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Torsion of micropolar elastic beams
lnt. J. Engng Sci.. 197 1, Vol. 9, pp. 1047-1060.
TORSION
Pergamon Press.
OF MICROPOLAR
Printed in Great Britain
ELASTIC
BEAMS
DORIN IESAN University of Jassy, Jassy, Romania Abstract-The present paper is concerned with the torsion problem of homogeneous and isotropic beams in the linear theory of micropolar elasticity [ 1, 21. The problem is solved in terms of three torsion functions. The existence theorem is derived by the method of potentials [3]. 1. INTRODUCTION
of the torsion of a cylindrical beam is very important and it was very much studied in the classical elasticity (see e.g. [4-61). In this paper we consider the torsion of isotropic beams in the linear theory of micropolar elasticity. Three micropolar torsion functions are introduced and their existence is derived by method of potentials. Initially, the problem of torsion of micropolar elastic beams was considered in [ 1O]t.
THE
PROBLEM
2. BASIC
EQUATIONS
Let V be a region of space occupied by a micropolar elastic material whose boundary is S. We assume that the body forces and the body couples are absent. The basic equations in the static theory of homogeneous and isotropic micropolar elastic solids are[l, 21: -the equilibrium equations tji.j =
09mjf,j+ EimJmn= 07
(2.1)
-the constitutive law tu=hETrSii+(EL+K)Eij+~Eji,mlj=ffcp,,,Gij+P~i,j+Y~j,i,
(2.2)
-the geometrical equations Eu=
Uj,*+
Eji&me
G.3
In these relations we have used the notations from [ 1,2]. The surface tractions and surface couples acting at a point x(xk) on the surface S are given by ti = tjinj, mi = F?lj$lj,
(2.4)
where nj = cos (n,, Xj) and II, is the unit vector of the outward normal to S at X. 3. TORSION
PROBLEM
We consider a cylindrical beam of homogeneous and isotropic micropolar elastic material bounded by plane ends perpendicular to the generators. The cross-sections I: ?The paper[ lo] had not been available to the present writer. After this note was submitted, the writer was kindly informed by a referee about it. 1047
1J.E.S. Vol. 9 No. 1 I -A
DORIN
1048
lE$AN
is assumed to be a simply connected region bounded by a closed Liapunov curve L. We suppose that the body forces and the body couples are absent and the lateral surface is free of applied forces and couples. Throughout this paper the axis OX,of our coordinate system will be directed parallel to the generators of the beam. The beam is assumed to be of length 1,and one of its bases is taken to lie in the x,0x,-plane, while the other is in the plane x3 = 1. We suppose the beam to be kept in equilibrium when the end xQ= 1 is twisted by a couple of magnitude M. On the plane x3 = 1 we have the following conditions
Ir t31 du = 0,
I ts2 do = 0, z
s
(x2fs3+ m3J da = 0,
z
I fS3dc = 0, 5:
(3.1)
I (x,t33-mz) da = 0, L
(3.2)
s
(xlfB2-x2t31 + m33)da = M.
(3.3)
z
The resultant forces and moments calculated across each cross-section satisfy the conditions of equilibrium, so that the conditions (3.1)-(3.3) must be satisfied for .Q = h(0 s h s 1)[6]. On the lateral surface of the beam we have the following conditions t&no = 0,
m&&=0,
(k= 1,2,3;a=
1,2).
(3.4)
The torsion problem consists in the solving of equations (2.1)-(2.3) with the conditions (3.1H3.4). We try to solve the torsion problem assuming that l41 = -7X2X3, 91
=
cpl (Xl,
112= x2)
7
7X1X3,
cpz=cpz(-G~~~),
u3=
(3.5)
Qb,rX*), (P3=7x3,
where the functions Cp(x,, x2), pa(xI, x2) and the constant T will be determined in the following. From (2.2) and (3.5) we get h=
fm=0,f33=0,
(3.6)
(~+K)@,lz-KQ~+/.‘TX~r
f32=ll~,3+K(P1+(CL+K)TX1,
~~I=@J-KQ~-(~+K)~X~,
(P+KK)@J+KQ~-~TX~, mv3= m3”=0, m33=~cp,,,+(a+p+y)7, = ff%.PSW + P%.v + Yso,,” + ~6m m, f13=
(v,q,p=
1,2).
The equilibrium equations (2.1) will be satisfied if the functions Q (x,, x2), cpo!(x,, x2) satisfy the following equations
(F+K)A@---(QI,~--Q~.~)= 0, YAQS + (Q + ~)Qv,yl)
+KEqp3@,p-2KQOp=KTXp,
(3.7) (r).V,p=
1,2),
Torsion of micropolar elastic beams
1049
throughout the cross-section of the cylinder, where A is the Laplacian operator. From (3.4) and (3.6) we obtain the following boundary conditions for the functions @,cp,
(P+KK)$h’ln,-~d
=/-d-W-X,~2),
r(~+P+Y)(Pl,l+~Y(P*,21~1+
m%l+y~1.21n*
=-wmlr
[9k1+ (a+p+r)(~*,~ln*=-(~7n~, 0nL.
[PPL~+Y~~.II~I+
(3.8)
Let us establish the uniqueness of the solution of the boundary value problem (3.7), (3.8). From (2.1)-(2.4) and the divergence theorem we obtain [ 11 j- (tiui + rn& du = 2 j- E dv, s V
(3.9)
where E is the internal energy density. Let us consider two solutions of the torsion problem lp=-Tx
2
x 39
L&j
soY~=P:*‘(&J2)~
=
7x1
x2,
up=
(3.10)
@)(x1,x2),
$e’=(pY?x1,x2),
&=7x3,
((y=
1,2).
In what follows it will be shown that the constant 7 is uniquely determined if M is given. We note f& =
4”
_
&
421,
=
-42’.
(3.11)
in V.
(3.12)
p$l’
Obviously ii1= ii2= (p3= 0,
On the lateral surface (S,) we have ti = 0, mi = 0. On the surface x3 = /(C,) we have t3=t33,~a=m3a(rx= 1,2).Fr om (3.6) it follows that on the & we have t3 = 0, m, = 0. Similarly, t3 = 0, m, = 0,on the surface x3 = 0 (C,). Thus, we get
J (Tiii,+ fi$icpl)dc = J (iiii, + A+i) do + J (iitii + %$i) do + J (?iii,+ fiGi) do = 0, s
SL
II
20
and using (3.9) we obtain s adv=O,
(3.13)
V
where Gis the internal energy density corresponding to the system (3.10). From (3.13) we obtain[l],
DORIN
1050
IESAN
From (3.12) and (3.14) weget 23 = c,
cpl = 0,
(3.15)
+3 = 0,
where c is an arbitrary constant, Using (3.9) we can establish a relation which is important in the following. Taking into account that on the X1 we have t, = taa, t3 = 0, m, = 0, m3 = ms3, u1 = -TX&, u2 = 7x11,p3 = rl and on Z,, we have u, = 0, t3 = 0, m, = 0, (p3= 0, we obtain J (tfui+
m33>da.
do = IT J (Xlt32 -~2131+ z
m$Qi)
s
(3.16)
Obviously E = E(x, , x2), so that (3.17)
From(3.9),(3.16),(3.17)weobtain 7
Iz
x2t31-t m33) da = 2s edcr. z
(w32-
In what follows it is convenient &(x1, xJ by the following relations Q, = rQ,
(3.18)
to introduce the torsion functions cp(x,, X.J, JI~(x~,x2),
(3.19)
$‘a = T(l(la-&XQ), (a = 1,2).
If we substitute (3.19) in (3.7) we get (~u+K)AQ--K(~LI,z-~I) =
yA&, + (a + ,@$v,vrl+ KEqpsQ,p-
2K$Jp= 0,
0,
(3.20)
(VT 9, P =
From (3.8) and (3.19) we obtain the following boundary functions
1~ 2).
conditions
for the torsion
Using (3.6) and (3.19) we get t31 = dPQ,l
-K$z-!i(2P+K)X,l,
t,, =~C~Q,~+KI(I,+)(~CL+K)XI~~ m33 =
d(*1(h+hd
+P+rl.
(3.22)
1051
Torsion of micropolar elastic beams
Taking into account (3.22), from (3.3) we obtain (3.23)
rD=M, where
D=j
[C~(X~(~,~-X~(P,I)+K(X,~~+.X~~~)+~(~CL+K)(X::+X~~)
z
+~(+m+hd
+P+yldu.
(3.24)
The constant D is the micropolar torsional rigidity of the beam. From (3.18) and (3.22) weget r2D = 2 I E du, P
(3.25)
D > 0.
(3.26)
functions (o, &, Qz are determined
the constant r can be calculated from
so that we conclude that
Thus, if the (3.23). Taking into cally satisfied. brium equations
IL
t31 da
=
j
z
account the relations (3.6) we see that the conditions (3.2) are identiLet us show that the conditions (3.1) are satisfied. Using the equiliwe can write [t13-
(m 12,1+ m22,2)
1du = j [t13+x1b1 + f23.2) - h2.1 + md 1do I
Making use of the Green’s theorem and of the boundary conditions (3.4), we obtain
Iz t31du = I
[xl (m
+ j2s2> - bw,
+
mz2n2>l ds = 0.
L
In a similar way it follows that the condition (3. 1)2 is satisfied. The last of the conditions (3.1) is identically satisfied on the basis of (3.6). 4. EXISTENCE
THEOREM
In what follows we consider the system (3.20) with the boundary conditions (3.21). Using a Galerkin representation we determine the fundamental solutions and introduce the elastic potentials. With the help of the method of potentials [3] we reduce the boundary value problem (3.20), (3.21) to singular integral equations for which Fredholm’s basic theorems are valid. The existence of the micropolar torsion functions is derived. The procedure is analogous with that used in the plane strain problem of a homogeneous and isotropic body [7].
DORIN IESAN
10.52
We introduce the matrix u=
(4.1)
(%$I,&)5
and the operators 2,~ = (~++K)h~--K(J11,2-~~,1), ~zu=?‘A’!‘~+
(4.2)
(~+P)~Y,v~+K(P,z-~K~~,
~P,U = rA$z + (a + p)‘!‘v,vz- K’f’,l - 2’@‘21 and
(4.3)
(4.4)
The system (3.20) can be written in the form _Yu=o.
(4.5)
The boundary conditions (3.2 1) can be written as 0nL
T-u=9,
(4.6)
where
fi = 3(2@+K) (WI
-XI%)
1
$2 = HP+r)nl?
f3 =HP+r)flz.
(4.7)
From (3.15) and (3.19) it follows that the solution of the system (4.5) with the homogeneous boundary conditions is given by Cp= k, where k is an arbitrary constant. We consider two matrices
6, = 0,
$2 = 0,
uCa)= (q@, I,@),I@)), (a = 1,2).
(4.8)
Using the Green’s
Torsion of micropolar elastic beams
1053
theorem, it is easy to verify that
l
[+P’T,u” + JI(12)T2d1) + ~$$%‘-~d~)] ds
(4.10) Let us consider the system _!TU=SY,
(4.11)
where h, SY= h, . 0hs Using the associate matrices method, we can obtain the following representation of Galerkin type (see [2]) q’=
[y(a+fi+y)Ab--2~(ct+p+2y)A+4~~]~~+~[(a+~+y)A--2~]~ 2
(4.12)
-K[((Y+p+y)d-2~]$ 1
$1 =
-K[(cz+~~+Y).&~K]~
+
y(p+K)AA+
(a+P)b+K)A$
2
--~K(~+K)A+K~-$
r2-
[(p+K)(a+@)A-K2]z,
1I
2
1
$~=K[(~+~~+Y)A-~K]+[(~+K)(L~~~)A-K~]+$ 2
~(~+K)AA+(~+~)(/L+K)A-$K(/A+K)A+K~-$
r,. 1
2I
1054
DORIN IESAN
The functions Ii (i = 1,2,3)
satisfy the equations .&IYi=hi,(i=
1,2,3),
K[2J’(P+Kj+
(~+fi+yj(2~+~jlA
(4.13)
where & = {Y(~+K)(Q!+/~+~)AA-
(4.14)
+~K~@L+K)}~.
Obviously, the operator & can be written in the form (4.15)
a=AA(A-k:j(h-kg), where ;=Y(P+Kj(o+fi+Y), and kf, kg are the roots of the equation k4-UK[2Y(pfK)+
(ar+~+~)(2~+K)]k2+2UK2(2~++K)
=o.
We assume that k: # k$ The other case can be treated similarly. If we have h,=hy=O,
h,=6(x-y),
where 6 is the Dirac measure, then I2 = I3 = 0 and Il satisfies the equation A(A-k;)(A-k;)x
= &(x-y).
hi = &,$(x-y),
(i=
(4.16)
In general, if 1,2,3),
then by putting IYj= &X(X, yj, (j= 1, 2, 3j, from (4.12) we obtain the functions @)(x, y), @~)(xz,y), ~,V~)(x,y), where x(x, y) is the solution of the equation (4.16). We have cp”‘(X,Y)
=
$?‘kY)
=-K[(a+B+y)a-2~]%,
(~‘~‘(x,y)
=
(4.17)
[Y((~!+P+~)~A--_K((Y+~+~~)A+~K*]x,
‘,%;‘(X,Y)
=
K[(~+@+Y)&-~K]&
~[(a+fl+y)A--2~]$f, 2
J1’,2’(X?Y) = [ y
( /.L+K)AA+
x,
(~+~)(~+K)~&~K(/L+K)A+K~-~$ 2
11
Torsion of micropolar elastic beams
(c’3’(-GY)
J/(‘)(X,Y) =-bfK)(a+P)A-~~l&,
@)(X,Y)
= [y
( ,u+~)hA+
1055
=-K[((Y+p+y)fbd~]
(~+@)(~+K)A$-~K(/L+K)A+K~-$ ::
Let us note that, if the functions gi (i = 1,2,3) Ag, = h,
= h,
(A-kf)g,
-$,
1
x.
2
satisfy the equations (A-&)g,
= h,
(4.18)
then the solution of the equation h,
A(A-&)(A--&)g=
(4.19)
can be written in the form
n=++
1
k;(k;--k;)
1 2
1 g2-
/qkf--kg)
(4.20)
g3.
The equations
Agl = u~@--Y),
(A-k:)g,
= aa(x-Y),
(A-k;)g,
= UC?(X-y),
(4.21)
have the solutions g1 =
e
In r,
s2=+(k,‘),
(4.22)
g3=+(kOr),
where r2 = (xl -Y~)~ + (x2 -Y~)~, K,,(x) denotes the modified Bessel function of zeroth order and k,, k2 are the roots with positive real parts. Using (4.20) we obtain the solution of the equation (4.16) in the form
x(x, Y) = & llnr+ k2k2
1 2
1 k:(k:-k;)
&(k,r)
1 - k,(k; _ ki) &(k,r)
1 .
(4.23)
We can write XkY)
=
ar41nr+o(x,y), 2,T
(4.24)
where w(x, y) and its derivatives when x = y, have a singularity of a lower order than the function r4 In Yand the corresponding derivatives [S]. From (4.17), (4.23) we get
DORIN IESAN
1056
b_
(4.25)
c%+p+2y 2y(a+p+r)’
(j= 1,2,3;v=
1,2),
&,,,,=O,
in which we have pointed out the terms with singularities at x = y. We consider the matrix of the fundamental solutions
in which we have pointed out the terms with singularities. It is seen from (4.17) that r(x,Y)
= r”(Y,x),
(4.27)
where I? is the transposed matrix of I?. We denote by I’@)(k = 1,2, 3) the columns of the matrix I’(x, y) and by Pk) the columns of the matrix r* (x, y). We introduce the matrix s,r*(X,y)
=
Tl-wi)
Tlr*(2)
T,r*(l)
T2r*(2)
/I
Tl-w3)
(4.28)
T,r*(i) T3r*(2)
where we have indicated that the operators are applied at the point y ( y,) . We denote by h(x, y) the matrix obtained from (4.28) by interchanging the rows and columns h(x,y) =
[.5qp(x,y)]*.
(4.29)
From (4.3), (4.17), (4.23), (4.26) and (4.28) we obtain alnr an, A(X,Y)
=
&
0 0
o
0
a Inr a In I dn,5& a lnr a lnr -Gas, an,
where c=
p(a+p+y) -cry 2y(cw+p+y) ’
+m,Y),
(4.30)
Torsion of micropolar elastic beams
1057
and lI(x, y) is a matrix with “weak” singularities (in comparison with the main one). It is easy to verify that each column of the matrix h(x, y) considered as matrix U, satisfies at x the system (4.5). We shall call the matrix u(x) a regular solution of the system (4.5) in the region I; if the formula (4.10) can be applied to U(X) and if it satisfies the system (4.5) in Xc.We denote by Xc,the complementary of Z +L to the entire plane. Let cr(y; E) be a circle with the centre in y, with radius E and circumference (T* ( y; E). Let y E 2 and let E be so small that u is entirely contained in 2:. Then formula (4.10) can be applied in 2 -u to some regular solution P(x) = u(x) of the system (4.5) and to the matrix P(x) = [F(x, y)] *. As in [3,7] we obtain
u* ( y)
= j { [.F,l-(x,
y)] “u”(x)
(x, y)~u(x)} ds,.
-P
(4.3 1)
L
Taking into account (4.27) and (4.29), we get U*(X) =
J
[A(X,Y)lF(Y)
(4.32)
--r(&Yv-4Y)ld&/.
L
Let p = ](~j((be a column-matrix in which the functions pj (j = 1,2,3) satisfy Hiilder’s condition. We introduce the potentials: -the potential of a single-layer
f’(x) = j- ~(x,Y)P(Y)
ds,,
(4.33)
ds,.
(4.34)
L
-the potential of a double-layer
W(x) = j- A(x,Y)P(Y) L
As in the classical theory of potential, we have -YV(x) = 0,
$PW(x) = 0,
x E L.
(4.35)
We can prove[3] Theorem 1. The potential of a single-layer is continuous throughout. Theorem 2. The potential of a double-layer tends to finite limits when the point x tends to x0 E L, both from within and from without, and these limits are respectively equal to Wi(xo>
=
b(xo)
+
/
A(xo,Y)P(Y)
L
ds,, (4.36)
W&o)
= -f~bo)
+
j- Aho,
L
Y)P(Y)
ds,,
the integrals on the right-hand side should be conceived as principal values.
1058
DORIN lEtjAN
Theorem 3. The F-operator of the single-layer potential tends to finite limits, as the point x tends to the boundary point x0 E L from within or from without, and these limits are respectively equal to
(4.37)
We seek the solution regular in Z of the system (4.5) satisfying the condition lim Fu(x)
5-uo
= 9(x0),
(4
where x E 2, x0 E L and 9 is given by (4.7). Let us consider the problem which consists in to find the solution regular in I&,,of the system (4.5) satisfying the condition
li+l&u(x>= G(xo),
09
where x E 2,, x0 E L and G is a given matrix satisfying Holder’s condition. We seek the solution of the problem (A) in the form of a single-layer potential and the solution of the problem (B) in the form of a double-layer potential [7]. On the basis of Theorems 2, 3 for unknown density we obtain the following singular integral equations
-in
+I
fJ(xo,
Y) P(Y)
ds, =
S(xo),
ds,=
Gbo).
L
-fpbo)+
j-
A(x,,Y)P(Y)
L
The equations (A) and (B) are mutually associate equations, as it is obvious from the form of their kernels. The homogeneous equations corresponding to equations (A), (B) for 9 = G = 0 will denoted by (A”), (B”) respectively. We note by t and to the affixes of the points y and x0. We have a In r dsds,+dr_
f-
U
f(J
i dv.
Taking into account (4.38) and pointing out the characteristic operator 191,the system (B) can be written in the form
%+X&o) 0
= -2G(r,).
(4.38) part of the singular
(4.39)
Torsion of micropolar elastic beams
1059
Let us denote by [f(r)lL the increment of the functionf(t) as the point t describes once the curve L in the direction leaving the domain Z on the left. The index[9] of the system (4.39) is
We assume c2 # 1. Since in our case we have n = 0, the system (4.39) is a system of singular integral equations of normal type for which Fredholm’s basic theorems are valid [9,7]. Taking into account (4.8) it follows that the matrix k
u,=
0
(4.40)
0, 0
satisfies the relations 2?u,* = 0,
Fug*
(4.4 1)
= 0.
From (4.32) we obtain
u,(x) = j- AC-x, y)Uo(y) ds,,
x E 2..
(4.42)
L
Passing to the limit in (4.42) as the point x approaches within, according to (4.36), we obtain -3Uoho)
+
I
Abo,y)
U,(Y)
the boundary point x0 E L from
ds, = 0.
(4.43)
L
Hence, the matrix U. satisfies the homogeneous that the matrix
equation (B”). From (4.40) it follows
1
v=
0 0,
(4.44)
0
is a solution of the equation (B”). According to Fredholm’s second theorem, the associate equation (A”) has at least a solution. As in [3] it can be shown that the equation (A”) has only one solution different from zero. Hence, the necessary and sufficient condition to solve the equation (A) has the form
s
vx9(x)
L
ds = 0.
(4.45)
1060
DORIN
lE$AN
From (4.7), (4.44), (4.45) it follows that the necessary existence of the torsion functions is I (xznI -x,n,) I,
and sufficient condition for the
ds = 0.
(4.46)
Obviously, this condition is satisfied. Thus, we have Theorem 4. The boundary value problem (3.20), (3.21) can be solved and the solution can be expressed as a single-layer potential and it is determined to within the additive matrix (4.40). REFERENCES Math. Mech. 15,909 (1966). [II A. C. ERlNGEN,J. PI A. C. ERINGEN, Theory of Micropolar Elasticity, Marhematical Fundamentals of Fracture, Vol. 2, edited by H. Liebowitz. Academic Press (1967). [31 V. D. KUPRADZE, Dynamical Problems in Elasticity. Progress in Solid Mechanics, Vol. 3. NorthHolland, Amsterdam (1963). Some Basic Problems of the Mathematical TheoryofElasticity. Groningen [41 N. I. MUSKHELlSHVlLl, (1953). Mathematical Theory of Elasticity. McGraw-Hill (1956). [51 1. S. SOKOLNIKOFF, Elastic Systems. Springer (1962). [61 L. M. MlLNE-THOMSON,Antiplane t71 D. IESAN, Int. J. Engng Sci. 8,877 (1970). PI E. E. LEVI, Rend. Circ. Mat. Palermo. 24,275 (1907). Singular Integral Equations. Groningen (195 1). 191 N. 1. MUSKHELISHVILI, [lOI A. C. SMITH, RecentAdvances in Engineering Science (editedby A. C. Eringen), Vol. 5, p. 129 (1970). (First received 22 October 1970; in revisedform
I March 1971)
R&suti - Le prksent article conceme le problkme de la torsion de poutres homogtnes et isotropes dans la theorie Ii&ire de 1’8asticit6 micropolaire [ 1,2]. Le probleme est rCsolu sous la forme de trois fonctions de torsion. Le thborkme d’existence est dCduit par la m&ode des potentiels [3]. Zusammenfassung- Die vorliegende Arbeit beschitigt sich mit dem Torsionsproblem homogener und isotropischer Balken in der linearen Theorie mikropolarer Elastizitat [ 1,2]. Das Problem wird in Ausdriicken von drei Torsionsfunktionen gelijst. Das Existenztheorem wird mit Hilfe der Methode von Potentialen abgeleitet [ 31. Sommario- Nel presente articolo I’A. tratta il problema della torsione dei cilindri omogenei ed issotropici nella teoria lineare dell’elasticit8 micropolare[l, 21. II problema viene risolto come tre funzioni di torsione. II teorema d’esistenza &derivato con il metodo delle potenziali [3]. AGcTpaIcT-
PaCCMOTpeHa npo6neMa Kpy'feHMSlOAHOpOAHblXii H3OTpOIlWIeCKeX 6anK BJlMHeiiHOiiTeOpHH ~~t~p0110~151p~0i%ynpyrocT&i [1,2]. nonyveH0 peLUeHueYepe3 rpcrr/)y~~Uwi KpyqeHHfl. BbIBOAciTCRTeopeMa CylUeCTBOBaHWR Ha OCHOBe MeTOAa IlOTeHUHaAOB [3].
TORSION
Pergamon Press.
OF MICROPOLAR
Printed in Great Britain
ELASTIC
BEAMS
DORIN IESAN University of Jassy, Jassy, Romania Abstract-The present paper is concerned with the torsion problem of homogeneous and isotropic beams in the linear theory of micropolar elasticity [ 1, 21. The problem is solved in terms of three torsion functions. The existence theorem is derived by the method of potentials [3]. 1. INTRODUCTION
of the torsion of a cylindrical beam is very important and it was very much studied in the classical elasticity (see e.g. [4-61). In this paper we consider the torsion of isotropic beams in the linear theory of micropolar elasticity. Three micropolar torsion functions are introduced and their existence is derived by method of potentials. Initially, the problem of torsion of micropolar elastic beams was considered in [ 1O]t.
THE
PROBLEM
2. BASIC
EQUATIONS
Let V be a region of space occupied by a micropolar elastic material whose boundary is S. We assume that the body forces and the body couples are absent. The basic equations in the static theory of homogeneous and isotropic micropolar elastic solids are[l, 21: -the equilibrium equations tji.j =
09mjf,j+ EimJmn= 07
(2.1)
-the constitutive law tu=hETrSii+(EL+K)Eij+~Eji,mlj=ffcp,,,Gij+P~i,j+Y~j,i,
(2.2)
-the geometrical equations Eu=
Uj,*+
Eji&me
G.3
In these relations we have used the notations from [ 1,2]. The surface tractions and surface couples acting at a point x(xk) on the surface S are given by ti = tjinj, mi = F?lj$lj,
(2.4)
where nj = cos (n,, Xj) and II, is the unit vector of the outward normal to S at X. 3. TORSION
PROBLEM
We consider a cylindrical beam of homogeneous and isotropic micropolar elastic material bounded by plane ends perpendicular to the generators. The cross-sections I: ?The paper[ lo] had not been available to the present writer. After this note was submitted, the writer was kindly informed by a referee about it. 1047
1J.E.S. Vol. 9 No. 1 I -A
DORIN
1048
lE$AN
is assumed to be a simply connected region bounded by a closed Liapunov curve L. We suppose that the body forces and the body couples are absent and the lateral surface is free of applied forces and couples. Throughout this paper the axis OX,of our coordinate system will be directed parallel to the generators of the beam. The beam is assumed to be of length 1,and one of its bases is taken to lie in the x,0x,-plane, while the other is in the plane x3 = 1. We suppose the beam to be kept in equilibrium when the end xQ= 1 is twisted by a couple of magnitude M. On the plane x3 = 1 we have the following conditions
Ir t31 du = 0,
I ts2 do = 0, z
s
(x2fs3+ m3J da = 0,
z
I fS3dc = 0, 5:
(3.1)
I (x,t33-mz) da = 0, L
(3.2)
s
(xlfB2-x2t31 + m33)da = M.
(3.3)
z
The resultant forces and moments calculated across each cross-section satisfy the conditions of equilibrium, so that the conditions (3.1)-(3.3) must be satisfied for .Q = h(0 s h s 1)[6]. On the lateral surface of the beam we have the following conditions t&no = 0,
m&&=0,
(k= 1,2,3;a=
1,2).
(3.4)
The torsion problem consists in the solving of equations (2.1)-(2.3) with the conditions (3.1H3.4). We try to solve the torsion problem assuming that l41 = -7X2X3, 91
=
cpl (Xl,
112= x2)
7
7X1X3,
cpz=cpz(-G~~~),
u3=
(3.5)
Qb,rX*), (P3=7x3,
where the functions Cp(x,, x2), pa(xI, x2) and the constant T will be determined in the following. From (2.2) and (3.5) we get h=
fm=0,f33=0,
(3.6)
(~+K)@,lz-KQ~+/.‘TX~r
f32=ll~,3+K(P1+(CL+K)TX1,
~~I=@J-KQ~-(~+K)~X~,
(P+KK)@J+KQ~-~TX~, mv3= m3”=0, m33=~cp,,,+(a+p+y)7, = ff%.PSW + P%.v + Yso,,” + ~6m m, f13=
(v,q,p=
1,2).
The equilibrium equations (2.1) will be satisfied if the functions Q (x,, x2), cpo!(x,, x2) satisfy the following equations
(F+K)A@---(QI,~--Q~.~)= 0, YAQS + (Q + ~)Qv,yl)
+KEqp3@,p-2KQOp=KTXp,
(3.7) (r).V,p=
1,2),
Torsion of micropolar elastic beams
1049
throughout the cross-section of the cylinder, where A is the Laplacian operator. From (3.4) and (3.6) we obtain the following boundary conditions for the functions @,cp,
(P+KK)$h’ln,-~d
=/-d-W-X,~2),
r(~+P+Y)(Pl,l+~Y(P*,21~1+
m%l+y~1.21n*
=-wmlr
[9k1+ (a+p+r)(~*,~ln*=-(~7n~, 0nL.
[PPL~+Y~~.II~I+
(3.8)
Let us establish the uniqueness of the solution of the boundary value problem (3.7), (3.8). From (2.1)-(2.4) and the divergence theorem we obtain [ 11 j- (tiui + rn& du = 2 j- E dv, s V
(3.9)
where E is the internal energy density. Let us consider two solutions of the torsion problem lp=-Tx
2
x 39
L&j
soY~=P:*‘(&J2)~
=
7x1
x2,
up=
(3.10)
@)(x1,x2),
$e’=(pY?x1,x2),
&=7x3,
((y=
1,2).
In what follows it will be shown that the constant 7 is uniquely determined if M is given. We note f& =
4”
_
&
421,
=
-42’.
(3.11)
in V.
(3.12)
p$l’
Obviously ii1= ii2= (p3= 0,
On the lateral surface (S,) we have ti = 0, mi = 0. On the surface x3 = /(C,) we have t3=t33,~a=m3a(rx= 1,2).Fr om (3.6) it follows that on the & we have t3 = 0, m, = 0. Similarly, t3 = 0, m, = 0,on the surface x3 = 0 (C,). Thus, we get
J (Tiii,+ fi$icpl)dc = J (iiii, + A+i) do + J (iitii + %$i) do + J (?iii,+ fiGi) do = 0, s
SL
II
20
and using (3.9) we obtain s adv=O,
(3.13)
V
where Gis the internal energy density corresponding to the system (3.10). From (3.13) we obtain[l],
DORIN
1050
IESAN
From (3.12) and (3.14) weget 23 = c,
cpl = 0,
(3.15)
+3 = 0,
where c is an arbitrary constant, Using (3.9) we can establish a relation which is important in the following. Taking into account that on the X1 we have t, = taa, t3 = 0, m, = 0, m3 = ms3, u1 = -TX&, u2 = 7x11,p3 = rl and on Z,, we have u, = 0, t3 = 0, m, = 0, (p3= 0, we obtain J (tfui+
m33>da.
do = IT J (Xlt32 -~2131+ z
m$Qi)
s
(3.16)
Obviously E = E(x, , x2), so that (3.17)
From(3.9),(3.16),(3.17)weobtain 7
Iz
x2t31-t m33) da = 2s edcr. z
(w32-
In what follows it is convenient &(x1, xJ by the following relations Q, = rQ,
(3.18)
to introduce the torsion functions cp(x,, X.J, JI~(x~,x2),
(3.19)
$‘a = T(l(la-&XQ), (a = 1,2).
If we substitute (3.19) in (3.7) we get (~u+K)AQ--K(~LI,z-~I) =
yA&, + (a + ,@$v,vrl+ KEqpsQ,p-
2K$Jp= 0,
0,
(3.20)
(VT 9, P =
From (3.8) and (3.19) we obtain the following boundary functions
1~ 2).
conditions
for the torsion
Using (3.6) and (3.19) we get t31 = dPQ,l
-K$z-!i(2P+K)X,l,
t,, =~C~Q,~+KI(I,+)(~CL+K)XI~~ m33 =
d(*1(h+hd
+P+rl.
(3.22)
1051
Torsion of micropolar elastic beams
Taking into account (3.22), from (3.3) we obtain (3.23)
rD=M, where
D=j
[C~(X~(~,~-X~(P,I)+K(X,~~+.X~~~)+~(~CL+K)(X::+X~~)
z
+~(+m+hd
+P+yldu.
(3.24)
The constant D is the micropolar torsional rigidity of the beam. From (3.18) and (3.22) weget r2D = 2 I E du, P
(3.25)
D > 0.
(3.26)
functions (o, &, Qz are determined
the constant r can be calculated from
so that we conclude that
Thus, if the (3.23). Taking into cally satisfied. brium equations
IL
t31 da
=
j
z
account the relations (3.6) we see that the conditions (3.2) are identiLet us show that the conditions (3.1) are satisfied. Using the equiliwe can write [t13-
(m 12,1+ m22,2)
1du = j [t13+x1b1 + f23.2) - h2.1 + md 1do I
Making use of the Green’s theorem and of the boundary conditions (3.4), we obtain
Iz t31du = I
[xl (m
+ j2s2> - bw,
+
mz2n2>l ds = 0.
L
In a similar way it follows that the condition (3. 1)2 is satisfied. The last of the conditions (3.1) is identically satisfied on the basis of (3.6). 4. EXISTENCE
THEOREM
In what follows we consider the system (3.20) with the boundary conditions (3.21). Using a Galerkin representation we determine the fundamental solutions and introduce the elastic potentials. With the help of the method of potentials [3] we reduce the boundary value problem (3.20), (3.21) to singular integral equations for which Fredholm’s basic theorems are valid. The existence of the micropolar torsion functions is derived. The procedure is analogous with that used in the plane strain problem of a homogeneous and isotropic body [7].
DORIN IESAN
10.52
We introduce the matrix u=
(4.1)
(%$I,&)5
and the operators 2,~ = (~++K)h~--K(J11,2-~~,1), ~zu=?‘A’!‘~+
(4.2)
(~+P)~Y,v~+K(P,z-~K~~,
~P,U = rA$z + (a + p)‘!‘v,vz- K’f’,l - 2’@‘21 and
(4.3)
(4.4)
The system (3.20) can be written in the form _Yu=o.
(4.5)
The boundary conditions (3.2 1) can be written as 0nL
T-u=9,
(4.6)
where
fi = 3(2@+K) (WI
-XI%)
1
$2 = HP+r)nl?
f3 =HP+r)flz.
(4.7)
From (3.15) and (3.19) it follows that the solution of the system (4.5) with the homogeneous boundary conditions is given by Cp= k, where k is an arbitrary constant. We consider two matrices
6, = 0,
$2 = 0,
uCa)= (q@, I,@),I@)), (a = 1,2).
(4.8)
Using the Green’s
Torsion of micropolar elastic beams
1053
theorem, it is easy to verify that
l
[+P’T,u” + JI(12)T2d1) + ~$$%‘-~d~)] ds
(4.10) Let us consider the system _!TU=SY,
(4.11)
where h, SY= h, . 0hs Using the associate matrices method, we can obtain the following representation of Galerkin type (see [2]) q’=
[y(a+fi+y)Ab--2~(ct+p+2y)A+4~~]~~+~[(a+~+y)A--2~]~ 2
(4.12)
-K[((Y+p+y)d-2~]$ 1
$1 =
-K[(cz+~~+Y).&~K]~
+
y(p+K)AA+
(a+P)b+K)A$
2
--~K(~+K)A+K~-$
r2-
[(p+K)(a+@)A-K2]z,
1I
2
1
$~=K[(~+~~+Y)A-~K]+[(~+K)(L~~~)A-K~]+$ 2
~(~+K)AA+(~+~)(/L+K)A-$K(/A+K)A+K~-$
r,. 1
2I
1054
DORIN IESAN
The functions Ii (i = 1,2,3)
satisfy the equations .&IYi=hi,(i=
1,2,3),
K[2J’(P+Kj+
(~+fi+yj(2~+~jlA
(4.13)
where & = {Y(~+K)(Q!+/~+~)AA-
(4.14)
+~K~@L+K)}~.
Obviously, the operator & can be written in the form (4.15)
a=AA(A-k:j(h-kg), where ;=Y(P+Kj(o+fi+Y), and kf, kg are the roots of the equation k4-UK[2Y(pfK)+
(ar+~+~)(2~+K)]k2+2UK2(2~++K)
=o.
We assume that k: # k$ The other case can be treated similarly. If we have h,=hy=O,
h,=6(x-y),
where 6 is the Dirac measure, then I2 = I3 = 0 and Il satisfies the equation A(A-k;)(A-k;)x
= &(x-y).
hi = &,$(x-y),
(i=
(4.16)
In general, if 1,2,3),
then by putting IYj= &X(X, yj, (j= 1, 2, 3j, from (4.12) we obtain the functions @)(x, y), @~)(xz,y), ~,V~)(x,y), where x(x, y) is the solution of the equation (4.16). We have cp”‘(X,Y)
=
$?‘kY)
=-K[(a+B+y)a-2~]%,
(~‘~‘(x,y)
=
(4.17)
[Y((~!+P+~)~A--_K((Y+~+~~)A+~K*]x,
‘,%;‘(X,Y)
=
K[(~+@+Y)&-~K]&
~[(a+fl+y)A--2~]$f, 2
J1’,2’(X?Y) = [ y
( /.L+K)AA+
x,
(~+~)(~+K)~&~K(/L+K)A+K~-~$ 2
11
Torsion of micropolar elastic beams
(c’3’(-GY)
J/(‘)(X,Y) =-bfK)(a+P)A-~~l&,
@)(X,Y)
= [y
( ,u+~)hA+
1055
=-K[((Y+p+y)fbd~]
(~+@)(~+K)A$-~K(/L+K)A+K~-$ ::
Let us note that, if the functions gi (i = 1,2,3) Ag, = h,
= h,
(A-kf)g,
-$,
1
x.
2
satisfy the equations (A-&)g,
= h,
(4.18)
then the solution of the equation h,
A(A-&)(A--&)g=
(4.19)
can be written in the form
n=++
1
k;(k;--k;)
1 2
1 g2-
/qkf--kg)
(4.20)
g3.
The equations
Agl = u~@--Y),
(A-k:)g,
= aa(x-Y),
(A-k;)g,
= UC?(X-y),
(4.21)
have the solutions g1 =
e
In r,
s2=+(k,‘),
(4.22)
g3=+(kOr),
where r2 = (xl -Y~)~ + (x2 -Y~)~, K,,(x) denotes the modified Bessel function of zeroth order and k,, k2 are the roots with positive real parts. Using (4.20) we obtain the solution of the equation (4.16) in the form
x(x, Y) = & llnr+ k2k2
1 2
1 k:(k:-k;)
&(k,r)
1 - k,(k; _ ki) &(k,r)
1 .
(4.23)
We can write XkY)
=
ar41nr+o(x,y), 2,T
(4.24)
where w(x, y) and its derivatives when x = y, have a singularity of a lower order than the function r4 In Yand the corresponding derivatives [S]. From (4.17), (4.23) we get
DORIN IESAN
1056
b_
(4.25)
c%+p+2y 2y(a+p+r)’
(j= 1,2,3;v=
1,2),
&,,,,=O,
in which we have pointed out the terms with singularities at x = y. We consider the matrix of the fundamental solutions
in which we have pointed out the terms with singularities. It is seen from (4.17) that r(x,Y)
= r”(Y,x),
(4.27)
where I? is the transposed matrix of I?. We denote by I’@)(k = 1,2, 3) the columns of the matrix I’(x, y) and by Pk) the columns of the matrix r* (x, y). We introduce the matrix s,r*(X,y)
=
Tl-wi)
Tlr*(2)
T,r*(l)
T2r*(2)
/I
Tl-w3)
(4.28)
T,r*(i) T3r*(2)
where we have indicated that the operators are applied at the point y ( y,) . We denote by h(x, y) the matrix obtained from (4.28) by interchanging the rows and columns h(x,y) =
[.5qp(x,y)]*.
(4.29)
From (4.3), (4.17), (4.23), (4.26) and (4.28) we obtain alnr an, A(X,Y)
=
&
0 0
o
0
a Inr a In I dn,5& a lnr a lnr -Gas, an,
where c=
p(a+p+y) -cry 2y(cw+p+y) ’
+m,Y),
(4.30)
Torsion of micropolar elastic beams
1057
and lI(x, y) is a matrix with “weak” singularities (in comparison with the main one). It is easy to verify that each column of the matrix h(x, y) considered as matrix U, satisfies at x the system (4.5). We shall call the matrix u(x) a regular solution of the system (4.5) in the region I; if the formula (4.10) can be applied to U(X) and if it satisfies the system (4.5) in Xc.We denote by Xc,the complementary of Z +L to the entire plane. Let cr(y; E) be a circle with the centre in y, with radius E and circumference (T* ( y; E). Let y E 2 and let E be so small that u is entirely contained in 2:. Then formula (4.10) can be applied in 2 -u to some regular solution P(x) = u(x) of the system (4.5) and to the matrix P(x) = [F(x, y)] *. As in [3,7] we obtain
u* ( y)
= j { [.F,l-(x,
y)] “u”(x)
(x, y)~u(x)} ds,.
-P
(4.3 1)
L
Taking into account (4.27) and (4.29), we get U*(X) =
J
[A(X,Y)lF(Y)
(4.32)
--r(&Yv-4Y)ld&/.
L
Let p = ](~j((be a column-matrix in which the functions pj (j = 1,2,3) satisfy Hiilder’s condition. We introduce the potentials: -the potential of a single-layer
f’(x) = j- ~(x,Y)P(Y)
ds,,
(4.33)
ds,.
(4.34)
L
-the potential of a double-layer
W(x) = j- A(x,Y)P(Y) L
As in the classical theory of potential, we have -YV(x) = 0,
$PW(x) = 0,
x E L.
(4.35)
We can prove[3] Theorem 1. The potential of a single-layer is continuous throughout. Theorem 2. The potential of a double-layer tends to finite limits when the point x tends to x0 E L, both from within and from without, and these limits are respectively equal to Wi(xo>
=
b(xo)
+
/
A(xo,Y)P(Y)
L
ds,, (4.36)
W&o)
= -f~bo)
+
j- Aho,
L
Y)P(Y)
ds,,
the integrals on the right-hand side should be conceived as principal values.
1058
DORIN lEtjAN
Theorem 3. The F-operator of the single-layer potential tends to finite limits, as the point x tends to the boundary point x0 E L from within or from without, and these limits are respectively equal to
(4.37)
We seek the solution regular in Z of the system (4.5) satisfying the condition lim Fu(x)
5-uo
= 9(x0),
(4
where x E 2, x0 E L and 9 is given by (4.7). Let us consider the problem which consists in to find the solution regular in I&,,of the system (4.5) satisfying the condition
li+l&u(x>= G(xo),
09
where x E 2,, x0 E L and G is a given matrix satisfying Holder’s condition. We seek the solution of the problem (A) in the form of a single-layer potential and the solution of the problem (B) in the form of a double-layer potential [7]. On the basis of Theorems 2, 3 for unknown density we obtain the following singular integral equations
-in
+I
fJ(xo,
Y) P(Y)
ds, =
S(xo),
ds,=
Gbo).
L
-fpbo)+
j-
A(x,,Y)P(Y)
L
The equations (A) and (B) are mutually associate equations, as it is obvious from the form of their kernels. The homogeneous equations corresponding to equations (A), (B) for 9 = G = 0 will denoted by (A”), (B”) respectively. We note by t and to the affixes of the points y and x0. We have a In r dsds,+dr_
f-
U
f(J
i dv.
Taking into account (4.38) and pointing out the characteristic operator 191,the system (B) can be written in the form
%+X&o) 0
= -2G(r,).
(4.38) part of the singular
(4.39)
Torsion of micropolar elastic beams
1059
Let us denote by [f(r)lL the increment of the functionf(t) as the point t describes once the curve L in the direction leaving the domain Z on the left. The index[9] of the system (4.39) is
We assume c2 # 1. Since in our case we have n = 0, the system (4.39) is a system of singular integral equations of normal type for which Fredholm’s basic theorems are valid [9,7]. Taking into account (4.8) it follows that the matrix k
u,=
0
(4.40)
0, 0
satisfies the relations 2?u,* = 0,
Fug*
(4.4 1)
= 0.
From (4.32) we obtain
u,(x) = j- AC-x, y)Uo(y) ds,,
x E 2..
(4.42)
L
Passing to the limit in (4.42) as the point x approaches within, according to (4.36), we obtain -3Uoho)
+
I
Abo,y)
U,(Y)
the boundary point x0 E L from
ds, = 0.
(4.43)
L
Hence, the matrix U. satisfies the homogeneous that the matrix
equation (B”). From (4.40) it follows
1
v=
0 0,
(4.44)
0
is a solution of the equation (B”). According to Fredholm’s second theorem, the associate equation (A”) has at least a solution. As in [3] it can be shown that the equation (A”) has only one solution different from zero. Hence, the necessary and sufficient condition to solve the equation (A) has the form
s
vx9(x)
L
ds = 0.
(4.45)
1060
DORIN
lE$AN
From (4.7), (4.44), (4.45) it follows that the necessary existence of the torsion functions is I (xznI -x,n,) I,
and sufficient condition for the
ds = 0.
(4.46)
Obviously, this condition is satisfied. Thus, we have Theorem 4. The boundary value problem (3.20), (3.21) can be solved and the solution can be expressed as a single-layer potential and it is determined to within the additive matrix (4.40). REFERENCES Math. Mech. 15,909 (1966). [II A. C. ERlNGEN,J. PI A. C. ERINGEN, Theory of Micropolar Elasticity, Marhematical Fundamentals of Fracture, Vol. 2, edited by H. Liebowitz. Academic Press (1967). [31 V. D. KUPRADZE, Dynamical Problems in Elasticity. Progress in Solid Mechanics, Vol. 3. NorthHolland, Amsterdam (1963). Some Basic Problems of the Mathematical TheoryofElasticity. Groningen [41 N. I. MUSKHELlSHVlLl, (1953). Mathematical Theory of Elasticity. McGraw-Hill (1956). [51 1. S. SOKOLNIKOFF, Elastic Systems. Springer (1962). [61 L. M. MlLNE-THOMSON,Antiplane t71 D. IESAN, Int. J. Engng Sci. 8,877 (1970). PI E. E. LEVI, Rend. Circ. Mat. Palermo. 24,275 (1907). Singular Integral Equations. Groningen (195 1). 191 N. 1. MUSKHELISHVILI, [lOI A. C. SMITH, RecentAdvances in Engineering Science (editedby A. C. Eringen), Vol. 5, p. 129 (1970). (First received 22 October 1970; in revisedform
I March 1971)
R&suti - Le prksent article conceme le problkme de la torsion de poutres homogtnes et isotropes dans la theorie Ii&ire de 1’8asticit6 micropolaire [ 1,2]. Le probleme est rCsolu sous la forme de trois fonctions de torsion. Le thborkme d’existence est dCduit par la m&ode des potentiels [3]. Zusammenfassung- Die vorliegende Arbeit beschitigt sich mit dem Torsionsproblem homogener und isotropischer Balken in der linearen Theorie mikropolarer Elastizitat [ 1,2]. Das Problem wird in Ausdriicken von drei Torsionsfunktionen gelijst. Das Existenztheorem wird mit Hilfe der Methode von Potentialen abgeleitet [ 31. Sommario- Nel presente articolo I’A. tratta il problema della torsione dei cilindri omogenei ed issotropici nella teoria lineare dell’elasticit8 micropolare[l, 21. II problema viene risolto come tre funzioni di torsione. II teorema d’esistenza &derivato con il metodo delle potenziali [3]. AGcTpaIcT-
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