is founded on the assumption that the above gradient
stress-strain relations can be generalized so that they will apply to every point of the
continuum representation of the discrete system [4 p.61]. Thus, by the inclusion of the
higher-order derivatives, the gradient elasticity stress-strain relations (7.4), (7.5) and
(7.6) seem to conclude Hookes stress-strain relation (5.6) as a special case.
Actually, in view of the analysis of par. 6, this conclusion is not correct. Since the
gradient elasticity stress-strain relations (7.4), (7.5) and (7.6) result from the gradient
equation of motion (1.2), which is a special case of the equation of motion resulting
from Hookes stress-strain relation (5.6), the gradient elasticity stress-strain relations
(7.4), (7.5) and (7.6) must also be a special case of Hookes stress-strain relation (5.6) .
It is worth noticing that due to the underlying Taylors expansions (2.1), the gradient
elasticity stress-strain relations (7.4), (7.5) and (7.6) require that the higher-order
derivatives 3 3 4 4i iu x , u x , L exist and are continuous for every point of the
displacement distribution ( )u x,t . In contrast, the continuum formulation (5.6) of
Hookes law requires for the existence of the derivative ( ) x,t x in Newtons 2nd
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axiom (5.7) that only the second-order spatial derivative ( )2 2u x,t x exist.
On the other hand, if Hookes stress-strain relation (5.6) were a special case of the
general gradient elasticity stress-strain relations (7.4), then the coefficient of the strain
iu x in the general gradient elasticity stress-strain relations (7.4) and the
corresponding coefficient in Hookes stress-strain relation (5.6) would coincide, which,
however, does not happen. Indeed, by means of the second of equations (6.3) and
substituting i im l A= in the coefficient of the strain iu x in equation (7.4), we
obtain
2 2
1 1 12 2 2
N N Nij ji
ij ji ji iji i ji ij j j
E A a k a a E
m l A a l = = =
= =
)
, (7.7)
with ijE denoting the elasticity modulus of the spring connecting the i and j mass
points of the discrete system,
A denoting a common cross-section area for all springs, and
il denoting a given length around i mass point, whose multiple with the mass
density and the cross-section area A gives the lumped mass im .
Further, the stress ( )i x ,t of the total of the springs with one end at the i mass point,
which represents the stress in the continuum formulation (5.6) of Hookes law, should
coincide with the sum of the stresses ( )ij i x ,t of the springs. Thus, owing to the
common strain iu x for all springs, the elasticity modulus E of the total of the
springs, which represents the elasticity modulus in the continuum formulation (5.6) of
Hookes law, should coincide with the sum of the elasticity moduli ijE of the springs,
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that is,
1
N
ij
j
E E=
= . (7.8)
Comparing equations (7.7) and (7.8) yields
2
12
N
ij jii j
k a Em = )
, (7.9)
which verifies that the coefficient of the strain iu x in the general gradient elasticity
stress-strain relations (7.4) and the corresponding coefficient in Hookes stress-strain
relation (5.6) do not coincide, and hence, Hookes stress-strain relation (5.6) is not a
special case of the gradient elasticity stress-strain relations (7.4), (7.5) and (7.6).
8. Conclusions
Confining our analysis to one-dimensional systems, we have derived the general
gradient equation of motion for both short and long-range interactions of discrete
systems and the corresponding gradient elasticity stress-strain relations. Besides, the
discrete formulation of Hookes law underlying the derivation of the gradient formulas
proves to be a special case of the classical continuum formulation of Hookes law as a
stress-strain relation. Owing to the fact that the general gradient elasticity stress-strain
relation can be derived from the classical continuum formulation of Hookes law and is
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founded on more limited assumptions than those of the classical continuum formulation
of Hookes law, the latter cannot be a special case of the former. Hence, the gradient
elasticity stress-strain relation can only be a special case and not a generalization of the
classical continuum elasticity stress-strain relation. As a consequence, the solution of a
gradient elasticity problem can only be a special case of the general solution of the
corresponding classical continuum elasticity problem.
References
[1] T. von Karman & M. A. Biot, Mathematical Methods in Engineering, McGraw-
Hill, 1940.
[2] Timoshenko S. P. and Goodier J. N. (1970), Theory of Elasticity, 3rd edition,
McGraw-Hill.
[3] A. Dimarogonas, Vibration for Engineers, 2nd edition, Prentice Hall, 1996.
[4] A. Askar, Lattice Dynamical Foundations of Continuum Theories, World
Scientific, 1986.
[5] G. Stephenson, Mathematical Methods for Science Students, 2nd edition,
Longman, 1973.
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