On the symmetry group properties of equations of nonlocal elasticity

Pergamon

Mechanics ResearchCommunications,Vol.26, No. 6, pp. 725-733, 1999 Copyright © 1999 ElsevierScience Ltd Printedin the USA. All fights reserved 0093-6413/99/S-see front matter

PII S0093-6413(99)00084-1

ON THE SYMMETRY GROUP PROPERTIES OF EQUATIONS OF NONLOCAL

ELASTICITY

T. C)zer

Istanbul Technical University, Faculty of Civil Engineering, Division of Mechanics, Maslak 80626, istanbul, Turkey ¢'Received8 April 1999; acceptedfor print 8 September 1998)

Introduction The early studies dealing with transformation groups have been started by Sophus Lie (see [1]) S. Lie used transformation groups in determining solution techniques of the ordinary differential equations. These kinds of groups depend on continuous parameters and they present contact transformations acting on the space of dependent and independent variables. A symmetry group of a system of differential equation is a group that transforms solutions of the system of differential equations to other solutions. He proved that if a differential equation remains invariant under a one-parameter Lie group of transformation then its order may be reduced by one. It has also been verified that these kinds of groups can be represented by their infinitesimals that contain dependent variables, independent variables and the derivatives of dependent variables as arguments. For partial differential equations these group invariant solutions are found by solving a reduced system of differential equations involving fewer independent variables than the original system. These groups created a deep effect on pure and applied mathematics, engineering, physics, and other mathematics-based sciences. The applications of Lie groups include such different fields as invariant theory, control theory, numerical analysis, classical mechanics, differential geometry and continuum mechanics. The determination of the symmetry groups of a system of differential equation provides the possibility of various types of applications. New solutions can be constructed for differential equations using their Lie groups. Thus the symmetry groups provides a means of classifying symmetry classes of solutions. Besides the symmetry groups can be used to determine the classification of differential equation families that depend on arbitrary parameters and functions. In that case the equations that have a high degree of symmetry should be preferred from the point of mathematical and physical reasons. One of the aims of these Lie group calculations is to determine the type of differential equations that admit a distinctive symmetry group. In literature typical examples are groups of rotations, translations and scaling symmetries. It is possible to calculate the fundamental solutions of some boundary value problems in theory of elasticity using their Lie symmetry groups. For instance, the solution of Boussinesq problem was calculated using its Lie point symmetries (see [2]). In addition to this, the Prandtl-Blasius problem for a fiat plate, nonlinear diffusion equation, fundamental solutions of the axisymmetric wave equation, an inverse Stefan problem and fundamental solutions of the heat equation are some of the examples discussed in literature as boundary value problems. During the last century these methods have been developed by various mathematical approaches based on infinitesimal transformations. L. Ovsiannikov (see [3]), G.W.Bluman (see [4]) and N. H. Ibragimov (see [5]) are some of the mathematicians who have huge number of studies in that field. 725

726

T. OZER

In this study, the aim is to investigate the Lie point symmetries of integro-differential equations. We try to find the symmetry groups of one-dimensional and two-dimensional equations of nonlocal elasticity. First of all, the determining equations are found, then the differential equations are obtained, which include kernel function. The symmetry groups are determined using these functions and solving determining equations. Then we get a classification with respect to kernel function. The classification of a problem with respect to Lie symmetries it accepts for different data is very important. The research techniques of Lie symmetries of an equation or a system of equations were improved in last twenty years. However the general methods cannot be given for a system of integro-differential equations. Nevertheless some mathematical formulations in mechanics produce integro- differential equations such as problems of visco-elasticity and nonlocal mechanics. The equations of visco-elasticity are handled and their classification with respect to Lie symmetries are done by S. V. Meleshko(see [6]). A. V. Bobylev(see [7]), S. I. Senashov(see [8]) and Yu. N. Grigoriev (see [9]) are some other mathematicians who have studied problems with integro-differential equations. The reader can refer to the comprehensive book by N H. Ibragimov cited above where fifth section of the third volume is about integro-differential equations (see [5]).

Fundamental equations of nonlocal elasticity

As is known in the linear theory of elasticity the stress tensor at a given point depends linearly on the strain tensor of the same point. Thus the classical theory of elasticity excludes the existence of long-range cohesive forces. In the nonlocal theory the stress at a point is assumed to depend on strain states of all points of the body. As a result the constitutive equation of nonlocal theory interrelating stresses and strains contains a term in the form of an integral over the whole of the body. The integral contains a kernel representing the effect of strain state of a point on the stress state of another. The theory of nonlocal elasticity effectively removes the inexplicable singularities encountered in the solutions of the boundary-value problems of classical elasticity and it is advantageously applied on the problems of dispersion of waves, dislocations, fracture mechanics etc. The theory were established by Eringen (see [10], [11], [13], [15]), Edelen (see [9]) and Kunin ( see [14]). The system of equations of nonlocal isotropic homogeneous elasticity can be given as follows (see [10]-[15]):

at~,(x,,x2,x3;t) &,

a2u~ :

= o - ~ i r - , x , , x ~,

tk,(X,,Xg,X,;t)=U~Sk, + 2 ~ k ,

..1_

3;t)

(1)

r t I(X.e,~fk, +21.t'e~,,)dv' V

where t u the nonlocal stress tensor, u k the displacement vector, ek~ the strain tensor, p the mass intensity, t the time, 7V,Ix are Lam6 constants, 7v',It' nonlocal elasticity moduli for homogeneous and isotropic bodies and e' = e(x',t) i) One-dimensional case: The governing equations of the problem for infinite one-dimensional medium are given as follows:

727

GROUP PROPERTIES NONLOCAL ELASTICITY 00

ct(x,t) = (X + 2~t)e(x,t) +

~K(x,x')e(x',t)dx' -oo

1~ - 2 ~, Ox co

~u v:--~,

~u e=-~,

(2) v~=et

where

K(x, x') = -~-~o~.'+ ;L+ z~t~t dy,dz,'

co -

p

We take up the following system of integro-differential equations that includes the onedimensional nonlocal elasticity as a special case:

vt =~x, et =vx 00

a(x,t)=e(x,t)+ ~K(x,x)e(%t)dx

(3)

-oo

The infinitesimal generator of the Lie groups is • X = ~v O + ~° ~ - + ~' ~e + ~ , ~ + rl~-~

(4)

The first prolongation is needed because the system of equations (3) has the derivatives of the first order. The first prolongation is

(Dt~ 0 - Otl]• t - D t ~ * x ) ~ - - ~ + ( D t ~

e- Dtqe .

-D,~e,)~-~+

(5)

(Dt,V - Dx~Vx - Dxlqvt ) ~--~+ (Dt," - Dx{ex -Dxrlet )~x + (Dt,° - Dtrlest - Dt~.x ) ~c Applying (5) to (3)1 we get the following equation

~,(c-~v,-

~vx)-~x(~ o - ~ox - ~o,)-- o

~6)

If (5) is applied to (3)~

Dt(~" - tier -~ex)- D~(~ ~ -~G - rlVt)= O

(7)

is obtained. By applying (5) to (3)3 we find:

[(~J-{ax-rl~t)-(~e-{ex-rlet)

- 7K(x,'t~e(x)-{(X)ex(X)-rl('c)et(x))dx I -oo

=0 (8) (sl

728

T. OZER

The existence o f the solution is assumed throughout the work. Thereby the equations are valid at a given moment t = t o . Hence we can examine the equations at the moment t = t o for the initial conditions v(X, to)=Vo(X ~

e(X, to)=eo(x)

(9)

The following initial conditions are obtainable using the system (3)

e~=e'o(X~

e,=Vo(X ~ vx=Vo(X )

v, =~x = e

x,x e o dx , (o(X)+ ! K x ()(~)1

~t = v

x,z v o z dz (o(X)+ if()()1

(10)

(11)

Substituting (10) and (11) in (6) and (7) we get vo (~: - ~ t - gl 1"1, - ;~)+ (Vo)2(__~ ) + g 2v° (flu _ ~ )+ g2 (~: + q x + eorl, )+ ; ; - e'o;: + eo~e (12)

- ~ + gl(~;-,~, - ~ +~x

+g,(-,~ +~o))=o

and + ~ . + g ~ o ) + (v; 12(~.-rb)-v'og2(rl,,)+ g2(~: -eo~,~)

v;(~:-rl,-~:

(13) .

,

~, -eo~,

-

v

~x

- e o' ~ , ~

+(¢

,

-eo~v - ~

gl

,

)

+rl~ +eorl , +g,rl,, = 0

where

gl = e'o(x)+ ;Kx(X,'Oeo(z)d~,

(14) Although merely initial conditions are substituted into (6) and (7), the results obtained below will produce the exact Lie symmetries since t o is lett arbitrary (see [6]). When (12) and (13) are decomposed in terms o f vo, v o' , vo' + f K ( x, z ) 'vo (x)dx and the coefficient of the each independent -oo

term is equated to zero we get

~, = ~ =~o =o,

~, =n~ =no =o

;~ - Cv~- ~, =o, ;;-q,-;;+~=o,

(16)

¢;+q

--o, ¢ ; = o

~,-4¢:-~: --g,(-~; +n, + ¢ : - ~ ) +;:)_;v

(is)

=

(-;:-

(17)

GROUP PROPERTIES NONLOCAL ELASTICITY

729

The decomposition of (17) can be achieved by observing that in a neighbourhood of the initial instant t = 0, such a fl (x) can be found that (18)

where f~ (x) is an arbitrary function. For the proof of the existence of a f~ (x) the reader is referred to Meleshko (see [5], [6]). Then, using the previous method of decomposition, the system of integral equations (17) can be decomposed to give the following system of determining equations:

~ - g ~ =o, ~ =o, ~ ; - n , - ~ ; + ~ =o, ~7-~; =o, ~,+q; =o, (; +2n~ =o ~ - ~ -~, : o, ~; - n , -~; +~x = o, ~ ; + n ~ = o , ~ ; = o

(19)

Using the determining equations, we find the functions of ~, 11,~', Q°, Q~ as follows:

= (alx+a2.)t +clx 2 - a s x + a 6 rl = (qt +c2)x +alt 2 +a3t + a 4 ~ =-(c,t +c2)a-(o,x +a2)e-ac, xv+ (- 2alt-a 3 + p)v+ ~,~, ~" = -2(c,t +c2)v+(a,t + 3c, x - a s - p)e + ~,=

(20)

;° = -2(alx +az)v+ (- 3alt- 2a 3 -c,x + p - a s ) a + ~ where ~,(x,t) is an arbitrary function. If we introduce zo, z~ as follows: z0 = ~° + 2v~,,, zl = ~" + 2vrl~

(21)

and rearrange (8) by using (21), we obtain c~

co

no

-ao

- I e° (xXKx (x, x)~(x)+ K, (x, x)~ (x)+ K(x, x)~ x (xlldx = 0 (22) If we rearrange (22) by integration by parts we get (23)

K(x, z)rlx(xl- K,(x, x~l(x)+ K,(x, xh(x)= O z o - z~- IK(x, xlz,(xldx- Ieo(xXKx(x,x)~(x)+ K,(x,x)~('c)+ K(x,x)~x(xl)dx=O -¢o

-~

From (23)1 we get a 1 = c1 = 0

(24)

730

T. OZER

If we introduce z 3 as a new variable we get

z3 = ( - a s x +a6)Kx(x,x)+(-a,T +a6)K~(x,x )

(25)

Here, a similar argument to the one given for expression (25) permits the decomposition

z~

=

f2(x)K(x,'c)

(26)

where fz (x) is an arbitrary function. For that property also, the reader is referred to Meleshko (see [5]), Using these properties equation (23)3 can be rearranged to get f2 = as,

a 2 ----c 2

(27)

The differential equations which include functions of ~, K are shown as follows:

a,K(x,x)= K.(x, xXa2t-asx +a6)+ K~(x, xXa2t-asx +a6)

(28)

Now, we will try to get a classification for (3) using (28). The calculations that were done for this purpose are shown as follows: For a general kernel

K = K(x,'c)

(29)

% = -as, ~,xt = m3

(30)

we get

and symmetries

X 1=0 t,

X 2 =Ov. X 5 =eO e+vov+ac3 o

(31)

For

we obtain the symmetries X~ = 8,, X2 = Or, X4 = xOx + tO,, X 5 = cO, + v8 v + aOo

(33)

For x(x

we find the following symmetries:

-

= x-

(34)

GROUP PROPERTIES NONLOCAL ELASTICITY

731

X l = O , . X2=Ov. X3=Ox X 5 = eO. + vOv + ~0o. X 6 = tc3x + xOt - (¢s + e)O v - 2vO. - 2vOo

(35)

Kernel(34) is valid under the following condition: lim xv(x,t) = 0

(36)

K = K ( x - x)

(37)

x~ao

For a difference kernel

the symmetries are

X~=O,,

X2=Ov,

X3 =0~,

X 5=eO.+vOv+oO~

(38)

All of the symmetries are shown in the following chart:

X 1, X 2 , X 5

X 1, X 2 , X 3 ,

X 1, X 2 , X 3 ,

X5

Xs. X6

X1, X 2 , X 4 , X 5

ii) Two-dimensional case: In this part we try to find the symmetry groups of two dimensional case using the same method. But we consider only the case of difference kernel here. It is clear from (1) that the system of equations for infinite two-dimensional medium for a difference kernel can be written as follows:

ey(X,y)= f~(x,y~ oo

gx(X,y)=hy(x,y)

oe

I IK'(x-r'Y-

sX~.(e(r,s)+ g(r,s))+ 2gg(r, s))drds +

-oO,-oO o',

m

II

--rt~oo

i iXx(x-r'y-s)(X(e(r's)+ g(r,s))+2

e(r, s))a a +

_e~--o~

(39)

i ] K e (x - r, y - s)(g(f(r,s)+ h(r, s)))drds = 0 ~o-oo

where eJ,,g,h are dependent variables and x,y,r,s are independent variables. Hence we can examine the equations at the moment y = Yo for the initial conditions

732

T. 0ZER

e(x,Yo)=eo(X~ f(x, yo)=fo(X ~ g(x, Yo)=go(X~ h(x,yo)=ho(x )

(40)

Then, using the method of decomposition (see [5]), we get the following determining equations for the compatibility equations (39)1

~:-~-~+~x

q, = q / = r l g =rlh =0, ~hg +~y =0,

e

=0, ~} +nx =o, ~ ey - ~

'Fix +Qgh =0,

=0,

Q) ~--0, ;~ _~h =0

(41)

Then using (39)2, (39)3 and (41) we get the following symmetry groups of two-dimensional problem ofnonlocal elastostafics for a difference kernel as Yl=XOx+YOy ,

Y2-~-Ox+Oy ,

X3=eO,+fOf+gOg+hO h

(42)

Conclusion

The symmetry groups of one-dimensional and two-dimensional elastodynamics problem of nonlocal elasticity are obtained. The author supposes that this study is a fundamental research directed towards the investigation of the symmetry groups of equations of nonlocal elasticity. As is known, the invariant solutions and solutions of some boundary value problems of partial differential equations could be found by using their symmetries. In the case of integrodifferential equations, the investigation of invariant solutions of these kinds of equations using their symmetries is not a trivial matter and the number of such researches is very few. Some solutions of integro-differential equations and nonlocal elasticity problems as well can be calculated using their symmetry groups. Thus, the results presented in this paper can be used as a preliminary tool in order to investigate solutions, symmetry groups of nonlocal elasticity problems and other problems including integro-differential equations as well. A specific field of application would be to get help from Lie symmetries in solutions of the boundary value problems of nonlocal elasticity whose classical solutions are singular. References

[1] [2] [3] [4]

S. Lie., Theorie der Transformationsgruppen, Vol. I, Vol. II, Vol. III, Leipzig, (reprinted by Chelsea Publishing Company, NewYork, (1970)) (1888, 1890, 1893). T. Ozer, Solution.of the Boussinesq Problem Using Lie Symmetries, Mathematical and Computational Applications, ( in press). L.V. Ovsiannikov, Group properties of differential equations, Academic Press (1982). GW. Bluman and S. Kumei, Symmetries and Differential Equations, SpringerVerlag, (1989).

GROUP PROPERTIES NONLOCAL ELASTICITY

733

[5]

N H.Ibragimov, Ed.,CRC Handbook of Lie Group Analysis of Differential Equations, Vol I, IL RI, (1994).

[6]

S.V. Meleshko, Group properties of equations of motions of a visco-elastic medium, Model. Mekh.2 (19), no.4, 114-126, (1988). A.V. Bobylev, The Boltzmann Equation and the Group Transformations, Mathematical Models and Methods in applied Sciences, Vol. 3, No. 4, (1993). S.I. Senashov, Lie symmetries and exact solutions for one-dimensional motion of elastoplastic media, Dokl. Akad Nauk S.S.C.R., (1990). Yu.N. Grigoriev and S.V. Meleshko, Group analysis of an integro-differential Boltzmann equation, Dokll. Akad Nauk S.S.C.R., 298(2), 323 (1987). A.C. Eringen and D.G.B. Edelen, On the Nonlocal Elasticity, Int. J. Engng. Sci., 10, 233-248, (1972). A.C. Eringen, Edge dislocation in nonlocal elasticity, Int. J. Engng Sci., 15, 177-183, (1977). g. Artan, Unsymmetrical Elastic Stamp on Nonlocal Elastic Half-plane, Computers and Structures, Vol. 63 No. 1, pp. 39-50, (1997). A.C. Eringen, C.G. Speziale, and BS. Kim. Crack-tip problem in non local elasticity, J. Mech. Phys. Solids, 25, 339-355, (1977). I.A. Kunin and A.M. Vaisman, On problems of the nonlocal theory of elasticity, in Fundamental Aspects of Dislocation Theory, Vol. 2, Ed. By J. Simmons and g. de. Wit, pp. 747-759, (1970). A.C. Eringen, Nonlocal continuum mechanics and some applications, in nonlinear equations in physics and mathematics, ed. By A. O. Barut, Reidel, Dordrecht, Holland, pp. 271-318, (1978).

[71 [8] [9] [lO] [111 [12] [13] [14] [15]