Jtam-A4.dvi


JOURNAL OF THEORETICAL SHORT RESEARCH COMMUNICATION

AND APPLIED MECHANICS

54, 2, pp. 671-674, Warsaw 2016
DOI: 10.15632/jtam-pl.54.2.671

ON GEOMETRICAL INTERPRETATION OF

THE FRACTIONAL STRAIN CONCEPT

Wojciech Sumelka

Poznan University of Technology, Institute of Structural Engineering, Poznań, Poland

e-mail: wojciech.sumelka@put.poznan.pl

In this paper, for the first time, the geometrical interpretation of fractional strain tensor
components is presented. In this sense, previous considerations by this author are shown in
a new light. The fractional material and spatial line elements concept play a crucial role in
the interpretation.

Keywords: fractional strain, fractional calculus, non-local models

1. Introduction

The fractional strain is a generalisation of the classical strain measure utilising the fractional
calculus (the branch of mathematical analysis which deals with differential equations of an
arbitrary order (Podlubny, 2002)). Such defined strain is non-local because of the fractional
derivative definition.
In the literature, there exist a fewconcepts of fractional strain.Onecanmentionhere thoseby

Klimek (2001), Lazopoulos (2006), equivalent concepts ofAtanackovic andStankovic (2009) and
Carpinteri et al. (2011) or, finally, that by Drapaca and Sivaloganathan (2012). It is important
that except for the concept presented in Drapaca and Sivaloganathan (2012), the previous ones
were defined for 1D problems and small strains. Of fundamental meaning is also the fact that
these authors consider different physical units of fractional strain tensor components, e.g. in
Klimek (2001), Atanackovic and Stankovic (2009), Carpinteri et al. (2011) we have [m1−α], in
Lazopoulos (2006) [m−α], or in (Drapaca andSivaloganathan, 2012) [m3−α1k−α2k−α3k]k=1,2,3,
where m denotes meter, and the parameter α is in general different than 1.
In thepaperbySumelka (2014c) adifferent concept of fractional strainwaspresented. In that

version, the fractional strain is without physical unit, as in the classical continuum mechanics,
and the length scale parameter is given explicitly and simultaneously related to the terminals of
the fractional differential operator.
In this paper, we follow the fundamental results given in the abovementioned paper (Sumel-

ka, 2014c), giving finally the geometrical interpretation of fractional strain tensor components.

2. Geometrical interpretation of fractional strain

The description is given in the Euclidean space in Cartesian coordinates. We refer to B as the
reference configuration of the continuum bodywhile S denotes its current configuration. Points
in B are denoted byX and in S by x.
The regular motion of the material bodyB can be written as

x=φ(X, t) (2.1)

thus φt :B→S is aC
1 actual configuration of B in S, at time t.



672 W. Sumelka

Taking the Taylor expansion of motion for dX, we have

φ(X+dX, t)=φ(X, t)+
∂φ(X, t)

∂X
dX+ |dX|r(X, t,dX) (2.2)

with the property of the residuum that lim|dX|→0 |r(X, t,dX)| = 0. Denoting dx = φ(X+
dX, t)−φ(X, t) and omitting higher order terms, one gets

dx=FdX (2.3)

and

dX=F−1dx (2.4)

where F(X, t) = ∂φ(X, t)/∂X denotes the deformation gradient, and F−1(x, t) = ∂ϕ(x, t)/∂x.

We introduce non-local effects throughmultiplication of Eq. (2.3) (left sided) by
α

F
X

andEq. (2.4)

(left sided) by
α

F
x
, thus

dx̃= F̃
X

dX (2.5)

and

dX̃= F̃
x
dx (2.6)

where (following thenotation in (Sumelka, 2014c)), dx̃=
α

F
X

dx is a fractional spatial line element,

dX̃ =
α

F
x
dX is a fractional material line element, while F̃

X

=
α

F
X

F and F̃
x
=
α

F
x
F−1 are fractional

deformation gradients defined as follows

F̃
X

(X, t)= ℓα−1
X
D
X

αφ(X, t) (2.7)

and

F̃
x
(x, t)= ℓα−1

x
D
x

αϕ(x, t) (2.8)

where ℓX and ℓx are length scales in B and S, respectively. In Eqs. (2.7) and (2.8), D
α is

the Riesz-Caputo fractional differential operator while α denotes the order of differentiation,
cf. Sumelka (2014c). Comparing Eq. (2.3) and Eq. (2.5) (or Eq. (2.4) and Eq. (2.6)), one can
also interpret such an assumption (by analogy to (Drapaca and Sivaloganathan, 2012)) as the
existence of motion of the order α, which means the motion accounting for non-local effects.
The situation is summarised in Fig. 1.
Notice that the length scales ℓX, ℓx preserve classical physical unit [m], and together withα,

they are additional material parameters. As an example, for metallic materials, they can be
identified as distances connected with non-homogeneous distribution of dislocations and cell
structures (Pecherski, 1983; Sumelka, 2014b).

We have now four ways to define the strain tensor (cf. Fig. 1). Denoting by
⋄
F deformation

gradients F or F̃
X

or F̃
x
or
α

F, one can obtain local/non-local classical/fractional strain tensors

through classical rules, namely

E=
1

2

(⋄
F
T
⋄
F− I

)
e=
1

2

(
i−

⋄
F
−T
⋄
F
−1
)

(2.9)



Short Research Communication – On geometrical interpretation of... 673

Fig. 1. Relations between thematerial and spatial line elements with their fractional counterparts

(
α

F= F̃
X

F−1F̃
x

−1

,
α

F
x
= F̃
x
F and

α

F
X

= F̃
X

F−1)

whereE is the classical Green-Lagrange strain tensor or its fractional counterpart, and e is the
classical Euler-Almansi strain tensor or its fractional counterpart.
It should be emphasised that appropriate mapping of terminals from a material to spatial

description (or inversely – cf. analogy in Sumelka (2014a)) that fulfil

α

F
X

=F
α

F
x

−1
F
−1 or

α

F
x
=F−1

α

F
X

−1
F (2.10)

assures that
α

F
X

=
α

F
x

−1, so then the operating on the pair dX̃→dx or dX→dx̃ is equivalent.

We can nowdrawapicture showing the geometrical meaning of fractional strain components

– cf. Fig. 2 (
⋄

(·) stands for classical or fractional line elements). It is clear that extension (normal

strain) of a (fractional) material line element d
⋄
X= |d

⋄
X|e is defined as

⋄
ε=
|d
⋄
x|− |d

⋄
X|

|d
⋄
X|

or
⋄
ε=

√
1+2e ·

⋄
Ee⇔ e ·

⋄
Ee=

⋄
ε+

⋄
ε2

2
(2.11)

where e is a unit vector along the fibre direction.

Fig. 2. Geometrical interpretation of the fractional extension and shear

The shear (shear strain) is defined by the deviation from orthogonality of two (fractional)

material line elements d
⋄
X1 = |d

⋄
X1|e1 and d

⋄
X2 = |d

⋄
X2|e2, namely (cf. Fig. 2)



674 W. Sumelka

sin
⋄
γ12 =

d
⋄
x1 ·d

⋄
x2

|d
⋄
x1||d

⋄
x2|

or sin
⋄
γ12 =

2e1 ·
⋄
Ee2+e1 ·e2√

1+2e1 ·
⋄
Ee1

√
1+2e2 ·

⋄
Ee2

(2.12)

where e1 and e2 are unit vectors along the fibres directions. In the case when initially the
material line elements are perpendicular, e1 ·e2 =0.

3. Conclusions

Geometrical interpretation of the fractional strain components is the same as that for classical
strain. It is because of its analogical definitionwhich is basedon fractional (’scaled’)material and
spatial line elements. Hence, the extension is the ratio of the difference of squares of current and
initial elemental lengths and squared initial elemental length. At the same time, shear defines
the deviation from orthogonality of two elemental line elements (in fractional picture theymust
not be initially perpendicular).
It is important that an analogous geometrical interpretation can also be applied to other

competitive formulations known in the literature (cf. Section 1 andpaper (Sumelka et al., 2015)),
where similarities between formulations are shortly listed) – however one should remember that
they operate on different physical units.

References

1. Atanackovic T.M., Stankovic B., 2009,Generalizedwave equation in nonlocal elasticity,Acta
Mechanica, 208, 1/2, 1-10

2. Carpinteri A., Cornetti P., Sapora A., 2011, A fractional calculus approach to nonlocal
elasticity,European Physical Journal Special Topics, 193, 193-204

3. Drapaca C.S., Sivaloganathan S., 2012, A fractionalmodel of continuummechanics, Journal
of Elasticity, 107, 107-123

4. Klimek M., 2001, Fractional sequential mechanics –models with symmetric fractional derivative,
Czechoslovak Journal of Physics, 51, 12, 1348-1354

5. Lazopoulos K.A., 2006, Non-local continuummechanics and fractional calculus,Mechanics Re-
search Communications, 33, 753-757

6. PecherskiR.B., 1983,Relationofmicroscopicobservationsto constitutivemodelling foradvanced
deformations and fracture initiation of viscoplasticmaterials,Archives ofMechanics, 35, 2, 257-277

7. Podlubny I., 2002,Geometric and physical interpretation of fractional integration and fractional
differentiation, Fractional Calculus and Applied Analysis, 5, 4, 367-386

8. Sumelka W., 2014a, A note on non-associatedDrucker-Prager plastic flow in terms of fractional
calculus, Journal of Theoretical and Applied Mechanics, 52, 2, 571-574

9. SumelkaW., 2014b,Applicationof fractional continuummechanics to rate independentplasticity,
Acta Mechanica, 255, 11, 3247-3264

10. Sumelka W., 2014c, Thermoelasticity in the framework of the fractional continuum mechanics,
Journal of Thermal Stresses, 37, 6, 678-706

11. Sumelka W., Zaera R., Fernández-Sáez J., 2015, A theoretical analysis of the free axial
vibration of non-local rods with fractional continuummechanics,Meccanica, 50, 9, 2309-2323

Manuscript received January 8, 2016; accepted for print February 5, 2016