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Plane deformations of incompressible isotropic elastic solids: an integral equation formulation
MECH. KES. COMM.
Voi.4(5), 347-352, 1977.
Pergamon Press.
Printed in USA.
PLANE DEFORMATIONS OF INCOMPRESSIBLE ISOTROPIC ELASTIC SOLIDS: AN INTEGRAL EQUATION FORMULATION
R.W. Ogden School of Mathematics, University of Bath, BA2 7AY, U.K. (Received 28 July 1977; accepted for print 6 August 1977.)
Introduction
In a recent paper [i] a new formulation of the plane-strain equations for compressible isotropic elastic solids was given, and for a particular class of strain-energy functions closed-form solutions to a number of boundary-value problems were derived. The method used was direct in that inverse or semiinverse techniques were not employed. Each of the boundary-value problems gave rise to an integral equation for the determination of a single complex function For incompressible isotropic elastic solids an analogous formulation was given in [2], and semi-inverse methods were used to illustrate the solutions of certain boundary-value problems. In this paper we present an integral equation formulation of the incompressible plane-strain problem which parallels that given in [I] for compressible elastic solids.
Description of the deformation
Let (XI,X2) be the rectangular Cartesian coordinates of a generic material point in the undeformed configuration and (Xl,X2) the coordinates of this point when material is deformed in such a way that no change occurs normal to the (l,2)-plane.
The coordinates x I and x 2 depend only on (XI,X2).
The (plane) deformation gradient 3xi/3X j is denoted by oij. In complex variable notation Z = X I + iX2, z = x I + ix 2. 3z 3--~ = ½{Oil + °22 + i(o21 - o12)},
Hence (I)
8z
~=
½{°i1 - ~22 + i ( ° 1 2 + ° 2 1 ) } '
(2)
where ~ is the complex conjugate of Z . Invariants p and q are defined by
½P = " ~ Scientific Communication
I
,
½q =
• 347
(3)
348
R.W.
OGDEN
These were central to the analysis used in [I] and [2]. For incompressible materials ¼(p2-
q2)E
-~
-
~
=
I
(4)
and we note the connections
3z ~ ~-~ = ~-~
The constitutive
~z
,
~-~ = -
~Z
~-~
(5)
law and ggverni~g equations
The (plane) components s.. of nominal stress are given by J~ ~W a ~. sji - ~.. j1 x3
f o r an i n c o m p r e s s i b l e
elastic
solid
in plane strain,
e n e r g y p e r u n i t v o l u m e , o i s an a r b i t r a r y For a homogeneous i n c o m p r e s s i b l e depends on o n l y a s i n g l e b u t any e q u i v a l e n t ,
invariant.
elastic
is equally
solid
in preference
With t h e h e l p o f ( 1 ) ,
equation
(2) and ( 3 ) ,
= Q~ij -
and e~k~kj = ~ i j '
in plane strain
acceptable.
is sometimes advantageous
( d e p e n d e n t on t h e p r o b l e m a t hand)
sji
stress
W
T h a t i n m o s t common u s e i s I ~ ¼(p2 + q ~ ) ,
s u c h as p o r q,
shown i n [1] and [2] i t
where W is the s t r a i n
hydrostatic
isotv~rp~
(6)
However, as we h a v e
to u s e one of t h e s e t o any o t h e r .
(6) may b e r e w r i t t e n
as
oe~.31 '
(7)
Q = wI = Wp/p = Wq/q
(8)
where
o r any e q u i v a l e n t s similarly
in terms of other
for the other
I n (8) WI d e n o t e s dW/dI, and
invariants.
As i n [1] and [2] we i n t r o d u c e equilibrium
invariants.
equations(with
the complex stress
no body f o r c e s )
function
are satisfied
h so t h a t
identically.
the We t h e n
have ~h ~=
• ½{Sll + s22 + 1 ( s 1 2 - s 2 1 ) }
,
3h • ~-~= ½{s22 - S l l - 1 ( s 1 2 ÷ s 2 1 ) } so t h a t ,
with the help of (1),
(2) and ( 7 ) ,
the governing equations
can be put
as 3_kh= ( Q 3z
o) ~~z
,
~h _ 3~
(Q + o )
~z ~. ~
(9)
AN INTEGRAL EQUATION FORMULATION
349
On the boundary we may prescribe either z or h, for example, specification of h being equivalent to specification of the (nominal) traction.
At this point,
in order to arrive at a suitable integral equation equivalent of (9), we - -
depart, temporarily, from our use of Z and Z as independent variables.
Change of independent variable: the .reciprocal equations
If the independent variables Z and ~ are replaced by z and E the equations (9) may be rearranged as ~h ~--~:
½pWp - @
use having been made of (5).
,
~h ~--~ = 2Q ~z ~ ~~
(lO)
For definiteness Q is regarded as a function
of p, which can now be expressed in the form ~P =
"~z
'
(ii)
noting (3) and (5). On integration the second equation in (IO) provides the solution for h(z,~) for any given Z(z,~) subject to (4), which is now put as ~zl
-
~
:
1.
(12)
The first equation in (I0) merely provides an expression for ~ once h and Z are determined.
Equations equivalent to (I0), but in other variables, can be
found in [3]. We write the solution as Z = f(z,~) .
(13)
According to Adkins [4], if (13) is a solution of (iO) and (12) then the 'inverse' z = f(Z,~)
(14)
is a solution of the same equations and for the same strain-energy function W. The proof given in [4] is rather inelegant, but a clearer and more general proof was given by Shield [5]. This inverse property of the equations can be seen immediately if the roles of Z and z are reversed and the function h*(Z,~) is defined by
~h* ~
~--~ ~--~. :
~h ~
~
~
•
(15)
Equations (IO) are replaced by ~h* - ~ : ½pWp - ~* '
~h*
~
~z DE
: 2Q ~
~
,
(16)
350
R . W . OGDEN
use having been made of (2), (5) and (II), where ~* + ~ = - W . Equations (16) have precisely the same forms as those in (IO), with the roles of Z and z reversed, but with p unchanged.
It follows immediately that if
(13) is a solution of (I0) then (14) is a solution of (16) and hence also oy (i0) ~n U ~ W o2~ ~
~q~u~en~
o~ (I0) ~n~ (16).
It should be pointed out,
however, that h* is no% the stress function associated with the defor~tion (14).
Once h* is calculated from the second equation in (16) the actual stress
function h is obtained from (15). Either (I0) or (16) ~ y boundary-value problem.
be taken as the starting point in the analysis of any Depending on the problem in question it ~ y
convenient to use one set of equations rather than the other.
be more
An advantage of
(16) is that ~n~%~Z oooP~n~%~8 Z and ~ are the independent variables, and the initial b o u n d a ~ geometry is k n o ~ .
~en
z and ~, on the other hand, are taken
as the independent variables the (current) boundary geometry depends on the (initially unknot)
deformation.
A disadvantage of (16) is that, in general,
(15) has to be used to calculate h in order to acco~odate any traction boundary condition. Equations (10)2 and (16) 2 are integrated to give h(z,~) = 2
~
and ~Z h*(Z,~) = 2
~_
Q{2fz(Z,~)~f~(~,z)~}f~(z,~)~(~,z)d~ + k(z)
(17)
- -
I_ Q{2fz(Z,~)~f~(~,Z)~}f~(Z,~)~(~,Z)d~ + k*(Z)
(18)
respectively, where k(z) and k*(Z) are arbitrary functions of z and Z respectively, and ~(~,Z) denotes the complex conjugate of f(Z,~). For any given boundary-value problem it r e ~ i n s to determine the u n k n o ~ functions k(z) and f(z,~), or, equivalently, k*(Z) and f(Z,~), subject to (12) or (4) when the boundary data are specified.
~ i c h e v e r of (17) or (18) is
adopted an integral equation for f arises when the boundary conditions are introduced.
We now illustrate the method by considering a specific problem,
adopting (18) as our starting point.
The inte.gral equation
It is well known that there can be two solutions to the zero-traction boundaryvalue problem for a hemi-spherical elastic shell.
The analogous problem in
AN INTEGRAL EQUATION FORMULATION
351
plane-straln is that of a seml-clrcular shell also under zero traction, and we now examine this. Suppose the (undeformed and unstressed) shell is defined by A ~ R ~ B
,
-½~ 6 @ ~ ½~
,
(19)
where Z = Re i@ , and let the surfaces R = A,B (-½~ ~ @ ~ ½~) and @ = ±½~ (A ~ R ~ B) be free of traction, so that we may take h = 0 on these boundaries. Equation (15) allows us to replace the boundary condition by h* = O, and we therefore have from (18) h*(Z,g) = 2
~ a/zQ{2fz(Z,~)~f~(~,Z)I}f~(Z,~)~C(~,Z)d~. ,_
(20)
The boundary condition on R = A is automatically satisfied and A~/Z k*(Z) = -2 J Q{2fz(Z,~)½~(~,Z)½}f~(Z,~)~(~,Z)d
~he b o u n d a r y c o n d i t i o n fAB2/Z 2/Z
Q{2fz(Z,~)½~(~,z)½}f~(z,~)~(~,Z)d~
evaluated for either boundary condition
fA
on R ~ B i s s a t i s f i e d
~ .
if
= 0
Z = Ae i 0 o r Be 10 (-½~ ~ @ ~ ½~) . on 8 = ±½~ i s
satisfied
(21)
(22) Additionally,
the
if
Q{2fz (Z' ~) ½~(~'Z) ½} f~ (Z' ~)~(~'Z) d~ = 0
(23)
~/z
evaluated for Z = ±JR. The solution (14) of (22) and (23) is subject to (4). solution is the trivial one f(Z,~) = Z. or not this solution is unique.
Clearly, one possible
The question now arises as to whether
Such a solution might be expected on
intuitive grounds, but the non-linearity of the equations seems to preclude the likelihood of an analytical solution being found.
For the neo-Hookean
solid Q = ~, a constant, but even in this case the basic problems remain. •
If the solution is assumed to be symmetrical about 8 = 0 the condition ~(Z,~) = f(Z,~) must be satisfied but this does not make solution of (22), (23) and (4) significantly easier. We do not take the problem any further at this stage, although if a second solution exists it should be feasible to find it numerically.
Our main
purpose is merely to give the integral equation formulation of the basic equations.
352
R . W . OGDEN
References
I. 2. 3. 4. 5.
D.A. R.W. A.E. J.E. R.T.
Isherwood and R.W. Ogden, Rheol. Acta 16, 113 (1977). Ogden, to be published. Green and J.E. Adkins, Large Elastic Deformations. Oxford (1970) Adkins, J. Mech. Phys. Solids ~, 267 (1958). Shield, Zeits. angew. Math. Phys. 18, 490 (1967).
Voi.4(5), 347-352, 1977.
Pergamon Press.
Printed in USA.
PLANE DEFORMATIONS OF INCOMPRESSIBLE ISOTROPIC ELASTIC SOLIDS: AN INTEGRAL EQUATION FORMULATION
R.W. Ogden School of Mathematics, University of Bath, BA2 7AY, U.K. (Received 28 July 1977; accepted for print 6 August 1977.)
Introduction
In a recent paper [i] a new formulation of the plane-strain equations for compressible isotropic elastic solids was given, and for a particular class of strain-energy functions closed-form solutions to a number of boundary-value problems were derived. The method used was direct in that inverse or semiinverse techniques were not employed. Each of the boundary-value problems gave rise to an integral equation for the determination of a single complex function For incompressible isotropic elastic solids an analogous formulation was given in [2], and semi-inverse methods were used to illustrate the solutions of certain boundary-value problems. In this paper we present an integral equation formulation of the incompressible plane-strain problem which parallels that given in [I] for compressible elastic solids.
Description of the deformation
Let (XI,X2) be the rectangular Cartesian coordinates of a generic material point in the undeformed configuration and (Xl,X2) the coordinates of this point when material is deformed in such a way that no change occurs normal to the (l,2)-plane.
The coordinates x I and x 2 depend only on (XI,X2).
The (plane) deformation gradient 3xi/3X j is denoted by oij. In complex variable notation Z = X I + iX2, z = x I + ix 2. 3z 3--~ = ½{Oil + °22 + i(o21 - o12)},
Hence (I)
8z
~=
½{°i1 - ~22 + i ( ° 1 2 + ° 2 1 ) } '
(2)
where ~ is the complex conjugate of Z . Invariants p and q are defined by
½P = " ~ Scientific Communication
I
,
½q =
• 347
(3)
348
R.W.
OGDEN
These were central to the analysis used in [I] and [2]. For incompressible materials ¼(p2-
q2)E
-~
-
~
=
I
(4)
and we note the connections
3z ~ ~-~ = ~-~
The constitutive
~z
,
~-~ = -
~Z
~-~
(5)
law and ggverni~g equations
The (plane) components s.. of nominal stress are given by J~ ~W a ~. sji - ~.. j1 x3
f o r an i n c o m p r e s s i b l e
elastic
solid
in plane strain,
e n e r g y p e r u n i t v o l u m e , o i s an a r b i t r a r y For a homogeneous i n c o m p r e s s i b l e depends on o n l y a s i n g l e b u t any e q u i v a l e n t ,
invariant.
elastic
is equally
solid
in preference
With t h e h e l p o f ( 1 ) ,
equation
(2) and ( 3 ) ,
= Q~ij -
and e~k~kj = ~ i j '
in plane strain
acceptable.
is sometimes advantageous
( d e p e n d e n t on t h e p r o b l e m a t hand)
sji
stress
W
T h a t i n m o s t common u s e i s I ~ ¼(p2 + q ~ ) ,
s u c h as p o r q,
shown i n [1] and [2] i t
where W is the s t r a i n
hydrostatic
isotv~rp~
(6)
However, as we h a v e
to u s e one of t h e s e t o any o t h e r .
(6) may b e r e w r i t t e n
as
oe~.31 '
(7)
Q = wI = Wp/p = Wq/q
(8)
where
o r any e q u i v a l e n t s similarly
in terms of other
for the other
I n (8) WI d e n o t e s dW/dI, and
invariants.
As i n [1] and [2] we i n t r o d u c e equilibrium
invariants.
equations(with
the complex stress
no body f o r c e s )
function
are satisfied
h so t h a t
identically.
the We t h e n
have ~h ~=
• ½{Sll + s22 + 1 ( s 1 2 - s 2 1 ) }
,
3h • ~-~= ½{s22 - S l l - 1 ( s 1 2 ÷ s 2 1 ) } so t h a t ,
with the help of (1),
(2) and ( 7 ) ,
the governing equations
can be put
as 3_kh= ( Q 3z
o) ~~z
,
~h _ 3~
(Q + o )
~z ~. ~
(9)
AN INTEGRAL EQUATION FORMULATION
349
On the boundary we may prescribe either z or h, for example, specification of h being equivalent to specification of the (nominal) traction.
At this point,
in order to arrive at a suitable integral equation equivalent of (9), we - -
depart, temporarily, from our use of Z and Z as independent variables.
Change of independent variable: the .reciprocal equations
If the independent variables Z and ~ are replaced by z and E the equations (9) may be rearranged as ~h ~--~:
½pWp - @
use having been made of (5).
,
~h ~--~ = 2Q ~z ~ ~~
(lO)
For definiteness Q is regarded as a function
of p, which can now be expressed in the form ~P =
"~z
'
(ii)
noting (3) and (5). On integration the second equation in (IO) provides the solution for h(z,~) for any given Z(z,~) subject to (4), which is now put as ~zl
-
~
:
1.
(12)
The first equation in (I0) merely provides an expression for ~ once h and Z are determined.
Equations equivalent to (I0), but in other variables, can be
found in [3]. We write the solution as Z = f(z,~) .
(13)
According to Adkins [4], if (13) is a solution of (iO) and (12) then the 'inverse' z = f(Z,~)
(14)
is a solution of the same equations and for the same strain-energy function W. The proof given in [4] is rather inelegant, but a clearer and more general proof was given by Shield [5]. This inverse property of the equations can be seen immediately if the roles of Z and z are reversed and the function h*(Z,~) is defined by
~h* ~
~--~ ~--~. :
~h ~
~
~
•
(15)
Equations (IO) are replaced by ~h* - ~ : ½pWp - ~* '
~h*
~
~z DE
: 2Q ~
~
,
(16)
350
R . W . OGDEN
use having been made of (2), (5) and (II), where ~* + ~ = - W . Equations (16) have precisely the same forms as those in (IO), with the roles of Z and z reversed, but with p unchanged.
It follows immediately that if
(13) is a solution of (I0) then (14) is a solution of (16) and hence also oy (i0) ~n U ~ W o2~ ~
~q~u~en~
o~ (I0) ~n~ (16).
It should be pointed out,
however, that h* is no% the stress function associated with the defor~tion (14).
Once h* is calculated from the second equation in (16) the actual stress
function h is obtained from (15). Either (I0) or (16) ~ y boundary-value problem.
be taken as the starting point in the analysis of any Depending on the problem in question it ~ y
convenient to use one set of equations rather than the other.
be more
An advantage of
(16) is that ~n~%~Z oooP~n~%~8 Z and ~ are the independent variables, and the initial b o u n d a ~ geometry is k n o ~ .
~en
z and ~, on the other hand, are taken
as the independent variables the (current) boundary geometry depends on the (initially unknot)
deformation.
A disadvantage of (16) is that, in general,
(15) has to be used to calculate h in order to acco~odate any traction boundary condition. Equations (10)2 and (16) 2 are integrated to give h(z,~) = 2
~
and ~Z h*(Z,~) = 2
~_
Q{2fz(Z,~)~f~(~,z)~}f~(z,~)~(~,z)d~ + k(z)
(17)
- -
I_ Q{2fz(Z,~)~f~(~,Z)~}f~(Z,~)~(~,Z)d~ + k*(Z)
(18)
respectively, where k(z) and k*(Z) are arbitrary functions of z and Z respectively, and ~(~,Z) denotes the complex conjugate of f(Z,~). For any given boundary-value problem it r e ~ i n s to determine the u n k n o ~ functions k(z) and f(z,~), or, equivalently, k*(Z) and f(Z,~), subject to (12) or (4) when the boundary data are specified.
~ i c h e v e r of (17) or (18) is
adopted an integral equation for f arises when the boundary conditions are introduced.
We now illustrate the method by considering a specific problem,
adopting (18) as our starting point.
The inte.gral equation
It is well known that there can be two solutions to the zero-traction boundaryvalue problem for a hemi-spherical elastic shell.
The analogous problem in
AN INTEGRAL EQUATION FORMULATION
351
plane-straln is that of a seml-clrcular shell also under zero traction, and we now examine this. Suppose the (undeformed and unstressed) shell is defined by A ~ R ~ B
,
-½~ 6 @ ~ ½~
,
(19)
where Z = Re i@ , and let the surfaces R = A,B (-½~ ~ @ ~ ½~) and @ = ±½~ (A ~ R ~ B) be free of traction, so that we may take h = 0 on these boundaries. Equation (15) allows us to replace the boundary condition by h* = O, and we therefore have from (18) h*(Z,g) = 2
~ a/zQ{2fz(Z,~)~f~(~,Z)I}f~(Z,~)~C(~,Z)d~. ,_
(20)
The boundary condition on R = A is automatically satisfied and A~/Z k*(Z) = -2 J Q{2fz(Z,~)½~(~,Z)½}f~(Z,~)~(~,Z)d
~he b o u n d a r y c o n d i t i o n fAB2/Z 2/Z
Q{2fz(Z,~)½~(~,z)½}f~(z,~)~(~,Z)d~
evaluated for either boundary condition
fA
on R ~ B i s s a t i s f i e d
~ .
if
= 0
Z = Ae i 0 o r Be 10 (-½~ ~ @ ~ ½~) . on 8 = ±½~ i s
satisfied
(21)
(22) Additionally,
the
if
Q{2fz (Z' ~) ½~(~'Z) ½} f~ (Z' ~)~(~'Z) d~ = 0
(23)
~/z
evaluated for Z = ±JR. The solution (14) of (22) and (23) is subject to (4). solution is the trivial one f(Z,~) = Z. or not this solution is unique.
Clearly, one possible
The question now arises as to whether
Such a solution might be expected on
intuitive grounds, but the non-linearity of the equations seems to preclude the likelihood of an analytical solution being found.
For the neo-Hookean
solid Q = ~, a constant, but even in this case the basic problems remain. •
If the solution is assumed to be symmetrical about 8 = 0 the condition ~(Z,~) = f(Z,~) must be satisfied but this does not make solution of (22), (23) and (4) significantly easier. We do not take the problem any further at this stage, although if a second solution exists it should be feasible to find it numerically.
Our main
purpose is merely to give the integral equation formulation of the basic equations.
352
R . W . OGDEN
References
I. 2. 3. 4. 5.
D.A. R.W. A.E. J.E. R.T.
Isherwood and R.W. Ogden, Rheol. Acta 16, 113 (1977). Ogden, to be published. Green and J.E. Adkins, Large Elastic Deformations. Oxford (1970) Adkins, J. Mech. Phys. Solids ~, 267 (1958). Shield, Zeits. angew. Math. Phys. 18, 490 (1967).