Inhomogeneous elastostatic problem solutions constructed from stress-associated homogeneous solutions

Journal of the Mechanics and Physics of Solids 52 (2004) 2207 – 2233

www.elsevier.com/locate/jmps

Inhomogeneous elastostatic problem solutions constructed from stress-associated homogeneous solutions M. Fraldia , S.C. Cowinb; c; d;∗ a Dipartimento

di Scienza delle Costruzioni, Facolta di Ingegneria, Universita di Napoli “Federico II”, Italy b The New York Center for Biomedical Engineering, New York, USA c Departments of Biomedical and Mechanical Engineering, School of Engineering of the City College, New York, USA d Graduate School of the City University, New York, USA Received 18 November 2003; received in revised form 7 April 2004; accepted 8 April 2004

Abstract In this paper, we present a theorem that provides solutions for anisotropic and inhomogeneous elastostatic problems by using the known solution of an associated anisotropic and homogeneous problem if the associated problem has a stress state with a zero eigenvalue everywhere in the domain of the problem. The fundamental property on which this stress-associated solution (SAS) theorem is built is the coaxiality of the eigenvector associated with the zero stress eigenvalue in the homogeneous problem and the gradient of the scalar function ’ characterizing the inhomogeneous character of the inhomogeneous problem. It is shown that most of the solutions of anisotropic elastic problems presented in the literature have this property and, therefore, it is possible to use the SAS theorem to construct new exact solutions for inhomogeneous problems, as well as to 5nd—using the SAS theorem—solutions for the shape intrinsic and angularly inhomogeneous problems. ? 2004 Elsevier Ltd. All rights reserved.

1. Introduction Finding solutions to anisotropic inhomogeneous elastic material problems is not easy (Chadwick and Smith, 1977; Cowin, 1987; Lekhnitskii, 1963; Lions, 1985; Maugin, ∗

Corresponding author. Departments of Biomedical and Mechanical Engineering, School of Engineering of the City College, New York, USA. Tel.: +1-212-799-7970; fax: +1-212-799-7970. E-mail address: [email protected] (S.C. Cowin). 0022-5096/$ - see front matter ? 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jmps.2004.04.004

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1993; Nemat-Nasser and Hori, 1993; Ting, 1996a; Alshits and Kirchner, 2001). A few very restricted classes of inhomogeneous problems are solved in a general way. One example of these solutions is for cylinders subjected to pure torsion and possessing cylindrical orthotropy, with a variation of the shear moduli with the local normal direction to the family of curves of which the lateral boundary is a member (Cowin, 1987). This solution is a generalization, to a set of arbitrary cross-sectional shapes, of a problem solved by Voigt (1886) for a circular cross-section with radial variation of its cylindrical anisotropy. These cylinders are said to possess shape intrinsic orthotropy since it is the boundary of the cylinder that establishes the possible directional variation of the elastic moduli. A second example was given by Ting (1989) and Chung and Ting (1995) who presented an exact solution for the case of an anisotropic half-space with elastic moduli dependent upon one coordinate, the angle , when the loads on the half-space are represented by a straight line of force. These kinds of problems were called angularly inhomogeneous problems by the authors. Closely related to these solutions is a third example called radially inhomogeneous problems (Alshits and Kirchner, 2001). As the name suggests, the variation of the elastic constants is in the radial direction in this case. A method is presented here that enables one to 5nd solutions for inhomogeneous, anisotropic elastostatic problems if two conditions are satis5ed: (1) a knowledge of the solution for a homogeneous elastic reference problem (the associated problem) whose solution has a stress state with a zero eigenvalue everywhere in the domain of the problem, and (2) an inhomogeneous anisotropic elastic tensor related to the homogeneous anisotropic elastic tensor of (1) by CI = ’(x)CH ; H

HT

’(x)| ∀x ∈ B; ’(x) ¿  ¿ 0;

 ∈ R+ ;

(1.1)

where C = C is the elasticity tensor of a generic anisotropic homogeneous elastic material of the reference problem, CI is the elasticity tensor of the corresponding anisotropic inhomogeneous elastic problem, B is the domain occupied by both the homogeneous object BH and the inhomogeneous one BI ,  ∈ R+ is an arbitrary positive real number, while ’(x) is a C 2 (B) scalar function. The assumption (2) means that the inhomogeneous character of the material is due to the presence of a scalar parameter producing the inhomogeneity in the elastic coeHcients. This method makes it possible to 5nd analytical solutions for an inhomogeneous anisotropic elastic problem if the elastic solution of the corresponding homogeneous anisotropic reference problem is known and characterized everywhere by a stress state with a zero eigenvalue. The solutions to the inhomogeneous anisotropic elastic problem are called the associated solutions of the homogeneous problem. We show how it is possible to use this method to construct exact solutions for several new inhomogeneous and anisotropic problems, as well as to rede5ne, summarizing in one uni5ed concept the three examples described in the opening paragraph, the shape intrinsic anisotropic materials, the angularly inhomogeneous materials and the radially inhomogeneous materials. We prove that, if the stress state in a given anisotropic and homogeneous problem is one with a zero stress eigenvalue and the structural gradient (the gradient of the scalar function ’ of Eq. (1.1)) is locally coaxial with the eigenvector of the zero eigenvalue, then there exists a corresponding inhomogeneous

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problem in which the strain–displacement 5eld solution is identical with the strain– displacement 5eld of the homogeneous reference solution, while the stress 5eld of the inhomogeneous problem is equal to ’(x) times the stress 5eld of the homogeneous problem. In fact, in either of these cases, the solutions are characterized by a local stress state with a zero eigenvalue and a structural gradient coaxial with the eigenvector of the zero-eigenvalue stress state. 1.1. Zero-eigenvalue stress and zero-eigenvalue strain
(det E = 0):

(1.2)

It is easy to show that a zero-eigenvalue stress (strain) state is a necessary condition for a plane stress (strain) state. The components of the stress tensor T (strain tensor E) are denoted by ij ( ij ). The strain tensor E is related to the displacement 5eld u by E = 12 [(∇ ⊗ u) + (∇ ⊗ u)T ] = sym ∇ ⊗ u

∀x ∈ B;

(1.3)

in which grad u = (∇ ⊗ u) and the symbol ⊗ represents the tensor product. In components we have

ij = 21 (ui; j + uj; i );

(1.4)

where the comma denotes diMerentiation and u is the displacement 5eld. 1.2. Stress-associated solutions (SASs) for inhomogeneous elasticity Consider the following mixed boundary-value elastostatic homogeneous and anisotropic problem P H in the absence of action-at-a-distance forces ∇ · T(u) = 0 in BH ;

T(u) · n = t on @BtH ;

u = u0 on @BuH ;

H

(1.5) H

={@BtH ∪@BuH }

where B is the domain occupied by the homogeneous elastic object, @B is its boundary and t and u0 are the traction 5eld and the displacements assigned on the corresponding partition of the boundary, respectively (Barber, 1992; Ciarlet, 1988; Gurtin, 1972; Muskhelishvili, 1953). The notation for the divergence of the stress tensor is ∇·T(u)=div T(u), where the del operator is a vectorial diMerential operator de5ned by ∇ ≡ @i ei , @i ≡ @=@xi = (∗); i is the partial diMerential operator and ei is the base unit vector of the i-axis. The anisotropic Hooke’s law is written T(u) = CH : E(u) = CH : sym(∇ ⊗ u) = CH : (∇ ⊗ u);

(1.6)

or, in components H H

ij = Cijhk

hk = Cijhk uh; k :

Let GH = {uH ; E H ; T H } be the solution of the homogeneous problem (1.5).

(1.7)

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Consider now an associated anisotropic elastic inhomogeneous problem P I , described by modifying the system (1.5), with t I = ’t representing the traction 5eld applied on @BtI and the inhomogeneous anisotropic elasticity tensor given by Eq. (1.1), thus T(u) · n = t I on @BtI ;

∇ · T(u) = 0 in BI ; H

u = u0 on @BuI :

(1.8)

I

The solid domains B and B , as well as their corresponding boundary partitions made on @BH and @BI , are geometrically the same in the homogeneous and inhomogeneous problems. Then, if we expand the 5rst equation (1.8) it is possible to write ∇ · T(u) = ∇ · [’(x)CH : E(u)] = ’(x)∇ · [CH : E(u)] + [C H : E(u)] · ∇’(x) = 0;

(1.9)

where ∇(∗) = grad(∗) is the gradient operator applied on a generic scalar-valued function (∗). Consider now the situation in which the displacements are equal for the homogeneous and inhomogeneous problems. Then, by substituting the displacement solution uH obtained for the homogeneous problem P H in (1.9) in place of the displacement vector u, we have that ∇ · T(uH ) = ’(x)[∇ · T H (uH )] + [T H (uH )] · ∇’(x) = 0:

(1.10)

But, since ∇ · [T H (uH )] = ∇ · [CH : E(uH )] = 0, it follows that [T H (uH )] · ∇’(x) = 0

∀x ∈ BI :

(1.11)

By excluding the trivial case in which ’(x) = constant, it follows that det T H = 0;

∀x ∈ BH :

(1.12)

This means that the stress state at x of the reference homogeneous problem is required to be a zero-eigenvalue stress state everywhere in the domain. To investigate the geometrical meaning of Eq. (1.11), since Eq. (1.11) must be true everywhere in BI , we consider, without loss of generality, the local principal stress reference system {1 ; 2 ; 3 }, in which the stress tensor T H takes the component form   H 0 0

1   (1.13) TH =  0 

H2 :  0 H 0 0

 3 Representing the gradient of the scalar function ’ as ∇’()T = [ ’; 1

’;  2

’; 3 ];

(1.14)

the three scalar equations implied by (1.11) are written as

H1 ’; 1 = 0;

H2 ’; 2 = 0;

H3 ’; 3 = 0:

(1.15) H

The system (1.15) is satis5ed if the stress tensor T for the reference homogeneous problem P H is, at each internal point x ∈ BH , a locally variable zero-eigenvalue stress state. If there is only one zero eigenvalue, say in the 3 direction, the only non-zero component of the vector ∇’, is ’; 3 at the corresponding points x ∈ BI . If there are two zero eigenvalues there can be two non-zero components of ∇’. The case of three

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inhomogeneous object I

3


 2

2


x


 1

H

equipotential surfaces of 

elementary volume in x


 3 = 0




 1
 2

B

1

B

I

I

stress plane in x

tangent plane at x to the equipotential surfaces of 

Fig. 1. Geometrical interpretation of the relationship between the equipotential surfaces of ’ and the distribution of the planes of stresses in the associated anisotropic problem.

zero eigenvalues of the stress tensor T H is trivial and will not be mentioned further. It follows that, at each internal point, the equipotential surfaces of ’ admit as a tangent plane the plane whose normal is coaxial with the eigenvector associated with the zero stress eigenvalue (or a direction, in the case of two zero stress eigenvalues). This is illustrated in Fig. 1 for the case of one zero eigenvalue of stress. The geometrical relationship (1.11) between the stress tensor T H and the vector ∇’ may be rewritten in the form {T H · ∇’ = 0} ⇔ {∀v ∈ V;

T H : (∇’ ⊗ v) = 0};

(1.16) 3

where v is any unit vector de5ned in the three-dimensional Euclidean space E and V represents the corresponding vector space. It follows that the stress vector on the plane whose normal is v is always orthogonal to the vector ∇’. Then, it is possible to establish the following theorem: Stress 1 -associated solution (SAS) theorem: Consider two geometrically identical elastic objects BH and BI , one homogeneous and the other inhomogeneous, respectively. Let CH and CI = ’(x)CH be the corresponding elasticity tensors (Fig. 2). The two elastostatic problems associated with the two objects are P H : {∇ · T(u) = 0 in BH ; T(u) · n = t on @BtH ; u = u0 on @BuH }; P I : {∇ · T(u) = 0 in BI ; T(u) · n = ’t on @BtI ; u = u0 on @BuI }; where ’(x) ∈ C 2 (B)| ∀x ∈ B; ’(x) ¿  ¿ 0; 1

 ∈ R+ :

In a subsequent paper, we will present a similar SAS theorem.

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M. Fraldi, S.C. Cowin / J. Mech. Phys. Solids 52 (2004) 2207 – 2233 n

n

B H

B I I

H

pH

H

=

pI

BtI

BtH

BI

BH

BuH

pH

B

H t

BuI

pI

B

I t

n

Homogeneous object

n

Inhomogeneous object

Fig. 2. The homogeneous and inhomogeneous bodies with their boundary conditions.

If uH is the solution of the homogeneous problem P H , then uI = uH if and only if {T H : (∇’ ⊗ v) = 0; ∀v ∈ V }, i.e. {∀x ∈ BI ; ∀v ∈ V; T H : (∇’ ⊗ v) = 0} ⇔ uI = uH : Proof. The necessary condition has been established in the preamble. To prove the su?cient condition: {∀v ∈ V; T H : (∇’ ⊗ v) = 0} ⇒ uI = uH ; we 5rst recall Eq. (1.16). Consequently, if uH is the displacement solution of the homogeneous problem P H , we can write {∇ · [T H (uH )] = 0; [T H (uH )] · ∇’ = 0} ⇒ ⇒ {∇ · [CH : (∇ ⊗ uH )] = 0; [CH : (∇ ⊗ uH )] · ∇’ = 0} ⇒ ⇒ ’∇ · [CH : (∇ ⊗ uH )] + [CH : (∇ ⊗ uH )] · ∇’ = 0; from which it follows that ∇ · [CI : (∇ ⊗ uH )] = 0; when Eq. (1.1) is considered. Then, if we rewrite the inhomogeneous elastostatic problem P I in terms of displacements, i.e. PI :

{∇ · [CI : (∇ ⊗ uI )] = 0

in BI ; [CI : (∇ ⊗ uI )] · n = ’t on @BtI ;

uI = u0 on @BuI }; we can observe that uH satis5es all these 5eld and boundary equations. Therefore, from the uniqueness theorem, it follows that uI = uH and, consequently, T I = ’ CH : (∇ ⊗ uH ) = ’T H . This proves the suHciency condition. It is convenient to increase the similarity between the elastic problems for the homogeneous and the inhomogeneous materials by writing the boundary conditions in the same way. Thus, we substitute for the prescribed boundary tractions a corresponding prescribed displacement 5eld; this converts the portion of the boundary upon which

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the surface tractions are prescribed to a portion of the boundary upon which the displacements are prescribed. Due to uniqueness of solution, this is always possible in a linear elastic problem. Then, the two problems may be written in the equivalent forms as PH : PI :

{∇ · T(u) = 0 in BH ; u = ut on @BtH ; u = u0 on @BuH }; {∇ · T(u) = 0 in BI ; u = ut on @BtI ; u = u0 on @BuI };

where ut represents the prescribed displacement on @Bt and where, now, the tractions t and ’ t represent the reactions of the constraints on @Bt speci5ed by ut . It follows that, when a solution GH ={uH ; E H ; T H } for an anisotropic homogeneous elastic problem P H is known, the SAS theorem yields the corresponding solution for an inhomogeneous problem P I as GI ={uH ; E H ; ’T H }, if and only if T H ·∇’ = 0 everywhere in the object and the displacement boundary conditions are the same for both the homogeneous and the inhomogeneous objects. Thus the solution GH = {uH ; E H ; T H } is used to construct a solution of the associated inhomogeneous problem. Finally, we note that the restriction (1.1) may be relaxed in many diMerent ways. For example, the associated solutions could involve only some selected elastic moduli of the homogeneous elasticity tensor, so that the solutions do not depend on all stiMness coeHcients. This means that it is possible to extend the validity of the proposed theorem by rewriting the assumption (1.1) in the weaker form Cˆ Iijhk = ’Cˆ H ijhk ; where Cˆ H ijhk represents only those elastic coeHcients explicitly involved in the speci5c anisotropic homogeneous problem used to construct the associated solution. In the next section, it is shown that components of the elasticity tensor not involved in the solution of the homogeneous problem will not be involved in the solution of the associated inhomogeneous problem. 1.3. Generalization of the SAS theorem to piecewise de
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we will make reference to some results obtained previously and formulate new hypotheses about the features of composite inhomogeneous bodies considered. In particular, for each phase p present of a composite material, we will assume here that the elasticity tensor can be written as H CH p = ’p C ;

p = {1; 2; : : : ; n} ⊂ N;

(1.17)

where CH is the elasticity tensor of a reference isotropic or anisotropic homogeneous material and ’p is a positive scalar parameter. This hypothesis does not constitute the most general case for describing the relation between the elastic tensors of the diMerent phases for a composite material, but it is widely utilized in the literature because many arti5cial and natural composites exhibit mechanical properties that are well represented by the proposed assumption (Lekhnitskii, 1963; Ting, 1996a; Fraldi and Guarracino, 2001; Nemat-Nasser and Hori, 1993). n Let us consider a partition of the inhomogeneous body {p (B)|B ≡ p=1 p (B)}, where @(p; q) represents the interface boundary between two generic sub-domains p H and q of the partition, with elasticity tensors CH p and Cq , respectively. If we assume that the solution for the anisotropic homogeneous reference problem is known, and the geometries of the homogeneous and composite material objects are the same, we can study the conditions under which the stress tensor for the inhomogeneous material (multi-phase material) assumes the form TpH = ’p T H ;

∀x ∈ p (B);

(1.18)

required by the SAS theorem. Note that the stress (1.18) satis5es the equilibrium equations in each sub-domain of the partition, ∇ · TpH = ’p ∇ · T H = 0;

∀x ∈ p (B):

(1.19)

Moreover, by virtue of the assumed constitutive relationships, −1

H H−1 H EpH = CH T = EH; p Tp = C

{∀p ∈ N; ∀x ∈ p };

(1.20)

the satisfaction of the compatibility condition on the surfaces of discontinuity between the diMerent materials of the composite object is automatic. From the force equilibrium on the interfaces between two adjacent phases, it follows that TpH · n(p; q) = TqH · n(p; q) ;

{∀{p; q} ∈ N; ∀x ∈ @(p; q) };

(1.21)

where n(p; q) is the unit normal vector to the interface between the phases p and q. By virtue of Eq. (1.18), Eq. (1.21) is satis5ed if T H · n(p; q) = 0;

∀x ∈ @(p; q) ;

(1.22)

Eq. (1.22) requires that for each point belonging to the interface surfaces between two phases, the stress tensor T H must possess at least one zero eigenvalue, that is {det T H = 0; ∀x ∈ @(p; q) }. This hypothesis is necessary in order to orient the plane of the stress on the interface surfaces such that the eigenvector associated with a zero eigenvalue of the stress tensor is coaxial with the unit normal vector to the tangent plane to the interface. For structures sometimes consistent with this hypothesis one can consider the interfaces between layers of certain plant structures, e.g., onions and leeks.

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In the literature of this subject examples that conform to this hypothesis include the piece-wise angularly inhomogeneous elastic wedges considered by Ting (1996a), the intrinsically orthotropic layered cylinders under torsion, described by Cowin (1987), as well as in other examples analyzed by Lekhnitskii (1963). To complete the elastic solution for the composite material (1.17) using the known solution of a homogeneous reference problem, we note the satisfaction of the compatibility and equilibrium conditions on the external boundary. The satisfaction of the compatibility conditions is easily veri5ed by virtue of Eq. (1.20). The equilibrium equation on the part of the external boundary where the tractions are prescribed is given by TeH · n = teH = ’e t H ;

∀x ∈ @Bt(e) ;

(1.23)

where @Bt(e) represents a typical element of the partition of the external boundary on which the tractions t H are prescribed in the homogeneous reference problem. The total k stress boundary is the sum over all the typical distinct boundaries, @Bt = e=1 @Bt(e) , where k represents the total number of phases that have a projection of their boundary on the external boundary on which the tractions are assigned. Then, if the conditions (1.22) and (1.23) are satis5ed, we can build the elastic solution of composite multi-phase materials from a knowledge of the displacements and the stresses for a homogeneous object with analogous geometry using the extension of the SAS theorem. Note that, in order to utilize the results of the proposed theorem for inhomogeneous materials in which ’ was assumed to be a continuous scalar function, the stress tensor T H had to exhibit a zero eigenvalue at each point of the body. However, in order to generalize the SAS theorem to composite materials where ’ is constant, but piecewise discontinuous, it is suHcient that the stress tensor T H related to the associated homogeneous problem possesses a zero eigenvalue (det T H = 0) only in the points belonging to the internal interfaces between the diMerent phases. This means that, in the case of materials where ’ is a constant, but piecewise discontinuous, T H can be a threedimensional stress 5eld in any other point of the solid domain. 1.3.2. Composite materials where ’ is piecewise continuous In this subsection, we consider the new and more general situation in which each phase p of the heterogeneous solid (composite material) can be represented by the following elasticity tensor H CH p = ’p (xp )C ;

∀xp ∈ p ⊂ B;

(1.24)

where CH is the elasticity tensor of a homogeneous reference material, while ’p is now a positive scalar function, not necessarily constant, but continuous inside each phase (or sub-domain de5ned by the partition described above). We relax some of the hypothesesfor the situation when ’ is constant, but retain the previous notation; n {p (B)|B ≡ p=1 p (B)} is again the partition of the inhomogeneous object, with @(p; q) representing the interface boundary between two generic adjacent sub-domains H p and q of the partition whose elasticity tensors are CH p and Cq , respectively, see

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  3 (B) 3 (B)  2 (B)  p(B)  1 (B) B

I

B

I

piecewise inhomogeneous object

Fig. 3. A representation of a possible spatial distribution of the phases inside a piecewise inhomogeneous material.

Fig. 3. The representation of the stress tensor of the phase p required by the SAS theorem is TpH = ’p (xp )T H ;

∀xp ∈ p (B):

(1.25)

Equilibrium is satis5ed if the divergence of the stress for each phase, Eq. (1.25), is zero ∇ · TpH = ’p ∇ · T H + T H · ∇’p = 0 ⇒ T H · ∇’p = 0; ∀xp ∈ p (B);

(1.26)

From this result it follows, using Eqs. (1.17) and (1.20), that EpH = E H , {∀p ∈ N; ∀xp ∈ p }. The equilibrium conditions (1.21)–(1.22) across the interface between two phases are then satis5ed as well the external boundary conditions (1.23) considered previously. This means that, in order to extend the SAS theorem to piecewise continuous composite materials, one has to 5rst establish two facts about the stress tensor T H , namely: (1) at each internal point of each phase p, the stress tensor T H possesses at least one zero eigenvalue and (2) at every point in the interface between two adjacent phases the normal to the tangent plane has to be coincident with the direction of the eigenvector associated with the zero eigenvalue. 2. Example applications The objective of this section is to present several examples in which the SAS theorem is employed to obtain both some known solutions and some new solutions for inhomogeneous anisotropic materials. In particular, it will be shown that the intrinsic orthotropic materials solutions (Cowin, 1987), the angularly inhomogeneous elastic space solutions (Ting, 1989), and the class of radially inhomogeneous problems (Alshits and Kirchner, 2001), may be viewed as examples of the application of the SAS

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theorem. These examples show that the possibility of obtaining closed-form solutions for inhomogeneous materials often does not depend on the relation between geometry of the solid domain and orientation of the planes of mirror symmetry, but explicitly upon the relation between the geometry of the stress distribution and the structural gradient of the inhomogeneous material. 2.1. Stress-intrinsic anisotropy in inhomogeneous materials 2.1.1. Torsion of trigonal inhomogeneous cylinders The Prandtl stress function for the problem of a homogeneous cylinder with trigonal material symmetry under a pure torsional loading is given by ∇ · ∇ = −2[c44 − (2c14 )2 =(c11 −c12 )] in the domain of the cross-section and =0 on the boundary of the cross-section. In the case of isotropy this equation reduces to ∇ · ∇ = −2G, where G is the shear modulus. In Fraldi and Cowin (2002) it is shown that, for elliptical and triangular cross-sections, a solution of these problems in terms of the stress function (x1 ; x2 ) in the form of a third order polynomial form is possible, so that H =−

23

; 1;

H

13 =

(2.1)

;2

and (x1 ; x2 ) = #0 (x2 ) + x1 #1 (x2 ) + x12 #2 (x2 ) =

2 

x1i #i (x2 );

(2.2)

i=0

where #0 (x2 ) = A5 + A4 x2 − A2 x22 − #1 (x2 ) = A0 + A1 x2 ;

1 HH !x22 ; A3 x23 − H H) 3 (c11 − c12

#2 (x2 ) = A2 + A3 x2

and 2

H H H H (c11 − c22 ) − 2c14 ] ¿ 0: H H = [c44

(2.3)

In Voigt notation (Lekhnitskii, 1963; Stroh, 1962; Voigt, 1910), the elastic tensor CH for a homogeneous trigonal material (Cowin and Mehrabadi, 1995) can be written as follows:  H  H H H c11 c12 c13 c14 0 0  H  H H H  c12  c11 c13 −c14 0 0    H  H H  c13  c c 0 0 0 13 33   H (2.4) [cij ] =  : H  cH −cH  0 c44 0 0 14  14    H H  0  0 0 0 c44 c14   0

0

0

0

H c14

1 2

H H (c11 − c12 )

The cylinder is assumed to be oriented so that its axis is coincident with the x3 -axis, while the cross-section belongs to the {x1 ; x2 } plane. As a result, the cross-section of

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the trigonal cylinder is not a plane of elastic symmetry. This produces a behavior in the displacements that is diMerent from those obtained in the same situation when the cross-sectional area lies in a plane of material symmetry (Fraldi and Cowin, 2002), u1H =

H c14 [A0 x1 + A1 x1 x2 + A2 (x12 − x22 ) + A3 x2 (x12 + x22 )] HH

H 2c14 2 x − x 2 x3 +! H − cH ) 2 (c11 12

H c14 1 [2A4 x1 +A0 x2 +2A2 x1 x2 +A3 x1 (x12 + x22 )+ A1 (3x12 −x22 )] + !x1 x3 ; H H 2  H H (c − c ) 1 u3H = − 11 H 12 −A4 x1 − A3 x1 (x12 − 3x22 ) + A0 x2 + 2A2 x1 x2 3 H 1 (2.5) − A1 (x12 − x22 ) − !x1 x2 ; 2

u2H = −

where the superscript “H” denotes all the quantities related to the anisotropic and homogeneous elastic associated solution. The stress tensor is given by   0 0 ;2   0 − ;1  (2.6) TH =    0 − ;1 0 ;2 from which it is easy to see that this is a zero-eigenvalue stress state, det T H = 0:

(2.7)

A consideration of the eigenvectors of T H shows that, at each point, the plane determined by the eigenvector associated with the zero eigenvalue is a plane orthogonal to the plane of the cross-section of the cylinder. We seek now a corresponding solution for an inhomogeneous trigonal cylinder under torsion, assuming the elasticity tensor in the form cijI = ’(x1 ; x2 ; x3 ) cijH :

(2.8)

In order to construct the elastic solution for the corresponding inhomogeneous trigonal cylindrical torsion problem using the SAS theorem, the conditions the scalar function ’ has to satisfy must be determined. By imposing the condition from the theorem between the stress tensor T H and the gradient of the scalar function ’ T H · ∇’ = 0;

∀x ∈ B

(2.9)

and then expanding it, we obtain ; 2 ’; 3

= 0;

; 1 ’; 3

= 0;

; 2 ’; 1

−

; 1 ’; 2

= 0;

(2.10)

The general solution of the system (2.10) is ’ = ’(x1 ; x2 ) = f ◦

(x1 ; x2 );

(2.11)

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where f may be an arbitrary composite function of the stress function (x1 ; x2 ). The SAS theorem ensures that, if the boundary conditions are satis5ed, the displacement and stress 5elds for the inhomogeneous trigonal cylinder will be uI = uH ;

T I = ’T H ;

(2.12)

where the superscript “I” denotes the quantities related to the inhomogeneous object. It will now be veri5ed that the displacements and stresses found in (2.12) satisfy the boundary conditions. Let B and @BL be, respectively, the region of the object occupied by both the homogeneous and inhomogeneous cylinders and the related lateral boundaries. It is easy to see that the conditions on the lateral boundary @BL are trivially satis5ed, given that T H · n = 0 ⇒ T I · n = ’T H · n = 0;

∀x ∈ @BL ;

(2.13)

where n is the outward unit normal vector from the lateral boundary. Finally, the other following weak boundary conditions at the ends of the inhomogeneous cylinder have to be satis5ed

(2.14) ’ ; 1 dA = 0; − ’( ; 1 x1 + ; 2 x2 ) dA = MtI ; ’ ; 2 dA = 0; A

A

A

where A is the cross-section of the cylinder and MtI is the total torque applied at the ends of the inhomogeneous cylinder. Consider the 5rst two equations of Eq. (2.14); these equations require the absence of shear resultants on the cross-sectional plane. From the associated homogeneous problem, it is known that  

 n dS = 0 (2.15) ∇ dA = ; 2 dA = 0; ; 1 dA = 0 ⇔ A

A

and

A

A

(

; 1 x1 +

; 2 x2 ) dA =

A

@A

∇ · x dA = −MtH :

(2.16)

From Eq. (2.13), using Eq. (2.1) and the fact that t · n = 0, it follows that ∇ ·t=0

on @BL

(2.17)

and thus the lateral boundary of the cylinder represents an equipotential surface for , as well as for any arbitrary continuous function F( ) = F ◦ . Using F( ) = F ◦ in Eq. (2.15) one obtains 



@F ∇ dA = 0; (2.18) (F ◦ ) n dS = 0 ⇒ ∇(F ◦ ) dA = @ @A A A therefore, recalling Eq. (2.11) and setting f = (@F=@ ), we obtain  

’∇ dA = 0 ⇔ ’ ; 2 dA = 0; ’ ; 1 dA = 0 : A

A

A

(2.19)

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Finally, by utilizing Eq. (2.16) and assuming that + := max {’} and , := min {’}, we x∈A

x∈A

can write the following inequality:             H      0 ¡ ,|Mt | = ,  ∇ · x dA 6  ’ ∇ · x dA 6 +  ∇ · x dA = +|MtH | A

A

A

from which it follows that

’∇ · x dA = − ’(y) MtH = −MtI ; A

(2.20)

where ’(y): , 6 ’(y) 6 + is an intermediate value of the function ’ calculated at the point y inside the domain A. We could obtain the same result (2.19) in a very straightforward way, rewriting the problem in terms of the shear stress vector T = [ 31 ; 32 ]:

(2.21)

In particular, if the following 5eld equilibrium equation is satis5ed: ∇·=0

(2.22)

and on the lateral boundary we have  · n = 0;

(2.23)

it is possible to prove that the shear resultant is always zero. In fact, we have (Romano, 2002)

 dA = ∇ · (x ⊗ ) dA − x ∇ ·  dA A

A

=

@A

(x ⊗ ) · n dS =

A

@A

x( · n) dS = 0;

that is true for all the vectors satisfying, respectively, the 5eld and the boundary conditions (2.21) and (2.22). Note that the solution to this torsion problem could also be approached in terms of the displacement 5elds, by utilizing the results of the SAS theorem. This could be done by requiring that the gradient of the scalar function ’ for the inhomogeneous cylinder is always orthogonal to the eigenvector associated with the zero stress eigenvalue in the trigonal homogeneous problem. In this situation the displacements for the homogeneous and inhomogeneous cases are identical and the stress tensor in the inhomogeneous object is given by T I =’T H . This means that if one imposed the boundary conditions in terms of displacements and not in terms of stresses—a possibility due to the uniqueness theorem—we would obtain the solution of the inhomogeneous trigonal cylinder under torsion directly. 2.1.2. Torsion of cylinders with shape intrinsic orthotropy The term shape intrinsic orthotropy has been introduced (Cowin, 1987) to describe the general situation in which the symmetry coordinate system for orthotropic symmetry of a cylinder is coincident with an axis parallel to the axis of the cylinder and with

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2221

x2

plane of stress

x1

MtI



n=k


MtI composite cylinder

x3 Fig. 4. A composite cylinder under pure torsion: note the family of the interface surfaces whose the lateral boundary is a member and the orientation of the plane of the stress.

local tangent and normal to a closed family of curves of which the lateral boundary is a member (see Fig. 4). It is possible to generalize the elastic torsion solutions for isotropic cylinders to cylinders with shape intrinsic orthotropy. It is also possible to generalize the solutions for combined torsion, pure bending and axial extension or compression of isotropic cylinders of arbitrary cross-section to cylinders of arbitrary cross-section with shape intrinsic orthotropy (Cowin, 1987). The solution to the torsion problem for cylinders with shape intrinsic orthotropy contains equations that are diMerent from the isotropic case only for Hooke’s law. In fact, in the local tangent coordinate system in which the material is orthotropic, the two shear elastic homogeneous moduli of interest H H are Gt3 and Gn3 , so that the elastic coeHcients in the Cartesian coordinate system can be expressed as follows: H H C2323 = Gt3 sin2  + Gn3 cos2 ; H H C1313 = Gt3 cos2  + Gn3 sin2 ; H H C1212 = (Gn3 − Gt3 ) sin  cos ;

where  represents the angle between the normal vector n to the generic curve of the family to which the lateral boundary belongs and the x2 -axis that lies on the plane of the cross-section of the cylinder. Then, after some algebraic manipulations, the only non-zero stresses are given in the form H H

23 = Gt3 (,; 2 + x1 );

H H

13 = Gt3 (,; 1 − x2 );

where ,(x1 ; x2 ) and  are, respectively, the warping function and a constant representing the twist per unit length of the cylinder. As a result, the displacement 5eld can be

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expressed in the usual way, that is u1H = −x2 x3 ;

u2H = x1 x3 ;

u3H = ,(x1 ; x2 ):

The case of varying the shear modulus is also considered in Cowin (1987); a solution was obtained for the torsion problem for shape intrinsic orthotropic materials when the shear modulus Gt3 was considered as a function of x1 and x2 . If Gt3 does not vary in the tangential direction, i.e. ∇Gt3 · t = 0; then it is possible to prove that the solution to the torsion problem for shape intrinsic orthotropy given above is still applicable, for any possible variation of the shear modulus in the direction normal to the curves to which the lateral boundary of the cylinder belongs. In this way a displacement 5eld analogous to that seen above is obtained, while the shear stresses will be formally represented by the stresses obtained for the homogeneous case, but where the shear modulus is now inhomogeneous in one direction. The same result is also obtained by invoking the SAS theorem. To see that this is the case note that I H Gt3 = ’(x1 ; x2 )Gt3 ;

{’(x1 ; x2 )|∇’ · t = 0};

then I ∇Gt3 · t = 0 ⇒ T H · ∇’ = 0;

where T H represents the stress tensor due to the torsion of the shape intrinsic homoH and geneous orthotropic cylinder in which the only non-zero stress components are 13 H

23 . The SAS theorem then ensures that, if the boundary conditions are satis5ed, the elastic solution for the inhomogeneous case is characterized by a displacement 5eld equal to that in the homogeneous case and a stress 5eld given by T I = ’T H . These results were obtained in another way in Cowin (1987); thus, the generalization of the SAS theorem to the composite materials has been illustrated already for the case of the cylinder under torsion where the shear modulus varies from one composite layer to another (see Fig. 4). 2.2. Axial extension and pure bending of composite cylinders with arbitrary anisotropy and generic cross-section in which the moduli do not vary along the generator Consider a homogeneous cylinder with generic cross-section and arbitrary kind of anisotropy subjected to a combined axial force and bending moment applied at its ends. The stress state for this situation is one with two zero eigenvalues   0 0 0   0  TH =  (2.24) : 0 0 H 0 0 33

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2223

The imposing of the equilibrium condition on this stress state restricts the functional dependence of the non-zero stress H H ∇ · T H = 0 ⇒ 33 = 33 (x1 ; x2 );

(2.25)

where {x1 ; x2 } is the plane of the cross-section of the cylinder. The strain components are given by E H = SH : T H

H H

33 ;

ijH = Sij33

or

(2.26)

where E H is the strain tensor and SH = CH−1 is the fourth rank compliance tensor. To satisfy compatibility it is required that ∇ × (∇ × E H ) = ∇ × [∇ × (SH : T H )] = 0;

(2.27)

where ∇×(∗)=curl(∗) which further restricts the functional dependence of the non-zero stress component, H

33 = a 0 + a 1 x1 + a 2 x2 ;

(2.28)

where ak represent constant coeHcients. These coeHcients can be easily determined by imposing the classical weak boundary conditions at the ends of the cylinder

H H H

33 dA = N;

33 x1 dA = −M2 ;

33 x2 dA = M1 ; (2.29) A

A

A

where A is the cross-section of the cylinder and N; M1 and M2 represent the total axial force and the two bending moments, respectively. Note that the solution of this anisotropic elastic problem exhibits the same stress 5eld as the corresponding isotropic problem, while the displacement 5elds are diMerent. In fact, recalling the strain–displacement relations E H = sym∇ ⊗ uH and integrating them to 5nd the displacements, 2 we have H H H H u1H = 12 S1133 a1 x12 + 21 (2a2 S1233 − a1 S2233 )x22 + S1133 (a0 + a2 x2 )x1 H H H + (2a0 S1333 − w1 )x3 − 12 a1 S3333 x32 + x2 [u0 + (2a2 S1333 − w0 )x3 ]; H H H H u2H = 12 S2233 a2 x22 + 21 (2a1 S1233 − a2 S1133 )x12 + S2233 (a0 + a1 x1 )x2 H H H + (2a0 S2333 − w2 )x3 − 12 a2 S3333 x32 + x1 [v0 + (2a1 S2333 − w0 )x3 ]; H H H u3H = S1333 a1 x12 + S2333 a2 x22 + S3333 (a0 + a1 x1 + a2 x2 )x3 + w0 x1 x2

+ w1 x1 + w2 x2 ;

(2.30)

where u0 ; v0 ; w0 ; w1 ; w2 are constants. 2

Under the hypothesis of internal connection of the solid domain, it would be convenient to utilize the CSesaro’s formula to explicitly integrate the displacement 5eld starting from an assigned admissible symmetrical second rank strain tensor (Romano, 2002).

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x2

N

composite cylinder with cylindrical anisotropic and inhomogeneous inclusions and voids

x1

cylindrical anisotropic sub-domains of the partition where the moduli do not vary along the generator

N cylindrical void with arbitrary shape cylindrical elastic fibers

x3

Fig. 5. Composite cylinder under axial forces and bending moments with arbitrary kind of cylindrical inclusions and voids.

The SAS theorem can now be used to build all the possible elastic solutions for corresponding inhomogeneous and anisotropic cylinders in which the elastic moduli vary arbitrarily with x1 and x2 and the cylinders are loaded by a combination of axial extension (or compression) and pure bending. It was observed above that the stress state for this situation is one with two zero eigenvalues. The elastic and compliance tensors for the corresponding inhomogeneous object are CI = ’(x1 ; x2 )CH ⇒ SH = ’(x1 ; x2 )SI ;

(2.31)

where ’ = ’(x1 ; x2 ) is a continuous positive scalar function depending only on x1 and x2 . This functional dependence combined with the known properties of the stress state yields T H · ∇’ = 0. By utilizing the results of the SAS theorem, it is possible to 5nd the solution for the anisotropic and inhomogeneous cylinder under axial extension and pure bending. In particular, for this inhomogeneous case, the stress and the displacement 5elds will be equal to T I = ’T H ;

uI = uH :

(2.32)

In order to generalize the solution to composite anisotropic cylinders, we consider a cylinder composed of one or more elastic anisotropic and inhomogeneous adjacent cylindrical matrices, embedded with cylindrical anisotropic and inhomogeneous elastic inclusions and voids (see Fig. 5). If, in this case, the stress solution of the anisotropic homogeneous cylinder is known and the constitutive equations of the phases and of the matrices of the heterogeneous cylinder can be described by Eq. (1.17) or Eq. (1.24), the stress tensor and the displacement 5eld in the generic point of the composite cylinder

M. Fraldi, S.C. Cowin / J. Mech. Phys. Solids 52 (2004) 2207 – 2233

2225

will be given by Tp = ’ p T H ;

up = uH ;

∀xp ∈ p ⊂ B;

where T H and uH are given by Eqs. (2.28) and (2.30), respectively. SuHciency is established by the fact that the conditions (1) about the existence of at least one zero eigenvalue of the stress tensor T H and (2) about the relation between the plane of the stresses and the interfaces present inside the composites—already established in Sections 1.3.1 and 1.3.2. above, are satis5ed. Furthermore, thanks to the one-dimensional character of the stress tensor T H , it is possible to assume any possible structural gradient inside the phases, restricted only by the hypothesis that ’p =’p (x1 ; x2 ); ∀x1 ; x2 ∈ p . 2.3. Stress-angularly and radially inhomogeneous anisotropic elastic spaces In Ting (1996a) the elastic solution was determined for problems of a composite of wedged-shaped materials subjected to a line force and/or a line dislocation, with elasticity tensors related by (m) H Cijhk = ’(m) Cijhk ;

m ∈ N;

(2.33)

(m) where Cijhk is the elasticity tensor of the mth material and the {(’(1) ; ’(2) ; : : : ; ’(m) ; : : : ; (n) ’ ), (n ¿ m; n ∈ N)} are positive scalar parameters. These problems have been generalized and included in the wider category of angularly inhomogeneous problems (Ting, 1989; Chung and Ting, 1995) in which the composite of wedges is subjected to a line dislocation and a line force at 2 = 0, as illustrated in Fig. 6a. This approach, due to Ting, is based on the observation that the surface traction on any radial plane of an elastic homogeneous space is invariant with , where {2; ; x3 } is the cylindrical reference frame. We summarize here only the salient equations from (Lekhnitskii, 1963; Ting, 1996a) in order to explain how the type of approach proposed by Ting is traceable to the original target of the present work. In the two-dimensional deformations in which an in5nite space is subjected to a line force f , dimensional analyses show that the stress must be proportional to 2−1 , in order to balance the applied force f . Therefore, the more general form of the stress components is the following:

ijH = 2−1 3H ij ():

(2.34)

H By recalling the equilibrium equations, ij; j = 0, and employing the relations suggested by Ting (1996a)

2; j = nj ;

; j = 2−1 mj ;

nj;  = mj ;

mj;  = −nj ;

where nT = (cos ; sin ; 0);

mT = (−sin ; cos ; 0);

one obtains H (3H ij mj );  = 0 ⇒ 3ij mj =

−1 gi ; 24

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plane of stress


3




x2

plane of stress


3


33

f x1



x

3

inhomogeneous anisotropic elastic half-space

n = k


θ

(a) x2 f

composite anisotropic space

n

x1

θ0 θ1

θn

1 x3 m

(b) Fig. 6. (a) Anisotropic half-space with “angularly inhomogeneous” variation of the elastic moduli. (b) Piecewise continuous anisotropic half-space in which the phases are “angularly” distributed.

where g is a constant vector. This implies that the surface traction vector on any radial plane is (Ting, 1996a) −1 tH = g: 242 In order to highlight some properties of the solutions for elastic inhomogeneous wedges, we 5rst consider the case in which the stresses are assumed to have the following form:

ijI = ’(2; ; x3 ) ijH = ’2−1 3H ij :

M. Fraldi, S.C. Cowin / J. Mech. Phys. Solids 52 (2004) 2207 – 2233

2227

Substituting these stresses in the equilibrium equations and recalling that ’;  ’;  ’; 1 = ’; 2 cos  − cos ; sin ; ’; 2 = ’; 2 sin  + 2 2 we obtain that H I H

ij; j = 0 ⇒ ( ij nj )’; 2 + ( ij mj )

’;  + ( ijH lj ) ’; 3 = 0; 2

(2.35)

where l T = (0; 0; 1). One special case of this result produces the class of radially inhomogeneous problems considered by Alshits and Kirchner (2001) and another special case produces the class of angularly inhomogeneous problems for elastic half-spaces or wedges considered by Ting (1996a). The former is obtained by restricting ’ to be a function of 2 only and the latter by restricting ’ to be a function of 2 only, thus {t2H = 0; ’ = ’(2)}

∀2 ∈ [0; +∞[; ∀ ∈ [ − 0 ; 0 ]

(2.36)

∀2 ∈ [0; +∞[; ∀ ∈ [ − 0 ; 0 ];

(2.37)

and {tH = 0; ’ = ’ ()} tH

H

t2H

H

= T · m and = T · n. respectively, where In order to illustrate a particular example of the application of the SAS theorem, consider the case in which the load f is normal to a plane that bounds the half-space and is also distributed uniformly along an in5nite straight line that is taken to be the x3 -axis. This special case is chosen because it represents a particularly useful case, although it is not the most general case possible. In the case of general anisotropic and homogeneous material, we can obtain the analytical solution for this elastic problem, with the stress components expressed in cylindrical coordinates as follows (Lekhnitskii, 1963):    f ·f H f ·f H f ·f H H H H

22 =− =− =− 322 (); 23 323 (); 33 333 (); 42 42 42 H H H

3 = 2 =  = 0;

(2.38)

where the superscript “H” indicates the quantities related to the homogeneous solution that will be here used as the SAS. From Eq. (2.38) it is easy to see that det T H = 0 and the eigenvector associated with the zero eigenvalue is coaxial with the normal to the planes  = constant. The local orthogonality condition that is required for the application of the SAS theorem then follows: T H · ∇’ = 0;

(2.39)

where ’ = ’(2; ; x3 ) is the now familiar positive scalar function. From Eqs. (2.37)– (2.39) it follows that ’ = ’(). This means that the elasticity tensor of the anisotropic and inhomogeneous elastic space has the representation I H = ’() Cijhk ; Cijhk

(2.40)

H is a constant tensor representing a completely where, due to the solution (2.38), Cijhk arbitrary kind of anisotropy. Using Eq. (2.40) and the SAS theorem it is possible

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to obtain the closed form solution of a corresponding anisotropic and inhomogeneous problem representing the case of a half-space subjected to a load distributed uniformly in a straight line. In particular, the displacement 5eld for this inhomogeneous problem will be identical to the homogeneous case (Lekhnitskii, 1963), while the stress tensor shall be equal to T I = ’T H . Then, using the extension of the SAS theorem to the piecewise discontinuous composite materials, it is easy to construct the solution for both an anisotropic half-space loaded with straight line of force and a composite space constituted by n wedges of diMerent anisotropic elastic materials. In the case of the n wedge problem each wedge occupies the region in the cylindrical coordinate system {2; ; x3 } speci5ed by Ting (1996a), m−1 6  6 m ;

m = 1; 2; : : : ; n;

0 6 0 ¡ 1 ¡ · · · ¡ n = 24 + 0 ; when the constitutive relation is given in the form (2.33), see Fig. 6b. This is the case analyzed by Ting (1996a) following a diMerent procedure. Ting relates these kinds of solutions to a class of particular elastic spaces, the angularly inhomogeneous elastic spaces (Ting, 1989; Chung and Ting, 1995) in which a similar hypothesis is made about the structure of the applicable Hooke’s law. 2.4. The problem of hollow transversely-isotropic cylinders, whose elastic moduli vary along the generator, and which are subjected to a radial pressure or assigned displacements on the inner and outer surfaces A homogeneous and transversely isotropic hollow cylinder, with pressures applied on both the external and the internal surfaces, is characterized respectively by the outer radius, Ro , and by the inner radius, Ri , see Fig. 7. The x3 -axis of a cylindrical coordinate system {2; ; x3 } coincides with the axis of the cylinder, thus, due to the loading and geometrical conditions, it is possible to assume that the deformations of the cylinder are axis-symmetric. The generalized Hooke’s law is

22



33

22 = H − 8H H − 8H ; 3 E E E3H

 =

22

33

 − 8H H − 8H ; 3 H E E E3H

33 =

( 22 +  )

33 − 8H ; 3 H E E3H

3 =

3 ; 2G3H

23 =

23 ; 2G3H

2 =

2 ; 2G H

(2.41)

where {E H ; E3H } and {G H ; G3H } are the Young and shear moduli of the homogeneous object, respectively, related to the plane of isotropy and to the planes in the radial directions, while 8H is the Poisson ratio in the plane of isotropy and 8H 3 is the Poisson coeHcient which characterizes the transverse contraction in the plane of isotropy for tension in a direction perpendicular to the plane.

M. Fraldi, S.C. Cowin / J. Mech. Phys. Solids 52 (2004) 2207 – 2233

to



Ro

Ri

ti

Ri

Ro



h0

h0

h h0

h h0

x3

ti

(a) Homogeneous hollow

(h + h 0 ) 2h

2229

x3

 (x ) 3

t o (h + h0) 2h

(b) Inhomogeneous hollow

cylinder

cylinder

Fig. 7. Homogeneous and inhomogeneous hollow cylinders under inner and outer pressures.

In the general case of axially symmetric deformation, the displacement components are u2 = u2 (2; x3 );

u = 0;

u3 = u3 (2; x3 );

(2.42)

and then the strain components become u2

22 = u2; 2 ;  = ; 33 = u3; 3 ; 2 2 23 = u2; 3 + u3; 2 ;

3 = 2 = 0:

(2.43)

By utilizing the compatibility and equilibrium equations, it is possible to prove that the stresses can be found by introducing a stress function 9 = 9(2; x3 ) as follows:

b

22 = − 9; 22 + 9; 2 + a 9; 33 ; 2 ;3

1

 = − b9; 22 + 9; 2 + a 9; 33 ; 2 ;3

c

33 = c 9; 22 + 9; 2 + a 9; 33 ; 2 ;3

1

23 = 9; 22 + 9; 2 + a 9; 33 ; (2.44) 2 ;2 where the constant a; b, and c depend on the elastic moduli and 9 must satisfy a fourth order diMerential condition, which—in the case of isotropy—requires the stress

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function to be biharmonic (Lekhnitskii, 1963). Then, for the homogeneous object with linear distributions of external pressures illustrated in Fig. 7a, the stress components can be expressed as  x3 to R2o − ti R2i (to − ti )R2o R2i 1 H ;

22 = − − h 22 R2o − R2i R2o − R2i  (to − ti )R2o R2i 1 x3 to R2o − ti R2i H ;

 + =− h 22 R2o − R2i R2o − R2i H H H H

33 = 23 = 3 = 2 = 0;

where h0 6 x3 6 h, to and ti are the coeHcients that characterize the intensities of the outer and inner pressure, respectively, while (h − h0 ) represents the total length of the hollow cylinder. The boundary conditions on the lateral surfaces are then satis5ed, i.e.,   to H H H 2 = Ro ⇒ 22 = − x3 ; 23 = 0; 2 =0 ; h   ti H H H = 0; 2 =0 ; = − x3 ; 23 2 = Ri ⇒ 22 h while both the upper and the lower ends of the cylinder (x3 = 0; x3 = h) are traction free. Consider next a similar cylinder that is inhomogeneous rather than homogeneous due to the variation of its elastic moduli that vary along the generator, i.e. ’ = ’ (x3 ), see Fig. 7b. From the equations above for the stress components in the homogeneous cylinder, it follows that det T H = 0 and thus the eigenvector associated with the zero stress eigenvalue is coaxial with the x3 -axis. Since the orthogonality condition (1.11) —required for the application of the SAS theorem—is also veri5ed, it is then possible to record the stress components in the inhomogeneous cylinder as follows:  x3 to R2o − ti R2i (to − ti )R2o R2i 1 I H

22 = ’(x3 ) 22 = −’(x3 ) ; − h 22 R2o − R2i R2o − R2i  x3 to R2o − ti R2i (to − ti )R2o R2i 1 I H ;

 = ’(x3 )  = − ’(x3 ) + h 22 R2o − R2i R2o − R2i I I I I

33 = 23 = 3 = 2 = 0:

It follows that the stresses acting on the lateral surfaces of the inhomogeneous cylinder are given by   to I I I 2 = Ro ⇒ 22 = −’(x3 ) x3 ; 23 = 0; 2 =0 ; h   t i I I I = −’(x3 ) x3 ; 23 = 0; 2 =0 : (2.45) 2 = Ri ⇒ 22 h This example demonstrates that, when the gradient of the scalar function characterizing inhomogeneity ’ is not orthogonal to the tangent plane of the surface on which the surface tractions are prescribed, it is not possible—in general—to have the same local traction boundary conditions for both the homogeneous and the inhomogeneous

M. Fraldi, S.C. Cowin / J. Mech. Phys. Solids 52 (2004) 2207 – 2233

2231

objects. In this case the SAS theorem can still be rigorously applied, the strain and the displacement 5elds in both the homogeneous and the inhomogeneous objects will still be locally identical, but the loads assigned on the surfaces of the two solids are only indirectly related. In the following two possible interesting cases are presented. Case a: Consider the case where one needs to seek the exact solution for an inhomogeneous transversely isotropic hollow cylinder under the inTuence of constant pressure applied on its inner and outer surfaces. This problem can be seen as an inverse problem, where the precise form of the loads applied on the inhomogeneous object is assigned (i.e., uniform internal and external pressures), and a SAS for a homogeneous object is known. Then—utilizing the SAS theorem—the problem is to 5nd the unknown scalar function ’. The scalar function ’ is determined by the requirement of uniform pressures on the inhomogeneous body and the boundary conditions (2.45), when the SAS for the homogeneous hollow cylinder is utilized. The result shows that the unknown scalar function ’ is hyperbolic: k ’(x3 ) = ; {x3 ∈ [h0 ; h]; k ∈ R+ }: (2.46) x3 This choice of the scalar function ’ creates a constant distribution of the pressures for the inhomogeneous hollow cylinder. The value of the constant k for which the total resultants of the outer and inner pressures (per unit length) are the same for both the homogeneous and inhomogeneous solids, is determined by the condition

h

h x3 d x 3 = ’x3 d x3 ; (2.47) h0

h0

where h0 6 x3 6 h, thus h + h0 h + h0 1 k= ; x3 ∈ [h0 ; h]: ⇒’= 2 2 x3 Case b: Due to the fact that elastic problems are often speci5ed by assigning only displacement boundary conditions (Ciarlet, 1988; Ting, 1996b, 1999; Alshits and Kirchner, 2001), a second interesting case is obtained using the SAS theorem. In this case, in the homogeneous problem, we substitute the corresponding prescribed displacement 5eld for the prescribed boundary tractions. As observed before, this converts the portion of the boundary upon which the surface tractions are prescribed to a portion of the boundary upon which the displacements are prescribed. Due to uniqueness of solution, this is always possible in a linear elastic problem. Then, with reference to the problem of Fig. 7b, it is observed that, if one replaces the pressures applied on the homogeneous hollow cylinder with the corresponding displacements, the SAS theorem may be used to construct all the solutions related to an inhomogeneous cylinder where the function ’(x3 ) is arbitrary. 2.5. Some remarks about cylindrically anisotropic inhomogeneous materials In some recent works (Ting, 1996b, 1999; Alshits and Kirchner, 2001), new solutions for cylindrically anisotropic homogeneous and inhomogeneous materials have been reported. In particular, these authors investigated the possibility of constructing exact

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solutions for cylindrically anisotropic circular tubes or bars under pressure, shearing, torsion and extension. Their analysis began from the Stroh and Lekhnitskii formalisms and generalized them for speci5c anisotropic problems in cylindrical coordinates (Ting, 1996b,1999). Other interesting results were also obtained for radially inhomogeneous elastic hollow cylinders with internal sources and under various boundary conditions (Alshits and Kirchner, 2001). We note that, by using the SAS theorem, it is possible to 5nd some other inhomogeneous solutions based the results obtained by Ting (1999) in which a local plane stress is found by assuming that the gradient of ’ is orthogonal to the stress plane and selecting appropriate classes of cylindrical anisotropy. Similarly, the SAS theorem might be utilized to recover some results obtained by Alshits and Kirchner (2001) for radially inhomogeneous elastic cylinders. 3. Conclusions The solution presented here for the torsion problem for trigonal cylinders highlights an interesting geometrical property of ’ that emerges by observing that its gradient must be coaxial to the gradient of the stress function . An analogous property is recognized when we illustrated the case of the torsion for shape intrinsic anisotropic problems (Cowin, 1987). Other particular examples of exact solutions for inhomogeneous problems are shown by Lekhnitskii (1963), for both the torsion and axis-symmetric problems in circular shafts in the case of cylindrical anisotropy. We have shown here how all these solutions can be obtained as particular applications of the SAS theorem. We have shown that the solution to the problem of an inhomogeneous cylinder possessing an arbitrary kind of anisotropy, subjected to an axial force and bending moments at the ends is also obtainable with the use of the SAS theorem. The solution to the problems of angularly inhomogeneous half-spaces and radially inhomogeneous problems is also obtainable by applying the SAS theorem in cylindrical coordinates. The fundamental property on which the SAS theorem is built is on the coaxiality of the eigenvector associated with the zero stress eigenvalues in the homogeneous problem and the gradient of the scalar function ’ associated with the inhomogeneous problem. Most of the solutions of anisotropic elastic problems recorded in the literature have this property. It is therefore reasonable to use this strategy to construct other exact solutions for inhomogeneous problems, if the solution for the associated homogeneous problem is known. The SAS theorem also appears to render the terms “shape intrinsic” and “angularly inhomogeneous” materials somewhat inaccurate and obsolete. These terms were introduced to distinguish the geometrical features of objects and the related anisotropy. They fail to describe the importance of the eigenvector associated with the zero stress eigenvalues and its signi5cant geometrical relationship with the gradient of the inhomogeneity. In fact these terms suggest solution properties that are not true, namely that the shape of the solid as well as the kind of anisotropy inTuence the possibility of 5nding the solution. The example of the cylinder, with arbitrary anisotropy and generic cross-section, under pure bending and axial extension is a particularly useful illustration of this. In that problem the type of anisotropy is arbitrary and the cross-section is

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generic and the eigenvector associated with the zero stress eigenvalue is coaxial with the gradient of the scalar function that describes the variability of the elastic moduli. References Alshits, V.I., Kirchner, O.K., 2001. Cylindrically anisotropic, radially inhomogeneous elastic materials. Proc. R. Soc. London A 457, 671–693. Barber, J.R., 1992. Elasticity. Kluwer Academic Publishers, Dordrecht, Boston, London. Chadwick, P., Smith, G.D., 1977. Foundations of the theory of surface waves in anisotropic elastic materials. Adv. Appl. Mech. 17, 303–376. Chung, M.Y., Ting, T.C.T., 1995. Line forces and dislocations in angularly inhomogeneous anisotropic piezoelectric wedges and spaces. Philos. Mag. A 71, 1335–1343. Ciarlet, P.G., 1988. Mathematical Elasticiy: Three-Dimensional Elasticity. North-Holland, Amsterdam. Cowin, S.C., 1987. Torsion of cylinders with shape intrinsic orthotropy. J. Appl. Mech. 109, 778–782. Cowin, S.C., Mehrabadi, M.M., 1995. Anisotropic symmetries of linear elasticity. Appl. Mech. Rev. 48, 247–285. Fraldi, M., Cowin, S.C., 2002. Chirality in the torsion of cylinders with trigonal symmetry. J. Elasticity 69, 121–148. Fraldi, M., Guarracino, F., 2001. On a general property of a class of homogenized porous media. Mech. Res. Commun. 28, 213–221. Gurtin, M.E., 1972. The Linear Theory of Elasticity, Handbbuch der Physik. Springer, Berlin. Lekhnitskii, S.G., 1963. Theory of Elasticity of an Anisotropic Elastic Body. Holden-Day, San Francisco. Lions, J.L., 1985. Les MSethodes de l’HomogSenSeisation: ThSeorie et Applications en Physique, Eyrolles Ed. Saint Germain, Paris. Maugin, G.A., 1993. Material Inhomogeneities in Elasticity. Chapman & Hall, London. Muskhelishvili, N.I., 1953. Some Basic Problems of the Mathematical Theory of Elasticity. Transl. by J.R.M. Radok, Noord-HoM, Groningen. Nemat-Nasser, S., Hori, M., 1993. Micromechanics: Overall Properties of Heterogeneous Materials. North-Holland, Amsterdam. Romano, G., 2002. Scienza delle Costruzioni—Tomo I—Cinematica ed Equilibrio, Hevelius Ed. Benevento, Italy. Stroh, A.N., 1962. Steady state problems in anisotropic elasticity. J. Math. Phys. 41, 77–103. Ting, T.C.T., 1989. Line forces and dislocations in angularly inhomogeneous anisotropic elastic wedges and spaces. Q. Appl. Math. 47, 123–128. Ting, T.C.T., 1996a. Anisotropic Elasticity—Theory and Applications. Oxford University Press, New York, Oxford. Ting, T.C.T., 1996b. New solutions to pressuring, shearing, torsion and extension of a cylindrically anisotropic elastic tube or bar. Proc. R. Soc. London A 455, 3527–3542. Ting, T.C.T., 1999. Pressuring, shearing, torsion and extension of a circular tube or bar of cylindrically anisotropic material. Proc. R. Soc. London A 452, 2397–2421. Voigt, W., 1886. Ueber die Elasticitatsverhaltnisse cylindrisch aufgebauter Kopper, Nachrichten v. D. Konigl. Gesellschaft der Wissenschaften und der Georg-Augustin Universitat zu Gottingen, n◦ 16. Voigt, W., 1910. Lehrbuch der Kristallphysik. B.G. Teubner (Leipzig and Berlin), p. 560.