p8-Gradient Elasticity in Comparison With Classical Continuum Elasticity

7/28/2019 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: [email protected]

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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p8-Gradient Elasticity in Comparison With Classical Continuum Elasticity