On large elastic deformation of prestressed right circular annular cylinders

International Journal of Non-Linear Mechanics 46 (2011) 96–113

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On large elastic deformation of prestressed right circular annular cylinders U. Saravanan Department of Civil Engineering, Indian Institute of Technology Madras, Chennai 600036, India

a r t i c l e in f o

a b s t r a c t

Article history: Received 18 January 2010 Received in revised form 23 July 2010 Accepted 26 July 2010

We formulate and study inflation, extension and twisting of prestressed cylindrical shells that are isotropic in the stress free configuration. We establish that if the prestresses vary only radially in the annular cylinder then a deformation field of the form r ¼ r^ ðRÞ, y ¼ Y þ OZ, z ¼ lZ is possible in annular cylinders made of any incompressible material and find sufficient conditions for the deformation to be possible when made of compressible materials. When the material is capable of undergoing large elastic deformations and has a non-linear constitutive relation, for the cases studied here, there is up to 26 percent variation in the boundary loads required to engender a given boundary displacement between the prestressed and stress free annular cylinders. On the other hand, the difference in the realized deformation field is only marginal (less than 2 percent). These are unlike the case wherein the material obeys Hooke’s law and undergoes small deformations. This study has some relevance to the deformation of blood vessels. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Prestresses Residual stress Inflation Extension Torsion Compressible Incompressible Elastic Large deformations

1. Introduction According to Withers and Bhadeshia [1] almost all manufactured components cannot be intact and stress free at the same instant because of the presence of misfits. These misfits can be intentionally designed as in a shrunk fit shaft or can arise due to the chemical, thermal or inelastic process that the component has been subjected to. The stress field in the configuration used as reference configuration, is called the prestress field. It is common in the literature to refer to what we call as prestress as residual stresses. However, we find that calling prestresses as residual stresses though apt in some cases it is not so for many others. As the nomenclature makes it clear, by residual stresses we understand that they are the stresses remaining at the end of some process that the body has been subjected to. But the stresses in a configuration need not arise only due to the process it can has been subjected to. For example, a body in static equilibrium while subjected to gravitational field cannot be traction free and hence stress free. Therefore, any static configuration used as reference configuration on the surface of the earth will have stresses in them which do not necessarily arise due to some process that they have been subjected to. Moreover since, here we are not interested in the cause for the stresses present in the configuration used as reference, we prefer to call these stresses in the reference configuration as prestresses. However, we do require that the prestress field result in no boundary traction. In other words, here we assume that the prestresses do not arise due to any E-mail address: [email protected] 0020-7462/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2010.07.007

body forces, like the gravitational field. Even though there exist no material which is inert to gravitational field, we still ignore them. The rationale behind this assumption, which requires validation, is that the prestresses due to gravitational forces would be very small compared to the applied stresses or the prestresses arising due to reasons other than body forces. It is common practice to ignore the influence of these prestresses in the stress analysis, just because they are difficult to include in the analysis. However, as the design of engineering components becomes less conservative, there is increasing interest in how prestresses affects the mechanical response of the components [1]. Superposition of these prestresses with the service stresses yields fairly accurate results when the material undergoes small deformations and the stress is linearly related to some measure of the strain. However, when the deformations are large and/or the stress is non-linearly related to stretch ratios or strain, one cannot superpose solutions and hence the effect of these prestresses is not known. Hence, here restricting ourselves to materials capable of undergoing large elastic deformations, like rubber and soft biological tissues, we investigate whether 1. The boundary load required to realize a given boundary displacement is the same from a prestressed and stress free configuration. 2. The interior deformation field is the same from a prestressed and stress free configuration for a given boundary condition, given that the prestresses are such that it does not result in any boundary traction. The importance of the requirement that the

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

prestresses result in no boundary traction cannot be overemphasized. As one would expect, unlike in the case of linearized elasticity, the boundary load required to realize a given boundary displacement varies with the prestress field. Our computations show that this variation can be as high as 26 percent for the cases considered in this study. Similarly, the interior deformation field also varies with the prestress field but by less than 1 percent for the various cases studied here. However, the difference in the gradient of the deformation field measured from the stress free and prestressed reference configurations is of the order of 2 percent for the different cases considered here. Thus, we suggest that for materials undergoing large deformations or whose constitutive relation is non-linear, the prestresses in the body could be estimated from standard experiments such as uniaxial extension, inflation of an annular cylinder at constant length and twisting of annular cylinders. However, the exact methodology of this procedure is not dealt with here. In biomechanics, usually (see [2,3]) to account for the prestresses in the intact body, a cut configuration, assumed to be free of stresses, is used as the reference configuration instead of the intact body. This allows one to continue to use the representation for stress from a stress free configuration and study the problem. However, it is known that in many cases, a single cut would not relieve the stresses and multiple cuts are required. Even if a single cut relieves the stresses there arise questions regarding the correct orientation of the cut. Also, the assumed deformation field from the cut configuration to the intact configuration would rarely be known a priori, especially in compressible materials because of the dearth of exact solutions. (Even the commonly assumed [2,3] deformation field for the blood vessels from the cut to the intact configuration does not ensure that the axial stresses are absent at the top and bottom surfaces; it could only ensure that no net axial load is applied [4,5].) Johnson and Hoger [6], Hoger [7] and Saravanan [8] realized that these difficulties can be overcome if the representation for stresses is available from a stressed reference configuration. While, Johnson and Hoger [6] outlined steps to obtain representations for Cauchy stress when the material is incompressible and isotropic in the stress free state, Hoger [7] considered the case wherein the incompressible material is transversely isotropic or orthotropic in its stress free state. Saravanan [8] derived a representation for Cauchy stress from a stressed reference configuration for compressible materials that are isotropic in the stress free state. In the framework propounded by Johnson and Hoger [6], Hoger [7] and Saravanan [8] the prestress field needs to be specified. The requirements on the prestress field are that it should satisfy the traction boundary condition and the equilibrium equations. It need not satisfy the compatibility condition, since prestresses could arise due to misfits. Hoger [9] outlined an approach based on Airy’s stress functions to prescribe the prestresses. Here we provide an alternative. We directly integrate the equilibrium equations subject to the no traction boundary condition and directly make assumptions on the spatial variation of certain stress components. We note that both the approach proposed here and that by Hoger are mathematically equivalent though the starting assumptions are different. The manufacturing process or knowledge of the orientation of the cuts to relieve the prestresses would suggest the coordinates on which the prestresses depend. For example, we show that if a radial cut relieves the prestresses in an annular cylinder, the prestresses would at most vary radially. We also find that there are many prestress fields that would be relieved by a radial cut. It is natural to expect that these different prestress fields result in different stress free configuration when cut radially; though each of them will have an opening (or closing) angle. Then, we examine

97

whether all these prestress fields that are relieved by a radial cut predict the same boundary load versus boundary deformation or the same deformation field from the prestressed configuration. We find that this is not the case and thus conclude that (small) deviations from the circular shape obtained when introducing a radial cut in blood vessels is important to be considered, as they correspond to different prestress distributions. In other words, we find that both the magnitudes of the prestresses as well as their spatial variation seem to determine the response of the body from its prestressed state. This finding has important ramifications in studying the growth and remodeling of biological soft tissues. Here we use these frameworks developed by Johnson and Hoger for incompressible materials and by Saravanan for compressible materials to study the effect of prestresses while inflating, extending and twisting a prestressed annular cylinder. This problem of inflation, extension and torsion of annular cylinders has been well studied. The case wherein the cylinder is made of incompressible isotropic materials has been studied by Rivlin [10]. The same problem for compressible isotropic materials has been studied by Green [11] and for incompressible anisotropic material by Green and Wilkes [12], Adkins [13], Green and Adkins [14] and finally for compressible anisotropic material by Zidi [15,16]. Inflation, extension and twisting of prestressed annular cylinder within the framework of cut stress free reference configurations has also been studied. While Chuong and Fung [17] studied this problem for prestressed annular cylinders made of incompressible materials, Zidi [18] considered compressible materials. Apart from studying the qualitative and quantitative features of the solution obtained for inflation, extension and torsion of prestressed cylinders for various radially varying prestressed cylinders, we are also interested in finding which of these cylinder admit deformations of the form: r ¼ r^ ðRÞ,

y ¼ Y þ OZ, z ¼ lZ,

ð1Þ

where O, l are constants and ðR, Y,ZÞ denotes the coordinates of a material particle in the reference configuration and ðr, y,zÞ denotes the coordinates of the same material particle after the deformation. We find that if the prestressed cylinder is made of incompressible material it always admit deformation of the form (1) provided the prestresses vary only radially. On the other hand if the annular cylinder is made of compressible material we find that if its stored energy function satisfies certain conditions then only deformation of the form (1) is possible. We emphasize that these are only sufficient conditions that ensure the existence of deformation of the form (1) and not necessary conditions. Ericksen [19] showed that the deformation field of the form (1) is possible in all annular cylinders made up of isotropic, incompressible materials. Later, Ericksen [20] showed that the same deformation field is not possible in all annular cylinders made up of isotropic, compressible materials. Here we are extending his results to include radially varying prestresses. We show that radially varying prestress field results in the symmetry group of the material in the reference configuration to change spatially. While a few material points are transversely isotropic, the rest of them have rhombic symmetry in the stressed configuration used as reference. Thus, our result here can be interpreted as showing the feasibility of deformation of the form (1) in a class of inhomogeneous bodies made up of anisotropic material with spatially varying symmetries. Moreover, it should be noted that if a prestressed configuration admits a deformation of the form (1) then the prestresses can vary at most radially. This is an important result for the following reason. If through an experiment it is ascertained that deformation is of the form (1) then we can be assured that the prestresses, if present, would at most vary radially, which is a significant simplification.

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U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

The arrangement of this article is as follows. In the next section we formulate the boundary value problem. In Section 3, we derive the general governing equations and discuss the solution methodology for various boundary conditions when the annular cylinder is made of compressible materials. We also establish the conditions under which such solutions exist for compressible materials. In Section 4, we study incompressible materials and establish that deformation that is being studied is possible in a prestressed annular cylinder made of any incompressible material provided the prestresses vary only radially. In the following section, we compare the boundary load required to realize a given boundary displacement and the deformation field for various prestressed cylinders and stress free cylinders.

2. Formulation of the boundary value problem 2.1. Geometry Here we are interested in a body that in the reference configuration is the annular region between two coaxial right circular cylinders: B ¼ fðR, Y,ZÞjRi r R rRo ,0 r Y r2p,0 r Z r hg,

ð2Þ

where ðR, Y,ZÞ denote the cylindrical polar coordinates of a point in the body and Ri, Ro and h are constants denoting the inner radius, outer radius and height of the cylinder, respectively. 2.2. Deformation field

y ¼ Y þ OZ, z ¼ lZ,

ð3Þ

where O, l are constants, ðR, Y,ZÞ denote the coordinates of a material particle in the reference configuration and ðr, y,zÞ denote the coordinates of the same material particle after the deformation. The function r^ ðRÞ describes the inflation or deflation of the annular region. The constant O denotes the angle of twist per unit length of the undeformed cylinder and the constant l the extension along the direction of the axis of the cylinder. Thus, here we study combined extension, inflation and twisting of an annular cylinder. 2.3. Kinematics For the assumed deformation field, the matrix components of the gradient of the deformation F in cylindrical polar coordinates is computed as 0

r,R B B F¼@ 0 0

0 r R 0

0

1

C rO C A,

0

1 B B r,R B B B 1 C ¼B 0 B B B @ 0

1

2

0

0

 2  O 2 R R þ r l





RO

l2

C C C RO C C  2 C: l C  2 C 1 C A

ð7Þ

l

Since, C is positive definite, we shall find it easier to express some of our results in terms of the following invariants  r 2 2 2 þ þðr OÞ2 þ l , ð8Þ J1 ¼ 1  C ¼ r,R R J2 ¼ 1  C1 ¼



1 r,R

2 þ

 2 R ðROÞ2 þ 1 þ , r l2

ð9Þ

r J3 ¼ ðdetðCÞÞ1=2 ¼ l r,R : R

ð10Þ

instead of the principal invariants of C. 2.4. General representation for Cauchy stress from a stressed reference configuration

In this study, we are interested in studying the response of the body subjected to a deformation of the form: r ¼ r^ ðRÞ,

Consequently, the right Cauchy–Green tensor, C and its inverse in cylindrical polar coordinate basis are given by 0 2 1 r,R 0 0 B C  r 2 B C r2 B 0 C t O C¼F F¼B ð6Þ C, R R B C 2 @ A r 2 2 0 O ðrOÞ þ l R

ð4Þ

l

The general representation for Cauchy stress from a stressed reference configuration depends on the invariants of C and the following invariants: J4 ¼ C  To , K1 ¼ 1  To ,

J5 ¼ C  ðTo Þ2 ,

J6 ¼ C1  To ,

K2 ¼ 12½ð1  To Þ2 1  ðTo Þ2 ,

J7 ¼ C1  ðTo Þ2 , K3 ¼ detðTo Þ,

ð11Þ ð12Þ

o

where T denote the prestress field, i.e., the Cauchy stress field in the reference configuration. The Cauchy stress depends on these invariants apart from the invariants of C because the symmetry of the material in the configuration used as reference is anisotropic by virtue of the presence of prestress fields that are not hydrostatic. Assuming that each point in the stressed reference configuration can be unloaded to a stress free configuration in which the material is isotropic we obtain the representation for Cauchy stress from a stressed reference configuration. Towards this, if Fo denote the gradient of the deformation field corresponding to the mapping between the stress free configuration to the stressed reference configuration, then left Cauchy–Green tensor Bo ( ¼FoFto) and To are related through To ¼ a0 1 þa1 Bo þa2 B1 o ,

ð13Þ

where ai ¼ i ðJ1r ,J2r ,J3r Þ, Jr1 ¼tr(Bo), Jr2 ¼tr(Bo 1),

a

where r,R denotes differentiation of r with respect to R. It then immediately follows that 0 1 1 0 C B r,R 0 B C B C R RO C B 1 F ¼B 0 ð5Þ C:  B r l C B C @ 1 A 0 0

l

a i are material response functions, J3r ¼ detðFo Þ, for compressible materials and for incompressible materials by To ¼ q1 þb1 Bo þ b2 B1 o ,

ð14Þ

where bi ¼ b i ðJ1r ,J2r Þ, b i are material response functions, Jr1 ¼tr(Bo), Jr2 ¼tr(Bo 1), q is the Lagrange multiplier that enforces the constraint of incompressibility. Eqs. (13) and (14) are well known (see Truesdell and Noll [21]) general constitutive representations for Cauchy stress in terms of the left Cauchy–Green tensor when

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

the body is subjected to an elastic process and the deformation is measured from a stress free reference configuration. Saravanan [8] showed that when E-inequalities hold1 then the Eqs. (13) (or (14)) can be semi-inverted to obtain 2

Bo ¼ d0 1 þ d1 To þ d2 ðTo Þ ,

ð15Þ 2

B1 o

¼ k0 1 þ k1 To þ k2 ðTo Þ ,

where di ¼ di ðJ1r ,J2r ,J3r Þ,  and di ¼ di ðJ1r ,J2r ,qÞ, i

k

ð16Þ

r r r i ¼ i ðJ1 ,J2 ,J3 Þ for compressible ¼ i ðJ1r ,J2r ,qÞ for incompressible

k

k

k

materials materials

and the form of these functions are known. Note that Jri ’s (and q in case of incompressible materials) are constants at a given material point and are related to the prestresses To by the requirements BoBo 1 ¼1 and T ¼To when F ¼ 1. Then, recognizing that the gradient of the deformation field from the stress free configuration to the current configuration is given by FFo and knowing the general representation for stress from the stress free configuration, the representation for the Cauchy stress from the stressed reference configuration can be obtained. Referring the reader to Saravanan [8] for details, here we present the final result. 2.4.1. Compressible material Thus, a general representation for stress from stressed reference configuration for a compressible material is o

o 2

T ¼ a0 1þ a1 F½d0 1 þ d1 T þ d2 ðT Þ F

t

þ a2 Ft ½k0 1 þ k1 To þ k2 ðTo Þ2 F1 ,

ð17Þ

where

ai ¼ a i ðJm1 ,Jm2 ,Jm3 Þ, Jm2 ¼ k0 J2 þ k1 J6 þ k2 J7 ,

Jm3 ¼ J3 J3r ,

d0 ¼

d1 ¼

1

D 1

D

ð18Þ #



J1r 3 a þð2a1 J2r a0 Þa22 ðJ2r J3r2 a21 þa20 Þa2 J3r2 a21 a0 , J3r2 2

k1 ¼ 

½2a2 a0 þ J3r2 a21 þ J2r a22 ,

d2 ¼ 

a2

D

ð19Þ

,

D

ð26Þ

r ¼ 3k0 þ K1 k1 þ ½K12 2K2 k2 , Jm2

ð27Þ

r Jm3 ¼ J3r :

ð28Þ Jri ’s

in the case of compressible materials are found such Thus, the that they satisfy the Eqs. (24) and (25). 2.4.2. Blatz–Ko constitutive relation We illustrate our ideas using a special form of the constitutive relation introduced by Blatz and Ko [22] to study the response of polyurethane. While they use a stress free configuration as reference, here we use a stressed configuration as reference. For this case

a0 ¼ 

m

1 ½2m þ 1

Jm3 3

a1 ¼

,

m1 Jm3

,

a2 ¼ 0:

ð29Þ

Now, we compute

d0 ¼

k0 ¼ k2 ¼

1 ðJ3r Þ2m3

1

d1 ¼ J1r

J2r 

m21

,

ðJ3r Þ2ðm3 þ 1Þ

J3r

m1 þ

,

d2 ¼ 0, 1

ðJ3r Þ2ð2m3 þ 1Þ

,

k1 ¼

1

m1

"

# J1r  , ðJ3r Þð2m3 þ 1Þ J3r 2

,

from the results presented in Eqs. (19)–(23). Then, the stress is given by " # m 1 m F r 21m 1þ J3r To Ft : ð30Þ T ¼  ½2m 1þ 1 1 þ Jm3 ðJ3 Þ 3 J 3

"

#

2a1 a0 þ J1r a21 þ

a22 , J3r2

D ¼ J3r2 a31 J1r a2 a21 þ J2r a22 a1 

k2 ¼

a1

D

In this case, the requirement that T¼To when F¼1, places no restriction. The condition (24) requires   ðJ r Þ2ðm3 þ 1Þ K3 r 2m3 þ 1 2 J1r ¼ r 2m þ ðJ3r Þ4m3 þ 2  3 2 ðJ3 Þ þ K2 , m1 m1 ðJ3 Þ 3 " # K3 J3r 1 r 2m3  þðJ Þ 1 , 3 m31 ðJ3r Þ2ðm3 þ 1Þ ðJ3r Þ2ð2m3 þ 1Þ J1r

ð20Þ

"

1

r ¼ 3d0 þK1 d1 þ½K12 2K2 d2 , Jm1

J2r ¼

! # J1r 2 1 r r2 3 r 2 2 2 k0 ¼ J J a þ ðJ1 a0 2a2 Þa1 þ a0 þ r2 a2 a1 þ r2 a0 a2 , D 23 1 J3 J3 1

r r r where ci ¼ a i ðJm1 ,Jm2 ,Jm3 Þ and

m3

Jm1 ¼ d0 J1 þ d1 J4 þ d2 J5 ,

"

99

ð21Þ

0 ¼ ðJ3r Þ2m3 þJ3r

K1

m1

þ

 ðJ3r Þ2ðm3 þ 1Þ K3

m

m1

2 1

 ðJ3r Þ2m3 þ 1 þ K2 ðJ3r Þ2ð2m3 þ 1Þ : ð31Þ

Jr3

,

1 3 a : J3r2 2

ð22Þ

ð23Þ

Then, the requirement BoBo 1 ¼ 1 reduces to requiring:

First, the Eq. (31c) is solved for and then substituted in (31a) and (b) to obtain Jr1 and Jr2, respectively. 2.4.3. Incompressible material A general representation for stress from stressed reference configuration for an incompressible material is T ¼ p1þ b1 F½d0 1 þ d1 To þ d2 ðTo Þ2 Ft 

1 ¼ d0 k0 þ K3 ½k1 d2 þ d1 k2 þ d2 k2 K1 ,





þ b2 Ft ½k0 1 þ k1 To þ k2 ðTo Þ2 F1 ,

ð32Þ

0 ¼ d1 k0 þ k1 d0 þ d2 k2 K3 K2 ½k1 d2 þ d1 k2 þ d2 k2 K1 , 0 ¼ d2 k0 þ d1 k1 þ d0 k2 d2 k2 K2 þ K1 ½k1 d2 þ d1 k2 þ d2 k2 K1 ,

ð24Þ

bi ¼ b i ðJm1 ,Jm2 Þ,

where Ki’s are the principal invariants of To. Finally, the requirement that T ¼To when F¼1 requires c0 þ c1 d0 þ c2 k0 ¼ 0,

c1 d1 þ c2 k1 ¼ 1,

c1 d2 þc2 k2 ¼ 0,

where p is the Lagrange multiplier,







Jm1 ¼ d0 J1 þ d1 J4 þ d2 J5 , ð25Þ

1

d0 ¼

D

d1 ¼

D

Jm2 ¼ k0 J2 þ k1 J6 þ k2 J7 ,

½J1r b32 þ ð2b1 þJ2r qÞb22 ðJ2r b21 þ q2 Þb2 þ b21 q,

ð33Þ ð34Þ

1

E-inequalities require the material response functions to satisfy the following inequalities: a 0 r 0, a 1 4 0, a 2 r 0 in case of compressible materials and b 1 40, b 2 r 0 in case of incompressible materials.

1

½2b2 q þb21 þJ2r b22 ,

d2 ¼ 

b2

D

,

ð35Þ

100

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

1

k0 ¼

2

½J2r b31 ðJ1r q þ 2b2 Þb21 þ ðq2 þJ1r b22 Þb1 qb2 ,

D

k1 ¼ 

1

D

½2b1 q þJ1r b21 þ b22 ,

k2 ¼

b1

D

ð36Þ

,

ð37Þ

D ¼ b31 J1r b2 b21 þ J2r b22 b1 b32 :

ð38Þ

Then, the requirement BoBo 1 ¼ 1 reduces to requiring: 1¼

d0

 0 ¼ d1

    0 þ K3 ½ 1 d2 þ d1

k

k

 0þ

k

   1 d0 þ d2

k





  2 þ d2

k

k

k

2.5. Representation for prestress

 2 K1 ,

    2 K3 K2 ½ 1 d2 þ d1



k



  2 þ d2

k





 2 K1 ,

k



0 ¼ d2 k0 þ d1 k1 þ d0 k2 d2 k2 K2 þ K1 ½k1 d2 þ d1 k2 þ d2 k2 K1 ,

ð39Þ

o

where Ki’s are the principal invariants of T . Finally, the requirement that T ¼ To when F ¼ 1 requires 



c1 d1 þ c2 k1 ¼ 1,

c1 d2 þc2 k2 ¼ 0,

ð40Þ

r r where ci ¼ b i ðJm1 ,Jm2 Þ and 





r ¼ 3d0 þK1 d1 þ ½K12 2K2 d2 , Jm1

ð41Þ

r ¼ 3k0 þK1 k1 þ ½K12 2K2 k2 : Jm2

ð42Þ

Thus, the and (40).

Jri ’s

and q are found such that they satisfy the Eqs. (39)

2.4.4. Exponential constitutive relation Next, for illustration, we consider a constitutive relation proposed by Mirsky [23] and Demiray [24] to study cardiovascular tissues, which is of the form

b1 ¼ m1 expðm2 ½Jm1 3Þ, b2 ¼ 0,

ð43Þ

where the material parameter m1 is a positive constant and the parameter m2 is any real constant. These restrictions arise from the requirement that the constitutive relations satisfy the E-inequalities. Then, from Eqs. (34) through (38) we compute

d0 ¼

 0

k

q , b1þ

1 , b1þ

d1 ¼

 2 Jr q q ¼ J2r  1þ  þ , b1 b1

k2 ¼

1 ðb1þ Þ2

d2 ¼ 0,

,

along with the appropriate traction boundary condition, where ro is the density of the body in the reference configuration and b is the body force per unit mass. Here the reference configuration is assumed to be a stationary configuration, because experimentally one always uses only stationary configurations as reference. Further to simplify, in keeping with the standard practice, body forces are neglected assuming that the stresses induced due to them will be very small in comparison with those developed due to the applied forces. Also if body forces are considered the body cannot be traction free and the orientation of the body with respect to the direction of body force will play a role. In order to avoid these difficulties body forces are not considered. Thus, now without any inconsistency, the body in the reference configuration is assumed to be traction free since the focus of this study is on the influence of stresses present in an apparently traction free body. Also, note that here the divergence is with respect to the reference coordinates and hence the capitalization of D in Div. Further, we require that the cylindrical polar basis components o of the prestress vary only radially i.e., T^ ij ¼ Toij ðRÞ because, as we shall see, the deformation field (3) is realizable only for this case. The requirement that the boundary of the annular cylinder be traction free necessitates that the surfaces defined by Z ¼ 0 and h be traction free, which in turn results in

o o ðRi Þ ¼ TRR ðRo Þ ¼ TRoY ðRi Þ ¼ TRoY ðRo Þ ¼ 0: TRR

r 2 ½3J1 Þ

expðm

m1

½qJ1 þ J4 :

ð46Þ

ð47Þ

On substituting (44) into (39), we obtain  2 bþ 2J r q q K3 , J2r ¼ 1þ þ þ þ 1  q b1 b1 qðb1þ Þ2 K3

#

ðb1þ Þ3

1 r þ 3½J1 b1 K1 :

" # J1r 1 2q q K2 þ  , þ þ  q ðb1þ Þ2 b1 b1 ðb1þ Þ3

dR

þ

o o ðTRR TYY Þ ¼ 0, R

ð52Þ

ð53Þ

ð48Þ

ð49Þ

ð50Þ

ð54Þ

o

dT RY 2TRoY þ ¼ 0: dR R

The requirement T ¼To when F¼ 1, translates into requiring Jrm 1 ¼Jr1 which yields q ¼ 13½J1r b1þ K1 :

o dT RR

ð45Þ

where

q¼

ð51Þ

Using (52), the balance of linear momentum (51) reduces to

T ¼ p1 þexpðm

"

DivðTo Þ þ ro b ¼ 0,

and that the surfaces defined by R ¼Ri and R ¼Ro be free of traction requires that ð44Þ

o t r 2 ½Jm1 J1 ÞF½q1 þ T F ,

0 ¼ 1

In this framework, we have to specify the prestresses, To, the Cauchy stress in the reference configuration. This prestress field as with every Cauchy stress field should satisfy the balance of linear momentum:

o o TZZ ðRÞ ¼ TRZ ðRÞ ¼ TZoY ðRÞ ¼ 0,

Jr 2q k ¼  1þ þ þ 2 , b1 ðb1 Þ  1

where b1þ ¼ m1 expðm2 ½J1r 3Þ. The general representation for Cauchy stress in incompressible materials (32) now reduces to

Jm1 ¼

Substituting for q from (50) in (49), we obtain a non-linear equation in Jr1 which is solved to obtain Jr1. Then, Eqs. (48) and (50) are used to compute Jr2 and q, respectively. Since Eqs. (47) and (50) are identical, the obtained values of q, Jr1 and Jr2 satisfying Eqs. (48)–(50) will also ensure equation (47) is met.

These equations can easily be integrated to obtain Z 1 R o o ¼ T dR, TRR R Ri YY TRoY ¼

k , R2

ð55Þ

ð56Þ

ð57Þ

where k is a constant. To get (56) the boundary condition (53a) has been used. Then, the boundary condition (53b) requires that Z Ro o TYY dR ¼ 0: ð58Þ Ri

Finally, the constant k in (57) is determined from the boundary condition (53c) as to be 0. Hence, TRoY ¼ 0 and the boundary condition (53d) is also simultaneously satisfied.

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

Thus, if the prestress field is to vary only radially, the matrix components of To in cylindrical coordinate basis are 0 o 1 0 0 TRR B C o 0 A, To ¼ @ 0 TYY ð59Þ 0 0 0 o through (56) and the radial variation of with ToRR related to TYY o TYY is such that its mean is zero. o Here the following variations of TYY is studied. # 8 " k1   X > n > > , k is even; ð1Þn H R > e1 12  > k < n¼0 o " # TYY ¼ PWC Variation k1  n > 2k X > n > > , k is odd; e 1 ð1Þ H R 1 > : ðk þ1Þn ¼ 0 k

ð60Þ o ¼ TYY

k X

en sinð2npRÞ, Sinusoidal Variation

ð61Þ

en cosð2npRÞ, Cosine Variation

ð62Þ

n¼1

o ¼ TYY

k X n¼1

o ¼ e1 ð12RÞ, TYY

Linear Variation

ð63Þ

condition (58) implies that there exists at least one material point o inside the body where TYY ¼ 0 and at this location ToRR cannot be zero in order to satisfy (56), especially since the point is inside the body. Therefore, there exist regions inside the body where the three principal values of the prestress are independent and regions where only two principal values are independent. At the boundary of the body either all three principal values are zero or at least two of them are zero (since ToRR has to be zero on the boundary). It then follows from the representations given in Coleman and Noll [25] that regions that have three independent principal values of To have rhombic symmetry while regions that have two independent principal values of To are transversely isotropic and regions where all the three principal values are the same remain as isotropic. While in the interior of the annular cylinder the fiber direction for transverse isotropy coincides with o ER, at the surface it coincides with EY depending on whether TYY o is zero or TRR is zero. Hence, the surface of the cylinder is either o transversely isotropic or isotropic, depending on whether TYY is not equal to 0 or equal to 0 on the surface, respectively. Thus, the radially varying prestress field results in different regions of the body possessing different material symmetries. Finally, we compute the invariants given in Eq. (11) as 2 o J4 ¼ C  To ¼ r,R TRR þ

 r 2 R

where 8 > <0



n ¼ H R > k :1

n k n, if R 4 k

2 o 2 J5 ¼ C  ðTo Þ2 ¼ r,R ðTRR Þ þ

if R o

is the heaviside function, R ¼ ðRRi Þ=ðRo Ri Þ, Rp ¼ Ri =ðRo Ri Þ, e1 and en are real constants and k is a positive integer. We observe that using any one or combination of the first three variations studied here any experimentally observed prestress field, if available, can be captured. Then, we compute ToRR from (56) as  8  1 1 m m > > k is even; > e1 2kðð1Þ ð2m þ 1Þ1Þð1Þ R > > þR ðR pÞ > > > < k  o ð1 þ ð1Þm ÞR , ¼ e1 R TRR ðk þ 1Þ > > > 

> > 1 1 > m > k is odd; > : 2kðð1Þ ð2m þ1Þ1Þ ðR þ R Þ, p

101

J6 ¼ C1  To ¼

1

J7 ¼ C

o TRR þ 2 r,R



 r 2 R

ð68Þ

o ðTYY Þ2 ,

"   2 # R 2 RO o þ , TYY l r

o TRR  ðT Þ ¼ r,R o 2

o TYY ,

2

"   2 # R 2 RO o þ þ Þ2 , ðTYY r l

ð69Þ

ð70Þ

ð71Þ

in addition to the principal invariants of To: o o K1 ¼ 1  To ¼ TRR þTYY ,

K2 ¼

ð72Þ

1 o o ½ð1  To Þ2 1  ðTo Þ2  ¼ TRR TYY , 2

ð73Þ

ð64Þ

K3 ¼ detðTo Þ ¼ 0,

ð65Þ

for the assumed deformation field (3) and radially varying traction free prestress field. We note that when the reference configuration is stress free To ¼0.

ð66Þ

2.6. Boundary conditions

ð74Þ

2

for PWC variation, where m ¼ floor(kR). k X en 1 o , TRR ¼ ½1cosð2npRÞ 2n p þR ðR pÞ n¼1

o ¼ TRR

k X en 1 sinð2npRÞ , 2np ðR þRp Þ n¼1

o ¼ e1 ð1RÞR TRR

1 , ðR þ Rp Þ

Sinusoidal Variation

Cosine Variation,

Linear Variation:

ð67Þ

We note that on introducing a radial cut, we produce two traction free surfaces defined by Y ¼ constant. Which means that, if the prestresses vary only radially, in addition to conditions (52) and (53), o TYY ðRÞ ¼ 0 and hence it follows from (56) that ToRR ¼0. Hence, we conclude that if the prestresses varies only radially in a right circular cylinder, then they would be relieved on introducing a radial cut. Next, we examine the material symmetry of the material particles in the stressed reference configuration. The three o principal values for this prestress state are 0, ToRR, TYY . The

Next, we record the boundary conditions. The boundary conditions that would be prescribed are a subset of the following: the deformed inner radius, ri, the deformed outer radius, ro, the ratio of the deformed length to original length, l, angle of twist per unit length, O, the radial component of the normal stress at the inner surface, P i , the radial component of the normal stress at the outer surface, P o , the axial load, L, defined as L ¼ 2p

‘floor(x)’ rounds x to the nearest integer towards 1.

ro

Tzz r dr ¼ 2p

Z

ri

Ro

Tzz rr ,R dR,

ð75Þ

Ri

and the torque, T , computed using T ¼ 2p

2

Z

Z

ro ri

Tyz r 2 dr ¼ 2p

Z

Ro

Ri

Tyz r 2 r,R dR:

ð76Þ

102

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

While for compressible materials any two out of fri ,ro ,P i ,P o g and two out of the remaining four conditions mentioned above can be prescribed, for incompressible materials either P i or P o has to be specified along with any three of the remaining seven conditions. This is expected, since the constitutive relation for incompressible materials contains the Lagrange multiplier, p, which has to be determined from boundary conditions and equilibrium equations and hence we need to specify either P i or P o .

The unknown function r^ ðRÞ, for compressible materials is to be found such that the balance of linear momentum is satisfied and in the case of incompressible materials it is found from the incompressibility condition. For an incompressible material, the Lagrange multiplier is found such that balance of linear momentum is satisfied. The balance of linear momentum ð77Þ

in the absence of body forces and dynamic loading, for the assumed deformation field (3) reduces to dT rr r,R þ ½Trr Tyy  ¼ 0, dR r

ð78Þ

on noting that the non-zero components of the stress, T depends only on R.

we seek to find f1 and f2. Towards this we compute Jm1,R ¼ 2r,R m1 r,RR þ h1 þ g1 ,

ð85Þ

2m3 r,RR þ h2 þg2 , 3 r,R

ð86Þ

Jm2,R ¼ 

First for the assumed deformation field (3) and radially varying prestress field, Eqs. (18a)–(c) simplify to  r 2 2 2 m1 þ m2 þ½ðr OÞ2 þ l d0 , ð79Þ Jm1 ¼ r,R R "   2 # 1 R 2 RO k0 þ ð80Þ m4 þ 2 , Jm2 ¼ 2 m3 þ r l r,R l ð81Þ

where o o 2 m1 ¼ ½d0 þ d1 TRR þ d2 ðTRR Þ ,

where

lh

J3,R ¼

R

 r 2 R

2

m4 ¼ ½k0 þ k1 TYY þ k2 ðTYY Þ :

2

m2,R þ½l þ ðr OÞ2 d0,R ,

k

m3,R h2 ¼ 2 þ 2 þ m4,R r,R l

"   2 # R 2 RO þ , r l

o o o2 o o þ d1 TRR,R þ d2,R TRR þ 2d2 TRR TRR,R , m1,R ¼ d0,R þ d1,R TRR o o o2 o o þ d1 TYY m2,R ¼ d0,R þ d1,R TYY ,R þ d2,R TYY þ 2d2 TYY TYY,R , o o o2 o o þ k1 TRR,R þ k2,R TRR þ 2k2 TRR TRR,R , m3,R ¼ k0,R þ k1,R TRR

g1 ¼ 2

r hr,R ri m2 þ 2d0 rr ,R O2 ,  R R R2

g2 ¼ 2m4

ð88Þ ð89Þ

"    2 # R 1 Rr ,R O  2 þR : r r l r

ð90Þ

Noting dT rr @a0 @a0 @a0 ¼ Jm1,R þ Jm2,R þ Jm3,R dR @Jm1 @Jm2 @Jm3   @a1 @a1 @a1 2 þ Jm1,R þ Jm2,R þ Jm3,R m1 r,R @Jm1 @Jm2 @Jm3   @a2 @a2 @a2 m3 þ Jm1,R þ Jm2,R þ Jm3,R 2 @Jm1 @Jm2 @Jm3 r,R " # 2 þ a1 r,R m1,R þ

a2

2 r,R

m3,R þ 2 a1 m1 r,R 

a2

3 r,R

m3 r,RR ,

ð91Þ

we find

o o 2 þ k2 ðTRR Þ , m3 ¼ ½k0 þ k1 TRR o

r i 2 r,R þrr ,RR  r,R , R

2 þ h1 ¼ m1,R r,R

o o m2 ¼ ½d0 þ d1 TYY þ d2 ðTYY Þ2 ,

o

ð87Þ

o o o2 o o þ k1 TYY m4,R ¼ k0,R þ k1,R TYY ,R þ k2,R TYY þ 2k2 TYY TYY,R ,

3. Governing equations and solution technique for compressible materials

r Jm3 ¼ J3r l r,R , R

ð84Þ

r J3 þ J3r J3,R , Jm3,R ¼ J3,R

2.7. Balance of linear momentum

divðTÞ þ rb ¼ ra,

Recognizing that the Eq. (78) would reduce to the form f1 r,RR þf2 ¼ 0,

ð82Þ

For the special boundary value problem being studied, the components of stress with respect to cylindrical polar basis computed using Eq. (17) are given by 9 8 1 > > 2 > > a þ a m r þ a m > > 0 1 1 ,R 2 3 2 > > > > r > > ,R > 9 > 8 > > > >     > > 2   T 2 > > > > rr r R > > > > 2 > > > > > > > > a0 þ a1 m2 > þ d ðr O Þ a m þ 0 2 4 > > > Tyy > > > > R r > > > > > > > > " # > > < > >   = = < 2 Tzz k0 RO 2 : ð83Þ ¼ a þ a d l þ a þm 0 1 0 2 4 T > > > > l > > l2 > > > > > ry > > > > > > > > > > > > T > > > 0 > > > > > rz > > > > ; > > :T > > > yz > > 0 > > > > > > > > 2 > > R O > > > > a d r Ol  a m ; : 1 0 2 4 r l

  @a1 2 3 @a1 @a0 @a0 Jm3 m1 r,R þ Jm3 m1 r,R þ2 a1 þ m1 r,R þ @Jm1 @Jm3 @Jm1 @Jm3 r,R   @a2 1 @a0 2 @a2 2 2 þ Jm3 m3 3 m3 a2 þ  m 5 3 @Jm3 @Jm2 r,R @Jm2 3 r,R r,R   @a2 @a1 2m1 m3 þ  , ð92Þ @Jm1 @Jm2 r,R

f1 ¼ 2

"

# @a0 @a1 @a2 m3 2 þ m1 r,R þ f2 ¼ ½g1 þ h1  2 @Jm1 @Jm1 @Jm1 r,R " # @a0 @a1 @a2 m3 2 þ m1 r,R þ þ ½g2 þh2  2 @Jm2 @Jm2 @Jm2 r,R " #   @a0 @a1 @a2 m3 r l 2 r r þ m1 r,R þ J þ J r r  þ J ,R ,R 3 3,R 3 2 R R @Jm3 @Jm3 @Jm3 r,R h  i m1  r 2 þr,R a1 r,R m1,R þ r,R m2 2 r O d0 r R

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

" þ a2

# 1  m3  R2  3 r,R m4 : m3,R þ r,R 2 r r r,R

ð93Þ

2 Jm3

a2 ¼ 

W1 ,

2 Jm3

W2 ,

ð94Þ

where Wi denotes @W=Jmi . For this case the expressions for f1 and f2 simplify to

ð95Þ

# 2 m1 r,R m3 ½g1 þ h1  2W12 f2 ¼ W13 þ 2W11 2 Jm3 Jm3 r,R " # 2 m1 r,R m3 2W22 þ W23 þ 2W12 ½g2 þh2  2 Jm3 Jm3 r,R " #



2 1 m1 r,R 1 m3 2 W23 W2 þ W33 þ 2 W13 W1 2 Jm3 Jm3 Jm3 r,R Jm3    l r r J3 þJ3r r,R r,R   J3,R R R i r,R h  m1  r m2 2 r O2 d0 r,R m1,R þ r,R þ 2W1 Jm3 r R " # 2 1  m3  R2 ð96Þ  3 r,R m4 , m3,R þ r,R W2 2 Jm3 r,R r r where Wij denotes @2 W=Jmi Jmj and we have assumed that the stored energy is a smooth function of Jm1, Jm2 and Jm3. As said before, we illustrate our ideas for the Blatz–Ko constitutive relation used to model polyurethane, recorded in (30). For this case the Eqs. (92) and (93) simplify to m1 m1 ð2m3 þ1Þ þ , ð2m þ 1Þ Jm3 r,R Jm3 3

ð97Þ

" #  r,R m1  r r O2 m2 2  r 2m r,R m1,R þr,R f2 ¼ m1 Jm3 r R ðJ3 Þ 3 " #   m ð2m þ1Þ 2 m1 r r  J3r J3,R J3 þ lr,R r,R  , m1 2 þ 1 2ðm 3þ 1Þ r,R R R Jm3 Jm3 3

r m1,R ¼ J3,R

r m2,R ¼ J3,R

l1 ¼ J3r

" Jm3 ¼ 

#1=ð2m3 Þ 2m3 þ 1 : 2m r,R 1

K1,R

m1

þJ3r

o TRR

m1

m1

þJ3r

o TYY

m1

o TRR

,

o TRR,R

m1

m2 ¼

2m3

o

þ J3r

þ ðJ3r Þ

TYY,R

m1

1 ðJ3r Þ2m3

þ J3r

ð98Þ

m31

m1

o then there cannot exist a region wherein TRR =m1 oðJ3r Þð2m3 þ 1Þ . o Towards this, we observe that if TRR =m1 ¼ ðJ3r Þð2m3 þ 1Þ , then Jr3 ¼ 0 o for Eq. (31) to hold irrespective of the value of Tyy . Since, J3r a 0, o =m1 aðJ3r Þð2m3 þ 1Þ . Hence, there cannot exist regions in TRR o o =m1 4 ðJ3r Þð2m3 þ 1Þ and TRR =m1 o  the same body where TRR

ðJ3r Þð2m3 þ 1Þ , since ToRR should always be a continuous function R. In all boundary traction free prestressed bodies ToRR(Ri) ¼ToRR(Ro)¼0 and J3r 4 0 and hence there exist regions where o TRR =m1 4 ðJ3r Þð2m3 þ 1Þ . Therefore, we conclude that whenever m3 41=2, f1 a 0, for the model proposed by Blatz–Ko for polyurethane. It is worthwhile to note that there is no restriction on the prestresses and the same restriction on the material parameters exist even when the reference configuration is stress free. 3.1. Solution of the governing equation We require any two out of the four conditions {ri, ro, P i , P o g to be specified for solving the second order ordinary differential Eq. (84). Mathematically, it is easier if ri and P i or ro and P o are specified. To determine the other constants in the deformation field either l or L and either O or T needs to be specified. We proceed first by assuming that ro, P o , l and O are specified. Then, we discuss the case when the prescribed boundary conditions are different from the above in Section 3.1.2. The governing Eq. (84) can be written as

r J3,R 3 r 2m3 þ 1 ðJ3 Þ

r J3,R ¼

,

K2,R

m21 "

l2 ¼ 2ð2m3 þ1ÞðJ3r Þ4m3 þ 1 ð2m3 þ 1ÞðJ3r Þ2m3 " ðJ3r Þ2m3 þ 1 2

where d ¼r,R, on assuming that f1 a 0. Now, we outline a solution procedure to solve the second order ordinary differential Eq. (101) over the domain Ri r R r Ro for given l, O and gðRo ,ro ,dðRo ÞÞ ¼ P o :

ð102Þ

K2

m1

m21

#

l1 , l2

o Jm1 ¼

J3r ,

K3

ðJ3r Þ2ðm3 þ 1Þ þ J3r

ð103Þ

where do ¼ d(Ro), d0 and k0 are known functions of Ro, o aoi ¼ a i ðJm1 , Jom 2, Jom 3),

#

m31

K3

ðm3 þ 1ÞðJ3r Þ2m3 þ 1 þ 3

ð101Þ

gðRo ,ro ,do Þ ¼ ao0 þ ao1 d0 d2o þ ao2 k0 =d2o ,

, ðJ3r Þ2m3 þ 1

ðJ3r Þ2ðm3 þ 1Þ þ

ð100Þ

which results in the problem becoming an initial value problem. Here

Jm3 ¼ J3r J3 ,

,

r J3,R

2m

" 2m3 þ 1 K3,R

o TYY

ð99Þ

Since, Jm3 4 0, either m3 o1=2 or m1 o0. It then follows from

rðRo Þ ¼ ro , 1

K3,R ¼ 0:

Next, we examine whether for model proposed by Blatz–Ko for polyurethane f1 ¼0. It follows from Eq. (97) that if f1 ¼0

r,RR ¼ f ðR,r,dÞ,

where ðJ3r Þ2m3

o o o o K2,R ¼ TRR,R TYY þ TRR TYY ,R ,

o continuous, we show next that if TRR =m1 4 ðJ3r Þð2m3 þ 1Þ for some R

"

m1 ¼

K3 ¼ 0,

o (99) that if m1 o 0, TRR =m1 o ðJ3r Þð2m3 þ 1Þ . Since, ToRR should be

m1 r,R Jm3 2 ½4W11 m1 r,R þ 4W13 Jm3 þ 2W1  þW33 f1 ¼ Jm3 r,R " # m3 6 4m3  3 W2 þ W22 þ 4W23 , 2 Jm3 r,R Jm3 r,R

f1 ¼ m1 r,R

o o K2 ¼ TRR TYY ,

o o þ TYY K1,R ¼ TRR,R ,R ,

For Green elastic materials the stress could be derived from a ^ ðJm1 ,Jm2 ,Jm3 Þ then potential called the stored energy, W ¼ W

a0 ¼ W3 , a1 ¼

o o K1 ¼ TRR þ TYY ,

103

K2

o ¼ Jm2

#

m21

þ 2m3 ðJ3r Þð2m3 þ 1Þ 

K1

m1



ro Ro

2

m2 þ ½d2o þðro OÞ2 þ l d0 ,

"    #   Ro 2 Ro O 2 1 1 þ m4 þ 2 þ 2 k0 , ro l do l

o ¼ J3r l Jm3

,

2

ro do , Ro

Jr3, m2 and m4 are also known functions of Ro.

104

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

We shall first solve the Eq. (102b) to obtain d(Ro) ¼do. Then, a solution to (101) is given by Taylor’s series 2

5 of dr=dRjR ¼ R1 ¼ d 1 is obtained by solving

  þ þ þ h1 ðR 1 ,r1 ,d1 Þ ¼ h1 ðR1 ,r1 ,d1 Þ,

ð109Þ

where the only unknown is d1 , since tion field requires r(R1  )¼r(R1+ ).

ðRRo Þ ðRRo Þ do þ f ðRo ,ro ,do Þ 1! 2!   ðRRo Þ3 d3 r  ðRRo Þm dm r  þ    þ þem , þ  m 3 3! m! dR R ¼ Ro dR R ¼ Ro

rðRÞ ¼ ro þ

continuity of the deforma-

Now, we solve the governing Eq. (101) for the boundary condition r(R1 )¼r1 and dr=dRjR ¼ R1 ¼ d over the domain 1 R2 oR o R1 . This process is continued until the other boundary of the body (i.e. R¼Ri) is reached.

where3 d3 r dR

3

d4 r dR

4

:¼ f1 ðR,r,dÞ ¼

3.1.2. Solution procedure for other boundary conditions If the specified boundary values are l, O, ri and P i , even then the problem is an initial value problem and the same technique as above could be used to solve with Ro replaced by Ri. On the other hand if the prescribed boundary values are l, O, ri and P o then the problem becomes a boundary value problem. Seeking to solve this boundary value problem through a series of initial value problem (IVP) we assume

@f @f @f þd þf , @R @r @d

  @2 f @2 f @2 f @2 f þ f þ fd þ 2 d @r@R @R@d @r@d @R2  2 2 2 @ f @ f @f @f @f @f @f @f þ þf , þd þ d2 2 þf 2 2 þ f @d @r @d @d @R @r @r @d

:¼ f2 ðR,r,dÞ ¼

r,R ðRi Þ ¼ dgi , d

mþ3

dR

r

mþ3

em ¼

@fm @fm @fm þd þf , :¼ fm þ 1 ðR,r,dÞ ¼ @R @r @d

 ðRRo Þm þ 1 dm þ 1 r   ðm þ1Þ! dRm þ 1 

,

ð104Þ

R¼x

Ri o x oRo . For the above series to converge we need lim em -0:

ð105Þ

m-1

It is still computationally costly to solve the differential equation using the above technique. Hence, for the simulations we solve the governing equation numerically. For this, we convert the second order ordinary differential equation, Eq. (101) to a system of two first order ordinary differential equations by the simple change of variables u ¼ r,

v ¼ r,R :

ð106Þ

The differential equations relating these functions are du ¼ v, dR

dv ¼ f ðR,u,vÞ, dR

ð107Þ

with the condition uðRo Þ ¼ ro ,

vðRo Þ ¼ do :

ð108Þ

This system of first order ordinary differential equations is integrated using ODE 45 in MATLAB.

3.1.1. Solution procedure for piecewise constant variation In the case where the prestress is only piecewise continuous, the governing Eq. (101) has to be solved in each sub-domain in which the prestress varies continuously. At the interface, the deformation field has to be continuous and tn ¼  t  n, where tðÞ denotes the Cauchy traction vector and n is the normal to the interface in the current configuration. Now, if R1, R2,y,Rn denote the radial locations4 where the prestress is discontinuous then we begin by solving the governing Eq. (101) with the boundary conditions (102) over the domain R1 o Rr Ro (instead of over the domain Ri rR r Ro ). Now the value of r(R1+ ) ¼r1+ and dr=dRjR ¼ R þ ¼ d1þ is known. Then, the value 1

3 We follow the standard notation that dðÞ=dR denotes the total derivative with respect to R and @ðÞ=@R denotes the partial derivative with respect to R. 4 R¼ Rj, a constant, denotes a surface across which the prestress is discontinuous and R1 4R2 4Rn .

ð110Þ

Then, solve the resulting initial value problem and evaluate the resulting Trr(r(Ro)) ( ¼ P comp ) for the assumed dgi . Unless it happens o comp that P o ¼ P o , another value has to be assumed for dgi and the resulting IVP solved. In order to arrive at an algorithm for improving the guesses for dgi , the present problem is cast into that of finding a zero of a non-linear function. Towards, this we define ðdgi Þ, eðdgi Þ ¼ P o P comp o

ð111Þ

dai

e(dai )¼0.

such that Here the bisection and seek to find technique is used to find the zero of the function e. This technique is chosen because the derivative of the function e with respect to dgi could not be computed. Further, this technique requires initial guesses (dgi )1 and (dgi )2 such that eððdgi Þ1 Þeððdgi Þ2 Þ r0. If e is a continuous function then this requirement on the initial guesses is sufficient to ensure existence of dai such that e(dai )¼0. It can be seen that e would be a continuous function of dgi provided the governing equation admits a converging Taylor’s series solution. Equivalently if l, O, ro and P i are specified then a similar procedure as above is used to solve the problem except that now r,R(Ro)¼dgo, has to be assumed. Similarly, if the axial load L is specified instead of stretch ratio, l, we begin by assuming l ¼ lg and solve the resulting initial value problem and evaluate the resulting axial load Lcomp . Unless it happens that Lcomp ¼ L, another value has to be assumed g for l and the resulting IVP solved. In order to arrive at an g algorithm for improving the guesses for l , the present problem is cast into that of finding a zero of a non-linear function. Towards, this we define g

g

lðl Þ ¼ LLcomp ðl Þ, a

ð112Þ a

and seek to find l such that lðl Þ ¼ 0. Any non-gradient based algorithm to find the zero of the function, like bisection technique could be employed to solve this equation. On the other hand if the torque T is specified a procedure similar to that used for the axial load is followed except that now the angle of twist O per unit length is assumed. Thus, if the specified boundary values are ri, P o , L and T , then there are two approaches to solve this problem. In the first g method, we assume that r,R(Ri) ¼dgi , l ¼ l , O ¼ Og and then solve the resulting initial value problem and evaluate the resulting Trr ðrðRo ÞÞ ¼ P comp , axial load, Lcomp and torque, T comp . o 5 For the assumed forms of deformation and hence the stress, the requirement þ tn ¼  t  n reduces to a scalar equation requiring Trr ðR 1 Þ ¼ Trr ðR1 Þ. Hence, h1 ðR,r,dÞ ¼ a0 þ a1 m1 d2 þ a2 m3 =d2 .

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

g g g ðdgi , l , Og Þ ¼ ½P o P comp ðdgi , l , Og Þ2 þ ½LLcomp ðdgi , l , Og Þ2 o g 2 comp g g þ ½T T ðdi , l , O Þ ,

e

ð113Þ

and unless e ¼ 0 we update our guess values and again solve the initial value problem. Note that here the minimum value that e can take is zero. Any non-gradient based minimization technique a could be employed to update the guess values and to find dai , l a a a a and O such that eðdi , l , O Þ ¼ 0. Alternatively, instead of solving a

for dai , l and Oa simultaneously, they could be solved sequentially. One of the ways of implementing this involves the following steps: 1. 2. 3. 4. 5.

6. 7.

8. 9. 10.

g

assume a value for l , assume a value for Og , assume a value for dgi , solve the resulting initial value problem, compute eðdgi Þ ¼ P o P comp ðdgi Þ, unless eðdgi Þ ¼ 0 update the o guess for dgi using some non-gradient based zero finding algorithm, steps 4 and 5 are repeated until we find dai such that e(dai )¼0, compute tðOg Þ ¼ T T comp ðOg Þ, unless tðOg Þ ¼ 0 update the guess for Og using some non-gradient based zero finding algorithm, steps 3 through 7 are repeated until we find Oa such that tðOa Þ ¼ 0, g g g compute lðl Þ ¼ LLcomp ðl Þ, unless lðl Þ ¼ 0 update the guess g for l using some non-gradient based zero finding algorithm, a steps 2 through 9 are repeated until we find l such that a lðl Þ ¼ 0.

However, it should be recorded that this method is computationally expensive. Here we note that the integration for the axial load and torque were performed using the trapezoidal rule. This first order method is believed to yield accurate enough results because adaptive meshing is used while solving the ordinary differential equation.

different materials, a case not considered here. Then, f(R,r,d) does not belong to C 1 ðoa Þ. However, in these cases if f ðR,r,dÞ A C 1 ðod Þ, where od ¼ fðR,r,dÞjRj r R rRj þ 1 ,0 or r ro ,0 o d o 1g for j A fi,1,2, . . . ,n,og, i.e. f(R,r,d) is infinitely differentiable in each subpart of the body in which To is continuous and there exist a real valued positive solution (dj )* for the interface condition (109) at each interface, then the deformation (3) is still possible. Next, we examine the case where the prescribed boundary conditions are different from those considered above. For this case, if f ðR,r,dÞ A C 1 ðoa Þ for some real positive values of ro and l and for some real values of O, the deformation of the form (3) is still possible. But one cannot ensure that the specified boundary conditions are met. In other words, suitable traction, which depends on the choice of material response function, must be applied to realize the deformation (3). But, this is the characteristic of controllable deformations.

4. Solution technique for incompressible materials Next, we consider the case where the prestressed annular cylinder is made up of incompressible material. The incompressibility constraint requires that lrr ,R =R ¼ 1. Integrating this simple

1

0.5

o T ΘΘ /μ1

Then we find

0

−0.5

−1

3.2. On the existence and uniqueness of the deformation (3)

0

0.2

0.4

0.6

0.8

1

(R − Ri)/(Ro − Ri) cs−1

cs−2

stsf

cs−3

cs−4

0.2 0.15 0.1 0.05 T oRR /μ1

It follows from Section 3.1 that if the prescribed boundary conditions are ro, P o , l and O then the existence and uniqueness of the deformation of the form (3) depends on the existence and uniqueness of do which satisfies the non-linear algebraic Eq. (102b) provided f ðR,r,dÞ A C 1 ðoa Þ, where oa ¼ fðR,r,dÞjRi r R r Ro ,0 r r rro ,0 od o1g. The latter restriction arises because if we seek the solution to the governing equation as a Taylor’s series, for the series to converge we require r,RR and its higher derivatives to be bounded on Ri r R rRo . By inspection, we find that r,RR and its higher derivatives would be bounded at all points except at R ¼0 and at points where f1 ¼0 assuming that ai ’s are smooth bounded functions of Jmi, i.e. ai A C 1 ðop Þ, where op ¼ fðJm1 ,Jm2 ,Jm3 Þj0 oJm1 o1,0 o Jm2 o 1, 0 o Jm3 o 1g. Thus, those material response functions, ai ’s which do not ensure that f1 a 0 when ðR,r,dÞ A oa or for those which do not belong to C 1 ðop Þ, one cannot be sure that it admits the deformation (3). Before proceeding further we observe that the material response functions would belong to C 1 ðop Þ when expressed as a polynomial series. On the other hand a rational function representation of the material response functions does not always ensure that it belongs to C 1 ðop Þ. Now, we consider the cases where To is only piecewise continuous or when the annular cylinder is made up of layers of

105

0 −0.05 −0.1 −0.15 −0.2

0

0.2

0.4 0.6 (R − Ri)/(Ro − Ri)

0.8

1

o o Fig. 1. Variation of prestress components (a) TYY =m1 (b) TRR =m1 with respect to (R  Ri)/(Ro  Ri) corresponding to cases 1–4.

106

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

first order differential equation between R and Ro, we obtain rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R2 R2 r ¼ ro2  o , ð114Þ

l

where o o 2 m1 ¼ ½d0 þ d1 TRR þ d2 ðTRR Þ , 



o o þ d2 ðTYY Þ2 , m2 ¼ ½d0 þ d1 TYY 

where ro is the deformed outer radius, i.e., ro ¼ r^ ðRo Þ. For the special boundary value problem being studied, the components of stress in the cylindrical polar basis computed using Eq. (32) are given by 9 8  2  2 > > > >  R  rl > > p þ b m þ b m > > 1 1 2 3 > > > > rl R > > 9 > 8 > > >     > > 2   T 2 > > > rr > r R > > > >  2   > > > > > > > > p þ b m þ d ðr O Þ b m þ > > > > > Tyy > 1 2 4 0 2 > > R r > > > > > > > > " # > > > >  2 = < Tzz = <  k R O  2  0 , ð115Þ ¼ p þ b1 d0 l þ b2 2 þ m4 T > > > > ry > l > > > l > > > > > > > > > > > > > > > Trz > > > > 0 > > > > > > > > ; > > :T > > > yz > > 0 > > > > > > > > 2 > > R O  > >  > > b1 d0 r Olb2 m4 ; : r l







o o 2 þ k2 ðTRR Þ , m3 ¼ ½k0 þ k1 TRR o o þ k2 ðTYY Þ2 : m4 ¼ ½k0 þ k1 TYY

ð116Þ

and the Lagrange multiplier p, is obtained by integrating (78) as pðRÞ ¼ Trre P o 

Z R

Ro

e e ½Trr Tyy 

R dR, r2 l

ð117Þ

where  2  R 2 rl þ b2 m3 , rl R  2   2  r R  e ¼ b1 m2 þ d0 ðr OÞ2 þ b2 m4 , Tyy R r

e Trr ¼ b1 m1



cs−1

cs−2

stsf

ð118Þ

cs−3

cs−4

3 1

2

1 L/μ1

o T ΘΘ /μ1

0.5

0

0

−0.5

−1

−2

−1 0

0.2

cs−5

0.4 0.6 (R − Ri)/(Ro − Ri) cs−6

stsf

0.8

0.8

0.9

1

1.2

1.3

1.4

1.5

λ cs−5

cs−7

1.1

1

cs−6

stsf

cs−7

cs−8

3

cs−8

0.4

2

0.3 0.2

1 L/μ1

o T RR /μ1

0.1 0

0

−0.1

−1

−0.2 −0.3 −0.4

−2 0

0.2

0.4

0.6

0.8

1

(R − Ri)/(Ro − Ri) o o Fig. 2. Variation of prestress components (a) TYY =m1 (b) TRR =m1 with respect to (R  Ri)/(Ro  Ri) corresponding to cases 5–8.

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

λ Fig. 3. Variation of axial load, L with stretch ratio l for polyurethane when Ri ¼0.5, Ro ¼1, m1 ¼ 1, m3 ¼ 6:25, P i ¼ P o ¼ 0, O ¼ 0 for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

and P o is the radial component of the normal stress at the outer surface. In the above, we assumed that the prescribed boundary conditions are ro, P o , l and O. When ri, P i , l and O are prescribed then a similar approach as outlined above can be used, except that now we have to integrate between Ri and R instead of integrating between R and Ro. When any other combination of four boundary values is specified we have to convert the problem into one of the two initial value problems discussed above by assuming the unspecified parameters, and then an iterative approach as outlined in Section 3.1.2 has to be adopted for updating the value of the assumed parameters. Thus, it follows that a deformation of the form (3) is possible in prestressed annular cylinders made of any incompressible material provided prestresses vary only radially. When the specified boundary conditions are {ro, P o , l, Og or {ri, P i , l, Og then the deformation of the form (3) satisfies the prescribed boundary conditions exactly. One cannot be sure that any other specified boundary condition can be met by a deformation of the form (3). In other words, suitable boundary tractions which depend on the choice of the material response functions have to be applied in order to realize the deformation (3).

cs−1

cs−2

stsf

cs−3

Before we conclude this section we specialize the expression (117) for the exponential constitutive relation (45). Now  2 R e o Trr ¼ ðq þTRR Þexpðm2 ½Jm1 J1r Þ , ð119Þ rl    r 2 e o Tyy ¼ expðm2 ½Jm1 J1r Þ ðq þTyy Þ þqðr OÞ2 , R 2

ð121Þ

Tyz ¼ qrOlexpðm2 ½Jm1 J1r Þ,

ð122Þ

where Jm1 ¼

q¼

m1

½qJ 1 þ J4 ,

ð123Þ

1 r o o ½J m expðm2 ½J1r 3ÞTRR Tyy , 3 1 1

cs−1

cs−4

0.6

0.82

0.4

cs−2

0.2

0.78

0

stsf

cs−3

cs−4

λ = 1.5

0.76

−0.2

0.74

−0.4 0.9

1

1.1

1.2

1.3

1.4

0.72 0.9

1.5

0.92

0.94

λ cs−5

cs−6

stsf

cs−7

cs−5

cs−8

cs−6

R

0.96

stsf

0.98 cs−7

1 cs−8

0.84

0.6

0.82

0.4

0.8

0.2 r

L/μ1

ð124Þ

0.8 r

L/μ1

expðm2 ½3J1r Þ

0.84

0.8

ð120Þ

e ¼ ql expðm2 ½Jm1 J1r Þ, Tzz

0.8

−0.6 0.8

107

0.78

λ = 1.5

0 0.76

−0.2

0.74

−0.4 −0.6 0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

λ Fig. 4. Variation of axial load, L with stretch ratio l for incompressible material when Ri ¼ 0.9, Ro ¼1, m1 ¼ 1, m2 ¼ 0:1, P i ¼ P o ¼ 0, O ¼ 0 for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

0.72 0.9

0.92

0.94

0.96

0.98

1

R Fig. 5. Variation of r with R for incompressible material when Ri ¼0.9, Ro ¼1, m1 ¼ 1, m2 ¼ 0:1, P i ¼ P o ¼ 0, O ¼ 0, l ¼ 1:5 for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

108

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

and Jr1 is obtained from solving the equation " # o o TRR Tyy J1r 1 2q q þ  ¼ 0, þ þ  2 þ þ q b1 b1 ðb1 Þ ðb1 Þ3

ð125Þ

where b1þ ¼ m1 expðm2 ½J1r 3Þ. Here we solve the non-linear Eq. (125) using the bisection technique and the integration is carried out numerically using quad, a built in function in MATLAB. We note that the integration has to be carried out numerically since the nonlinear Eq. (125) does not have an analytical solution.

5. Comparison of the response of various prestressed and stress free annular cylinders We assume the prestressed annular cylinder to be thick walled with Ri ¼0.5 and Ro ¼1 when made up of a compressible polyurethane material and thin walled with Ri ¼ 0.9 and Ro ¼1 when made up of an incompressible material. Following Blatz and Ko [22], we assume that m3 ¼ 6:25 in (30) and use parameter m1 to make stresses dimensionless. For the exponential model, (45) we assume m2 ¼ 0:1, for illustration and use parameter m1 to make the stresses dimensionless.

cs−1

0.818

cs−2

stsf

cs−3

Next, we specify the assumptions made with regard to the prestress fields. Towards this, we list the different prestress variations studied along with the value of the parameters used in these variations: cs-1 cs-2 cs-3 cs-4 cs-5 cs-6 cs-7 cs-8 stsf

Cosine variation: e1 ¼ 0:4, e2 ¼ e3 ¼ e4 ¼ 0:2, Linear variation: e1 ¼ 1, Linear variation: e1 ¼ 1, Cosine variation: e1 ¼ 0:4, e2 ¼ e3 ¼ e4 ¼ 0:2, PWC variation: e1 ¼ 1, k ¼ 2, Sinusoidal variation: e1 ¼ 1, Sinusoidal variation: e1 ¼ 1, PWC variation: e1 ¼ 1, k ¼ 2, Stress free reference configuration.

Figs. 1 and 2 shows the variation of the non-zero components of the prestress fields with respect to (R  Ri)/(Ro Ri), for the different variations listed above. Though we study eight prestress fields, there are only four independent prestress fields; the other four differ only in their sign. Then, the purpose of

cs−1

cs−4

cs−2

stsf

cs−3

cs−4

0.7 0.6

0.816 0.5 −Trr (ri)/μ1

r, R

0.814 0.812 λ = 1.5

0.81

0.3 0.2 0.1

0.808 0.806

0.4

0

0.9

0.92

0.94

0.96

0.98

1

1

cs−6

stsf

cs−7

1.2

1.3

1.4

1.5

ro

R cs−5

1.1 cs−5

cs−8

cs−6

stsf

cs−7

cs−8

0.8 0.7

0.816

0.6 −Trr (ri)/μ1

0.814 r, R

λ = 1.5 0.812

0.5 0.4 0.3

0.81

0.2

0.808

0.1

0.806 0.9

0

0.92

0.94

0.96

0.98

1

R Fig. 6. Variation of r,R with R for incompressible material when Ri ¼ 0.9, Ro ¼1, m1 ¼ 1, m2 ¼ 0:1, P i ¼ P o ¼ 0, O ¼ 0, l ¼ 1:5 for prestress distributions corresponding to cases (a) 1–4 (b) 5–8.

1

1.1

1.2

1.3

1.4

1.5

ro Fig. 7. Radial component of the normal stress at the inner surface, Trr(ri) required to inflate an annular cylinder made of polyurethane, so that the outer diameter is ro when Ri ¼ 0.5, Ro ¼ 1, m1 ¼ 1, m3 ¼ 6:25, P o ¼ 0, O ¼ 0, l ¼ 1 and for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

studying these prestress fields that differ only in the sign is to examine how the global response of the body differs when the hoop stresses at the boundary is compressive in nature versus tensile in nature. Specifically, we are interested in examining if the tensile hoop prestress at the boundary results in, say, the axial load required to realize a given axial stretch to be more than the case when the reference configuration is stress free or when the hoop prestress is compressive in nature. Next, we outline the rationale for choosing these four different variations in the prestress, namely cs-3, cs-4, cs-5 and cs-7. Prestresses arising due to plastic deformation by subjecting the annular cylinder to excessive radial component of the normal stress at the inner surface, would resemble that given by cs-3. When thermal gradients during cooling process are the cause for the prestresses, the core will be subjected to tensile hoop prestresses while the surfaces will be experiencing compressive hoop prestresses and this state of hoop prestress corresponds to the case cs-4. Recognize that while cooling the surfaces solidify faster than the core giving raise to these type of prestresses. cs-5

cs−1

cs−2

stsf

cs−3

109

represents the prestress fields that are likely to arise in a shrink fit annular cylinder, made from two annular cylinders. Though we claim that cs-3 through cs-5 represent prestress fields arising due to different reasons, there is no experimental evidence for rubbery materials or for soft tissues that either of these is indeed the transmural variation of the prestress fields. It is more based on our understanding of the prestress fields that arise in metallic structures. cs-7 is introduced to see what happens if the hoop as well as radial prestresses are zero at the surface, purely from an academic perspective. Note that in all the cases studied the maximum hoop stress is constant. This we feel is necessary to allow us to compare the responses of the various prestress distributions, since the extent of the change in response of the body due to these prestress fields depends on the magnitude of the prestresses. Using the developed framework various boundary value problems can be studied. However, in this article we restrict ourselves to those boundary value problems for which experimental setups can be developed with minimum effort or already exists.

cs−4

cs−1

1.5

cs−2

stsf

cs−3

cs−4

0.75 0.7

1

0.65 ro = 1.5

r, R

L/μ1

0.6 0.55

0.5

0.5 0.45

0

1

1.1

1.2

1.3

1.4

0.4 0.5

1.5

ro cs−5

cs−6

0.6

0.7

0.8

0.9

1

R

stsf

cs−7

cs−8

cs−5

1.5

cs−6

stsf

cs−7

cs−8

0.75 0.7

1

r, R

L/μ1

0.65 0.6

ro = 1.5

0.55

0.5

0.5 0.45

0

1

1.1

1.2

1.3

1.4

1.5

ro Fig. 8. Axial load, L required to maintain a constant stretch ratio l ¼ 1, as the annular cylinder is being inflated for polyurethane when Ri ¼ 0.5, Ro ¼1, m1 ¼ 1, m3 ¼ 6:25, P o ¼ 0, O ¼ 0 for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

0.4 0.5

0.6

0.7

0.8

0.9

1

R Fig. 9. Variation of r,R with R for polyurethane when Ri ¼0.5, Ro ¼ 1, m1 ¼ 1, m3 ¼ 6:25, ro ¼ 1.5Ro, l ¼ 1, O ¼ 0, P o ¼ 0, for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

110

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

5.1. Pure axial extension of the annular cylinder Here the specified boundary conditions are P i ¼ P o ¼ 0, O ¼ 0 and l ¼ ls , some specified value. Then, we compare the axial load L required to engender a given axial stretch ratio ls for various prestressed annular cylinders made of compressible polyurethane material (30) in Fig. 3 to find that the boundary axial load required to engender a given boundary displacement does depend on the prestresses present in the cylinder. The magnitude of the axial load required to realize a given stretch ratio, in prestressed cylinder is more than that required in stress free cylinder but the percentage increase seems to depend on the prestress variations; the maximum being 13 percent. Studying the variation of r,R with R for the eight prestress fields listed above we conclude that the deformation field measured from a prestressed configuration does not vary from that measured from a stress free configuration, though the evidence for the same is not presented here. Fig. 4 portrays the axial load required to engender a given axial stretch ratio in the prestressed annular cylinders made of

cs−1

0.12

cs−2

stsf

cs−3

exponential incompressible material. In Fig. 5, we study the variation of r^ ðRÞ with R and in Fig. 6 the variation of r,R with R for the eight prestress fields being studied. As in the case of cylinders made of certain classes of compressible materials, the axial load required to engender a given axial stretch increases due to presence of prestresses. However, unlike for polyurethane the deformation field and the gradient of the deformation field also varies due to the presence of prestresses. There is a 12 percent variation in the axial load and 1.5 percent difference in the radial component of the displacement and norm of the deformation gradient between the stress free and prestressed cylinders.

5.2. Inflation of the annular cylinder at constant length In this case, the specified boundary conditions are P o ¼ 0,

O ¼ 0, l ¼ 1, and ro ¼rso, some specified value. Then, we compare the radial component of the normal stress at the inner surface of the annular cylinder, P i required to inflate the cylinder so that its outer diameter becomes rso and the axial load, L, required to

cs−4

cs−1

0.4

cs−2

stsf

cs−3

cs−4

0.35

0.1

0.3 0.25 L/μ1

−Trr (ri)/μ1

0.08 0.06

0.2 0.15

0.04

0.1 0.02 0

0.05 1

1.1

1.2

1.3

1.4

0

1.5

1

1.1

1.2

ro/Ro cs−5

0.12

cs−6

1.3

1.4

1.5

ro/Ro

stsf

cs−7

cs−8

cs−5

0.4

cs−6

stsf

cs−7

cs−8

0.35

0.1

0.3 0.25 L/μ1

−Trr (ri)/μ1

0.08 0.06

0.2 0.15

0.04

0.1 0.02 0

0.05 1

1.1

1.2

1.3

1.4

1.5

ro/Ro Fig. 10. Radial component of the normal stress at the inner surface, Trr(ri) required to inflate an annular cylinder made of incompressible material, so that the outer diameter is ro when Ri ¼ 0.9, Ro ¼ 1, m1 ¼ 1, m2 ¼ 0:1, P o ¼ 0, O ¼ 0, l ¼ 1 and for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

0

1

1.1

1.2

1.3

1.4

1.5

ro/Ro Fig. 11. Axial load, L required to maintain a constant stretch ratio l ¼ 1, as the annular cylinder is being inflated for an incompressible material when Ri ¼ 0.5, Ro ¼1, m1 ¼ 1, m2 ¼ 0:1, P o ¼ 0, O ¼ 0 for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

maintain it at its original length as the cylinder is being inflated for various prestress distributions. While it is seen from Fig. 7 that P i required to realize a given outer radius increases (for the cases studied here, the maximum increase is 14 percent) due to the presence of prestresses, the axial load, L required to maintain constant length seem to increase by as much as 20 percent for some prestress distributions and decrease by a maximum of 0.7 percent for other prestress distributions, as seen in Fig. 8. Contrary to what we observed for uniaxial extension the deformation field and its gradient seem to vary due to the presence of prestresses. However, for the cases studied here, while the difference between the prestressed and stress free cylinders for the radial component of the displacement is less than 0.1 percent that for the norm of the deformation gradient is less than 2 percent. It can be seen from Fig. 9 that this difference in r,R between the prestressed and stress free cylinder is the largest at the inner surface of the cylinder or at the interfaces where the prestress field is not continuous. Fig. 10 portrays the variation of P i required to realize a given outer radius for various prestressed incompressible cylinders.

cs−1

cs−2

stsf

cs−3

111

From the figure, it is clear that P i required to realize a given outer radius increases due to the presence of prestresses. We also find that for the cases considered here P i could increase by as much as 26 percent due to the presence of prestresses. In Fig. 11 we plot the variation of the axial load L required to maintain constant length as the cylinder is being inflated for various prestressed cylinders. Irrespective of the variation of the prestresses, due to the presence of prestress, the applied axial load to maintain constant length increases from that required in the case of stress free cylinders. For the cases studied here, this increase is as much as 12 percent, for some cases. Unlike in the case of uniaxial extension of incompressible cylinders, now the prestressed and stress free annular incompressible cylinders are subjected to the same deformation field, since the value of ro is prescribed.

5.3. Pure twisting of annular cylinder Here the specified boundary conditions are P i ¼ P o ¼ 0, l ¼ 1 and O ¼ Os , some specified value. Then, we compare the torque, T

cs−1

cs−4

1

cs−2

stsf

cs−3

cs−4

0

−0.05

0.5

L/μ1

T/μ1

−0.1 0

−0.15 −0.5

−0.2

−1 −0.5 −0.4 −0.3 −0.2 −0.1 cs−5

cs−6

0 Ω

0.1

stsf

0.2

0.3

cs−7

0.4

−0.25 −0.5 −0.4 −0.3 −0.2 −0.1

0.5

cs−8

cs−5

1

cs−6

0 Ω

0.1

stsf

0.2

0.3

cs−7

0.4

0.5

cs−8

0

−0.05

0.5

L/μ1

T/μ1

−0.1 0

−0.15 −0.5

−1 −0.5 −0.4 −0.3 −0.2 −0.1

−0.2

0 Ω

0.1

0.2

0.3

0.4

0.5

Fig. 12. Torque, T required to induce a angle of twist per unit length, O in an annular cylinder made of polyurethane when Ri ¼ 0.5, Ro ¼ 1, m1 ¼ 1, m3 ¼ 6:25, P o ¼ P i ¼ 0, l ¼ 1 and for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

−0.25 −0.5 −0.4 −0.3 −0.2 −0.1

0 Ω

0.1

0.2

0.3

0.4

0.5

Fig. 13. Axial load, L required to maintain a constant stretch ratio l ¼ 1, as the annular cylinder is being twisted for polyurethane when Ri ¼ 0.5, Ro ¼ 1, m1 ¼ 1, m3 ¼ 6:25, P o ¼ P i ¼ 0 and for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

required to twist the cylinder so that its angle of twist per unit length becomes Os and the axial load, L, required to maintain the cylinder at its original length as the cylinder is being twisted for various prestress distributions. Fig. 12 portrays the variation of torque in polyurethane annular cylinder as it is twisted for various prestress distributions and Fig. 13 depicts the variation of the axial load L required to maintain the cylinder at constant length as the cylinder is being twisted. We find that the torque as well as the axial load varies depending on the distribution of the prestress field. Unlike, in the previous two boundary value problems certain prestress distributions causes the magnitude of the torque as well as the axial load to be more than that in the case of stress free cylinders and for other prestress distributions to be less. The maximum deviation in torque as well as axial load is about 26 percent. Then, studying the variation of r with respect to R for various prestress fields introduced in Figs. 1 and 2 and when O ¼ 0:5, we find that the maximum difference between the stress free and prestressed configuration is only 0.05 percent. However, the maximum difference in r,R is 0.5 percent.

0

cs−1

cs−2

stsf

cs−3

cs−4

−0.01 −0.02 −0.03 L/μ1

112

−0.04 −0.05 −0.06 −0.07 −0.08 −0.5 −0.4 −0.3 −0.2 −0.1

0

cs−5

cs−6

0 Ω

0.1

stsf

0.2

0.3

cs−7

0.4

0.5

cs−8

−0.01

0.3

cs−1

cs−2

stsf

cs−3

cs−4

−0.02 −0.03 L/μ1

0.2

T/μ1

0.1

−0.05 −0.06

0

−0.07

−0.1

−0.08 −0.5 −0.4 −0.3 −0.2 −0.1

−0.2

−0.5 −0.4 −0.3 −0.2 −0.1

0.3

cs−5

cs−6

0 Ω

0.1

stsf

0.2

0.3

cs−7

0.4

0.5

0.1 0 −0.1 −0.2

−0.5 −0.4 −0.3 −0.2 −0.1

0 Ω

0.1

0.2

0 Ω

0.1

0.2

0.3

0.4

0.5

Fig. 15. Axial load, L required to maintain a constant stretch ratio l ¼ 1, as the annular cylinder is being twisted for incompressible material when Ri ¼ 0.9, Ro ¼ 1, m1 ¼ 1, m2 ¼ 0:1, P o ¼ P i ¼ 0 and for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

cs−8

0.2

T/μ1

−0.04

0.3

0.4

0.5

Fig. 14. Torque, T required to induce a angle of twist per unit length, O in an annular cylinder made of incompressible material when Ri ¼ 0.9, Ro ¼ 1, m1 ¼ 1, m2 ¼ 0:1, P o ¼ P i ¼ 0, l ¼ 1 and for prestress distributions corresponding to cases (a) 1–4, (b) 5–8.

Fig. 14 portrays the variation of torque in an annular cylinder made up of an incompressible material as it is twisted for various prestress distributions and Fig. 15 depicts the variation of the axial load L required to maintain the cylinder at constant length as the cylinder is being twisted. We find that the torque as well as the axial load varies depending on the distribution of the prestress field. Unlike for polyurethane cylinders but consistent with the previous two boundary value problems all prestress distribution studied here causes the magnitude of the torque as well as the axial load to be more than that in the case of stress free cylinders. The maximum deviation in torque is about 18 percent and that for the axial load is about 9 percent. In this case too, we find that the deformation field though depends on the prestress field, the maximum difference in the radial component of the deformation field between the stress free and prestressed configuration is 0.3 percent, for the prestress fields considered here and when O ¼ 0:5. The maximum difference in r,R is also about 0.4 percent. Thus, we find that prestresses cause a significant difference in the boundary load required to engender a given boundary displacement which can be utilized to detect the prestresses. The maximum difference in the boundary load is when the

U. Saravanan / International Journal of Non-Linear Mechanics 46 (2011) 96–113

deformation field from both the prestressed and stress free configuration is the same as in the case of inflation of annular cylinders made of incompressible materials and pure torsion or axial extension of compressible cylinders. Even if there is a difference in the radial deformation field inferred from a prestressed and a stress free reference configuration, it is less than 1 percent and hence becomes hard to detect from a practical point of view. However, the difference in the norm of the gradient of the deformation field inferred from a prestressed and a stress free reference configuration is about 2 percent.

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