- Email: [email protected]
A simple approach to the problem of defect localization in an elastic body
International Journal of Non-Linear Mechanics 36 (2001) 515}521
A simple approach to the problem of defect localization in an elastic body Paolo Cermelli*, Franco Pastrone Dipartimento di Matematica, Universita% di Torino, Via Carlo Alberto 10, 10123 Torino, Italy Received 17 July 2000; accepted 21 July 2000
Abstract We discuss a simple approach to the problem of the localization of screw dislocations in an elastic cylinder, given measurements of the strain at the boundary. We work in the context of anti-plane shear and linear elasticity, in order to reduce the problem to two dimensions. Since in the presence of dislocations the displacement is multi-valued, we formulate and solve the problem in terms of stress functions and, equivalently, in terms of meromorphic functions with assigned boundary data on the unit disk in the complex plane. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Screw dislocations; Inverse problems; Localization of defects
1. Introduction Dislocations are by far the most common imperfections in crystalline solids. Their presence dramatically a!ects the mechanical behavior of the material: two outstanding examples being dislocation motion, which is the basic microscopic mechanism underlying plastic #ow, and the modi"cation of the dispersion relation for sound waves (see, e.g., [1]). But the role of dislocations in other, complex phenomena is also important: since stress is generally high in a neighborhood of these defects, they a!ect the nucleation of new-phase particles and their spatial localization in alloys, the pinning of vortices in type II superconductors, and they can also play the role of channels for atomic
* Corresponding author. E-mail address: [email protected] (P. Cermelli).
di!usion in general, or for charge transport in semiconductors. On the other hand, the spatial distribution of dislocations inside a specimen is not easily accessible by external measurements, and may only be determined quantitatively by its indirect e!ect on the broadening of di!racted X-rays peaks (TEM observations do not yield quantitative results). Thus, it would be interesting to have at hand a method of locating the dislocations by measurements of the displacement "eld at the boundary of the body. A number of techniques to solve this kind of inverse problems have been developed to localize faults in structures and trusses [2]. For instance, a possible approach is to consider a defect as a center of force, and use Betti's theorem to correlate the actual displacement at the boundary to the "ctictious force at the defect and to a control displacement of the body in the absence of the defect. The position of the fault and the intensity of the associated force may then be determined by
0020-7462/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 0 8 1 - 0
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P. Cermelli, F. Pastrone / International Journal of Non-Linear Mechanics 36 (2001) 515}521
measurements of the displacement at the boundary only. In this work we investigate whether a similar approach is feasible in order to locate dislocation defects, in the context of linear elasticity. To keep the presentation simple, we work in the framework of anti-plane shear deformations of a cylindrical body, a natural simpli"ed context to study screw dislocations [3]. We assume that the body is in equilibrium under zero shear stress at the boundary, and that the shear strain may be measured at the boundary. Since the displacement "eld is multi-valued in the presence of dislocations, and no localized force is associated to them [3], the technique based on Betti's theorem cannot be used directly. We thus develop a dual formulation of the problem in terms of stress functions, which are well-de"ned harmonic functions away from the defects [5]. Since singularities of the stress functions do correspond to localized (dual) forces, the above techniques may be applied. Indeed, in this framework Betti's theorem reduces to Green's identity [6], and a standard relation in harmonic function theory yields an explicit formula for the position of a single dislocation in terms of the shear strain at the boundary. More explicit results, in the presence of many dislocations, may be obtained in the complex-variable formulation, since the problem is two-dimensional. We obtain a formula relating the position of a "xed number of dislocations to the Fourier coe$cients of the boundary strain. This relation allows, as a bonus, to give an explicit characterization of the coe$cients of the data in the presence of a single dislocation, of a symmetric dislocation dipole, or simple dislocation arrays. A deeper investigation of this issue, not developed here, would give simple criteria to decide the form of the dislocation distribution immediately in terms of the structure of the Fourier expansion of the strain at the boundary. The formulas we obtain below allow us to determine the location of the dislocations given boundary measurements of the strain. This strain, The forces associated to dislocations are purely con"gurational, see [3] and Gurtin's monography on the role of con"gurational forces in continuum physics [4].
however, is measured with respect to a stress-free (and thus defect-free) reference con"guration, and is thus really a plastic strain, which is hardly available to direct measurements. Hence our results should be viewed as general relations between the defect location and the plastic strain at the boundary, rather than as useful criteria for experimental procedures. On the other hand, any localization method based on measurements of the incremental displacements, which would be experimentally feasible, would fail here as a consequence of the linearity of the model, a fact pointing out the strong intrinsic limitations of linear elasticity when defects are present. 2. Screw dislocations In this paper we restrict attention to anti-plane shear, a special class of deformations of a cylindershaped body: let (x, y, z) be Cartesian coordinates in 1, with basis (i, j, k), and ) a regular simply connected region in the (x, y)-plane, and consider the cylinder with vertical axis );1: an anti-plane shear deformation of );1 is an invertible map of the form (x, y, z) C (x, y, z#u(x, y)),
(1)
with u, the shear displacement, a real function on ). Since the displacement gradient corresponding to anti-plane shear is k u, the Cauchy stress T for a linear isotropic material has the form T"k(k u# u k),
(2)
with k the shear modulus. Thus div T"k(*u)k, and the equilibrium problem for anti-plane shear of a linear isotropic material may be formulated in terms of u"u(x, y) only, a scalar function satisfying *u"0 in ), u"u in *) , S *u "p in *) , N *n
(3)
Here and * are the two-dimensional gradient and laplacian.
P. Cermelli, F. Pastrone / International Journal of Non-Linear Mechanics 36 (2001) 515}521
with *) 6 *) "*), p an assigned shear at the S N boundary and *u/*n"n ) u the normal derivative of u, with n (a plane vector) the outward unit normal to *). Anti-plane shear provides a natural simpli"ed setting for the study of screw dislocations; these are multi-valued displacement "elds, with singlevalued gradient, singular at the so-called dislocation lines (the original de"nition goes back to Volterra [7]). For anti-plane shear, the dislocation lines are parallel to the axis of the cylinder: their traces on the (x, y)-plane are points, which are singularities for the displacement gradient (a plane vector "eld). Consistently (see [3]), we characterize a system of screw dislocations at +z ,2, z ,L) by a two , dimensional vector "eld g"g(z), z3)!+z ,2, z ,, , such that
(4)
curl g"0 in )!+z ,2, z ,, (5) , where curl g"* g !* g for g"g i#g j. W V Here, g plays the role of an incompatible singular strain or stress "eld on ). The Burgers moduli of the system of dislocations are the real numbers de"ned by
b " I
g ' t ds, k"1,2, N
(6)
/" for suitably small e'0 (D (z ) is the e-disk C I centered at z , t the unit tangent vector to *D (z ) I C I and ds is the arc element on *D (z )). Recall that C I there exists a displacement "eld u on )!+z ,2, z , , such that g" u if and only if all the b are zero. I If we restrict attention to the homogeneous Neumann problem, and choose p "0 and k"1, the equilibrium problem for a system of screw dislocations [5] is z C I
E Given a set of Burgers moduli b and points z in I I ), k"1,2, N, "nd a vector "eld g on )!+z ,2, z , which is a solution of the bound , ary-value problem div g"0 in )!+z ,2, z , , g ' n"0 on *)
517
and curl g"0 in )!+z ,2, z ,, ,
(8)
together with the conditions
g ' n ds"0, k"1,2, N,
(9)
g ' t ds"b , k"1,2, N, I /"C zI for a suitable e'0.
(10)
/"C zI
and
Notice that (8) and (10) may be replaced by , (11) curl g"! b dzI I I in the sense of distributions, with dt (x)"d(x!z ) I for x"(x, y). The relations (9) may be motivated upon noting that, by (7),
, g ' n ds"0, (12) I /"C zI but this does not exclude the possibility of nonvanishing forces localized at a single dislocation, a possibility we indeed rule out by means of the N conditions (9). 2.1. A dual formulation Consider "rst a usual linear problem without singularities [5]. Let g be a smooth plane vector "eld on ): if curl g"0, let u be a function on ) such that g" u. This is the displacement associated to g. On the other hand, if div g"0, write w for the function on ) such that g" w;k; we refer to w as the stress function associated to g. The following identities hold: g" u 0 curl g"0 0 div g"*u,
(7)
g" w;k 0 div g"0 0 curl g"*w.
(13)
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P. Cermelli, F. Pastrone / International Journal of Non-Linear Mechanics 36 (2001) 515}521
Moreover, denoting by t"k;n and n the unit tangent and (outward) unit normal to *), we have g ' n" u ' n" w ' t and g ' t" u ' t"! w ' n, i.e., *u dw g ' n" " *n ds
2.2. Complex variable formulation First of all, notice that, identifying 1 to the complex plane by letting z"x#iy, denoting by z the complex number associated to z , and introI I ducing the complex function on )!+z ,2, z , , g(z, z )"g !ig , (17) where g"g i#g j, the conditions div g"0 and curl g"0 coincide with the Cauchy}Riemann equations *M g"0 for g, so that (7) and (8) imply that
and du *w g ' t" "! , ds *n
g"g(z), (14)
on *). Hence Neumann data for u correspond to Dirichlet data for w, and conversely. Contrary to the displacement "eld, the stress function is globally de"ned also in the presence of dislocations: let g be given on )!+z ,2, z ,, and , assume that div g"0 on )!+z ,2, z ,, with , C zI g ' n ds"0 for k"1,2, N. Then PoincareH 's /" Lemma (see e.g., [8]) ensures that there exists a function w on )!+z ,2, z , such that w;k"g, , and by (13) and (14) relations (7) , (8) and (10) are equivalent to the equilibrium problem E Given a set of Burgers moduli b and points I z in ), k"1,2, N, "nd a function w on )! I +z ,2, z , which is a solution of the boundary , value problem *w"0 in )!+z ,2, z ,, , w"0 on *),
and g is analytic in )!+z ,2, z ,. Moreover, by (9) , and (10), and the identity g ' t ds#ig ' n ds"g dz, we have
g dz"b , I /"C XI so that the residue of g at z is b /2pi. I I We assume from now on that
(15) /"C zI Notice that (15) may be equivalently rewritten as
so that g ' n"Re(e 0g(e 0)) on *), with Re(z) the real part of z, and 03[0,2p). The equilibrium problem for a system of screw dislocations may be rephrased in terms of g: E Given a set of Burgers moduli b and points z in I I ) for k"1,2, N, "nd a meromorphic function g on ) such that
in the sense of distributions. Thus, the stress function is a linear combination of Green's functions for the Laplacian.
(20)
and g has simple poles in +z ,2, z , with given , residues +b /2pi,2, b /2pi,. , The solution to this problem for the unit disk is well known, and is
(16)
(19)
) is the unit disk,
Re(e 0g(e 0))"0, 03[0,2p),
*w ds"!b , k"1,2, N. I *n
, *w"! b dzI I I w"0 on *)
(18)
, b 1 z I g(z)" I # (21) 2pi z!z 1!z z I I I which may be obtained, for instance, by the method of images (cf. e.g., [1]). Notice that the equilibrium problem does not have a unique solution, unless some extremality or regularity condition is imposed (for a complete
P. Cermelli, F. Pastrone / International Journal of Non-Linear Mechanics 36 (2001) 515}521
discussion of this issue, see [5]); here, we assume that such conditions have been tacitly imposed on the equilibrium problems listed above, in order to guarantee uniqueness.
3. Localization of defects The problem of the localization of defects is a classical inverse problem: assign traction boundary data for the body, measure the displacement at the boundary, and try to determine the location of the dislocations and their Burgers moduli. A "rst approach to this problem might be to impose an incremental shear stress p to a dis located cylinder and measure the incremental displacement at the boundary. However, as a consequence of linearity, any incremental shear stress just superposes to the prestressed state, and the resulting incremental displacement is insensitive to the location of the defects. In fact, the solution of (7)}(10) with g ' n"p O0, has the form g"g # u, with g the solution of (7)}(10) such that g ' n"0, and u the solution of the problem *u"0 in ), *u "p *n
on *).
Moreover, since by (9) the dislocations have no associated localized forces, the method based on Betti's theorem (see e.g., [2]) cannot be used directly. Since incremental displacements do not help, let us examine another class of boundary data which allow to determine the location of the defects. Note that, in our simpli"ed context, g may be identi"ed with the strain, since in the absence of dislocations g" u. Thus g ' t is the tangential strain at the boundary, and, in the absence of defects, is uniquely determined by the displacement at the boundary, and conversely. This argument suggests that, also in the presence of dislocations, we choose g ' t on *)
(22)
as the basic boundary measurement. Since g ' n is assigned, being the boundary traction, then g may be assumed to be known on *): the inverse problem we will try to solve in this paper is thus
519
P1 Given a continuous vector "eld g on *) such that g ' n"0, does there exist +z ,2, z , in ), , +b ,2, b , and a "eld g in )!+z ,2, z , sat , , isfying the equilibrium problem (7)}(10), such that g"g on *)? P2 If the answer to P1 is positive, determine the location and the Burgers moduli of the dislocations in ). Recalling that, by (14), Neumann data for u correspond to Dirichlet data for w, and conversely, the inverse problem may be restated equivalently in terms of the stress function w: P 1
P 2
Given a continuous function q on *), does there exist +z ,2, z , in ), +b ,2, b , and , , a function w in )!+z ,2, z , satisfying the , equilibrium problem (15), such that *w/*n"q on *)? If the answer to P1 is positive, determine the location and the Burgers moduli of the dislocations in ).
Notice that g ' t and g ' n correspond in this context to u and *u/*n or to *w/*n and w. Thus, to assign g at the boundary is equivalent to assign both Dirichlet and Neumann data on the whole *), and it is well known in the theory of elliptic equations that they may not be assigned arbitrarily, for instance, in order to obtain a smooth solution of the Laplace equation in ). Here we emphasize that no a priori compatibility requirement is imposed between the tangential and normal components of g on *), just to be able to deal with the presence of singularities in ). We "nally reformulate the inverse problem in terms of complex functions: P 1 Given a continuous function g on *) such that Re(e 0g(e 0))"0, does there exist a meromorphic function g in ) with a "nite number of simple poles in ), such that g"g on *)? P 2 In case the answer to P 1 is positive, determine the location of the poles of g and the corresponding residues. It may be shown that, if there exists a solution to P 2, this is unique by consequence of the Analytic Continuation Principle (see [8] or [9]). Moreover,
520
P. Cermelli, F. Pastrone / International Journal of Non-Linear Mechanics 36 (2001) 515}521
the condition that the poles are simple is related to the minimum-energy principle of Buzano and Cermelli [5]. 3.1. Solution in terms of the stress function We do not attempt to solve problems P1, P 1, and P 1 here, but rather assume that a solution indeed exists, with a known number of singularities in ), and determine the location and Burgers moduli of the defects, solving P2, P 2, and P 2. Our "rst approach is based on the stress functions. Assume thus that, given g on *), a vector "eld g on )!+z ,2, z ,, solution of P1, does exist. , This is equivalent to assume that, given q on *), a function w on )!+z ,2, z , solution of P 1 does , exist. Since, by (15), the stress function w associated to g is the sum of Green's functions for the Laplacian, Green's identity yields, for any harmonic function u on ), continuous on ) M ,
, *w *u b u(z )"! u !w ds, I *n *n / I and since w"0 and *w/*n"!g ' t on *), we have
, b u(z )" ug ' t ds I / I for any u harmonic in ).
(23)
For instance, if we know by some means that a single dislocation is present in ), the above formula allows to determine its location by
(g ' t)x ds, / recalling that bz "
(24)
g ' t ds. (25) / If there are N dislocation in ), it is necessary to apply (23) 3N times to di!erent harmonic functions u, in order to determine the 2N coordinates of z , I and their Burgers moduli. Notice that the fact that (23) holds for every harmonic function u may be interpreted as a necessary condition for P 1 to have a solution.
b"
3.2. Solution in terms of the Fourier coezcients of the data Our second approach is in terms of the complex function g. As in the preceding section, assume that, given a complex-valued function g on *) as in P 1, a meromorphic function g does exist in ) with a given number N of simple poles, located at +z ,2, z ,. Expanding g in Fourier series we ob , tain > g (0)" c e L0, (26) L L\ with 03[0, 2p). The solution g to P 1 (which is assumed to exist) is given explicitly by (21), and may be expanded in Laurent series in a neighborhood of *):
, b > zL\ > g(z)" I I # z L>zL . (27) I zL 2pi I L L for max "z "("z"(min 1/"z ". EvaI 2 , I I 2 , I luating this series on *), imposing that g"g and using (26), we "nd that
b b z\L>#2# , z\L>, n)!1, 2pi , 2pi c " L b b z L>#2# , z L>, n*0. 2pi 2pi , (28)
Hence, given the Fourier coe$cients c , the comL plex numbers corresponding to the location of the dislocations are solutions the in"nite set of algebraic equations (28). On the other hand, if we know that only N poles are present, it is of course su$cient to solve a "nite number of such equations. In the hypothesis that the function g has N simple poles, it is su$cient to solve a "nite number of such equations. Notice that only the coe$cients corresponding to either positive or negative indices are needed to solve the problem since, by (26) > e 0g(e 0 )" (c e L>0#c e\ L>0 ), L \L> L\ and, taking the real part of this expression, we see that the boundary condition Re(g(e 0 )e 0 )"0
P. Cermelli, F. Pastrone / International Journal of Non-Linear Mechanics 36 (2001) 515}521
implies that c #c "0, n*!1, L \L> i.e., c
\
is imaginary, c "!c , \
(29)
3.3. Examples Inspection of the fundamental relation (28) allows to give explicit formulas for the location and Burgers moduli of the dislocations, at least for some simple cases. These in turn yields su$cient conditions for the existence problem P 1 to have a solution. (i) A single dislocation. Assume that the coe$cients c satisfy the iterative relations L c \L> "j for n*1 and Re(c )"0, \ c \L (30) with j a "xed complex number such that "j"(1. Then there exists a single dislocation in ), located at the point c (31) z "j" \ with b "2pic , \ c \ a relation which follows upon noting that, in the presence of a single dislocation, then (28) becomes b c " zL\ for n*1. \L 2pi (ii) A symmetric dislocation dipole: Assume that the odd terms in the Fourier expansion of g vanish, and the even terms satisfy an iterative relation of the form (30), i.e., c "0 for n*0, \L> c \L> "j for n*1 c \L and
c Im \ "0, c \
with j a "xed complex number such that "j"(1. Then, letting (c )/(c )" !k, with \ \ k a real number, then P 1 admits a solution with two poles, located at z "$j, with Burgers moduli
and so on.
(32)
521
(33)
b "$pk, (34) relations which follow upon noting that, in the presence of a symmetric dislocation dipole (z #z "0, b #b "0), then (28) becomes b c " zL\, n*1. \L pi
Acknowledgements We acknowledge valuable discussions with Ernesto Buzano. This work was completed under the support of the Italian MURST project `Modelli Matematici per la Scienza dei Materialia.
References [1] J.P. Hirth, J. Lothe, Theory of Dislocations, Second edition, McGraw-Hill, New York, 1982. [2] H.G. Natke, Fault detection and localization in structures: a discussion, Mech. Systems Signal Process. 5 (1991) 445}456. [3] P. Cermelli, M.E. Gurtin, The motion of screw dislocations in materials undergoing anti-plane shear: glide, crossslip, "ne cross-slip, Arch. Rat. Mech. Anal. 148 (1) (1999) 3}52. [4] M.E. Gurtin, Con"gurational Forces as Basic Concepts of Continuum Physics, Springer, Berlin, 1999. [5] E. Buzano, P. Cermelli, A singular variational problem in dislocation theory. Z.A.M.P. 51(6) (2000). [6] S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer, Berlin, 1992. [7] V. Volterra, Sur l'eH quilibre del corps eH lastiques multiplement connexes, Ann. Ec. Norm. 24 (1907) 401}517. [8] J.B. Conway, Functions of One Complex Variable I and II, Springer, Berlin, 1978 and 1995. [9] M. Heins, Complex Function Theory, Academic Press, New York, 1968.
A simple approach to the problem of defect localization in an elastic body Paolo Cermelli*, Franco Pastrone Dipartimento di Matematica, Universita% di Torino, Via Carlo Alberto 10, 10123 Torino, Italy Received 17 July 2000; accepted 21 July 2000
Abstract We discuss a simple approach to the problem of the localization of screw dislocations in an elastic cylinder, given measurements of the strain at the boundary. We work in the context of anti-plane shear and linear elasticity, in order to reduce the problem to two dimensions. Since in the presence of dislocations the displacement is multi-valued, we formulate and solve the problem in terms of stress functions and, equivalently, in terms of meromorphic functions with assigned boundary data on the unit disk in the complex plane. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Screw dislocations; Inverse problems; Localization of defects
1. Introduction Dislocations are by far the most common imperfections in crystalline solids. Their presence dramatically a!ects the mechanical behavior of the material: two outstanding examples being dislocation motion, which is the basic microscopic mechanism underlying plastic #ow, and the modi"cation of the dispersion relation for sound waves (see, e.g., [1]). But the role of dislocations in other, complex phenomena is also important: since stress is generally high in a neighborhood of these defects, they a!ect the nucleation of new-phase particles and their spatial localization in alloys, the pinning of vortices in type II superconductors, and they can also play the role of channels for atomic
* Corresponding author. E-mail address: [email protected] (P. Cermelli).
di!usion in general, or for charge transport in semiconductors. On the other hand, the spatial distribution of dislocations inside a specimen is not easily accessible by external measurements, and may only be determined quantitatively by its indirect e!ect on the broadening of di!racted X-rays peaks (TEM observations do not yield quantitative results). Thus, it would be interesting to have at hand a method of locating the dislocations by measurements of the displacement "eld at the boundary of the body. A number of techniques to solve this kind of inverse problems have been developed to localize faults in structures and trusses [2]. For instance, a possible approach is to consider a defect as a center of force, and use Betti's theorem to correlate the actual displacement at the boundary to the "ctictious force at the defect and to a control displacement of the body in the absence of the defect. The position of the fault and the intensity of the associated force may then be determined by
0020-7462/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 0 0 ) 0 0 0 8 1 - 0
516
P. Cermelli, F. Pastrone / International Journal of Non-Linear Mechanics 36 (2001) 515}521
measurements of the displacement at the boundary only. In this work we investigate whether a similar approach is feasible in order to locate dislocation defects, in the context of linear elasticity. To keep the presentation simple, we work in the framework of anti-plane shear deformations of a cylindrical body, a natural simpli"ed context to study screw dislocations [3]. We assume that the body is in equilibrium under zero shear stress at the boundary, and that the shear strain may be measured at the boundary. Since the displacement "eld is multi-valued in the presence of dislocations, and no localized force is associated to them [3], the technique based on Betti's theorem cannot be used directly. We thus develop a dual formulation of the problem in terms of stress functions, which are well-de"ned harmonic functions away from the defects [5]. Since singularities of the stress functions do correspond to localized (dual) forces, the above techniques may be applied. Indeed, in this framework Betti's theorem reduces to Green's identity [6], and a standard relation in harmonic function theory yields an explicit formula for the position of a single dislocation in terms of the shear strain at the boundary. More explicit results, in the presence of many dislocations, may be obtained in the complex-variable formulation, since the problem is two-dimensional. We obtain a formula relating the position of a "xed number of dislocations to the Fourier coe$cients of the boundary strain. This relation allows, as a bonus, to give an explicit characterization of the coe$cients of the data in the presence of a single dislocation, of a symmetric dislocation dipole, or simple dislocation arrays. A deeper investigation of this issue, not developed here, would give simple criteria to decide the form of the dislocation distribution immediately in terms of the structure of the Fourier expansion of the strain at the boundary. The formulas we obtain below allow us to determine the location of the dislocations given boundary measurements of the strain. This strain, The forces associated to dislocations are purely con"gurational, see [3] and Gurtin's monography on the role of con"gurational forces in continuum physics [4].
however, is measured with respect to a stress-free (and thus defect-free) reference con"guration, and is thus really a plastic strain, which is hardly available to direct measurements. Hence our results should be viewed as general relations between the defect location and the plastic strain at the boundary, rather than as useful criteria for experimental procedures. On the other hand, any localization method based on measurements of the incremental displacements, which would be experimentally feasible, would fail here as a consequence of the linearity of the model, a fact pointing out the strong intrinsic limitations of linear elasticity when defects are present. 2. Screw dislocations In this paper we restrict attention to anti-plane shear, a special class of deformations of a cylindershaped body: let (x, y, z) be Cartesian coordinates in 1, with basis (i, j, k), and ) a regular simply connected region in the (x, y)-plane, and consider the cylinder with vertical axis );1: an anti-plane shear deformation of );1 is an invertible map of the form (x, y, z) C (x, y, z#u(x, y)),
(1)
with u, the shear displacement, a real function on ). Since the displacement gradient corresponding to anti-plane shear is k u, the Cauchy stress T for a linear isotropic material has the form T"k(k u# u k),
(2)
with k the shear modulus. Thus div T"k(*u)k, and the equilibrium problem for anti-plane shear of a linear isotropic material may be formulated in terms of u"u(x, y) only, a scalar function satisfying *u"0 in ), u"u in *) , S *u "p in *) , N *n
(3)
Here and * are the two-dimensional gradient and laplacian.
P. Cermelli, F. Pastrone / International Journal of Non-Linear Mechanics 36 (2001) 515}521
with *) 6 *) "*), p an assigned shear at the S N boundary and *u/*n"n ) u the normal derivative of u, with n (a plane vector) the outward unit normal to *). Anti-plane shear provides a natural simpli"ed setting for the study of screw dislocations; these are multi-valued displacement "elds, with singlevalued gradient, singular at the so-called dislocation lines (the original de"nition goes back to Volterra [7]). For anti-plane shear, the dislocation lines are parallel to the axis of the cylinder: their traces on the (x, y)-plane are points, which are singularities for the displacement gradient (a plane vector "eld). Consistently (see [3]), we characterize a system of screw dislocations at +z ,2, z ,L) by a two , dimensional vector "eld g"g(z), z3)!+z ,2, z ,, , such that
(4)
curl g"0 in )!+z ,2, z ,, (5) , where curl g"* g !* g for g"g i#g j. W V Here, g plays the role of an incompatible singular strain or stress "eld on ). The Burgers moduli of the system of dislocations are the real numbers de"ned by
b " I
g ' t ds, k"1,2, N
(6)
/" for suitably small e'0 (D (z ) is the e-disk C I centered at z , t the unit tangent vector to *D (z ) I C I and ds is the arc element on *D (z )). Recall that C I there exists a displacement "eld u on )!+z ,2, z , , such that g" u if and only if all the b are zero. I If we restrict attention to the homogeneous Neumann problem, and choose p "0 and k"1, the equilibrium problem for a system of screw dislocations [5] is z C I
E Given a set of Burgers moduli b and points z in I I ), k"1,2, N, "nd a vector "eld g on )!+z ,2, z , which is a solution of the bound , ary-value problem div g"0 in )!+z ,2, z , , g ' n"0 on *)
517
and curl g"0 in )!+z ,2, z ,, ,
(8)
together with the conditions
g ' n ds"0, k"1,2, N,
(9)
g ' t ds"b , k"1,2, N, I /"C zI for a suitable e'0.
(10)
/"C zI
and
Notice that (8) and (10) may be replaced by , (11) curl g"! b dzI I I in the sense of distributions, with dt (x)"d(x!z ) I for x"(x, y). The relations (9) may be motivated upon noting that, by (7),
, g ' n ds"0, (12) I /"C zI but this does not exclude the possibility of nonvanishing forces localized at a single dislocation, a possibility we indeed rule out by means of the N conditions (9). 2.1. A dual formulation Consider "rst a usual linear problem without singularities [5]. Let g be a smooth plane vector "eld on ): if curl g"0, let u be a function on ) such that g" u. This is the displacement associated to g. On the other hand, if div g"0, write w for the function on ) such that g" w;k; we refer to w as the stress function associated to g. The following identities hold: g" u 0 curl g"0 0 div g"*u,
(7)
g" w;k 0 div g"0 0 curl g"*w.
(13)
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Moreover, denoting by t"k;n and n the unit tangent and (outward) unit normal to *), we have g ' n" u ' n" w ' t and g ' t" u ' t"! w ' n, i.e., *u dw g ' n" " *n ds
2.2. Complex variable formulation First of all, notice that, identifying 1 to the complex plane by letting z"x#iy, denoting by z the complex number associated to z , and introI I ducing the complex function on )!+z ,2, z , , g(z, z )"g !ig , (17) where g"g i#g j, the conditions div g"0 and curl g"0 coincide with the Cauchy}Riemann equations *M g"0 for g, so that (7) and (8) imply that
and du *w g ' t" "! , ds *n
g"g(z), (14)
on *). Hence Neumann data for u correspond to Dirichlet data for w, and conversely. Contrary to the displacement "eld, the stress function is globally de"ned also in the presence of dislocations: let g be given on )!+z ,2, z ,, and , assume that div g"0 on )!+z ,2, z ,, with , C zI g ' n ds"0 for k"1,2, N. Then PoincareH 's /" Lemma (see e.g., [8]) ensures that there exists a function w on )!+z ,2, z , such that w;k"g, , and by (13) and (14) relations (7) , (8) and (10) are equivalent to the equilibrium problem E Given a set of Burgers moduli b and points I z in ), k"1,2, N, "nd a function w on )! I +z ,2, z , which is a solution of the boundary , value problem *w"0 in )!+z ,2, z ,, , w"0 on *),
and g is analytic in )!+z ,2, z ,. Moreover, by (9) , and (10), and the identity g ' t ds#ig ' n ds"g dz, we have
g dz"b , I /"C XI so that the residue of g at z is b /2pi. I I We assume from now on that
(15) /"C zI Notice that (15) may be equivalently rewritten as
so that g ' n"Re(e 0g(e 0)) on *), with Re(z) the real part of z, and 03[0,2p). The equilibrium problem for a system of screw dislocations may be rephrased in terms of g: E Given a set of Burgers moduli b and points z in I I ) for k"1,2, N, "nd a meromorphic function g on ) such that
in the sense of distributions. Thus, the stress function is a linear combination of Green's functions for the Laplacian.
(20)
and g has simple poles in +z ,2, z , with given , residues +b /2pi,2, b /2pi,. , The solution to this problem for the unit disk is well known, and is
(16)
(19)
) is the unit disk,
Re(e 0g(e 0))"0, 03[0,2p),
*w ds"!b , k"1,2, N. I *n
, *w"! b dzI I I w"0 on *)
(18)
, b 1 z I g(z)" I # (21) 2pi z!z 1!z z I I I which may be obtained, for instance, by the method of images (cf. e.g., [1]). Notice that the equilibrium problem does not have a unique solution, unless some extremality or regularity condition is imposed (for a complete
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discussion of this issue, see [5]); here, we assume that such conditions have been tacitly imposed on the equilibrium problems listed above, in order to guarantee uniqueness.
3. Localization of defects The problem of the localization of defects is a classical inverse problem: assign traction boundary data for the body, measure the displacement at the boundary, and try to determine the location of the dislocations and their Burgers moduli. A "rst approach to this problem might be to impose an incremental shear stress p to a dis located cylinder and measure the incremental displacement at the boundary. However, as a consequence of linearity, any incremental shear stress just superposes to the prestressed state, and the resulting incremental displacement is insensitive to the location of the defects. In fact, the solution of (7)}(10) with g ' n"p O0, has the form g"g # u, with g the solution of (7)}(10) such that g ' n"0, and u the solution of the problem *u"0 in ), *u "p *n
on *).
Moreover, since by (9) the dislocations have no associated localized forces, the method based on Betti's theorem (see e.g., [2]) cannot be used directly. Since incremental displacements do not help, let us examine another class of boundary data which allow to determine the location of the defects. Note that, in our simpli"ed context, g may be identi"ed with the strain, since in the absence of dislocations g" u. Thus g ' t is the tangential strain at the boundary, and, in the absence of defects, is uniquely determined by the displacement at the boundary, and conversely. This argument suggests that, also in the presence of dislocations, we choose g ' t on *)
(22)
as the basic boundary measurement. Since g ' n is assigned, being the boundary traction, then g may be assumed to be known on *): the inverse problem we will try to solve in this paper is thus
519
P1 Given a continuous vector "eld g on *) such that g ' n"0, does there exist +z ,2, z , in ), , +b ,2, b , and a "eld g in )!+z ,2, z , sat , , isfying the equilibrium problem (7)}(10), such that g"g on *)? P2 If the answer to P1 is positive, determine the location and the Burgers moduli of the dislocations in ). Recalling that, by (14), Neumann data for u correspond to Dirichlet data for w, and conversely, the inverse problem may be restated equivalently in terms of the stress function w: P 1
P 2
Given a continuous function q on *), does there exist +z ,2, z , in ), +b ,2, b , and , , a function w in )!+z ,2, z , satisfying the , equilibrium problem (15), such that *w/*n"q on *)? If the answer to P1 is positive, determine the location and the Burgers moduli of the dislocations in ).
Notice that g ' t and g ' n correspond in this context to u and *u/*n or to *w/*n and w. Thus, to assign g at the boundary is equivalent to assign both Dirichlet and Neumann data on the whole *), and it is well known in the theory of elliptic equations that they may not be assigned arbitrarily, for instance, in order to obtain a smooth solution of the Laplace equation in ). Here we emphasize that no a priori compatibility requirement is imposed between the tangential and normal components of g on *), just to be able to deal with the presence of singularities in ). We "nally reformulate the inverse problem in terms of complex functions: P 1 Given a continuous function g on *) such that Re(e 0g(e 0))"0, does there exist a meromorphic function g in ) with a "nite number of simple poles in ), such that g"g on *)? P 2 In case the answer to P 1 is positive, determine the location of the poles of g and the corresponding residues. It may be shown that, if there exists a solution to P 2, this is unique by consequence of the Analytic Continuation Principle (see [8] or [9]). Moreover,
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the condition that the poles are simple is related to the minimum-energy principle of Buzano and Cermelli [5]. 3.1. Solution in terms of the stress function We do not attempt to solve problems P1, P 1, and P 1 here, but rather assume that a solution indeed exists, with a known number of singularities in ), and determine the location and Burgers moduli of the defects, solving P2, P 2, and P 2. Our "rst approach is based on the stress functions. Assume thus that, given g on *), a vector "eld g on )!+z ,2, z ,, solution of P1, does exist. , This is equivalent to assume that, given q on *), a function w on )!+z ,2, z , solution of P 1 does , exist. Since, by (15), the stress function w associated to g is the sum of Green's functions for the Laplacian, Green's identity yields, for any harmonic function u on ), continuous on ) M ,
, *w *u b u(z )"! u !w ds, I *n *n / I and since w"0 and *w/*n"!g ' t on *), we have
, b u(z )" ug ' t ds I / I for any u harmonic in ).
(23)
For instance, if we know by some means that a single dislocation is present in ), the above formula allows to determine its location by
(g ' t)x ds, / recalling that bz "
(24)
g ' t ds. (25) / If there are N dislocation in ), it is necessary to apply (23) 3N times to di!erent harmonic functions u, in order to determine the 2N coordinates of z , I and their Burgers moduli. Notice that the fact that (23) holds for every harmonic function u may be interpreted as a necessary condition for P 1 to have a solution.
b"
3.2. Solution in terms of the Fourier coezcients of the data Our second approach is in terms of the complex function g. As in the preceding section, assume that, given a complex-valued function g on *) as in P 1, a meromorphic function g does exist in ) with a given number N of simple poles, located at +z ,2, z ,. Expanding g in Fourier series we ob , tain > g (0)" c e L0, (26) L L\ with 03[0, 2p). The solution g to P 1 (which is assumed to exist) is given explicitly by (21), and may be expanded in Laurent series in a neighborhood of *):
, b > zL\ > g(z)" I I # z L>zL . (27) I zL 2pi I L L for max "z "("z"(min 1/"z ". EvaI 2 , I I 2 , I luating this series on *), imposing that g"g and using (26), we "nd that
b b z\L>#2# , z\L>, n)!1, 2pi , 2pi c " L b b z L>#2# , z L>, n*0. 2pi 2pi , (28)
Hence, given the Fourier coe$cients c , the comL plex numbers corresponding to the location of the dislocations are solutions the in"nite set of algebraic equations (28). On the other hand, if we know that only N poles are present, it is of course su$cient to solve a "nite number of such equations. In the hypothesis that the function g has N simple poles, it is su$cient to solve a "nite number of such equations. Notice that only the coe$cients corresponding to either positive or negative indices are needed to solve the problem since, by (26) > e 0g(e 0 )" (c e L>0#c e\ L>0 ), L \L> L\ and, taking the real part of this expression, we see that the boundary condition Re(g(e 0 )e 0 )"0
P. Cermelli, F. Pastrone / International Journal of Non-Linear Mechanics 36 (2001) 515}521
implies that c #c "0, n*!1, L \L> i.e., c
\
is imaginary, c "!c , \
(29)
3.3. Examples Inspection of the fundamental relation (28) allows to give explicit formulas for the location and Burgers moduli of the dislocations, at least for some simple cases. These in turn yields su$cient conditions for the existence problem P 1 to have a solution. (i) A single dislocation. Assume that the coe$cients c satisfy the iterative relations L c \L> "j for n*1 and Re(c )"0, \ c \L (30) with j a "xed complex number such that "j"(1. Then there exists a single dislocation in ), located at the point c (31) z "j" \ with b "2pic , \ c \ a relation which follows upon noting that, in the presence of a single dislocation, then (28) becomes b c " zL\ for n*1. \L 2pi (ii) A symmetric dislocation dipole: Assume that the odd terms in the Fourier expansion of g vanish, and the even terms satisfy an iterative relation of the form (30), i.e., c "0 for n*0, \L> c \L> "j for n*1 c \L and
c Im \ "0, c \
with j a "xed complex number such that "j"(1. Then, letting (c )/(c )" !k, with \ \ k a real number, then P 1 admits a solution with two poles, located at z "$j, with Burgers moduli
and so on.
(32)
521
(33)
b "$pk, (34) relations which follow upon noting that, in the presence of a symmetric dislocation dipole (z #z "0, b #b "0), then (28) becomes b c " zL\, n*1. \L pi
Acknowledgements We acknowledge valuable discussions with Ernesto Buzano. This work was completed under the support of the Italian MURST project `Modelli Matematici per la Scienza dei Materialia.
References [1] J.P. Hirth, J. Lothe, Theory of Dislocations, Second edition, McGraw-Hill, New York, 1982. [2] H.G. Natke, Fault detection and localization in structures: a discussion, Mech. Systems Signal Process. 5 (1991) 445}456. [3] P. Cermelli, M.E. Gurtin, The motion of screw dislocations in materials undergoing anti-plane shear: glide, crossslip, "ne cross-slip, Arch. Rat. Mech. Anal. 148 (1) (1999) 3}52. [4] M.E. Gurtin, Con"gurational Forces as Basic Concepts of Continuum Physics, Springer, Berlin, 1999. [5] E. Buzano, P. Cermelli, A singular variational problem in dislocation theory. Z.A.M.P. 51(6) (2000). [6] S. Axler, P. Bourdon, W. Ramey, Harmonic Function Theory, Springer, Berlin, 1992. [7] V. Volterra, Sur l'eH quilibre del corps eH lastiques multiplement connexes, Ann. Ec. Norm. 24 (1907) 401}517. [8] J.B. Conway, Functions of One Complex Variable I and II, Springer, Berlin, 1978 and 1995. [9] M. Heins, Complex Function Theory, Academic Press, New York, 1968.