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Finite deformations of slightly compressible materials—on the finite screw dislocation
hf. J. Engng Sci., 1972, Vol. 10, pp. 953-961.
Pefgamon Press.
Printed in Great lkitain
FINITE DEFORMATIONS OF SLIGHTLY MATERIALSON THE FINITE SCREW
COMPRESSIBLE DISLOCATION
M. SHAHINPOORt School of Engineering, Pahlavi University, Shiraz, Iran (Communicated by A. J. M. SPENCER) Abstract- Employing the known universal, controllable solutions of hnite deformations of incompressible, isotropic and elastic materials, a perturbation scheme is developed to extend the above controllable solutions to slightly compressible, isotropic materials. For a general strain energy function the governing equations of a class of finite screw-dislocation are obtained. The solutions of these governing equations are then found for some slightly compressible materials such as the Blatz, Rlatz-Ko, Polynomi~ and Muds materials. 1. INTRODUCTION
to Ericksen’s notable exposition [ 11,there exists only one class of deformations that are possible in all homogeneous compressible elastic materials, namely the homogeneous deformations. In the case of incompressible, isotropic elastic materials exact solutions for five families of deformations have so far been obtained. Four of these deformations were enunciated by Ericksen [2] and the fifth one (azimuthal shear, extension and flexture of a cuboid) was found by Klingbeil and Shield [3] and independently by Singh and Pipkin [4]. It is worthwhile to mention that the counterpart dynamical (quasi-equilibrated) solutions of these five families of defo~ations were first classified by Truesdell[5]. The existence of such known universal solutions suggests the feasibility of finding solutions to the problem of finite deformations of slightly compressible materials. It has been felt for some time, that there is a definite need for a rational theory to analyse the finite deformations of slightly compressible materials. This is because the idealization of incompressibility does not have experimental justification. For example, even rubbers are not, according to experiments[6], completely ~comp~ssible. For these reasons one needs to lay down a theoretical foundation for such materials. Oldroyd [7] and independently Spencer[& 91 have originally proposed a theory for slightly compressible materials. This theory depends on inventing an associated incompressible material and was applied recently to the finite dynamic deformations of spherical shell by Faulkner [ lo]. In the present work we propose an approach to the problem of finite defo~ations of slightly compressible materials. However, our approach is different in the sense that the isochoric part of the deformation is assumed to fall in one of the five families of controllable deformations found in [2-41, and it is assumed to be known a priori. To shed some light on the present theory, a special class of controllable deformation of finite screw-dislocation type is considered and then applied to some typical slightly compressible materials such as the Blatz, Blatz-Ko, Polynomial, and Mumaghan materials. ACCORDING
tChairman: Department of Mechanical Engineering. 953
954
M. SHAHINPOOR
Governing formulae Since subsequently we will be using the theory of Green, Rivlin and Shield [ 1 l] for superimposed additional displacements it will be convenient to use the convected coordinate formalism as presented in [ 121. There is however no diaculty in expressing the results in spatial coordinate formalism as presented by Truesdell and Toupin [ 131. A very brief summary of the results required in the present problem is now given, but for details of the notation and for the derivation of these formulae the reader is referred to the original expositions [8,9, 11, 121. The body in its undeformed and finitely deformed states is denoted by B,,, and B respectively. The displacement vector of a point in B is denoted by d + EW,where E is a small arbitrary constant. Since the material is slightly compressible we can assume that there is a small additional displacement vector w superposed on some large displacement vector d and furthermore d + EWis such that, under the same loading condition, EW+ 0 as the material becomes incompressible. Therefore d actually corresponds to a state of strain which would be produced in the material if it were incompressible. Thus, one can assume that the perturbation displacement w plays the role of a small correction to the displacement vector d which corresponds to the incompressible case under the same prescribed loading. Note that the deformation corresponding to d must fall into one of the five classes of deformations possible in every incompressible, isotropic and elastic material and it is assumed to be known a priori. Taking the metric tensor in B,, to be gti, g”, we then have the following quantities corresponding to the displacement field d + EW: Metric tensors in B; Gij + EG;, Gij+ EG’~~,where G; = willj + wj(li; Gtij = - GirGj8G;8 .
The determinant
(1.1)
of GU + EGL in B becomes G + EG’ with G’
=
GGijG!.
(1.2)
II -
The strain invariants in B are
i-1= II + el: iz= Zz+d;
= g’“G,, + q”G:s, = g,,G’SZ3+EgrS(G’TSZB+GT8Zj), (1.3)
In this problem the material is homogeneous, a strain energy density function w=w(z;,z;,z;,
isotropic and therefore
= w(z~+Ez;,zz+Ez~z,+6z’3).
it possesses
(1.4)
The stress tensor in B then becomes ,p=Tij+,rti=
4gij+~Bij+pG”+E(4’gil+~‘B’ii+p’Gij+pG’ii+~B’ij),
(1.5)
Finite deformations of slightly compressible
materials
95.5
(l-6) (3.7) FZ;+BZ;+Dz;-$lj
(1.8) 3
pi- EpI = 21’12aw*+r az 3
Z~(~Z~+DZ~+CZ~)+~Z~
3
1 ,
(1.9)
with A= --2 azw*,B_
2 _, a2w* C - 2 hv” zp art I:” azg ’
Ii’2 az:
D = --2 d2W” E= --2 a2w* ,F= zpaz2az3’ zg2 azdz,
--2 a2w*
z:f2 az,az 2 ’
(1.10)
where w* (I,, Z2,Z3) is the energy stored in the body due to the deformation vector d and
p+ l p’ is a known hydrostatic pressure. The equation of equilibrium in B is
where Fi is the body force per unit mass. Finally, at the boundaries, the components of the surface tractions must satisfy the relations aF,
Y-$&7
=PJ
(i,j=
1,2,3)
where F (&, f&,19,)= 0 is the equation of the surface of the body and y is a constant such that y (a F/a&) are the components of a unit normal vector to the surface. SU~~Q~ of the procedure (1) Depending on the nature of deformation consider a finite deformation d corresponding to the isochoric case of Z3= 1. This will create a strain energy in the body w* = w (II, Zz,Z3), enabling one to find the system of stresses +k corresponding to d. Since the material is indeed compressible this system of stresses does not uniquely determine the solution and does not, in general, satisfy the equilibrium equations and boundary conditions. If the compressib~ity is su~ciently small, the correct displacement may be expected to differ from d by a small amount EWwhere Eis a small arbitrary constant. Such a correctional displacement vector EWproduces incremental stresses ~7’ij in the stress system #j. (2) ~iif~~‘~ must satisfy the equilibrium equations and boundary conditions and this determines EWthe correction vector. Now d + EWgives the complete solution to the problem. Note here that E may be a mathematical artifice which could be incorporated into w itself or may be taken as connected to the property of small compressibility of the material(i.e.Z,-1 =E:e % 1).
M. SHAHINPOOR
956 2. ILLUSTRATIVE
SPECIAL
CLASS
OF
FINITE
DEFORMATIONS
Let us investigate the problem of a finite quasi-screw dislocation of some slightly compressible materials. Here by quasi-screw dislocation one means that the isochoric part of the deformation corresponds to that of a finite screw dislocation which falls in family 3 of Ericksen’s problem [2]t . Let us introduce in B the cylindrical coordinate system (Ii= (r, 8, z). Then taking the initial isochoric deformation, corresponding to d, to be a finite screw dislocation of strength a, we have R=r,O=e,Z=z+f&e,
(2.1)
where R, 0 and Z are initial coordinates of the point 8’. The metric tensors for B. and B are, respectively inBo+
with b = a/2m. The strain tensor becomes (2.4) The corresponding strain invariants are f Z3= 1.
ZI=Z2=3+b2r2
(2.5)
The deformation corresponding to d produces in the body B the stress tensor TU such that + = Q-t (2+b2r-2)++p, r%22= @+2$++,
(2.6)
733= (1+b2r-‘)(Q)+JI)+$+p, m23= -(b/r) (Q,+ I&), 713 =
712 =
0
(2.7)
,
where @==2$$
JI=2!$, 1
Wee also Truesdell and Noll [ 141.
pz2$.$ 2
(2.8) 3
Finite deformations of slightly compressible materials
957
Now the correct displacement is expected to differ from that described by (2.1) by a small radial displacement u. The corresponding incremental stresses become? 7’11= 4’ + (2 + Pr-2) JI’+p’ + 2Jlu/r - 2pur, r2r’22 = 4’ + T’33=
2JI’+p’ + 2Jru, - 2pu/r,
(1+b2F2)+‘+
(2+b2r-2)$‘+p’+2$(ut+u/r+b2r-2ur),
rr’““=-(&)(+‘+$I’)-2(b/r)Jlu,, ,f13
=
7’12
=
0
(2.9)
9
with 4’= (2A+2E-~)(u,+u/r)+2F[(2+b2r-2)(u,+u/r)-(bZr-2)(u/r)],
(2.10)
$‘= (2F+20-_)(u,+u/r)+2B[(2+bzr-“)(u,+u/r)-(b2r-*)(u/r)],
(2.11)
P’=
(2.12)
(2E+2C+P)(u,+u/r)+20[(2+b2r-2)(ur+u/r)-(b2r-2)(u/r)], &-2?Y,
*=2?5
Dz2if&
c=2!!5
.=,&,
.=,-i&
(2.13) 1
1
3
2
In the absence of body forces the equations of equilibrium become $
+ Ijm+mi+
I j,+
#~=T~+T’~; i,j,m=
= 0, 1,2,3,
(2.14)
where the only non-zero Christoffel symbols I;,,, are r:, = -r,
r-F2 = l/r.
(2.15)
Equations (2.14) now become
+24,(u,+u,/r-u/r2)+2~[(2+b2r-2)un.+ur/r-(1+b2r-2)u/r2]+2pu,,=0.
(2.16)
The above equilibrium equation is valid for arbitrary forms of the function w(Zl,Z2,Z3), provided that the material is slightly compressible. The solution u of the above equation determines the correct deformation field for the problem under consideration. 3. SPECIAL
SLIGHTLY
COMPRESSIBLE
MATERIALS
(a) Blatz [ 151proposed the following strain energy function for slightly compressible material W~=~[(Z,-3)-(&--)lnZj~2+(&)(Zj”-1)] tNote here that Eis incorporated in all incremental quantities here after.
(3.1)
958
M. SHAHINPOOR
where CL,,and I+, are shear modulus and Poisson’s ratio for the material when the deformations become vanishingly small. With (3.1) based on (3.1) equation (2.16) together with (2.6)-(2.13) reduces to (3.2) whose solution (Euler’s equation) is simply u
=Alr+A,r-’
(3.3)
where A 1and A 2 are appropriate integration constants. (b) Blatz and Ko [ 161 introduced the following strain energy function
(3.4) where cc0 and v0 are defined as before and f is a material constant such that 0 < f < 1. Again based-on (3.4) and together with (2.6)-(2.13) equation (2.16) reduces to u,,+ (u,/r)[1+2a*b2/r2]
- (u/r2)[1+4a*b2/r2]
= a*b2/P
(3.5)
where a* _-(1-f)(l-2%?) 2(1--v,)
’
Note that for slightly compressible materials (1 - 2~~) 4 1 and for ah practical purposes one can find a good approximation to u by putting 1 + 2a *b2r-2 = 1 + 4a *b2rp2 = 1 provided r > 0 i.e. the region around the z-axis is excluded. Thus one can finally obtain: U= C,(r/b)+C2(b/r)(u*/2)(b2/r) In (r/b), (3.6) where C1 and C, are integration constants. (c) Levinson and Burgess [ 171 introduced the following strain energy function for the simplest rational polynomial representation of the strain energy density for a compressible hyperelastic material which reduces to the Mooney-Rivlin material as Vg-, 4: w,=(~u,/2)[f(~,-33)+(1--f)((~2/~,)-33)+22(~--2f)(z:‘2-1)
+(2f+~)(z;‘2-1)2] where po, v. and fare defined as before. Again based on (3.7) and together with (2.6)-(2.13)
(3.7)
equation (2.16) reduces to (3.8)
where a * is defined as before.
Finite deformations of slightly compressible
959
materials
The solution of (3.8) is also given by equation (3.6) for r > 0. (d) Finally assume that the strain energy is that proposed by Mumaghan I18] : W,,g= (~+2mf24)(ZI-33)3+((hf2~+4m)/8)(Z~-3)2+((8~+n)/8)(Z,-3) -~~/4~t~,-3)(1,-3)-(~4~~~)/~)(fz-3)+~~/~)(z3-1),
(3.9)
where P, A, q, m, n are elastic constants. Again based on (3.9) and together with (2.6)-(2.13) one arrives, from (2.16), at IC, [i + aIb2T2 + (qf4)b*P]
+ (u,/r) [i + azb2rT2+ adb4rs*]
- (u/r*) [i - a3b2rw2- (3/4)qb4re4] = a5b2re3+ aSb4re5,
(3.10)
where i=
(A+~P), a, = ((2h+2m+4q-n)/4),
u3= ((2h-l2q-6m-n)l4),
a,=
((6h+6m-4q+8p-3n)/4),
u4= ((4m+5q)/4),
aa=h+p+m--(n/4),
(3.11)
a6 = q+ (m/2).
The solution of equation (3.10) for small (b2/r2)
X-t-p+m-
(n/4)
2@+2P)
In(rib) 1b’LrF2
-+ CIr + C2re1, r > 0,
(3.12)
where CI and C2 are appropriate integration constants. However a general series solution of (3.10) can be obtained by Frobenius’ Method and is u = 2 Cylrp’+u+ Cy2rPPtYf Cv3rP3+y+ Cv4rp4+v+ 5 Ag’, V=O i=O where for even V’S _-~~~-z,nf2tPn+~--2)-~~v--4~nf4tPn+~-44)
c Ylt
-
.f&Pn+ 4
, (n = 1,2,3,4)*
(3.13)
(3.14)
for odd v’s and v < 0, c,, = 0 and Ply P2 =
-2((m/q)+l)t[((mlg)+1)2-(3/4)11’2,
p3=3,p4=
1.
(3.15) (3.16)
Cal, Cozare integration constants, and .6(x) = x(x-
1) fb2q/4) +xa,b*+
_&(x) = x(x-
l)b2al+xb2a,+a3b2,
h(n) =x(x-
l)T;+xh--,
(3/4)qb4,
(3.17)
960
M. SHAHINPOOR
and Ai = 0, for all even i’s and i < 0, A. = [lI
(i-4)2]A~--4[i(i(i= 5,7,9,11,
[(i-2)(i-3)a,b2+ 1)(~/4)~*+~~~~4+
13,. . .),
(i--2)~~b~+a&~]A~_~ , (3/4)qb*]
(3.18) (3.19)
4. CONCLUSION
A theory of finite defo~ations for slightly compressible materials was formulated. This theory was extended to the class of finite screw-dislocation, and the governing equations were obtained for a general strain energy function. The solutions of these governing equations were then found for some slightly compressible materials such as the Blatz, Blatz-Ko, Polynomial, and Murnaghan materials. In each case the entire deformations field d-t w was obtained explicitly from which the stresses may be obtained under appropriate boundary conditions. Ackno~ledgmenf
- I am greatly indebted to Professor Dr. A. 3. M. Spencer and to my colleague Professor Dr. G. Ahmadi for their valuable remarks concerning this paper.
REFERENCES [l] J. L. ERICKSEN, J. math. Phys. 34(1955). 121 J. L. ERICKSEN, ZAMP 5 (1954). [3] W. M. KLINGBEIL and R. T. SHIELD, .Z. angew. math. Phys. 17 (1966). [4] M. SINGH and A. PIPKIN,Arch. r&on. Me& Analysis 21(1966). 151 C. TRUESDELL,Arch. ration. Me&. Anafysis il(l962). [6] P. J. BLATZ and W. L. KO, Trans. Sot. Rheol. 5,223 (1962). [7] OLDROYD, Proc. R. Sot. A202,407 (1950). [8] A. J. M. SPENCER, Proc. 8th. Int. Symp. On Second Order Effects in Elasticity, Plasticity and Fluid Mechanics (Edited by REINER and ABIR). Pergamon Press (1964). [9] A. 3. M. SPENCER, J. Inst. Math. Applies. 6 (1970). [lo] T. R. FAULKNER, fnt. J. Engng Sci. 19, (197 1). [ 1l] A. E. GREEN and R. S. RIVLIN and R. T. SHIELD, Proc. R. Sot. A211,128 (1952). [ 121 A. E. GREEN and W. ZERNA, Theoretical Elasticity. Oxford (1954). [ 131 C. TRUESDELL and R. TOUPIN, Handbook of Physics, 3, Springer (1960). [!4] C. TRUESDELL and W. NOLL, Encyclopedia of Physics, vol. I/3 (Edited by FLUGGE). Springer (1965). [ 151 P. J. BLATZ, Calif. Inst. of Tech. CaLCIT SM6Q-25, (1960). [16] P. J. BLATZand W. L. KO, Trans. Sot. Rheol. 5,223 (1962). [17] M. LEVINSON and I. W. BURGESS, lnt. J. Mech. Sci. 13 (1971). [IS] F. D. MURNAGHAN, Finite Deformations of an Elastic Solid. New York (195 1). [19] A.SEEGER~dE.MANN,Z.~af.l~, 154(1959). [20] Z. WESOLOWSKI and A. SEEGER, IUTAM Symposium, Fredenstadf. Stutgart (1967). (Received
18 April 1972)
R&u&-En utilisant les solutions universelfes et controlables connues des deformations finies de materiaux incompressibles, isotropes et Bastiques, un sch6ma de perturbation est developpe pour Ctendre les solutions ci-dessus aux matCriaux isotropes legerement compressibles. Pour une fonction &n&ale de l’lnergie de contrainte, les equations regissant une classe de dislocations finies de vis sont obtenues. Les solutions de ces equations rogissantes sont alors trouvees pour quelques mat&iaux leg&ement compressibles teis que les mat&iaux Blatz, Btatz-Co, polynomiaux et de Murnaghan.
Finite deformations of slightly compressible
materials
961
Zusammenfassung-Unter Beniitzung der bekannten universellen kontrollierbaren Losungen endlicher Deformationen von inkompressiblen, isotropen und elastischen Stoffen wird ein Stiirungsschema entwickelt urn die obigen kontrollierbaren Losungen auf ein wenig kompressible, isotrope Stoffe auszudehnen. Die regierenden Gleichungen einer Klasse von endlicher Schraubenverdrlngung werden fur eine allgemeine Belastungsenergiefunktion erhalten. Es werden dann die Losungen dieser regierenden Gleichungen fiir einige ein wenig kompressible Stoffe gefunden, wie Blatz-, Blatz-Ko-, polynomische und Murnaghan-Stoffe. Sommario-Impiegando
le soluzioni controllabiii
comprimibili,
ed
isotropici
elastici,
si
sviluppa
e universali note di deformazioni uno
schema
dip
perturbazione
per
finite di materiali estendere
le
in-
soluzioni
di cui sopra a materiali isotropici leggermente comprimibili. Si ottengono le equazioni regolatrici di una classe di spostamento a vite finito nei riguardi di una funzione generale d’energia di sollecitazione. Si trovano quindi le soluzioni di queste equazioni regolatrici nei riguardi di alcuni materiali leggermente comprimibili quali i materiali di Blatz, Blatz-Ko, polinomiali e di Mumaghan. controllabili
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Pefgamon Press.
Printed in Great lkitain
FINITE DEFORMATIONS OF SLIGHTLY MATERIALSON THE FINITE SCREW
COMPRESSIBLE DISLOCATION
M. SHAHINPOORt School of Engineering, Pahlavi University, Shiraz, Iran (Communicated by A. J. M. SPENCER) Abstract- Employing the known universal, controllable solutions of hnite deformations of incompressible, isotropic and elastic materials, a perturbation scheme is developed to extend the above controllable solutions to slightly compressible, isotropic materials. For a general strain energy function the governing equations of a class of finite screw-dislocation are obtained. The solutions of these governing equations are then found for some slightly compressible materials such as the Blatz, Rlatz-Ko, Polynomi~ and Muds materials. 1. INTRODUCTION
to Ericksen’s notable exposition [ 11,there exists only one class of deformations that are possible in all homogeneous compressible elastic materials, namely the homogeneous deformations. In the case of incompressible, isotropic elastic materials exact solutions for five families of deformations have so far been obtained. Four of these deformations were enunciated by Ericksen [2] and the fifth one (azimuthal shear, extension and flexture of a cuboid) was found by Klingbeil and Shield [3] and independently by Singh and Pipkin [4]. It is worthwhile to mention that the counterpart dynamical (quasi-equilibrated) solutions of these five families of defo~ations were first classified by Truesdell[5]. The existence of such known universal solutions suggests the feasibility of finding solutions to the problem of finite deformations of slightly compressible materials. It has been felt for some time, that there is a definite need for a rational theory to analyse the finite deformations of slightly compressible materials. This is because the idealization of incompressibility does not have experimental justification. For example, even rubbers are not, according to experiments[6], completely ~comp~ssible. For these reasons one needs to lay down a theoretical foundation for such materials. Oldroyd [7] and independently Spencer[& 91 have originally proposed a theory for slightly compressible materials. This theory depends on inventing an associated incompressible material and was applied recently to the finite dynamic deformations of spherical shell by Faulkner [ lo]. In the present work we propose an approach to the problem of finite defo~ations of slightly compressible materials. However, our approach is different in the sense that the isochoric part of the deformation is assumed to fall in one of the five families of controllable deformations found in [2-41, and it is assumed to be known a priori. To shed some light on the present theory, a special class of controllable deformation of finite screw-dislocation type is considered and then applied to some typical slightly compressible materials such as the Blatz, Blatz-Ko, Polynomial, and Mumaghan materials. ACCORDING
tChairman: Department of Mechanical Engineering. 953
954
M. SHAHINPOOR
Governing formulae Since subsequently we will be using the theory of Green, Rivlin and Shield [ 1 l] for superimposed additional displacements it will be convenient to use the convected coordinate formalism as presented in [ 121. There is however no diaculty in expressing the results in spatial coordinate formalism as presented by Truesdell and Toupin [ 131. A very brief summary of the results required in the present problem is now given, but for details of the notation and for the derivation of these formulae the reader is referred to the original expositions [8,9, 11, 121. The body in its undeformed and finitely deformed states is denoted by B,,, and B respectively. The displacement vector of a point in B is denoted by d + EW,where E is a small arbitrary constant. Since the material is slightly compressible we can assume that there is a small additional displacement vector w superposed on some large displacement vector d and furthermore d + EWis such that, under the same loading condition, EW+ 0 as the material becomes incompressible. Therefore d actually corresponds to a state of strain which would be produced in the material if it were incompressible. Thus, one can assume that the perturbation displacement w plays the role of a small correction to the displacement vector d which corresponds to the incompressible case under the same prescribed loading. Note that the deformation corresponding to d must fall into one of the five classes of deformations possible in every incompressible, isotropic and elastic material and it is assumed to be known a priori. Taking the metric tensor in B,, to be gti, g”, we then have the following quantities corresponding to the displacement field d + EW: Metric tensors in B; Gij + EG;, Gij+ EG’~~,where G; = willj + wj(li; Gtij = - GirGj8G;8 .
The determinant
(1.1)
of GU + EGL in B becomes G + EG’ with G’
=
GGijG!.
(1.2)
II -
The strain invariants in B are
i-1= II + el: iz= Zz+d;
= g’“G,, + q”G:s, = g,,G’SZ3+EgrS(G’TSZB+GT8Zj), (1.3)
In this problem the material is homogeneous, a strain energy density function w=w(z;,z;,z;,
isotropic and therefore
= w(z~+Ez;,zz+Ez~z,+6z’3).
it possesses
(1.4)
The stress tensor in B then becomes ,p=Tij+,rti=
4gij+~Bij+pG”+E(4’gil+~‘B’ii+p’Gij+pG’ii+~B’ij),
(1.5)
Finite deformations of slightly compressible
materials
95.5
(l-6) (3.7) FZ;+BZ;+Dz;-$lj
(1.8) 3
pi- EpI = 21’12aw*+r az 3
Z~(~Z~+DZ~+CZ~)+~Z~
3
1 ,
(1.9)
with A= --2 azw*,B_
2 _, a2w* C - 2 hv” zp art I:” azg ’
Ii’2 az:
D = --2 d2W” E= --2 a2w* ,F= zpaz2az3’ zg2 azdz,
--2 a2w*
z:f2 az,az 2 ’
(1.10)
where w* (I,, Z2,Z3) is the energy stored in the body due to the deformation vector d and
p+ l p’ is a known hydrostatic pressure. The equation of equilibrium in B is
where Fi is the body force per unit mass. Finally, at the boundaries, the components of the surface tractions must satisfy the relations aF,
Y-$&7
=PJ
(i,j=
1,2,3)
where F (&, f&,19,)= 0 is the equation of the surface of the body and y is a constant such that y (a F/a&) are the components of a unit normal vector to the surface. SU~~Q~ of the procedure (1) Depending on the nature of deformation consider a finite deformation d corresponding to the isochoric case of Z3= 1. This will create a strain energy in the body w* = w (II, Zz,Z3), enabling one to find the system of stresses +k corresponding to d. Since the material is indeed compressible this system of stresses does not uniquely determine the solution and does not, in general, satisfy the equilibrium equations and boundary conditions. If the compressib~ity is su~ciently small, the correct displacement may be expected to differ from d by a small amount EWwhere Eis a small arbitrary constant. Such a correctional displacement vector EWproduces incremental stresses ~7’ij in the stress system #j. (2) ~iif~~‘~ must satisfy the equilibrium equations and boundary conditions and this determines EWthe correction vector. Now d + EWgives the complete solution to the problem. Note here that E may be a mathematical artifice which could be incorporated into w itself or may be taken as connected to the property of small compressibility of the material(i.e.Z,-1 =E:e % 1).
M. SHAHINPOOR
956 2. ILLUSTRATIVE
SPECIAL
CLASS
OF
FINITE
DEFORMATIONS
Let us investigate the problem of a finite quasi-screw dislocation of some slightly compressible materials. Here by quasi-screw dislocation one means that the isochoric part of the deformation corresponds to that of a finite screw dislocation which falls in family 3 of Ericksen’s problem [2]t . Let us introduce in B the cylindrical coordinate system (Ii= (r, 8, z). Then taking the initial isochoric deformation, corresponding to d, to be a finite screw dislocation of strength a, we have R=r,O=e,Z=z+f&e,
(2.1)
where R, 0 and Z are initial coordinates of the point 8’. The metric tensors for B. and B are, respectively inBo+
with b = a/2m. The strain tensor becomes (2.4) The corresponding strain invariants are f Z3= 1.
ZI=Z2=3+b2r2
(2.5)
The deformation corresponding to d produces in the body B the stress tensor TU such that + = Q-t (2+b2r-2)++p, r%22= @+2$++,
(2.6)
733= (1+b2r-‘)(Q)+JI)+$+p, m23= -(b/r) (Q,+ I&), 713 =
712 =
0
(2.7)
,
where @==2$$
JI=2!$, 1
Wee also Truesdell and Noll [ 141.
pz2$.$ 2
(2.8) 3
Finite deformations of slightly compressible materials
957
Now the correct displacement is expected to differ from that described by (2.1) by a small radial displacement u. The corresponding incremental stresses become? 7’11= 4’ + (2 + Pr-2) JI’+p’ + 2Jlu/r - 2pur, r2r’22 = 4’ + T’33=
2JI’+p’ + 2Jru, - 2pu/r,
(1+b2F2)+‘+
(2+b2r-2)$‘+p’+2$(ut+u/r+b2r-2ur),
rr’““=-(&)(+‘+$I’)-2(b/r)Jlu,, ,f13
=
7’12
=
0
(2.9)
9
with 4’= (2A+2E-~)(u,+u/r)+2F[(2+b2r-2)(u,+u/r)-(bZr-2)(u/r)],
(2.10)
$‘= (2F+20-_)(u,+u/r)+2B[(2+bzr-“)(u,+u/r)-(b2r-*)(u/r)],
(2.11)
P’=
(2.12)
(2E+2C+P)(u,+u/r)+20[(2+b2r-2)(ur+u/r)-(b2r-2)(u/r)], &-2?Y,
*=2?5
Dz2if&
c=2!!5
.=,&,
.=,-i&
(2.13) 1
1
3
2
In the absence of body forces the equations of equilibrium become $
+ Ijm+mi+
I j,+
#~=T~+T’~; i,j,m=
= 0, 1,2,3,
(2.14)
where the only non-zero Christoffel symbols I;,,, are r:, = -r,
r-F2 = l/r.
(2.15)
Equations (2.14) now become
+24,(u,+u,/r-u/r2)+2~[(2+b2r-2)un.+ur/r-(1+b2r-2)u/r2]+2pu,,=0.
(2.16)
The above equilibrium equation is valid for arbitrary forms of the function w(Zl,Z2,Z3), provided that the material is slightly compressible. The solution u of the above equation determines the correct deformation field for the problem under consideration. 3. SPECIAL
SLIGHTLY
COMPRESSIBLE
MATERIALS
(a) Blatz [ 151proposed the following strain energy function for slightly compressible material W~=~[(Z,-3)-(&--)lnZj~2+(&)(Zj”-1)] tNote here that Eis incorporated in all incremental quantities here after.
(3.1)
958
M. SHAHINPOOR
where CL,,and I+, are shear modulus and Poisson’s ratio for the material when the deformations become vanishingly small. With (3.1) based on (3.1) equation (2.16) together with (2.6)-(2.13) reduces to (3.2) whose solution (Euler’s equation) is simply u
=Alr+A,r-’
(3.3)
where A 1and A 2 are appropriate integration constants. (b) Blatz and Ko [ 161 introduced the following strain energy function
(3.4) where cc0 and v0 are defined as before and f is a material constant such that 0 < f < 1. Again based-on (3.4) and together with (2.6)-(2.13) equation (2.16) reduces to u,,+ (u,/r)[1+2a*b2/r2]
- (u/r2)[1+4a*b2/r2]
= a*b2/P
(3.5)
where a* _-(1-f)(l-2%?) 2(1--v,)
’
Note that for slightly compressible materials (1 - 2~~) 4 1 and for ah practical purposes one can find a good approximation to u by putting 1 + 2a *b2r-2 = 1 + 4a *b2rp2 = 1 provided r > 0 i.e. the region around the z-axis is excluded. Thus one can finally obtain: U= C,(r/b)+C2(b/r)(u*/2)(b2/r) In (r/b), (3.6) where C1 and C, are integration constants. (c) Levinson and Burgess [ 171 introduced the following strain energy function for the simplest rational polynomial representation of the strain energy density for a compressible hyperelastic material which reduces to the Mooney-Rivlin material as Vg-, 4: w,=(~u,/2)[f(~,-33)+(1--f)((~2/~,)-33)+22(~--2f)(z:‘2-1)
+(2f+~)(z;‘2-1)2] where po, v. and fare defined as before. Again based on (3.7) and together with (2.6)-(2.13)
(3.7)
equation (2.16) reduces to (3.8)
where a * is defined as before.
Finite deformations of slightly compressible
959
materials
The solution of (3.8) is also given by equation (3.6) for r > 0. (d) Finally assume that the strain energy is that proposed by Mumaghan I18] : W,,g= (~+2mf24)(ZI-33)3+((hf2~+4m)/8)(Z~-3)2+((8~+n)/8)(Z,-3) -~~/4~t~,-3)(1,-3)-(~4~~~)/~)(fz-3)+~~/~)(z3-1),
(3.9)
where P, A, q, m, n are elastic constants. Again based on (3.9) and together with (2.6)-(2.13) one arrives, from (2.16), at IC, [i + aIb2T2 + (qf4)b*P]
+ (u,/r) [i + azb2rT2+ adb4rs*]
- (u/r*) [i - a3b2rw2- (3/4)qb4re4] = a5b2re3+ aSb4re5,
(3.10)
where i=
(A+~P), a, = ((2h+2m+4q-n)/4),
u3= ((2h-l2q-6m-n)l4),
a,=
((6h+6m-4q+8p-3n)/4),
u4= ((4m+5q)/4),
aa=h+p+m--(n/4),
(3.11)
a6 = q+ (m/2).
The solution of equation (3.10) for small (b2/r2)
X-t-p+m-
(n/4)
2@+2P)
In(rib) 1b’LrF2
-+ CIr + C2re1, r > 0,
(3.12)
where CI and C2 are appropriate integration constants. However a general series solution of (3.10) can be obtained by Frobenius’ Method and is u = 2 Cylrp’+u+ Cy2rPPtYf Cv3rP3+y+ Cv4rp4+v+ 5 Ag’, V=O i=O where for even V’S _-~~~-z,nf2tPn+~--2)-~~v--4~nf4tPn+~-44)
c Ylt
-
.f&Pn+ 4
, (n = 1,2,3,4)*
(3.13)
(3.14)
for odd v’s and v < 0, c,, = 0 and Ply P2 =
-2((m/q)+l)t[((mlg)+1)2-(3/4)11’2,
p3=3,p4=
1.
(3.15) (3.16)
Cal, Cozare integration constants, and .6(x) = x(x-
1) fb2q/4) +xa,b*+
_&(x) = x(x-
l)b2al+xb2a,+a3b2,
h(n) =x(x-
l)T;+xh--,
(3/4)qb4,
(3.17)
960
M. SHAHINPOOR
and Ai = 0, for all even i’s and i < 0, A. = [lI
(i-4)2]A~--4[i(i(i= 5,7,9,11,
[(i-2)(i-3)a,b2+ 1)(~/4)~*+~~~~4+
13,. . .),
(i--2)~~b~+a&~]A~_~ , (3/4)qb*]
(3.18) (3.19)
4. CONCLUSION
A theory of finite defo~ations for slightly compressible materials was formulated. This theory was extended to the class of finite screw-dislocation, and the governing equations were obtained for a general strain energy function. The solutions of these governing equations were then found for some slightly compressible materials such as the Blatz, Blatz-Ko, Polynomial, and Murnaghan materials. In each case the entire deformations field d-t w was obtained explicitly from which the stresses may be obtained under appropriate boundary conditions. Ackno~ledgmenf
- I am greatly indebted to Professor Dr. A. 3. M. Spencer and to my colleague Professor Dr. G. Ahmadi for their valuable remarks concerning this paper.
REFERENCES [l] J. L. ERICKSEN, J. math. Phys. 34(1955). 121 J. L. ERICKSEN, ZAMP 5 (1954). [3] W. M. KLINGBEIL and R. T. SHIELD, .Z. angew. math. Phys. 17 (1966). [4] M. SINGH and A. PIPKIN,Arch. r&on. Me& Analysis 21(1966). 151 C. TRUESDELL,Arch. ration. Me&. Anafysis il(l962). [6] P. J. BLATZ and W. L. KO, Trans. Sot. Rheol. 5,223 (1962). [7] OLDROYD, Proc. R. Sot. A202,407 (1950). [8] A. J. M. SPENCER, Proc. 8th. Int. Symp. On Second Order Effects in Elasticity, Plasticity and Fluid Mechanics (Edited by REINER and ABIR). Pergamon Press (1964). [9] A. 3. M. SPENCER, J. Inst. Math. Applies. 6 (1970). [lo] T. R. FAULKNER, fnt. J. Engng Sci. 19, (197 1). [ 1l] A. E. GREEN and R. S. RIVLIN and R. T. SHIELD, Proc. R. Sot. A211,128 (1952). [ 121 A. E. GREEN and W. ZERNA, Theoretical Elasticity. Oxford (1954). [ 131 C. TRUESDELL and R. TOUPIN, Handbook of Physics, 3, Springer (1960). [!4] C. TRUESDELL and W. NOLL, Encyclopedia of Physics, vol. I/3 (Edited by FLUGGE). Springer (1965). [ 151 P. J. BLATZ, Calif. Inst. of Tech. CaLCIT SM6Q-25, (1960). [16] P. J. BLATZand W. L. KO, Trans. Sot. Rheol. 5,223 (1962). [17] M. LEVINSON and I. W. BURGESS, lnt. J. Mech. Sci. 13 (1971). [IS] F. D. MURNAGHAN, Finite Deformations of an Elastic Solid. New York (195 1). [19] A.SEEGER~dE.MANN,Z.~af.l~, 154(1959). [20] Z. WESOLOWSKI and A. SEEGER, IUTAM Symposium, Fredenstadf. Stutgart (1967). (Received
18 April 1972)
R&u&-En utilisant les solutions universelfes et controlables connues des deformations finies de materiaux incompressibles, isotropes et Bastiques, un sch6ma de perturbation est developpe pour Ctendre les solutions ci-dessus aux matCriaux isotropes legerement compressibles. Pour une fonction &n&ale de l’lnergie de contrainte, les equations regissant une classe de dislocations finies de vis sont obtenues. Les solutions de ces equations rogissantes sont alors trouvees pour quelques mat&iaux leg&ement compressibles teis que les mat&iaux Blatz, Btatz-Co, polynomiaux et de Murnaghan.
Finite deformations of slightly compressible
materials
961
Zusammenfassung-Unter Beniitzung der bekannten universellen kontrollierbaren Losungen endlicher Deformationen von inkompressiblen, isotropen und elastischen Stoffen wird ein Stiirungsschema entwickelt urn die obigen kontrollierbaren Losungen auf ein wenig kompressible, isotrope Stoffe auszudehnen. Die regierenden Gleichungen einer Klasse von endlicher Schraubenverdrlngung werden fur eine allgemeine Belastungsenergiefunktion erhalten. Es werden dann die Losungen dieser regierenden Gleichungen fiir einige ein wenig kompressible Stoffe gefunden, wie Blatz-, Blatz-Ko-, polynomische und Murnaghan-Stoffe. Sommario-Impiegando
le soluzioni controllabiii
comprimibili,
ed
isotropici
elastici,
si
sviluppa
e universali note di deformazioni uno
schema
dip
perturbazione
per
finite di materiali estendere
le
in-
soluzioni
di cui sopra a materiali isotropici leggermente comprimibili. Si ottengono le equazioni regolatrici di una classe di spostamento a vite finito nei riguardi di una funzione generale d’energia di sollecitazione. Si trovano quindi le soluzioni di queste equazioni regolatrici nei riguardi di alcuni materiali leggermente comprimibili quali i materiali di Blatz, Blatz-Ko, polinomiali e di Mumaghan. controllabili
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