Bifurcational instability of an atomic lattice

J. Mech. Phys. Solids, 1975, Vol. 23, pp. 21 to 37. Pergamon Press.

BIFURCATIONAL By Department

Printed in Great Britain.

INSTABILITY OF AN ATOMIC LATTICE

J. M. T. THOMPSON and P. A.

SHORROCK

of Civil and Municipal Engineering, University College, London (Received 24th Jury 1974)

IN A recent article N. H. Macmillan and A. Kelly (1972) have confirmed on the basis of a linear eigenvalue analysis that a m~hani~lly stressed perfect crystal can exhibit a bifur~tioM1 instability at stresses ranging to 20 per cent below that of the limiting maximum of the primary stress-strain curve. The question thus arises as to whether the branching point is in a non-linear sense. either stable or unstable. In the former case, perfect and slightly imperfect crystals would be. capable of sustaining stresses over and above the eigenvalue critical stress. In the unstable case, however, this eigenvalue stress would represent the ultimate strength of a perfect solid, while an imperfect crystal would fail at a limiting stress substantially below the eigenvalue. At 20 per cent below the limit point such a branching point is essentially distinct, and the nonlinear stability analysis needed to answer this question is provided by a recently established general branching theory for discrete conservative systems. Often, however, the two critical equilibrium states are much nearer than this, and the branching theory is here suitably extended to cover the case of near-compound instabilities. An illustrative study of a close-packed crystal under uniaxial tension is next presented. A kinematical&admissible displacement field is employed and a bifurcation point is located on the primary equilibrium path just before the limiting maximum, the eigenvector being associated with a transverse shearing strain. Under these conditions a corresponding small transverse shearing stress would represent an ‘imperfection’, and the non-linear branching problem is next studied using the new general theory. This shows (in excellent quantitative agreement with an ad hoc numerical solution) that the branching point is non-linearly unstable with a quite severe imperfectionsensitivity which manifests itself as a sharp cusp on the failure-stress locus.

THE RECENT work

by MACMILLAN and KELLY (1972) is typical of a continuing study of the mechanical properties of perfect crystals within a static Newtonian approximation for the atoms. Semi-empirical inter-atomic potentials of the Lennard-Jones and Born-Mayer type are assumed, and the equilibrium and the stability of the lattices are examined under prescribed conservative loading. Of particular interest is the confirmation by Macmillan and Kelly on the basis of a linear eigenvahze analysis that a mechanically stressed perfect crysta1 can exhibit a bifurcational instability at stresses ranging to 20 per cent below that of the limiting maximum of the primary stress-strain curve. Such an instability typically occurs under a direct tensile stress ~7when, at a finite direct strain E, the bifurcation destroys some basic symmetry of the system by allowing the development of a shearing strain. To date, these instabilities have only been studied under a linear eigenvalue approximation, and the question naturaliy arises as to whether the associated branching point is in a non-linear sense either stable or unstable. In the former case, perfect and slightly imperfect systems would be capable of sustaining stresses over and above 21

22

J. M. T. THOMPSON and P. A. SHORROCK

the eigenvalue critical stress. In the unstable case, however, this eigenvalue stress would represent the ultimate strength of a perfect system, while an imperfect system would fail at a limiting stress substantially below the eigenvalue. In the type of situation that we have outlined we can observe that a small shearing stress would represent an imperfection destroying the basic symmetry of the system. Within the framework of the static Newtonian approximation, the aforementioned studies are concerned with the equilibrium and stability of a discrete conservative mechanical system described by a total potential energy function V(Qi, A) where the Qls are a set of n generalized coordinates describing admissible displacements of the atoms and A is a loading parameter describing the overall stress level. They can thus be viewed in a framework of a recently completed general theory of elastic stability (THOMPSON and HUNT, 1973) which offers established perturbation procedures for examining the non-linear features of the branching points. At 20 per cent below the limit point such a point of bifurcation is essentially distinct, and closed-form general results are available to delineate the non-linear features. In the crystals field, however, the two critical equilibrium states on the primary stress-strain curve are often much nearer than this, and the branching theory is here extended to cover these near-compound instabilities. As an illustration, the general theory is next applied in a pilot study of a closepacked sheet of atoms under uniaxial tensile stress eil perpendicular to a closepacked direction (see Fig. 5). A planar, kinematically-admissible homogeneous displacement field with only two degrees of freedom is employed, and a bifurcation point is located on the primary equilibrium path just before the limiting maximum. The eigenvector is associated with a shearing strain slz, and a small shearing ‘imperfection’. The non-linear problem is stress gi2 is introduced as a corresponding then solved using the new branching theory for near-compound failures and the bifurcation is shown to be non-linearly unstable with a quite severe ‘imperfectionsensitivity’ on the failure-stress locus. The predictions of the theory are shown to be in excellent agreement with the results of an ad hoc numerical solution. It is not suggested that the results of this preliminary and illustrative qualitative study have quantitative relevance for any particular actual three-dimensional lattice. It might be of interest to note, however, that our planar and homogeneous displacement field for the single close-packed sheet of atoms can be used to generate an admissible homogeneous field for a face-centred cubic crystal or an admissible but strictly non-homogeneous field for a close-packed hexagonal crystal. The general branching theory for distinct critical points is finally invoked to demonstrate the existence of sharp cusps on the failure-stress loci of perfect crystals. 2. GENERAL THEORY 2.1 Quantum mechanical foundations A suitably general starting point for a discussion of the mechanical properties of many solids is provided by the steady-state quantum mechanical theory. In this we can eliminate the electrons from consideration by means of the Born-Oppenheimer adiabatic approximation (BORN and HUANG, 1954) to obtain the Schrodinger equation for the nuclei alone, and we can include in the effective nuclear potential the energy of any applied conservative forces whose mechanical influence we wish to examine.

Bifurcational instability of an atomic lattice

23

Now the solutions of the quantum mechanical problem are intimately related to the solutions of the corresponding Newtonian problem. Within the harmonic approximation, for example, the quantum energy levels are related in a simple fashion to the classical vibration frequencies, so many Newtonian results can be readily quantized (see, for example, DEAN, 1967). In particular it is clear that a steady-state quantum mechanical solution can only be expected in the vicinity of a stable Newtonian equilibrium state. A logical first step towards a quantum mechanical solution is thus to make a static Newtonian stability analysis, and indeed this step will often supply all the information needed for a bulk mechanical description of the solid. MacMillan and Kelly, for example, make an analysis of this type using semi-empirical interaction potentials to construct the effective nuclear potential. The appropriate tool for this Newtonian analysis is the general theory of elastic stability for discrete conservative systems, and in line with this we can write the effective potential Y as a single-valued function of n generalized coordinates Q, defining admissible positions of the atoms and a loading parameter A defining, for example, the intensity of the applied forces.

2.2 General branching theory If, as seems likely and as confirmed by our pilot study, the branching points located by MacMillan and Kelly correspond to un~tabie-symme~c points of bifurcation we are faced with two separate problems depending on whether the eigenvalue critical stress is either close to or at some distance frown the limiting maximum of the fundamental path. If there is an appreciable separation we can appeal directly to the general theory for a distinct branching point on a rising primary path to predict the type of behaviour shown schematically in Fig. 1. Here, A might, for example, represent a direct tensile stress 6x 1, Q, a direct strain e1 r, and Q, a shearing strain a12. The primary equilibrium path lying in the (A, Q&plane loses its initial stability at the eigenvalue critical point C, distant from B, at which it intersects a falling secondary equilibrium path. On the plot of A against the critical coordinate Q, the secondary path is locally parabolic and an initial imperfection (such as an initial value of Q,, equal say to Qy) would give rise to a continuous path rounding-off the corner as shown. This makes the failure load AM sensitive to small values of the imperfection and the theory predicts that a plot of AM against Qy will follow locally a two-thirds power-law giving a sharp cusp as shown in the Figure. If, however, the branching point C is very close to the maximum B, this distinct branching theory, although still valid very close to C, may give misleading results. In this circumstance it is safest to approximate the true analysis (THOMPSONand HUNT, 1973) by assuming that the critical points are in fact absolutely coincident, and we shall now extend the general theory to embrace this coincident situation, We shall then use the new results directly in our crystallographic study. 2.3 The Hill-top br~ch We consider then a discrete conservative system with the energy function V{Qi, A) and it will be sufficient for our purpose to suppose that we have only two (active) coordinates Q, and Q,. Taking V to be symmetric in Q, the equilibrium equations

24

J. M. T. THOMPSONand P. A. SHORR~CK A

cfi

Q,

FIG. 1. A distinct branching point on the primary stress-strain

curve of a crystal.

Di = 0 are assumed to give equilibrium paths with the form of those shown in Fig. 2, and to study the hill-top branch in detail we introduce the new incremental coordinates ul= QI, u2= Q,-Q:. In terms of these fixed coordinates we have the energy function

(1)

D(Ui, A) E ‘V(U~,Q(~‘+u,, A)(2) To establish the local form of the secondary equilibrium path we now write it in the parametric form u2 = UZ(Ul), A = Nu,), (3) and we can substitute these expressions into the equilibrium equations Di = 0 to give the identities (4) Mu,, uz(t49 Nu,)) = 0, D,(u,, uz(uI), N~I)) = 0. We next generate a perturbation scheme (THOMPSON and HUNT, 1973, p. 127) to determine derivatives of u2(u1) and A(ui) by differentiating these identities repeatedly with respect to u1 to obtain D,,+D,,u$~)+D;A”’ = 0 (5) D,, +D,,u$‘)+D;A(‘) = 0’ (6) (D,,,+D,,,U~~~+D;,A’~~)+(D,,,+~~~~U~~~+D;~A~~~;U~~~+D~~U~~~+ (l)+D;I\(l))A(l)+D;A(z) = 0, (7) +(%1+%2u2 (D2~~+D2l2u(:~+D;,A~1~)+(D22l+D222u(:~+D;2A~1~)u’,‘~+D22u~~+ +(%1+%2u2

(1)

+

D”A”‘)A’l’+D;A’2’ 2

= 0,

(8)

Bifur~tional

25

instability of an atomic lattice

1

FIG. 2. A hill-top branching point on the primary stress-strain

curve of a crystal.

where a bracketed superscript indicates the number of differentiations with respect to u1 and a prime denotes differentiation with respect to A. We now evaluate these equations at the critical equilibrium state C, noting that because of the assumed symmetry of the energy function we have etc.,

D$ = 0;” = 0,

Dtz2 = 0’;’ = D;“2 = 0,

D:,,

= 0.

(9) In particular, 0; is .diagonal, the coordinates U, being thus principaI coordinates, and because we are postulating a compound critical equilibrium state we have the two zero st~~il~t~ ~~e~cients DC 11 =0 ,

DC 22=

0*

(10)

With these zero values, (5) gives on evaluation no information. For a limit point in the (A, Q&plane the general theory shows that we must have 0;” # 0, and with this assumption, (6) yields the first derivative of interest, viz. A(l)C = ()

(11)

(12)

In the situation envisaged we have D:12 # 0, so (7) gives us the second derivative of interest, viz. (IN = 0 , u2

(13) (14)

26

J. M. T. THOMPSON and P. A. SHORROCK

and finally (8) yields Dzlr +D;A’*‘/c = 0

(15)

A(Z)C= -D, 12/D;IC>

(16)

so that which is an expression for the initial curvature of the secondary equilibrium path at the branching point, corresponding to the first non-zero term of the Taylor series expansion of h(u,) about u1 = 0. So far we have only considered a ‘perfect’ system, and we now identify a corresponding imperfect system by the energy function D(u, A, E) where E is an imperfection parameter, the vanishing of which will serve to retrieve the perfect system of our earlier consideration. The equilibrium solutions for such a system with E # 0 will have the form of the light lines in Fig. 3 and we are primarily interested in the maxima of these paths for which we write A= AM, Ui= UM, E= EM.

(17)

FIG. 3. The hill-top branching point for perfect and imperfect systems. We would like to construct a series solution for A”(aM) along the trace of maxima (MA4 in Fig. 3) and to do this we write this locus in the parametric form A = AM(@),

u2 = u~(z.$),

E = s”(z$).

(18)

The maxima are firstly equilibrium solutions and secondly critical equilibrium solutions, so clearly they must satisfy D, = 0 and A = /D,[ = 0, and characterizing identities are written D,(UF, u~(u”;“), A”(Ur”,, &M(#F))E 0,

(19)

D,(UK GUY), AYuY), EYUY)) = 0,

(20)

A(& uj+y),

A”(ur), Ed)

= D,1D22--L)122 = 0.

(21)

Repeated differentiation of these identities with respect to UF and evaluation at the critical point of the perfect system, r,$ = 0,

A” = AC, sU =: 0,

(22)

Bifurcationalinstabilityof an atomic lattice gives after some algebra the required derivatives EM1Oc= 0 , AMG)C= 0 sMP)C=

27

(23) (24) (25)

T2(~,,2,D,)(d,,,;D,2,)‘jC,

AMVK = -2D,,,/L&jC.

(26)

Here, we have assumed, as is the case in our later application, that @#O,

b$=o,

(27) where a superimposed dot denotes differentiation with respect to E. It is instructive to note that AM@fC= 2A’VC_ (28) If we now consider the Taylor series for A~(~~) and e”(@) about RF = 0 and retain only the first relevant terms, then AM = AC-t &A~(2)c(U~)2, (29) EM

--

_r E”(2qUy)2,

(30) where the leading derivatives AMC2jCand &“(2)Chave been determined by our perturbation scheme. Eliminating u “: between (29) and (30) gives the required local form of the imperfection-sensitivity relationship h”(cM) as AM-AC = rt(~~/L);C)(D~22/L)~12)~EM. (31) The details of this compound critical point are now apparent and are shown in Figs. 2 and 3. The main difference from the distinct branching point is that the AH or A”{Qt) imperfection-sensitivity is now less severe since the two-thirds power-law cusp has been replaced by a locally-linear relationship. Now in the crystals field (MACMILLANand KELLY, 1972) we often encounter a branching point which is strictly distinct but which is nevertheless quite close to the maximum of the primary stress-strain curve. In this circumstance the general results with the two-thirds power-law will apply very close to the branching point, so that there will indeed be a sharp cusp. Once we are some distance from the two critical points, however, it will seem as if they are coincident and we can expect the linear law to be more valid. We shall indeed see in our application that although the critical points are strictly distinct, an a~~roxi~ate analysis in which they are considered strictly coincident gives extremely good agreement with a numerical solution for the whole range of practical interest. We might finally observe that the compound theory is actually easier to apply since it requires energy derivatives of lower-order than those in the distinct theory. The former does, for example, only require the third-order partial derivatives D$,, and D:,, while the distinct branching-point involves the fourth-order partial derivative Dyl 11. 3. IMPERFECTION-SENSITIVITY IN CRYSTALFRACTURE We consider a close-packed crystal of iV identical atoms so that we have initially 3N degrees of freedom corresponding to the coordinates X, of the atoms.

28

J. M. T. THOMPSONand P. A. SHORROCK

3.1 The single bond We shall assume that the potential energy of interaction between any two atoms is some function u(s), where s is the distance between centres, so that we have only spherically-symmetric central-force two-body potentials. Considering further only nearest-neighbour interactions there will be no inter-atomic forces in the unloaded crystal which can thus be viewed as a close-packed array of spheres of diameter d where dv 0 z s_* = . In particular, we shall take the Lennard-Jones KELLY (1972) for A in which

(32) potential used by MACMILLAN and

v(s) = A(Bs-i2 -s-6),

(33)

where A and B are physical constants. Using (32) we find immediately that B = td6 2

(34)

and writing s = yd we obtain v(y) = Ad+($y-12

-f6),

(35)

which relation is shown in Fig. 4. The inter-atomic tensile force T is given by T = dv/ds = (6A/y’d’)(l

FIG. 4. The interatomic

-y+)

(36)

potential and cohesive force.

and this relation (shown in Fig. 4) attains its maximum value given by Td7/6A = (6/13)(7/13)“”

(37)

for y = (13/7)“6.

(38) Derivatives of v with respect to y which will be required later can finally be written down as vY= 6Ad-+(-y-13+y-‘), vyy= 6Ad-6(13y-‘4-7y-8), (39) etc.

29

Bifurcational instability of an atomic lattice

3.2 The close-packed sheet

In our close-packed crystal we erect a set of three rectangular axes OXi orientated so that Ox, and Ox, lie in a close-packed plane of centres of atoms with Ox, lying along a close-packed column as shown in Fig. 5. Here it will be seen that we have chosen to locate the coordinate origin mid-way between centres of atoms.

t

3

Unlaaded state p=\15/2,&O,P=O

-

FIG. 5. The close-packed sheet of atoms in the unloaded state.

We consider next a displacement field with components Ur(Xj) SO that an atom originally at xi will after deformation have the coordinates Xi= Xi+Ui(Xj)

(40) (with, as elsewhere, use of the usual abbreviated notation for subscripts). If we restrict attention to homogeneous deformations of the crystal, we can now write Ui = CijXj,

(41)

so that Xi = X,+CijXj. (42) Choosing now the same finite strain definition as MACMILLAN and KELLY (1972) we can write Eij= +(Cij+Cji+C,iC,), (43) but as they observed it is convenient in calculations of this type to retain the Cij’S themselves as our deformation measures.

With our present restriction to homogeneous deformations we now have only nine degrees of freedom for our crystal which we associate with the Cij coefficients. Of these nine degrees of freedom three will be associated with rigid-body rotations and will be eliminated by suitable constraints. We propose to load the crystal by a uniaxial tensile stress cl1 along the x,-axis, and we shall constrain the atoms in the xg = 0 plane to remain in that plane so that we must set c31 = c32 = 0. (44)

30

J. M. T. THOMPSONand

P. A. SHORROCR

Secondly, atoms lying on the x,-axis wiI1 be constrained to remain on that axis so that c21 -- 0. (45) We are left now with six degrees of freedom corresponding to the six non-zero Cij coefficients. Considering the close-packed plane x3 = 0, the applied stress cl, will tend to separate the close-packed columns as shown in Fig. 6. It would then seem plausible that at large direct strain ~~~ these columns would become rotationaliy unstable so that a shearing strain .s12would develop as shown in Fig. 7. To explore this possibility firstly for just a single sheet of atoms we have the three non-zero coefficients err, cl2 and cZ2 but we shall allow the plane only two degrees of freedom by assuming (perhaps somewhat arbitrarily) that the close-packed columns experience no change of length. The deformation can then be described conveniently by the resolved distance between the columns which we write as pd and

Fundamental

State

p= I.O,B=O, PQ40

FIG. 6. The

close-packed sheet of atoms in the fundamental state with p = I *O. Post-critical state p=l.O,6’=9”. P-O.325

FIG. 7.

The close-packed sheet of atoms in the post-critical state with p = I.0 and 0 =: 9”.

Bifurcational

instability of an atomic lattice

31

rotation of the columns 8 as shown in Fig. 7. The planar displacement coefficients can then be written in terms of our two generalized coordinates 8 and p as cl1 =(2/J&-l,

cl2 = sin 8,

cz2 = cos 8-1,

with czl = 0 as already specified. The sheet is loaded primarily by the direct stress (~r~, and as an ‘imperfection apply additionally a small shearing stress 612,

(46) we

3.3 The four-atop unit cell It is easy to see that with our assumptions the total potential energy of the infinite sheet is simply a multiple of that of the unit cell (of four atoms) shown in Fig. 8. It will therefore be useful to focus our attention on this unit cell so that our system is precisely defined.

FIG. 8. The four-atom unit cell in the undeformed and deformed states.

With our consideration of only nearest-neighbour interactions the unloaded unit cell rests as shown in Fig. 8(a) with no inter-atomic forces. The two central atoms are assumed to maintain their unloaded distance d and the two independent generalized coordinates 8 and p serve to define the deformation of the unit cell. The two convenient dependent variables K and fl of Fig. 8 can be expressed in terms of 8 and p as a2 = &+p’-p

sin 6,

p” = $+p2+p

sin 8.

(47) The unit cell is loaded by the direct axial forces F, which are related in a simple fashion to CT~ 1 and by the shearing forces L which are related in a simple fashion to pi 2. The potential energy of these forces is -2Fpd-Ld

sin 0

(48)

which is approximated here by -2Fpd-Ld8

(4%

J. M. T. THOMPSONand

32 so

P. A. SHORROCK

the total potential energy of the unit cell can be written as V@, P, F, M) = 2u(c@, P)) + 2v@(B, P)) - 2FPd- MB.

(50) Here, we have written Ld = M to emphasize the fact that our energy expression is exact if we care to think of the central atoms as loaded by a moment A4 rather than by the forces L. Derivatives of V can now be written down as VP = 2v,a,+2vaj?,-2Fd,

V, = 2v,a,+2v&--M,

(51)

etc., where aa Zp-sin ap= ap = ~._._ 2a

&

Q

,

-pcos@ = ~__2tt ,

ct, G z

(52)

etc. For the ‘perfect’ system for which M = 0 the equilibrium equations V, = 0

v, = 0,

(53)

yield a primary (or fundamental) solution with 0 = 0 given by 4v,y, = 2Fd,

(54)

where y denotes either a or /I, for which P = Fd7/6A = y-8(1 -y-6)(4$-

1)f.

(55)

In terms of p, equation (55) becomes (561

p = 2P~(p2+S~-4-~P2+~~-7~,

which relation is shown in Fig. 9. Now on this primary path the symmetry of the system ensures that the derivative V,, is zero, so that the st&lity of the path is-determined by the two stability coefficients V$ and V,‘, where F denotes evaluation on the path. The vanishing of F;,“,simply

0.’4-

Primary

3-

path

Secondary path 2-

Jmperfd system 0.

I-

I 06

I o+l

I

I

I.0

I.2

I

1.4

I

I.6

P

EIG.9. The

numerical solutions for the infinite sheet of atoms showing the response of the perfect system and one imperfect system.

Bifurcationalinstabilityof an atomic lattice

33

corresponds to the maximum of the path while the vanishing of V$ locates a point of bifurcation at which a o-type deformation can develop. Then, setting v& = 0, (57) we find the critical value of y, viz.

yc = (7/4)“6,

(58) and we observe that the bifurcation occurs just before the limiting maximum of the primary path as shown in Fig. 9. Solving the equilibrium equations numericalIy for the ‘perfect’ system with no shearing stress and for ‘imperfect’ systems with N zz ~d616A # 0 we have obtained the curves given in Figs. 9 and 10. The deformation of the sheet of atoms is drawn to scale for one of the predicted post-critical equilibrium states in Fig. 7.

I

I

I

,20

0

-10

FIG. 10. The unstable-symmetric

I

/

IO

20

branching point on a plot of the direct stress parameter P against the shearing strain parameter B.

3.4 Application of the general theory Identifying A with P, u, with 0, and E with AJ, the asymptotic results of our extended general theory can be written in the terms of our crystallograp~c study as d2P ’ =de2 PM -PC = + (

v,s,l Jq”,

ri,/VJ( VbPP/VBOy)fNMIC.

(59) (60)

Equation (59) gives an expression for the initial curvature of the secondary equilibrium path, while (60) gives a failure-stress locus on a graph of PM versus NM. Equations (59) and (60) apply strictly to a compound critical point at which the energy derivatives should be evaluated, but it is our intention to approximate the analysis and to proceed as if there were such a compound point. To do this, we can simply adopt these general expressions and evaluate the derivatives either at the ~r~~c~~~gpoint or at the limit point. 3

J. M. T. TH~MPWN and P. A. SHORR~CK

34

We have chosen to make the evaluations at the real branching point denoted by C, and the required values can be listed as

a3v c -aezaf = v&

a2v c

=

-87*78A,d”,\

apap

= I’;” = -12A/d6,

-ggyy

dljec

a2v c

1 = - 6A/d6,

a3vc -= = - 351*1A/d6. ap3 - 5% I Using (61) we obtain immediately the required solutions which are compared with the results of the numerical analysis in Figs. 11 and 12. The agreement is seen to be exceptionally good. 3.5 The three-dimensional lattice If we care to make the rather drastic assumption that all bonds between the closepacked planes xj = constant retain their original length d we find that we can stack identical deformed sheets to generate kinematically-admissible displacement fields for a three-dimensional lattice. In the case of a face-centred cubic crystal the displacement field will be homogeneous, but in the case of a close-packed hexagonal crystal the field will be strictly non-homogeneous. For the face-centred cubic crystal the displa~ment coefficients would be

Cij =i

I

43 $100

FIG. 11. The post-critical

sin0 6

cos e-1

&(3

- (p2 4%~’ - *)2/p’ - S)/~P cos2 O)&- 11 ’ -4j tanO(p’--;))/2p

(62)

response of the perfect system as predicted by the general theory and the numerical

solution.

Bifurcational instability

of an atomic Iattice

3.5

0.1 -

N 0 FIG. 12. The failure-stress

1 0.2

I 0.1

I 03

locus on a plot of the direct stress against the shearing stress as predicted by the general theory and the numerical solution.

so that on the primary path we have

9 0

(63)

&%3-(P”-a,‘/P’,f-11

while the eigenvector of our distinct branching point is given by (64) 3.6 Further work The work reported in the present paper is essenti~~y exploratory in nature aiming to delineate phenomena rather TV to derive quantitative results, and for this reason we have made some quite severe approximations. We have, for example, considered oniy central forces, and have restricted attention to nearest-neighbour interactions. Since preparing this paper we have continued our study of the singie sheet of atoms and have now eliminated the approximation in passing from (48) to (49); we have relaxed the lateral constraint to allow for the full Poisson’s ratio contraction, and have studied the related problem in which c21 is allowed to be non-zero. In all cases the numerical results are very close to those of the present pilot study.

4. CONCLUSIONS The general theory of elastic stability for discrete conservative systems has been extended to yield local information about a hill-top branching point, and its predictions are shown to be in excellent agreement with the results of a numerical study.

36

J. M. T. THOMPSON and P. A. SHORROCK

This suggests that the theory, which is easily extended for n degrees of freedom, can be used with confidence in problems of crystal fracture, and it is worth noting that the hill-top branching analysis involves energy derivatives of lower-order than those in a conventional distinct analysis. In the pilot study of an atomic lattice loaded by uniaxial tension we have shown the existence of an unstable-symmetric point of bifurcation which precipitates the development of an unexpected shearing strain violating the basic symmetry of the system. We can thus expect the Macmillan and Kelly eigensolutions to be likewise non-linearly unstable, so that the stress eigenvalues will represent the ultimate loadcarrying capacities of the perfect crystals. A perfect crystal would thus ‘snap’ dynamically from the unstable bifurcation, and because of its extremely high strainenergy density it can be expected to shatter immediately with great violence. The final ‘imperfection-sensitivity’ curve of Fig. 12 is essentially part of a failure-stress locus with PM representing crl 1 and N representing C, 2. The general theory for distinct branching points shows that this failure-stress locus must exhibit locally a sharp cusp on the direct stress axis. These conclusions could be highly relevant to fracture mechanics studies, since an unstable bifurcation in the tensile zone at the tip of a crack could be a mechanism for destroying the symmetry of a plane propagating crack. Because we have here restricted attention to homogeneous deformations we have employed a small symmetry-breaking shearing stress as an imperfection. It is clear, however, that a dislocation would be equally valid as a ‘trigger’ for the imperfectionsensitivity, which can thus be seen as an asymptotic manifestation of the weakening action of a dislocation.

Note Added in Proofs Since the present paper was completed, the writers’ attention has been drawn by Professor R. Hill to a forthcoming paper (see R. Hill, On the elasticity and stability of perfect crystals at finite strain, Math. Proc. Camb. Phil. Sot. 7 (1975). In press). His paper emphasizes the need, in calculations of the present type, to consider carefully the tractions applied to the homogeneous element, since the instability loads naturally depend on their precise definition. In the work reported here, we have prescribed dead loads and to avoid a bifurcation in the unloaded state we have added what seems to be an intelligent rotational constraint. As an alternative to dead loads one could consider various types of follower stresses, although it is not clear how these could be realized in an experimental test. A long-term aim is clearly to use the calculated strengths to predict failure in a position of stress concentration in a loaded body, but since the elastic environment in such a body will vary from problem to problem, it is far from clear which is the most realistic loading to use for our homogeneous calculations.

REFERENCES BORN, M. and HUANG, K.

1954

DEAN, P.

1967

Dynamical Theory of Crystal Lattices. Oxford University Press. J. Inst. Maths Applies 3, 98.

Bifurcational instability of an atomic lattice MACMILLAN,N. H. and KELLY, A. THOMPSON,J. M. T. and HUNT, G. W.

37

1972

Proc. Roy. Sot. A 330,291.

1973

A General Theory of Elastic Stability. Wiley,

London.