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".ImL‘CLITC: 1 “MI 5* -~11zwh:hf1a§l-mf€& “2:: 1131 ~-- :5» 111 $11 ‘.?ccr“1‘mw~“~.~a “ “v 1‘1“: .‘ b% 1* ‘ ~. m- ‘1: «a. g‘gwfi - 53??! L; 75.1 - {:4}; Y 1'. l ifr I if; :7???" LIBRARY Michigan 51:61::+ University l This is to certify that the dissertation entitled ESTIMATION OF THE FORCE ON SCREW DISLOCATIONS IN FINITE ELASTICITY USING THE J-INTEGRAL presented by Bijan Khatib-Shahidi has been accepted towards fulfillment of the requirements for PAD degreein MeCin‘ai/H‘QS Web ii Qhflno / Date 5 5%? 'K / / \. MSUis an Affirmative Action/Equal Opportunity Institution 0-12771 MSU LIBRARIES " RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped below. "T _-. ___.__._ ESTIMATION OF THE FORCE 0N SCREW DISLOCATIONS IN FINITE ELASTICITY USING THE J'INTEGRAL by Bijan Khatib-shahidi A DISSERTATION Submitted to Michigan State University in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in Mechanics Department of Metallurgy. Mechanics and Material Science 1987 ABSTRACT ESTIMATION OF THE FORCE ON SCREW DISLOCATIONS IN FINITE ELASTICITY USING THE J‘INTEGRAL by Bijan Khatib-shahidi This study is concerned with estimating the value of the force between two screw dislocations in an elastic solid. Since the linear theory of elasticity predicts unbounded strains at a dislocation. the results predicted by such a theory are suspect. The present study is carried out within the fully nonlinear theory of elasticity. Since the force between two dislocations is defined as the rate of change of energy with respect to the separation between the dislocations. one way to calculate this force is to first calculate the total energy stored in the body. This is an almost impossible task to carry out exactly in a fully nonlinear theory. However. we observe that the value of the J-integral evaluated along a path surrounding one dislocation is precisely equal to the magnitude of this force. We exploit the path-independence of the J-integral in order to obtain accurate estimates of the force between two dislocations in the presence of nonlinearity. Bijan Khatib-shahidi We first carry this out in the context of the linear theory. for illustrative purposes. and then carry it out in the context of nonlinear theory. a. To my father and mother for the example and inspiration they have provided throughout my life. ACKNOWLEDGMENTS I would like to express my sincere appreciation to my teachers: Professor Rohan Abeyaratne of Massachusetts Institute of Technology for his guidance throughout this work and for his inspiration and teaching of finite elasticity. nonlinear fracture mechanics and instabilities while he was at Michigan State University. I would also like to thank Professor Cornelius O. Horgan who taught me continuum mechanics. elasticity. variational calculus and discussed many challenging problems in nonlinear continuum mechanics. Grateful thanks are extended to Professor David ILY. Yen who taught me applied mathematics in support of the elasticity theory. and Professor Nicholas J. Altiero who taught me elastic stability and computational mechanics. Finally. I would like to thank Dean David L. Sikarskie of Michigan Technological University. formerly chairman of the Metallurgy. Mechanics and Materials Science Department at Michigan State University for his guidance. teaching and support throughout my entire education at the University of Michigan and Michigan State University. A final note of appreciation must go to U.S. National Science Foundation for their support of this work under Grant MEA 86-96034. TABLE OF COITEITS List of Figures Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Appendix Introduction Preliminaries on finite elasticity and the dislocation problem Linear problem Nonlinear problem Summary and discussion List of references vi Page ii 25 43 65 7O 80 .LIST OF FIGURES Figure 3.1 Cross section D of cylinder with dislocation on x2=0. x1; 0 3.2 Dislocation cores in two dislocation regions Ds 3.3 Dislocation in a traction free half-space 3.4 Two dislocations centered at x1=£ and x1=-£ 4.1 Two dislocations in a non-linear domain D 4.2 Representation of two dislocations with one and small perturbation a 5.1 Shear response curves for pure power law materials 5.2 Variation of J* with 2 vii Page 72 73 74 75 76 77 78 79 ESTIMATION OF THE FORCE ON SCREW DISLOCATIONS IN FINITE ELASTICITY USING THE J-INTEGRAL viii CHAPTER I The theory of dislocations has received a large amount of attention in the materials science literature. In particular. solutions to the problems of straight dislocations in an infinite continuous medium have been found within the linear theory of elasticityu The displacement field and the stress field have both been obtained in the presence of a single dislocation and multiple dislocations. The results of these investigations are unsatisfactory*within the linearized theory since the results predict infinite strains at the dislocations. contrary to the assumption upon which the theory rests. These results can be found in the theory of dislocations literature. eug. Hirth and Lothe [1] and Lardner [2]. cite these findings. We recapitulate these results in Chapter 3. In recent years investigators have employed some nonlinearities in the formulation of the problem of dislocations. In particular. Kachanov [3]. solves a screw dislocation problem in an infinite medium under the assumption that the stress-strain relations are nonlinear; however. he assumes that the displacement gradients are small. The problem of dislocations in nonlinear elasticity is also described in an article by Gairola [4]. The formulation is described in general tensorial notation and the problem of a screw dislocation in an infinitely long and straight cylinder bounded by two circles is treated by the method of successive approximation (Signoriniis method E43). There is also an interest among material scientists to obtain the force on a dislocation. The force on a dislocation. or in general the "force on.a defect". is defined to be the rate of change of the total energy of the body thich.is composed of the elastic strain energy of the material containing the "defect" and the potential energy of any external loading mechanism.) with respect to a change of the position of a defect. In the case of the screw dislocations this is the force tending to make the screw glide in the x-direction.(chapter 3). We also note that the force acting on a dislocation is not a: true mechanical force but it has the dimensions of a force. It may be viewed as a "driving force" in the thermodynamic sense. This is discussed in detail in Eshelby's article "The Continuum Theory of Lattice Defect s" [S]. The J-integral is a path independent integral which plays a central role in nonlinear fracture mechanics [6]. As shown by Rice [7]. one can readily verify that J reduces to the energy release rate. which in turn leads to the stress intensity factor in linear elastic fracture mechanics. It is also stated by Esbelby [8] and Rice [9] that. in general. the J-integral can be used to define the "force on a defect". the defect being the dislocation in the present case. In the present study. the theory of dislocations is studied within the context of finite elasticity. and attention is restricted to screw dislocations. The exact displacement field and stress field are obtained analytically for a single screw dislocation in an infinite region. Further. the results are extended in an approximate manner to two screw dislocations in a nonlinearly elastic region by using the theory of "small deformations superposed on a large deformation". The force between the dislocations is calculated via J-integral. Even though the analysis is approximate. the results for the force are expected to be good. for the reasons explained below. Consider. for example. the case of two screw dislocations in an infinite medium. If one were to assume that the presence of a second dislocation only causes a small perturbation. one can then carry out an analysis based on the theory of small deformations superposed on a large deformation. Clearly this assumption is invalid in the vicinity of the dislocation but it is expected to be reasonable at points far from the dislocation. Our interest here lies solely in evaluating the value of J which represents the force on a dislocation. Since the J-integral is path-independent. we can evaluate it at points distant to the dislocation. Weexpect to get reasonable results for our purpose. We will note that for a screw dislocation in an infinite medium. the value of J becomes zero. but for the two screw dislocations in an infinite medium J is calculated to be nonzero. The chapters and discussions are presented in the following sequence: Chapter 2 summarizes results from finite elasticity theory. the notion of the J-integral and finally the notion of a screw dislocation. In Chapter 3 the formulation and results for a single screw dislocation - ' and two screw dislocations in an infinite linear elastic region are discussed. The calculation of the force on a dislocation. using two different techniques. is then carried out and finally the results of the two techniques are compared to one another. Chapter 4 covers similar issues in the nonlinearly elastic case. Chapter 5 summarizes and discusses the results. Finally. we note that after the first draft of this dissertation was written. we received a paper by P.J. Rosakis and A.J. Rosakis [24] in which they independently solved the problem of a single screw dislocation in incompressible finite elastostatics. Their results for this problem are more general than ours. since ours are restricted to generalized neo-Eobkean materials and theirs arenfiu However. they do not address the problem of two dislocations nor do they consider the force between dislocations. CHAPTER‘Z RRELIMINARIES QN.EINITE.ELASIIQIIX.AND.THE DISLQCAIIQN.£ROBLEM In Section 2.1 of this chapter. certain pertinent results from the equilibrium theory of finitely deformed. homogeneous and isotropic elastic solids are summarized. The complete theory and general results from the continuum theory of finite elasticity may be found in [10]. Herein the theory is first presented in 3-dimensions. Then in Section 2.2. it is specialized in two ways: The case of finite anti-plane shear deformation (which provides the setting for discussing the screw dislocation) is considered first and then. this is further Specialized to the case of infinitesimal anti-plane shear deformation. The J-integral. with some historical notes. is described in Section 2;3 of this chapter. The proof for the path independency of the J- integral is omitted from the section but may be found in [7]. The relationship between the path independent integral J and the notion of a force on a defect is discussed. Finally. in section 2.4 we formulate and discuss the notion of a screw dislocation. 2.1 E. I E] O 0 Let R be an open region occupied by the interior of a body in its undeformed configuration and denote by x the position vector of a material point in R. A deformation is described by y = y (x) = ~ ~ which is a mapping x + u (x) for all x E R. (2.1)1 ~ of R onto a domain R*; g (x) represents the diSplacement field associated with this deformation. The transformation (2d) is assumed to be invertible and suitably smooth. Let g be the deformation gradient tensor field. the determinant of F by J. J (x) = det The deformation of an preserving. hence J (E) = det (x) for all x c R: (2.2) is called the Jacobian determinant and is denoted F (x) > O for all x s R. (2.3) incompressible material must be locally volume F (§) = 1 for all § 6 R. (2.4) ~ Define the tensor fields 9 and g by 1 and tensors. Letters underlined by a tilde represent three-dimensional vectors 9=§§- §=§f. - V (2.8)1 The stresses are denoted by T (y) for the actual (Cauchy) stress tensor field on R* and g (x) for the corresponding nominal (Piola) 11 stands for the idem tensor. stress tensor field on R. They are. in general. related to each other by 1:1-01-‘1‘. - J-.. (2.9) swift- In the absence of body forces. the local equilibrium equations are div 1 = 0 on R*. ” " (2.10) div 9 = g on R. together with I = ET on R*. and gift = {gr on R. To continue. let s be the nominal surface traction andt:the corresponding true surface traction. Then 3 = o n on S , " “ “' (2.11) E = I 2* on 5*s . . . * . . where n 18 a unit normal to the surface S in R. and n is a unit normal to 8*. the image of 5 after deformation. It can be shown that O on S in II if and only if f = 9 on S. (2.12) This is useful in the case of a traction-free surface 8*. since it enables the boundary condition on the unknown deformed surface to be specified on the known undeformed surface 8. Let W be the elastic potential. characteristic of the given elastic material. the value of W represents the stored strain energy per unit undeformed volume. For homogeneous. isotropic and incompressible elastic materials. W can be shown to depend on'the deformation solely through the invariants 11’ and 12 i.e. W = W (11' 12). The corresponding constitutive law takes the form aw an 3 = 2 -— g + -— (I1 1 - g) g -‘P 1. (2.13) 311 312 or. equivalently. aw aw _ o = 2 -— F + —— (I1 1 - c) F - P F T. (2.14) “ 311 ~ 312 “ " ~ ” where P is a scalar field required to maintain the constraint of incompressibility (2.4). 2.2 Manama: Next. we consider the special case of finite anLi;niene sheer defgzxuujguuh_ The deve10pment that follows is based on the analysis by 10 Knowles [11]. Assume the region R occupied by the undeformed body to be cylindrical and a fixed cartesian coordinate frame is chosen so that the x3 - axis is parallel to the generators of R. Let D be the cross- section of R in the plane x3 = O. The deformation (2J3 on R is said to be an anti-plane shear if it is of the form = = + ( ) 1 ya X 0. s Y3 X3 U Xls X2 o (2.15) Thus in an anti-plane shear deformation. particles in the body are displaced purely in the axial direction by an amount u(x1.x2). In the absence of body forces. the deformation (2.15) can. in general. be sustained only in materials for which the strain energy density takes a more restricted form. (Knowles [123% A.class of such materials are those characterized by. A w = a (11) for all I1 3 3. w (3) = 0. (2.16)2 and referred to as ”Generalized Neo-Hookean Materials". We assume that W conforms to the Baker-Ericksen inequality (see [13]): 1Greek subscripts take the range 1. 2 while Latin subscripts assume the values 1. 2. 3. Repeated subscripts are summed. 2Note that I1 = 3 in the undeformed state. 11 {31111) > o for all 11 > 3. (2.17) where W' is the derivative of W with respect to the argument. This can be shown to be equivalent to requiring a positive modulus of shear at all deformations. The infinitesimal shear modulus is u = 2 w'(3) > O. (2.18) For the class of deformations given by (2.15). the components of the deformation gradient tensor are calculated from (2.2) to be : (S , FaB a8 Fa3 : 0. (2.19)l F33 : u. 9 F33 = 1 on Do In viewing (2.19). one can conclude that J. the determinant of f. 1 becomes 1 automatically. Now calculating E‘— . its components denoted by Fgl ’ we find 1 1Subscripts preceded by a comma indicate partial differentiation with respect to the corresponding material cartesian coordinate. Next the tensor End a as Ga3 : 6(18 . = 0, = " Us s a = 1 on D. a8 a C3o = u'a 1 on D, dB ' 630. = “on 1 +u.a u. on D. 12 (2.20) (2.21) (2.22) 13 The principal scalar invariants 11’ I2. and 13 are given by (2.6) and for anti-plane shear deformations are ‘ ‘- 2 Il IZ=3+ tVUI 9 (2.23) 13:1 onD. The components of the Lagrangian strain tensor are obtained from equations (2.8). (2.21 ): 1 E = - u. u. (no sum). as a a 2 2 E33 = - Ivul . 2 (2.24) l E = - U. u. 9 a82a8 a Recalling from equation (2.16) that W is only a function of 11’ and using equations (2.13) " (2.14). the constitutive equation becomes l4 2 w‘(11) g - P 1. H H (2.25) 2 w‘(11) g - P F 10 H For anti-plane shear deformations. the components of I and g are found from (2.25). (2.22) and (2.21) to be a H as { 2 W (II) ’ P }'508 . Ta3 2 &'(Il) u’a‘. (2.26) A i 2 2 W'(Il) (l + [Vul ) - P. d U) U) I Similarly. Q I 08- {ZW'(11) ‘P } 6GB 0 0013 = Pu.“ ’ (2.27) AI 2 W (11) “ea s 033 = 2 W'(Il) ‘ P e The equilibrium equations in the absence of body forces are given by (2.10). Substituting (2.27) into (2.10) gives the three equations ‘{2 &'(Il) ' P } a + P93 Usa : 0 on D (2.28) 15 and x. i .- {2W(Il)u.c1 }a 'P.3 -0 on D. 9 (2.29) From equations (2.29). (2.15) and (2.23) one concludes that P.3 must be a function of x1 and x2 only; therefore Ps3 = fi (Xls XZ) o Using (2.30) in (2.28) and differentiating with respect to x3. that Psa3 : Os which implies that P must be a constant-valued function. say. *6) (X1. x2) = P1 = constant. Therefore 0 (X1. X2. X3) = P1 X3 + P (X1, X2). RM (2.30) indicates (2.31) (2.32) (2.33) 16 Substituting (2.33) into (2.28) and (2.29). leads to the conclusion that. ‘2 '- P(x1s st KB) : Z'W.(Il) + P1 [U + X3] + P20 (2.34) and s [2 {1.111) (1.011,“ = P1 (2.35) where P1 and P2 are constants. Equation (2.35) is the gexerning differential equatign for an anti-plane shear deformation. Once it is solved.(with the appropriate boundary conditions) for the displacement field u. the stress field can then be obtained from (2.26) and (2.27). If the body contains a traction-free surface 8*. then * . t = O on S (2.36) and from equation (2.11) one concludes that on S. (2.37) 10 (D n 0 Note from equation (2.34) and the first of (2.27). that l7 ' ' E P1 (u (x10 x2) + X3) + P2]. (2.38) (I N N I O = 0. By applying the boundary condition (2.37) and using (2.38). one finds that the constants P1 and P2 must vanish P1 : P2 = O, (2.39) Finally. on substituting (2.39) into (2.26) and (2.27). the stresses become T08 = 00.8 = Os To3 : T33 : 0a3 = 0301 = 2 wl(3 + [Vu|2) “’u ' (2°40) _ A1 2 '2 133 - 2 w ( 3 + [Vu| ) {Vul . and the differential equation (2.35) can be written as [2 W'(3 + quIZ) “'o ],o = 0 on D . (2.41) We now consider the particular solution u (x1. x2) = Ka “a on D. (2.42) 18 which satisfies the differential equation (2.41): a deformation of this type is termed simple shear. We let K and r be 1 )c = IV“| = (KaKa )2, (2.43) l T: ( T3o T31:1)2 ’ so that T is the resultant shear stress. and K is the resultant amen; of shear. From equation (2.40) we can write I : T (K) 5 2 W'(3 + K2) K, (2.44) The function r (K) characterizes the material reSponse in shear. In summery. the displacement field u(x1. x2) associated with a finite anti-plane shear deformation is to satisfy the governing differential equation (2.41). The corresponding stress components are given by (2.40). NeXt consider the special case of aninfinitesimalanttnlaneshear .defnrmatinn In this case. a linearization of the nonlinear equations is carried out under the assumption that |Vu[. the gradient of the displacement is small. i.e.. l9 —- < < 1.‘ (2.45) On linearizing equation (2.24). the tomponents of the Lagrangian strain tensor become 8:18 = 0 . E33 '- 0. _ _ 1 E3“ " E03 " '2- Uga . (2.46) Next. the stored energy function W approximates to A u w (3 + [(7142) ~ - [17142. (2.47) 2 so that by (2.40). the stresses are calculated to be T = 0 s T33 = Os as (2.48) T03 : TBQ : nu.“ . on D. The governing differential equation (2.41) reduces to Laplace's equation. V u =() on D. (2.49) 20 Thus in the case of infinitesimal anti-plane shear. the displacement field u(xl. x2) is to satisfy the differential equation (2.49). The corresponding stress and strain components are given by (2.48) and m‘ V (2.46) respectively. 2.3 Lingual The concept of the path independent J-integral was first introduced by Eshelby [8] in the course of determining the "energetic force" on an "elastic defect" such as a void or an. inclusion. It was subsequently exploited by Rice [9] in the context of fracture mechanics. Additional path independent integrals were discovered by Knowles and Sternberg [14]. and more recently by Abeyaratne and Knowles [15]. In a general three-dimensional setting. J is given by the surface-independent integral J=f(Wn-Fon)dA. (2.50) where W is the elastic potential. 3 is the outward normal to the regular closed surface S. I: is the deformation gradient tensor and g is the nominal (Piola) stress tensor. If the body does not contain any holes or inclusions within S. the value of J is zero. ,If it does. the value is non-zero. and this value. can be interpreted as a "force" on the hole or defect (see below). Similarly. Rice [7]. Budiansky and Rice [16] 21 have shown that a similar situation occurs if the region inside 8 contains a crack-tip. In this thesis we consider the case where the region inside S contains a dislocation. Let the total energy of a body under the prescribed loads be denoted by E. Suppose that the body contains a "defect" (a singularity in the elastostatic field) at the position 5- Then. the change in the total energy with respect to a change of the position of a defect can be shown to be (Eshelby [7]. Budiansky and Rice [15]) A E = J1 A E is (2.51) where Ji is the ith component of J taken on a surface S enclosing the defect. Since AE has the dimensions of work and [151 the dimensions of length. it follows that Ji may be thought of as a "fence en the defegh". For example. in fracture mechanics. Ji is referred to as the force on a crack-tip. From (2.51) we can write Ji =35/ 361. This latter expression leads to the alternative interpretation of Ji as an anew release rate. According to most theories of fracture mechanics. crack prOpagation occurs when the energy release rate J1 reaches a critical value. In an entirely analogous manner. if we have a dislocation in an elastic material located at E. J becomes the forge an the dialecetinn due to the elastic field. It is known to be an important and useful parameter in predicting dislocation motion. 22 For the case of finite anti-plane shear deformations.the x component of the force on a defect is obtained from. Jo: =[(Wna - nBOB3u'a) ds. (2.52) I' which is the appropriate restricted form of (2.50) in anti-plane shear: ds is arc length along a closed contour I‘ enclosing the defect. our purpose in this thesis is to emphasize the fact that the force on a dislocation is a concept analogous to. and possibly as important as. the 'force on a crack-tip: that it can be calculated using the path- independent J-integral. and to calculate explicitly the force between two screw-dislocations in the linear and the non-linear theory of elasticity. We will eXploit the path independence of (2.52) for this - latter purpose. 2.4 Dislncatian Consider an infinite two-dimensional region D and view D as the cross-section of an infinite body in its undeformed (reference) configuration. Suppose that a cut is made on the half line x2 = 0. x1 > 0. the top face of the cut is'denoted by 1' 4’. while the bottom face by F I. The upper face F+ is now displaced by an amount b (relative to I") in the x3-direction and the two faces are fused back together. If 11 (x1. 0+) and u (x1. 0-) denote the x3-components of displacement on F + and 1' ' then (see fig. 3.1). 23 u (x1. 0+) -u (x1. 0') = b for x1 > 0. (2.53) The vector b e3. where e3 is a unit vector in the x3-direction is known as the breeze meter associated with this dislocation. Traction continuity across the line x2 = 0. x1 > 0 requires T32 (Xls 0+) ' T32 (Xls 0-) = 0 for X1 > Us (2.54) In W. let Dodenote the entire (x1. x2)-plane with the line x2 =0. x1 3 0 deleted. We wish to find a displacement field u (x1. x2). twice continuously differentiable on Dosuch that 4 (2 w' (3 + (NH) ma)“: = 0 on Do. (2.55) 0 (X1. 0*) “.u (x1. 0‘) = b for x1 > 0. (2.56) T32 (11- 0+) ' T32 (11. 0") = 0 for x1 > o _ (2.57) u. + 0 as x2 + x2 + co . (2.58) 0 1 This is a "weak formulation" of the equations of (anti-plane) elastostatics. since it involves a discontinuity in the displacement (and possibly in the displacement gradient In and the shear stress T31) across the line x2 = 0. x1 > 0. 24 An alternative weak formulation. which is in fact lees, singular than the preceding one. pertains to elastostatic fields involving equilibrium shocks. Here one permits discontinuities in Bu but net in u itself (see. for example. Knowles and Sternberg [l7] ). It has been shown ([18]. [19]) that if the response curve in shear is monotonically increasing (T'(|<) > 0 for all <) then equilibrium shocks cannot exist. (The condition T'(K) > 0 ensures the ellipticity of the partial differential equation (5.55). see Zee and Sternberg [20] )- A minor modification of the arguments used in [18]. [19] shows that (in contrast to the situation with equilibrium shocks). even if T'(K) > 0 one gen have weak solutions involving dislocations. However. in and T31 are continuous across the dislocation-line (even though 11 itself is discontinuous). We will assume throughout that T‘(K) > 0 for all K . (2.59) cums LINEAR mm In this chapter. the linearized problem for a single screw dislocation in an infinite medium is discussed first. This is then extended to the case of two screw dislocations. In section 3.2. the force on a dislocation is found by two different techniques. first by ' calculating the total elastic energy and taking the rate of change of the energy with respect to the dislocation position: second. by application of the J-integral. Naturally. the results coincide. 3-1 £1118an Consider a screw dislocation in an infinite region R made of homogeneous isotropic elastic material. Let us suppose that a cut is made along the semi-infinite line x2 = 0. x1 > 0. the top face of the cut (x2 = 0+) being held fixed and the bottom face displaced in the x3- direction by the constant amount b: bg3 is the Burger's vector. See Figure 3.1. The dislocation is said to be a Lightzhanded screw disloc- ation if b is positive. Let D be the cross section of R in the plane x3 = 0. for a screw dislocation described above. the deformation is one of anti-plane shear and the governing linearized differential equation for the displacement field is Laplace's equation (2.49). We use (r.6) as polar [coordinates (r>0. 043211) and formulate the problem in terms of these polar coordinates. There exists a jump in the 25 26 displacement field across the dislocation line. equation (2.56). but the tractions are continuous'across the dislocation line. equation (2.57). We also note that as r4» . the stress field and the displacement gradients vanish. Therefore the boundary value problem (2.55) - (2.58) for a single screw dislocation in a linear elastic field is expressed as azu 1 3n 1 32u . ‘) —5+-—+—2--—2= 0. forr>0.0<9<21T 3r r 3r r 36 u (r.. 0+) - u (r. 211') = b". for r > 0 }- (3.1) au/ae (r. 0+) - au/ae (r. 2n“) = 0. for r > o 1 u.a=0(;) as r+uo. J We seek a solution to the boundary value problem (3.1) which is a function of 6 alone: u = u(6). Upon integration. the general solution for u(6) is found to be. u = A6 + B on D. . (3.2) Note that the condition at infinity (3.1)4 is automatically satisfied. By applying the boundary conditions (3.1)2. (3.1 )3. we evaluate the 27 constants of integration‘A and B. The resulting solution is found to be. b u = -'6 on D . (3.3) Zn _-ub x2 , 31'— 2 2 4. 2n x1 x2 on D. r #0. (3.4) x UBsz—l—b. 1 9 2n xi + xi all others being zero. In polar coordinates. o3r = 0 on D . (r #0). (3.5) ub‘ U :— 36 an The fares on a single screw dislocation can be calculated by incorporating the stresses found in.(3.4) and the displacement gradients 28 obtained from (3.3). in the J-integral of (2.52). It can then be calculated by taking a closed circular contour centered at the dislocation. One readily finds Ja -= 0. The force on a single screw dislocation in an infinite elastic medium due to the elastic field turns out to be zero. This is due to the fact that there are only the internal stresses (3.4) or (3.5) which are caused by that dislocation itself and there are no other means of raising or lowering the total elastic energy in the elastic field. In order to obtain a force on this dislocation. one can either introduce another dislocation into the body or place the existing screw dislocation at a finite distance from a boundary. Finally. we discuss the total strain energy in the body. While we. have no immediate need for this here. it helps to make clear the need to introduce the notions of gene 111111.113. and mate; radius, These will be utilized in the subsequent sections. In order to calculate the total strain energy associated with the deformation. (3.3). we must first A obtain the strain energy density. W. From (2.47). it becomes. .. .. 2 w = ub . . ' (3.6) 8fl2r2 Note that there is a singularity at the origin of the order r-2. Thus the total strain energy stored in a disk of radius r centered at the origin becomes unbounded. We follow standard practice (e.g. [1]) 29 and eliminate the singularity at the origin of the dislocation by considering a "Volterra core" assumption that there existsa small radius of ro centered at the dislocation in which we do not calculate the total energy. Therefore. we consider a here regign such that 6 = {(x1. x2) I O 5 x (3.7) ”N + N NN IA '1 N 55...; . and integrate the strain‘ energy density on the remaining material Ds =ID - 5. Similarly. the total stored energy has a logarithmic singularity at ihfiinihy. This forces one to cut out the far-field by some large radius R ( >r° ). In summary. when calculating the total energy. one must restrict attention to the region between two finite radius cylinders. the smaller one being the Volterra core and the larger one. a circular cylinder of radius R. see [1]. The total energy is. therefore. written as 2'11 .‘ 2 3 = 4" rdrde . (3.8) 8fi2rz 0 to 30 using (3.6). Integrating (3.8) gives ln . (3.9) It is important to note that the choice of ro and R does he; affect the energy release rate or dislocation force. This is zero independent of the choice of r and It. This feature continues to be 0 true for two dislocations and. therefore. one is able to get useful results for J mm; having to specify the values of to and R. 3.2 In Dialncatiens Consider now two screw dislocations in an infinite region R. made of ‘homogeneoms isotropic linear elastic material. The first of the two dislocations is located at (2. 0) with the Burger‘s vector b1 33 and the second one located at (-2. 0) with the Burger vector b2 93. We set up three polar coordinate systems. first set (r1. 91). (r1>0. 0<91§2fl) centered at (2. 0) and the second set of polar coordinates (r2.62). (r2>0. ”(@2510 at ('2. 0) and finally (r. 9). (r>0. 0<952fi) centered at the origin. This is illustrated in Figure 3.2. The displacement field ‘for the two screw dislocations is found immediately by applying the principle of superposition to the displacement field for a single screw dislocation. equation (3.3). The displacement field is therefore given by 31 b b 1 2 u = ‘- 91 + -" 92 . (3.10) 211' 211 c ‘h Observe that u is discontinuous across the lines x2=0. x1>£ and x2=0. x1<-2. . The strains may then be calculated from (2.46) and (3.10) El 361 (X1. X2) + 132 362 (Kls x2). 2 I“331 = u'1 = 2n 3x1 2n 3x1 (3.11) b1 361 (Xls X2) b2 392 (Kl. X2)s 2E32=U92=—_ +_— 2n 3x2 2n 3x2 which reduce to b sine b sine _ l l 2 2 2 E31 - Usl = ' - — s 2n r1 2n r2 (3.12) b1 cosel b2 cosez 2 E = u. = -— + — s 32 2 2N r1 2“ r2 Next we calculate the strain energy density associated with this deformation from A 14:; [Vulz . (3.13) 32 . . . . . ' 2 where u is the infinitesimal shear modulus and [W] = u.“ uh. By substituting (3.12) into (3.13). we can readily show that. 2 2 =‘ b1 b2 + 2 blbz + cos ( 6 - 6 ) . (3.14) Anzri 4n2r 1 2 i 4n2t1r2 In order to calculate the total strain energy E in the body. we must integrate (3.14) over D. However. it is first necessary to eliminate the singularity at the origin of dislocations and infinity from D which cause the total strain energy to become unbounded as in the case of a single screw dislocation. To do so. we consider two hollow- cores each with a radius of ro at the center of the dislocations ( 2.0 ) and ( -£.0 ). See Figure 3.2. Let 61 and 6 2 denote the so- cal led core regions . 01 I. 2 2 (x1-£)2+x2_ }. [A A *1 { (x1. 2(2)] 0 A H O N 55,—; o 2 {(31, X2)! 0 (X1 + 2.)2 + X2 -_. [A N I Next. let R ( >1 + ro ) be a large fixed number and let DR denote the region inside the circle of radius R: 2 2 1 DR={(x1.x2)| 05x1+x2§R} . 33 Let Ds denote the material inside DR but outside 51. 52. A The total strain energy in D8 is finite. The strain energy density. w from equation (2.14) is now integrated over D5 5 = Ira dA . (3.15) Ds Substituting equation (3.14) into equation (3.15) and writing the integral as three separate terms. gives a total strain energy. E=jw1dA+ [wsz*fw34A° (3.16) Ds Ds Ds where “‘—a .32, 3? ginlfrs ““~a I... .g 2 D D r1 3 S 2 ub l &2 dA = __2' If -—-' dA , (3.17) 87f 2 r22 Ds Ds of: % ubb 2 “ - 12] cos (91-52)dA' 872 rlrz 34 In order to evaluate the three integrals in CLJ7). we first relate (1'1. 61). (r1>0, 0<€52r) to (r2. 52). (22>0. «$29) by geometry: r2 sinez = r1 sinel , r2 C0862 = 22. "‘ r1 C0861 , (3.18) 1 1' r12 2 r = 21 l + -- cose +‘-- . 2 1 1 49.2 We use (r1.61) as the independent variables chosen to perform the integration over Ds.'The element of area dA is then rldrldel. The integrals (3.17) are, evaluated with the aid of (3.18). The first term of the equation (3.16) is written as * 2w r1 2 a ‘ [. dA f] vb, 1 uh; a 19.2 22 W = —- :1 d9 =-—-— ———— +6 —- 1 1'1 r1 1 1“ 8.2 r12 4. ‘ r0 4 a2 . 32 DS () to (3.19) where at r1 = r1 (61) 5 )JR? - 22 sin2 61 - 2cosel is the equation of the large circle xi + xi = R2 . 35 In evaluating the integral in (3.19) we have used the fact that R is large in comparison to I. and r0. Thus. the final eXpression in (3.19) hold-sf'asymptotically. for large R. The second term of the equation 5' (3.16) is essentially identical to the first and so 277 1'2 b2 2 IW:I{:21 ~.b2 a 1 2.2 12 — _ rzdrzdaz 3 _ 1 n — - '- _ + O - s :2 43 r a a2 32 D 2 o _ S (3.20) where * r2 = r2 ( 62) E Vlz- 9.2sin2 62 + icos 92 . Finally the third term of the equation (3.16) is integrated over Ds by expressing the. integrand in terms of :2. 92. Carrying out this integration. the third term of (3.16)‘becom as 2r r; . Zubl b2 ["3“ =6fl -"'—_ CO. (61' 62) rzdrzdez 8172 :13: D5 = (lo—~21n—+- --)+0 — . (3.21) t.- to ro R2 112 36 Thesumlaation of the three integrals (3.19) - (3.21). will give the total strain energy of the elastic field in the presence of two screw dislocations. with the Burger‘s vectors b1 and b2 respectively. ' )4 2 R 2.2 L'blbz . 21 2,2 E'—(b1-b2) [41n —-—] 4- 1n—+0 7 ‘ 161! to R2 41v to R ' (3.22) on ”8' If the two screw dislocations have the same Burger's vectors. bl.= b2 = b. then the total strain energy E reduces to 2 ub 2% 2 E = "" ln "' + 0 -' (3.23) 4w to R2 Equation (3.23) is in the same form as it is described in [1] and [2]. however in the general case (b1 # b2). (3.22) holds. 37 Our primary interest lies in finding the force on a dislocation. say. the right-hand one in Figure 3.2. For this purpose. one calculates the rate of change of the total strain energy with respect to the dislocation position. as described :previously. One can readily show ‘ that from (3.23) and (2.51) the force on a dislocation becomes -BE ublbz F = - . (3.24) 8(22) 4st Equation (3.24) is a classical result which is cited in [1] and [2]. It provides the force on a screw dislocation in a homogeneous. isotropic. linear elastic field in the presence of another screw dislocation. With bl = b2. it is also the same result as a force on a single screw dislocation which is placed at a distance 2. from a traction free surface as is illustrated in Figure 3.3. We now describe an alternative way of finding the force on a dislocation by utilizing the equivalence between the force on the dislocation and the path independent integral J. The formulation of the J-integral was described andidiscussed in Chapter 2. section 2.3. For reasons of algebraic convenience we only consider the case bl=b2 here. In order to calculate the force on one of the dislocations. a path of integration for J must be chosen. which encloses that dislocation but not the other. Since the integral is path-independent. we will choose a 38 convenient path to eliminate cumbersome calculation. In this case. a circular path containing the dislocation located at ( i . 0) is chosen. (Figure 3.4). centered at the dislocation. with radius :1 which is taken to be infinitesimal. The element of arc length ds is r1661. Since the dislocations are located on the xl-axis. by symmetry the force between them must also be in the xl-direction. Thus. it is the J1- integral that we must consider. From (2.52) and the path independence 2n 0 Next. we will approximate the integrand of J for small r1 by using-the binomial expansion in (3.18). The result is. as :1 a 0 at fixed :11. r2 ~ 22 + :1 cosel . rlsinel sine; ~ —. (3.26) 21 21 + rlcosel cosez ~ = l. 22 + rlcosel The displacement gradients of equation CLIZ) are likewise approximated and they are. as r1 e 0 at fixed 81 39 ~ b_' - sine1 + rlsinel “.1 2n r1 (2£)2' . (3.27) b cosel 1 - u.2 ;’ r1 " 22' ° By using equation (3.13) and (3.27). we can now calculate the corresponding value of the strain energy density W . it is. given as rlsinel cosel + sinel ~ 2 A ub w — 1 29. ' r1 r1 (22) Similarly the stresses 033 are approximated to the appropriate order by using equations (2.48); they become . ub sin 61 rlsinel 0 ~ -- c- + —— . 31 Zn r1 (22 )2 0 ~ ub C08 61 - :— 32 2“ r1 22 . 0n substituting equations (3.27) - (3.28) into the J-integral (equation (3.25)) and separating the integrals into three parts. the result becomes. 40 J = Ja + Jb + Jc ’ (3.30) where c 21: J8 = f W‘nlrldel , 0 27v J - 2 . _ b ' '”1““’1’1d51 ’ (3.31) 2 Jc : f ‘nzuu.1u.2r1d61 . 0 Here 111 = cos 61. n2 = sin 61. Using equation (3.28) and the first of (3.31). Ja becomes. 21! 2 _ f ubz 1 Zlcosel + rlsin 61 C... I _ e - C08 1 2 2 2 c0691 rldel . (3.32) 8" r1 21'12. 0 ub2 n =—' 0+E-0 , 81r2 2 11b 8112. 41 To calculate 3b' the second of the equations (3431) we first obtain u2.1 from equation (3.27). 2 b2 sxnzel rlzsin 9! 231a .1 0.1 -— "———-+-——-—Z—'——-——2— o (3.33) 4.2 r12 <22) (22) On substituting this into Jb. we find Zn -,,b2 I sinzel rlzsin 31 Zsinze 3" _ "'——*——'——— coserde . (331.) b . (“1,2 0 :12 (22%. (2£)2 1 1 1 = 0 Finally the third equation in.(3.31L. Jc. is calculated by first obtaining u.1u.2 from equation (3.27). u . b2 sinelcosel + sinel + rlsinalcosel (3 35) 4772 r12 211']. 21(22. 2 We now substitute (3.35) into (3.31) and evaluate it 42 2w - _'-ub2 -sin61cosel sinel sinelcosel . Jc ' 2 + + 2 Sinelrldel , (3.36) 471' 0 r12 (22-)1'1 (22) .2 = ub O + 1— + O , 4"2 21 _ ubz But We then obtain J by combining (3.32) - (3.34) and (3.36). ub 411 2. (3.37) which is. of course. identical to the result (3.24) (when b1=b2=b) obtained by the previous direct method. m4 NQN:LDWMHLEBQBLEM The boundary value problem for a single screw dislocation in an infinite nnnlineazlx elastic medium is posed in section«kJ.of this chapter. It is observed that the displacement field has the same form as in the linear elastic case. found in Chapter 3. However. the stress field depends on the particular choice of the constitutive law and therefore. it is different from the one obtained in the previous chapter. In section 4.2. two screw dislocations are introduced in an infinite nonlinearly elastic medium. The boundary value problem is posed in terms of the field equation (ZJHJ and the appropriate boundary conditions for the two screw dislocations. An exact solution to this nonlinear boundary value problem is not determined. instead we obtain an approximate solution as follows: our analysis is based on the assumption that the introduction of a second dislocation causes only a small perturbation in the pre-existing nonlinear field induced by the first dislocation. This permits us to use the theory of small deformations superposed on a large deformation to solve the problem. Clearly. this approximation is very good at points far from the second dislocation and is poor near that dislocation. However. since.nnz primarxpnmsshezeiamfindthefmennnnanfthedialmatinnsand since this can he stiffen in terms of. the nathzindanendent Lingual 43 44 mamammmummnmammmmmmum Length. we expect that this approximation will provide accurate results for the force. A similar procedure was used by Abeyaratne [21] to calculate the energy release rate in fracture mechanics. Consider a screw dislocation at the origin of an infinite region R made of homogeneous isotropic and incompressible elastic material. Suppose that a cut is made on the half-‘plane x2 = 0. x1>0. the tap face of the cut (x2 = 0+) is held fixed and the bottom face displaced by b in the x3-direction. Let D be the cross-section of R. in the undeformed state. in the plane x3 = O and Do is D with the line x2=0. xlip deleted. The resulting deformation is assumed to be one of £ini;g,anti;plang shear. In order to seek a solution. we set up (r. 6) as polar coordinates (r>0. 0<6f2n) at the origin. The boundary value problem for a single screw dislocation is then described by 3 A 3 A 2 “'6 —(.-w' (3+IVuI2) n.1,) +-—(w‘(3+ 1v“; )—> :0 onDo. a: 36 r u (r. 0+) ‘ U (to 2"-) = be r>0. (4.1) 1 “9a +0(;) 38r‘*°°s 032 (to 0+) ‘ 032 (to Zn.) = 0. f>0. 45 The first of the boundary conditions in (4.1) insures that there exists a jump in the displacement equal to the amount b; the second boundary condition states that the displacement gradient vanishes as we further move away from the dislocation; the third states that the tractions must be continuous across the dislocation line. Hotivated by the nature of the displacement field in the corresponding linearized problem. we again seek a solution of the form. n = G (9)0 (4.2) Substituting equation (4.2) into the boundary-value problem (4.1). reduces it to d - . 6.6 -— w',(3+ |Vu|2)—— =0, (4.3) G (0+) - 8 (2n-) = b . Traction continuity condition is automatically satisfied. Integrating (4.3). yields A u'(9) ‘3 C (1') s (4.4) 46 the constant of integration C possibly being a function of n. In (4.3) and (4.4) we have 2 s 1 IVUI2 = ‘3'3‘(6). r : (4.5) Recall the definition of the material shear response T (K) T22n“w'<3+n2). -~0 and -W<6§fl) centered at the dislocation at (-i. 0) and (5.41). (E>0.0<¢52W) centered at the dislocation at (9.. 0) Figure 4.1. The notation here is different from that used in the linearized problem. The equilibrium equation (2.41) for u(r. 9) is given as a . 3u 2w'au 1 a . an --(2w'-)+--—+---(2w'-)=o (4.14) 3r 3r r 3r :2 86 66 on Do where no is D with the twoiline segments x2=0. x12}; and x2=0. x1 _<_-2. deleted. The boundary conditions are u(r.1r)-u(r. -W)=b' forr>£ (4.15) 9 (on the dislocation line emanating from ('9. .0).) and u (r. 0*) - u (r. o‘) = b _ for r > 2 2 . (4.16) 50 (on the dislocation line emanating from (9. .0)). we also require that the displacement gradient vanish at infinity. We further impose the continuity of tractions across the twp dislocation lines: 1 u. *0(-)as r-u. a, T’ 032 (to 7) ‘ 632 (to ‘11): 0 for 1' > 2. s (4.17) C32 (to 0+) ' 032 (to 0" ) = O for r > 22. o In order to solve the boundary value problem (4.14) ' (4.17). we employ the technique of small deformations superposed on large deformations. The general theory of this technique in finite elasticity is discussed in Green and Zerna [22] and Ogden [23]. The displacement field in the presence of the two screw dislocations is assumed to be approximately the same as the sum of the displacement of the single screw dislocation in the non-linear field found in section (4.1). plus some small perturbation J (r.6) caused by the presence of the second ' screw dislocation. Accordingly. the displacement field u (n99 is now assumed to be of the form b ~ ~| I ‘ u (r. 9) = *2" '3 + u (r. 6) . Wu; < < 1. (4.18) where the first term is the displacement field if there was only the single dislocation 81’- (‘2,0 ). Our aim is to substitute (4.18) into the 51 boundary value problem (4.14)- (4.17) and linearize it based on lvu| < < 1. In order to do this we first linearize the various quantities involved in our problem. ‘From (4.18) we have Bu 3% — g , 3r 3r (4.19) au b 36 “" = -'+ -' on D. 36 2H 86 The first principal scalar invariant I1’ is calculated to leading order from equation (2.23) 11=3+|m|2=10+1, (4.20) where I0 and I are defined as I ' 3 + b2 - O o 4w2r2 (4.21) ~ 2b 31': I = . 2n r2 39 The strain energy density W(Il) and its derivatives with respect to 11’ are now represented by a series expansion about Io. 52 G (11) = s (10 +.f) = Q (10) + §'(I°> f , fi' (11) = fi'(zo) + d" (10) f , (4.22) ‘3" (II) = ‘31! (Io) + ém (Io);f . We can show by substituting (4.20) - (4.22) into equation (4.14) and retaining the leading order terms. that the governing differential equation (4.14) takes the form 2~ A! ~ AI 2" . 3 u W b2 .. au 2W 1 2b2 .. 3 u 2w'—+[2-- 3211”] —+[—+— 2211“] —-2-=0 (4.23) a:- r an 3r r2 r2 4n2r 36 where 11' and 1;" are evaluated at 10’ This is a linear differential equation with mug coefficients depending on r. We define the mantaheax modulus MK) by u(K) = 2 4'<3+<2>. ~ (4.24) Differentiating the material shear response function (2.44) with reapect to <. gives the tangent shear modulus. T' (K) as T'(~<)=2W'+4K2W". (4.25) Solving for W' and W'8 in terms of T‘(~<) and M(»:) from the above equations and then substituting them into the differential equation (4.23). leads to the simpler form. 326' - 136 1 326 M(<) 2 + [2M0 - r'(r<)] - — + r‘(<) --— = 0. (4.26A) 3r r 3r r2 862 Here we have h K : —- (4.263) Zwr The linear differential equation (4.26A) for Ci (r.6) has coefficients which depend on r. In the particular case of a pure power-law material T(K)= AK“. this cancels out and (4.26A) becomes a re-scaled Laplace _' equat ion . If we keep in mind the fact that our goal is simply to calculate the dislocation force J. and that to do this we will evaluate the J- integral on an infinitesimal contour surrounding the dislocation at (2,0). it becomes clear that we are only interested in the value of a near (2,0 ). Therefore we now specialize the differential equation to the region near the second dislocation located at (9 ,0). Letting r=22 in (4.263) gives S4 < = = K (= known constant) , (4.27) ‘1 and therefore the shear moduli. near (2. O). are I" (K) r' ((0), (4.28) M (K) M (‘0) . Thus. we replace (4.26) by 326 1 ea 1 32.”: 140(0) —-2-+ [2 a (no) - 10%)] -T-+ : (<0) 7 ..2 = c. (4.29) 3r r cr r s: in the vicinty of (i. 0). we now write this equation in terms of polar coordinates (E. o). (E>O. O<¢<2W)centered at the second dislocation. i.e. at (Z. 0). Figure 4.2. We change the variables from (r. a) to (5.6 ) with the aid of the transformation equations. 55 35 g+£ coso 3r \[62 + £2 + Zlgcoso 3¢ laino 3r ‘5ng + 22 + 2£Ecos¢ BE 36 laino , 3¢ 5 + £cos¢ 36 E _ 325 22 sin2¢ - ’ 3:2 4(52 + 12 + 2£€cos¢) 3 <1) 22 (Esind) + zsindacoso) 52(52 + 22 + 2250081)) (4.30) 56 The partial derivatives of 6 are then derived as a (£+2cos¢)2 Zsino(€+£coso ) -( ~s) = ~s '2 ~s + a, rut _ as “54 V €2+£2+252cos¢ : €Vf2+£2+251cos¢ lzsin2¢ 52+ Elcos¢ + izsinzo 1'). + E. + 00 g £2\[Ei+122+22£ coso 5VE2+ £2+252 coso 5151M +2 zzsino cos¢ ‘ ¢ £2 ’.J€_2+22+2£ (com ‘ (4.31) a 13-9 2251:1243 A ~ asinugncoscb) - — — = u. +2 11. + 36 r 2 2 £5 5: ' \[2 +2 +25 2cos¢ E \lc—2+22+251cos¢ (5+ £cos¢)2 £cos¢(£+1coso) as ‘1' Us + ¢¢ E 52 V €2+22+2€2cos¢ €sz+£2+2€1cos¢ -£sin¢ (5+2cos¢)-£2sin¢ case i. . ¢ 52 V 52+ 22+ZE£cos¢ an 33 33 Raine ' r - = -'(E+ icos¢) ' - .- a: 35 Bo E 57 Keeping in mind that g is small in the immediate vicinity of (t. 0). the partial derivatives in (4.31) are. to the leading order. ‘ a 2 29. 51213 2099 '—'(r B. ) ~ Zoos p E - '. E. + at r ’66 5 5° . 2 . 2 . lain ¢ ~ £31n o 2131n¢coso u. + 5. + a. 9 2 ¢¢ E 2 a e 5 ¢ sin¢ fig: ~C08¢ figE " 6 80¢ , (4.32) 3 u.e 2 22 sin¢ cos¢ -- —— ~ 9. sin ¢ 5. + E. + 36 r 55 E E¢ l coaz¢ ~ 2 cosz¢ ~ 22 sin¢ cos¢ ~ 2 "W + “’E ' 2 “’15 ’ E E E 32': 9. _ = 9‘ Sin¢ f1. + - COB¢ Ti. 0 68 E g Incorporating equations (4w32) into the differential equation (4~29) and regrouping the terms; we get 58 Z2[H(*°)c052¢ + t'(ro)sin2¢]fi. + 2:[t‘(21 . (4.39) which in viev of (4.34) requires 60 - v (0*) + v (25') = 1:. (4-40) (Recall that -TT<6<1r and O<¢<27r.) :. In order to impose (4.40) we need V(0) and V(2'rf), when ¢ = 0 (4.383) requires that we take p' = O and so (4.38A) gives v (0*) = v. (4.401.) When :9 = 2n . (4.383) requires that we take p.'= 2 and so. ZWC V (2n-) = D + (4.4013) {u(Ko) 1"(Ko) From equations (4.40A). (4.403). (4.40). we obtain b mum.) ' = \r" .° . > (4.41) C . 211‘. Therefore. the solution V(¢) takes the form b u(Ko) p'b v (M = — tan-1 —— tan¢ + -—-+ D. (4.42) 2n T'(Ko) 2 . or from equation (4.34) 2r Vr'( 0 and m >() are constitutive parameters. Figure SJ” An m ‘* alternative class of power law materials is the one proposed by Knowles [11). The second model has the advantage that it linearizes well about <=0. On the other hand. for large values of <. they are both of the same form. 67 For the pure power law materials described by the shear response (5.4). one can readily show that the secant Shear modulus and the tangent shear modulus take the form , m-l no.0) = j- ..o , (5.5) i and I m-l T'(Ko) = 'i'm ‘0 = m M(Ko) . (5.6) ' K i where K0 is described in (4.27) as KO = b/4n2 . The force on a dislocation in pure power law form of materials can now be calculated by substituting (5.5) in (4.52). it becomes -1 21 ' m J = b * KO . (5e7) 4N£KE In dimensionless form. (5.7) becomes 68 J b“ —— = —— .. (508) T b (4n£)me * * 3* = A graph of J* vs 2 is shown.in Figurs 5.2. Note that as 2 increases the force on a dislocation decreases which is expected since the effect of the second dislocation is negligible when the separation is large. Since the results here would be more accurate for larger values of 2. the range $2); > 1 is of primary interest here. We also note that for fixed separation 1. the force between the dislocations decreases with increasing hardening exponent m. In particular. this means that for softening materials (m < 1) the force is greater than that predicted by the linear theory; for hardening materials (m > 1) it is less. 69 Harmer: This study was concerned with estimating the value of the force between two screw dislocations in an elastic solid. Since the linear theory of elasticity predicted unbounded strains at a dislocation. the results predicted by such a theory were suspect. The present study was carried out within the fully nonlinear theory of elasticity. Since the force between two dislocations was defined as the rate of change of energy with respect to the separation between the dislocations. one way to calculate this force was to first calculate the total energy stored in the body. This was an almost impossible task to carry out exactly in a fully nonlinear theory. However. we observed that the value of the J-integral evaluated along a path surrounding one dislocation was precisely equal to the magnitude of this force. We exploited the path-independence of the J-integral in order to obtain accurate estimates of the force between two dislocations in the presence of nonlinearity. APPENDIX A In. ghapter 4 we have integrated We) as follows: Renal]. that. . C . 17(0) 3 s (A.1) (T'+M)+(T'-M)cos2¢ where C is a constant. By using trigonometry. M ... __ . (4.2) . c v(¢) =- 2 Integrating (A. 2) gives . 4 - 4 . cl-g 1 c v(¢)-v(o) = V(s)ds = - f d.+ — 4 . (A.3) 2' 2. 2M 0 o Thegprincinal value of the integral in (A.3) is i. ‘E tan-1 ( L1? HIM) ' (A.4) -1 r and therefor e . ‘70 71 1--‘—' \lli‘. C H ' _ / v(v)-V(0) = -'-- T tan 1 ( g7 tans) -‘-JL' (A.5) ‘ 2 1" M . .‘E -——-1 ---l -l :1 TI P|fit.l T. C + +_@ 9 M - 1' 2n where the integer p' is tO-be chosen from 1 "I (2p'-1)-<¢<(2 p‘ +1)- - (A.6) 2 2 Simplifying (A.5) shows that C 1 -1 M C P." V(¢) = V(0) + - tan ( -7 tans ) +‘- ' (A.7) 2 V Mt' T 2 V Mr' We now can integrate. 2: 2" M cos a - tan C anus] d3 = .1 2 ' (11.8) 0 d¢ T 0 1+ 3- tan ¢ T. Using (A.7). (A.8) becomes, 23\( r' ________. (A.9) 72 o (a A x .oumx .8 5.33035 5;) .35.? he o .3303 $20 :2” Saw: 73 m a 9.2mm. 3:323“. 95 5 «88 523220 . ma 2%: 74 mosquito; as: .828: m 5 5:32.20 .m.m Sam: 75 anfix 6:0 QHHX um Dohoucoo 383322“. of. é.m o .T mh=w_m 76, a 52:6 52:15:. a E 223836 of. .Hé Ezmi 77' m 533:.th :25 new 28 5:; 25.3335 2.: E cozflcomoamm .mé 83m; m cozonstma + A _._DEm db 78 22.39: 25. xi EA— ..oon 22. .3 $33 @2833 :35 45 Sam; _nE _AE EC‘ » 5;; S 5:25 .3 2a.“. eel-D X E _ n E 7/ 3 79 . _ _ |I\ . _v_._._ .11--- 4.. BxEA Viv; 11. 12. 13. 14 15. K.D. J.P. HIRTH and J. LOTHE RAJ. LARDNER H. KACHANOV GAIROLA J.D. ESHELBY JJI. HUTCHINSON J.R. RICE J.D. ESHELBY J.R. RICE C. TRUESDELL and W. NOLL J.K. KNOWLES J.K. KNOWLES M. BAKER and J.L. ERICKSON .LK. KNOWLES and ELI STERNBERG R. ABEYARATNE and J.K. KNOWLES B. BUDIANSKY and J.R. RICE J.K. KNOWLES and ‘ ELI STERNBERG REFERERCBS 1982 W J. Willey and Son. New York 1974 W :2 D. 1 o I E . University of Toronto Press 1982 W 1979 W edited bx Roberto. 221L_L;_22i 1955 5W 1979 W Mania; The Technical University of Denmark 1968 1.31.431an ‘ 3.1.4.212 1974 Wanna—2Q}. 1973 MW ' 1965 W I 1 . E 1 l' E p] . W Springer-Verlag 1979 W 1977 WWW Series Bo 13.. P. 400 1954 MW 1971-72 W HEW 1986 WALL—All 1973 WWALMQL 1980 l‘_Q£_£135Li£iLxA—n&—&l:lln 18. 19.’ 20. 21. 22. 23. 24. 81 J.K. KNOWLES and ELI STERNBERG 'R. ABEYARATNE L. ZEE and ELI STERNBERG R. ABEYARATNE A.E. GREEN and W. ZERNA R.W. OGDEN PBOEBUS J. ROSAKIS and ARES J. ROSAKIS 1981 1980 'fl979 1983 1954 1984 1986 W. Oxford University Press 11 -1. E] . I E . John Wiley and Son. New York WW ”7111111711leer31131111111111“