Fractional nonlocal elasticity and solutions for straight screw and edge dislocations Текст научной статьи по специальности «Математика»

УДК 539.3

Дробная нелокальная теория упругости и решения для прямых винтовых и краевых дислокаций

Y. Povstenko

Университет им. Яна Длугоша в Ченстохове, Ченстохова, 42-200, Польша

Нелокальная теория упругости использует определяющее интегральное уравнение напряжений, учитывает дальнодействующие силы межатомного взаимодействия, сводится к классической теории упругости в пределе длинных волн и к теории атомной кристаллической решетки в пределе коротких волн. В статье предложена нелокальная теория упругости, ядром (весовой функцией в определяющем интегральном уравнении напряжений) которой является функция Грина задачи Коши для уравнения дробной диффузии, выраженная через функции Май-нарди и Райта в виде обобщений экспоненциальной функции. В рамках предложенной теории получены решения для прямых винтовых и краевых дислокаций в бесконечном твердом теле.

Ключевые слова: нелокальная упругость, дробное исчисление, производная Капуто, производная Рисса, винтовая дислокация, краевая дислокация

DOI 10.24411/1683-805X-2020-12004

Fractional nonlocal elasticity and solutions for straight screw and edge dislocations

Y. Povstenko

Faculty of Science and Technology, Jan Dlugosz University in Czestochowa, Czestochowa, 42-200, Poland

Nonlocal elasticity assumes integral stress constitutive equation, takes into account interatomic long-range forces, reduces to the classical theory of elasticity in the long wave-length limit and to the atomic lattice theory in the short wavelength limit. In this paper, we propose the nonlocal elasticity theory with the kernel (the weight function in the integral stress constitutive equation) which is the Green function of the Cauchy problem for the fractional diffusion equation and is expressed in terms of the Mainardi function and Wright function being the generalizations of the exponential function. The solutions for the straight screw and edge dislocations in an infinite solid are obtained in the framework of the proposed theory.

Keywords: nonlocal elasticity, fractional calculus, Caputo derivative, Riesz derivative, screw dislocation, edge dislocation

1. Introduction

At the present time, it is generally recognized that process of deformation occurs at various structural levels with their own scale lengths [1-5]. Therefore, it is necessary to use specific physical concepts and mathematical tools at every structural level, while events progressing at different scale levels are interdependent.

Real crystals contain a large amount of defects. The problem of modeling stress and strain fields pro-

duced by imperfections in a crystal has attracted much attention of researchers. Imperfections in an elastic solid, as a rule, are modeled by certain distribution of concentrated forces or by certain distribution of the tensor of plastic strain incompatibility, see, e.g., [611]. A limitation of classical elastic solutions is that the stress fields have nonphysical singularities and the elastic energy diverges if one does not cut the defect core. The classical solutions cannot describe the situation in the strongly distorted region in the vicinity of

© Povstenko Y., 2020

the defect. Moreover, the classical elasticity is not valid in the immediate vicinity of the imperfection and fails to explain the phenomena at the atomic scale. This situation has given rise to the regular attempts to improve elastic models of crystal defects.

Recently, great advances have been made in crystal defect research by application of nonlocal elasticity. Several versions of nonlocal continuum mechanics based on various suggestions have been proposed (see [2, 3, 12-20], among others). Starting from interrelated equations describing elasticity and diffusion, Pod-strigach [12] excluded the chemical potential from the constitutive equation for the stress tensor and obtained the stress-strain relation containing spatial and time derivatives. In the infinite medium this relation can be integrated using the Fourier and Laplace integral transforms, and the final result has the nonlocal integral form with exponential kernel. Kroner [13], Krum-hansl [14] and Kunin [15, 19, 20] started from a discrete lattice and interpolated functions of discrete argument by special continuous functions. In this case, the stress at a reference point in the body depends not only on the strain at this point but also on the strains at all other points of the body. Thus, the relation between the stress tensor and the strain tensor has the nonlocal integral form. The interested reader is also referred to the recent book [21] and references therein.

Nonlocal elasticity takes into account interatomic long-range forces, reduces to the classical theory of elasticity in the long wave-length limit and to the atomic lattice theory in the short wave-length limit. The nonlocal theory of elasticity makes it possible to apply a unified approach to the study of phenomena at different structural levels. In contrast to many generalized models that improve the description of only one specific type of defects, in the context of nonlocal theory of elasticity one can consider various types of defects from a single point of view. A number of problems solved in the framework of nonlocal theory indicate the power of the theory. More justified results can be obtained, as the nonlocal elasticity is effective in removing nonphysical singularities occurring at dislocations, disclinations, points of applications of singular forces, cracks, etc. It should be emphasized that the theory of nonlocal elasticity indicates its power in the study of point defects [22, 23], straight edge [24] and screw [25-27] dislocations, straight wedge and twist disclinations [28], circular prismatic and glide dislocation loops [29], circular twist, rotation and wedge dis-clination loops [30], line cracks [3, 31, 32], and concentrated forces [33-36].

The fractional calculus (the theory of integrals and derivatives of noninteger order) has numerous applications in mechanics, physics, geophysics, rheology, engineering, chemistry, geology, biology, bioengineering, finance, and medicine, see [37-46] among many others. Very interesting line of investigation in the field of nonlocal theories consists in using fractional calculus. Di Paola and Zingales [47] formulated the governing equations in terms of the Marchaud fractional derivative of the displacement field and considered an elastic bar with long-range interaction. In the case of one spatial dimension, Challamel et al. [48] postulated the fractional differential relation between the uniaxial stress and uniaxial strain with symmetric Caputo fractional derivative and studied harmonic wave propagation in a bar. Carpinteri et al. [49] used the fractional Riesz integral in the constitutive equation relating uniaxial strain to the uniaxial stress in a bar. Tarasov [5053] considered long-range interactions of lattice particles, introduced the lattice fractional integro-differential operators and transformed a lattice field of some function into the corresponding continuum field function. The generalization of the Frenkel-Kontorova model of dislocation based on fractional-order differences was obtained in Ref. [54]. Tarasov and Aifantis [55] studied fractional gradient elasticity based on fractional Lapla-cian in the Riesz form. The results of Podstrigach [12] were generalized in Ref. [56] to space-time nonlocal dependence between stresses and strains based on the fractional calculus approach. Instead of the exponential kernel in the nonlocal constitutive equation for stresses, the kernel being the Wright function was obtained. Su-melka [57, 58] introduced fractional deformation gradients and fractional strains and used this approach in nonlocal elasticity [59].

Eringen [2, 3, 17, 24-26] proposed several forms of nonlocal modulus, in particular the nonlocal kernel being the Green function of the Cauchy problem for the diffusion equation. In Refs. [60, 61] the exponential Helmholtz nonlocal kernel was used to solve nonlocal elasticity problems for nanobeams. Lazar et al. [62] employed Eringen's model of nonlocal elasticity with bi-Helmholtz type kernels which are Green functions of partial differential equations of fourth order in the case of one, two and three spatial dimensions. They also obtained the solutions for straight dislocations. In the present paper, we propose the nonlocal elasticity theory with the kernels being the Green functions of the Cauchy problem for the fractional diffusion equation with the Caputo and Riesz derivatives and obtain the solution for the straight screw and edge disloca-

tions in an infinite solid in the framework of the considered theory.

2. Fractional nonlocal elasticity

For the static case with vanishing body forces, the basic equations for a linear isotropic nonlocal elastic solid are [3, 17, 26]:

V-t = 0, (1)

t(x, X) = Jy(|x -x'|, X)a(x')de(x'), (2)

V

ct(x') = 2^e(x') + Xtre(x')I, (3)

e(x') = -2[V'u(x') + u(x')V'], (4)

where x and x' are the reference and running points, t and ct are the nonlocal and classical stress tensors, u is the displacement vector, e the linear strain tensor, X and ^ are Lamé constants, V and V' denote the gradient operators with respect to x and x', respectively, I stands for the unit tensor. The volume integral in Eq. (2) is over the region occupied by the solid.

The nonlocal kernel y(|x - x'|, x) depends on the distance |x - x'| between the reference and running points and includes the parameter x connected with a characteristic length ratio a/l, where a is an internal characteristic length and l is an external characteristic length [2, 3]. The nonlocal kernel y(|x - x'|, x) is a delta sequence and in the classical elasticity limit x ^ 0 becomes the Dirac delta function.

For example, the nonlocal kernel y(|x - x'|, x) can be considered as the Green function of the Cauchy problem for the diffusion operator [3, 26]

Cy(x, x)

cf

--Ay(x, x) = 0,

x = 0: y(x, x) = S(x),

which results in the kernel

1

y(x, x) =

(2v^x )

-exp

|2\

4x

(5)

(6)

(7)

for n = 1, 2, 3 space variables.

In this case, the nonlocal stress tensor is a solution of the corresponding Cauchy problem [3, 26]

at(x, x)

cf

-At (x, x) = 0,

x = 0: t(x, x) =ct(x).

(8) (9)

In the present paper, we consider the nonlocal kernel y(|x - x'|, x) as the Green function of the Cauchy problem for the fractional diffusion operator

aay (x, x)

+ (-A)P/2y(x, x) = 0, cf

x = 0: y (x, x) = ô(x).

(10) (11)

Here Cf (x)/ôxa is the Caputo fractional derivative [37, 63]:

C" f (x) =_

ôxa T(m -a) 0

m -1 <a <m

1 f (x- h)m-a-1dmff, (12) J dum

L

T(a) is the gamma function. The Laplace transform rule for the Caputo derivative (12) has the following form:

f} = saf *(s) - I f(*}(0+ )sa-1-k, (13) dxa J k=0

m -1 <a<m,

where the asterisk denotes the transform, s is the Laplace transform variable.

The Riesz derivative (-A)p/2f (x) being the fractional generalization of the classical Laplace operator, obtained for p = 2, can be defined by its Fourier transform [63]

F{(-A)^2 f (x)} = | $ |p F{f (x)}, p > 0, (14)

where x is a vector of space variables, $ is a vector of transform variables.

In the framework of the proposed fractional nonlocal elasticity, instead of the Cauchy problem (8), (9) we obtain

+ (-A)P/2t(x, x) = 0,

axa

x = 0: t(x, x) =a(x). In Eq. (15), we assume 0 < a < 1 and 1 < p < 2.

2.1. One-dimensional case

Equations (10) and (11) are rewritten as

aay(x, x) apy(x, x)

(15)

(16)

Cxa ô|x|p x = 0: y(x, x) = S(x).

= 0,

(17)

(18)

Applying to the Cauchy problem (17), (18) the Laplace transform with respect to the parameter x and the

Fourier transform with respect to the spatial coordi- The two-fold Fourier transform in the case of axial nate x, we get symmetry is reduced to the Hankel transform of the

1 s a-1 zeroth order with respect to the radial coordinate r

Y ^ s) = ЖTTjlF, (19) [45, 64]- Hence

1 „«-i

where the tilde denotes the Fourier transform, E is the Y (r s) = ——a—"Tip", (29)

transform variable. - s +iri

Inverting the integral transforms, we get the nonlo- where the hat denotes the Hankel transform, and

cal kernel i 7

1 7 B a Y(r,T) = WEa ("iriP^a )^0(rE)EdE. (30)

Y (x, X) = — J Ea (-irip xa )cos(xE)dr. (20) 2- 0

For a = 1 and p = 2, taking into account integral Here Ea(z) is the Mittag-Leffler function in one para- (B1) from Appendix B, we get meter a (see Appendix A). 1 f 2 \

When p = 2, taking into account the relation [45] y(r, x) =-exp--. (31)

4-x

4t

v V

2» 2 fa

■ J Ea(-r )cos(xE)dr = M [ —;1 x 1J, (21) When a ^ 0 and p = 2, evaluating the corresponding

integral (see Eq. (B2)), we arrive at

where M(a; x) is the Mainardi function (see Appen-

1

dix A), we obtain the nonlocal kernel in the form y(r) = — K0 (r). (32)

21 f a i X i \

Y(x, x) =-MI —; —-2 J, 0 < a < 1, (22) Both kernels (31) and (32) were considered by Erin-

2x [ 2 X ^ gen [3, 26].

and

1 7 M fa i x - xX i ( ,)d , (23) 2.3. Three-dimensional central symmetric case

In the spherical coordinates in the central symmetric case

tv (^ т)=^^ iM (f;^ Ь ((23)

Due to Eq. (A10), for a = 1

Y(x, т) = —-^exp 2v лт

( x2 >

4т

(24) ах

+ ("A)^2Y(r, т) = 0, (33)

and x = 0: Y(r, x) = ^T8(r). (34)

7 4-r2

t'J (x, x) = 27-X J expf-i xX i Jci>(x')dx', (25) After employing the Laplace transform and three--7 fold Fourier transform (see Appendix C), we obtain

which coincides with the corresponding equations of the expression for the nonlocality kernel

[3].

Another particular case of (22) is obtained for a ^ , _ 2

0. Due to Eq. (A9), we get 2-Z r

1 For p = 2, taking into account that Y (x) =1 e-ixi. (26)

2 2 7 fa.

Th fi Hlhltk 1 • - J Ea (-r2)sin( xE) r dr = W [--,1 -a; x|, (36)

The exponential Helmholtz kernel was considered in - 0 [ 2 '

Refs. [60, 61].

i да

Y(r, т) J Ea (- | та )sin (r£) £ d^ (35)

0

0 < a< 2,

2.2. Two-dimensional axisymmetric case where W(a, P; x) is the Wright function (see Appen-

In the polar coordinates in the axisymmetric case dix A), we obtain the nonlocality kernel in the form

^ + (-A)^2Y(r, X) = 0, (27) Y(r, X) = ^W^f,1 -a; \. (37)

Putting in (37) a = 1 and accounting for Eqs. (A8)

dr1

1

т = 0: Y(r, т) =-5(r). (28) and (A10), we arrive at the nonlocality kernel

2%r

y(r, X) =-

1

8( rcx)3/2 proposed by Eringen [3, 26].

exp

2

r 1 4x

(38)

3. Screw dislocation in the framework of fractional nonlocal elasticity

In what follows, we restrict ourselves to the case P = 2. The classical elasticity stress tensor for the straight screw dislocation has one nonzero component [6, 7]

ctze= -, A = ^, (39)

where b is the Burgers vector, p is the Lamé constant.

In the framework of the proposed fractional nonlocal elasticity, taking into account the expressions for components of the Laplacian of the stress tensor (see Appendix D) we obtain that the component tze of the nonlocal stress tensor satisfies the equation

Cat,

z e

^C2 tze 1 Ct

Cxc

+ —

z e

Cr2 r Cr

1

~~ tze r

= 0

j

under initial condition

f = 0: tze=-.

A r

(40)

(41)

The Laplace transform with respect to the parameter x and the Hankel transform of the first order with respect to the radial coordinate r give

Ce fê,s) =

A

a-1

Ç sa+^2'

After inverting integral transform we get

t..e (r, x) = - JE" (-£,2fa )Mr

0

(42)

(43)

Consider several particular cases of the solution (43). For a = 1, taking into account Eqs. (A2) and (B3), we arrive at the solution

tze( r, X) = —

1 - exp

2

r

4x

y j

(44)

which coincides with that obtained by Eringen [25, 26] for the exponential nonlocal modulus.

When a ^ 0, taking into consideration Eqs. (A3) and (B5), we have

tze ( r ) = A

1 - K1( r )

r

(45)

The solution (45) was also considered by Eringen [3, 26] for the nonlocal modulus being the modified Bes-sel function.

Figure 1 shows the dependence of nondimensional stress component

_ xa/2t,

tze =-^

ze A

ze

on nondimensional spatial coordinate p = r/xa/2. We also present the classical local solution (1/p).

4. Edge dislocation in the framework of fractional nonlocal elasticity

The stress tensor associated with the straight edge dislocation in an infinite solid in polar coordinates has the nonzero components [6, 7]

B B B

a rr =--sin 0, a00 =--sin 0, ar 0 = — cos 0, (46)

r r r

B =

pb

2rc(1 -v )

where b is the Burgers vector, v is the Poisson ratio. Let

trr = Trr sin 0, t00= T00 sin 0, tr0= Tr0 cos 0. (47)

From the Cauchy problem (15), (16) with p = 2, taking into consideration the components of the Lapla-cian of the stress tensor in polar coordinates (see Appendix D), we obtain the following system of equations for determining Trr, T00 and Tr0:

Fig. 1. Dependence of nonlocal stress caused by the screw dislocation: classical (1), a = 0.00 (2), 0.25 (5), 0.50 (4), 0.75 (J), 1.00 (6)

daT v J-rr "d 2T ^ ±rr

dxa _ dr2

" + il rr 2 r 2 Tr e 2 ' r

daTee "d 2Tee

dxa _ dr2

„ 4 ee r 2 ■r + -2- Ere+ 2 r 2

daT o ire "d 2Tr e

dra dr2

da g

.+1 STr

r dr (Err _ Tee )

= 0,

1 ST

ee

r dr

= 0, (48)

1 ST

- + —

r e

r dr

5 2

~Tre + ~(Trr _Tee) r r

= 0,

under initial conditions

t = 0: Trr = -B, Tee=--, TrQ= B. (49) r r r

The system of coupled equations (48) can be split-ted into independent equations by introducing the combinations of the components

f(r, t) = Trr(r , t) + Tee(r , tX

g(r , t) = Trr(r , t) - Tee(r , t) - 2Tr e(r , t) (50)

h( r, t) = Trr(r , t) - Tee(r , t) + 2Tre(r, t).

For the auxiliary functionsf g and h, we have a system of equations

dff s2f, i df i

dxa

. - +- .

dr2 r dr r2

f

= 0,

Fig. 2. Dependence of nonlocal stress component trr caused by the edge dislocation: classical (7), a = 0.00 (2), 0.25 (3), 0.50 (4), 0.75 (J), 1.00 (6)

Sra vdr 2

da h

dxa vdr 2

^d2g , 1 Sg_g

n 2 &

r dr r

= 0,

(51)

- +----2 h

r dr r2

= 0,

under initial conditions

x = 0: f = _2B, g = 2B, h = _2B.

r r r

(52)

The Laplace transform with respect to the parameter t and the Hankel transform (of the first order for the functions f and g and the third order for the function h) with respect to the radial coordinate r give

a-1

/(%, s ) = _2

B sL

% sa+%2

¿(%, s ) = 2

B s

a_1

h (%, s) = _2

% sa+%2

B sa_1 % sa+%2

(53)

Carrying out the inverse integral transforms, we finally obtain

B ^

trr = - - J Ea (2 Ta )[ Ji( r £) + J3( r £)] d^ sin e, 2 0

B ^

tee = J Ea K2 Ta )[3Ji( r£) - rQ] d^ sin e, (54)

B ^

tre = - J Ea (-^2Ta )[JiK) + J3(r^)]d^ cos e. 20

For a = 1, taking into account integrals (B3) and (B4) from Appendix B, we arrive at

B l 4t

trr =--0 —T

B K u 4t

tee = _ 7 j_1 + l 2 + ^

B_

r

tre=-<1

4r

' 2

1 _ exp

1 _ exp 1 _ exp

2

r

4t

y j

2

r

4t

sin e,

sin e, (55)

j a

2

r

4r

cos e.

Assuming l2 Ik2 = 4t, Eqs. (55) coincide with the corresponding equations of Eringen [3, 24].

For a ^ 0, using integrals (B5) and (B6), we get

B

trr 2

2 _ A _ K1( r ) + K3( r ) rr

sin e,

_ B

tee- y

_ B

tre _ г

2 + A-3Ki(r)-Кз( r) r Г

_ -Ki(r) + Кз(Г )

sin e, (56)

cos e.

Ex (^)

, a > 0, z e С,

(A1)

k-o T(ak +1)

provides a generalization of the exponential function

Ei( z) -I _ ez. k-o k!

(A2)

Figure 2 shows the dependence of nondimensional stress component Trr = xa/2trr/B on nondimensional distance p = r/xa/2 for 9 = л/2 and different values of the order a of the derivative. We also present the classical local solution (1/p).

5. Concluding remarks

In the present paper, we have considered the generalization of Eringen's theory of nonlocal elasticity in which the nonlocal modulus in integral constitutive equation for the stress tensor is the Green function of the Cauchy problem for fractional diffusion equation with the Caputo derivative of the order 0 < a < 1 with respect to the nonlocality parameter x and the fractional generalization of the Laplace operator with respect to the spatial coordinates of order 1 < p < 2. Eringen proposed several nonlocal moduli which in the framework of fractional nonlocal elasticity are obtained as particular cases of the Wright function and the Mainardi function. The stress fields caused by the straight screw and edge dislocations in an infinite solid are investigated. The studied solution for the screw dislocation in the case 0 < a < 1 interpolates the appropriate results of Eringen obtained for different nonlocal kernels associated with the values a = 0 and a = 1. The solution (56) for the edge dislocation corresponds to the solution (45) for the screw dislocation. The components of the stress tensor reach a maximum at some distance from the dislocation line. The point of the maximum and the maximum value of stress depends on the order of fractional derivative a; the values of maximum are consistent with estimations of Eringen. In the general case, the proposed theory can be used for better matching the theory of elasticity and the atomic lattice theory.

Appendix A

The Mittag-Leffler function in one parameter a [37, 63, 65]

Another particular case of the Mittag-Leffler function is

Eo(-z) _-+-.

1+z

(A3)

The Mittag-Leffler function Ea(z) appears in the inverse Laplace transform

„a—1 1

L

■i J s

s a+ b

_ Ea (-bxa).

(A4)

The Wright function in two parameters a and p is defined as [37, 63, 65]

W(a, P; z) _ I

-, a > -1, z e С. (A5)

k=0 k !T(ak + p)

For the derivative of the Wright function the following equation is satisfied: dW(a, p; z)

dz

_ W(a, a + P; z).

(A6)

The Mainardi function in one parameter a is a special case of the Wright function

M(a; z) = W(-a, 1 - a; - z), 0 < a < 1,

or

M(a; z) _— W(-a, 0; - z). az

(A7) (A8)

In particular,

M(0; z) _ W(0,1; - z) _ e-z,

M12; z)_ W Й; - z H exp

( z2 >

(A9) (A10)

Appendix B

Integrals (B1)-(B3) are taken from Ref. [66]; integrals (B4)-(B6) are evaluated by the author:

Ю _ 2 1 f xe px J0(qx)dx _— exp

0 2p

( „2\

4 p

f

2 . 2 0 x + p

J0(qx)dx _ K)( pq),

f e- px2 J1 (qx)dx _ -

2

1 - exp

4 p

(B1) (B2) (B3)

f e-px2 J3 (qx)dx

1 -

8 P

8 P

2

exp

4 p

(B4)

k

k

z

x

k

7 i ii

J-2-2 Ji(qx)dx = —2--Ki(pq), (B5)

-2 ■ "2 qp2 p

2 2 o x2 + p2

7 1

J-2-2 J3(qX)dx

0 x + p

1 8 1 ^ , ^

——tt +_ K3( pq)-

qp2 q p p

(B6)

Here Jn(z) is the Bessel function of the first kind, Kn(z) is the modified Bessel function.

Appendix C

The three-fold Fourier transform in the case of spherical symmetry can be reduced to the sin-Fourier transform of the special type [45, 64]:

F {f (r)} = f ($) = A P J rf (r )si^dr, (C1)

F-1{f (£)> = f (r) = J2 J £ f (^^^d^. (C2)

In this case

F { f +1 iM W f («. (C3)

I dr2 r dr I

Appendix D

The Laplacian of the symmetric tensor in polar coordinates has the components [28]:

(At)rr = Atrr -44(trr -100), (D1)

r2 50 r

(At )ee = Ate,

4 5tre -4(t„. - tee), (D2)

ae

(At)re=Atre- 4 tre + 4 , (D3)

r2 r2 ae

1 2 atr

(At) ze = Atze tze +

ae '

where

Af =

a2 f , 1 af , 1 a 2f

dr2 r ar r2 ae2'

(D4)

(D5)

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Received 22.01.2020, revised 22.01.2020, accepted 27.01.2020

Сведения об авторе

Yuriy Povstenko, Prof., Jan Dlugosz University in Czestochowa, Poland, j.povstenko@ujd.edu.pl