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Edges and Wedges
G. Saada
To cite this version:
G. Saada. Edges and Wedges. Journal de Physique, 1989, 50 (18), pp.2505-2517.
�10.1051/jphys:0198900500180250500�. �jpa-00211077�
Edges and Wedges
G. Saada
Laboratoire P.M.T.M., C.N.R.S., Université Paris XIII, 93430 Villetaneuse, France (Reçu le 28 février 1989, accepté le 17 avril 1989)
Résumé. 2014 Nous étudions le champ de contraintes dû à une inclusion polyhédrale et montrons
qu’au voisinage des arêtes, le champ de contraintes est singulier et varie comme Ln (U/U0) où
U est le carré de la distance à l’arête et U0 une constante déterminée par les conditions aux limites.
Nous discutons un certain nombre d’applications pratiques.
Abstract.
2014We study the stress field of a polyhydral inclusion and show that close to the edges
the stress field varies like Ln (U/U0). U is the square of the distance from the wedge and U0 a constant determined by boundary conditions. Applications are discussed.
Classification
Physics Abstracts
46.30C - 46.30J
-61.70G
-62.20F
1. Introduction.
The calculation of the internal stress field resulting from non uniform stress free deformation is crucial in many problems of solid state physics : crystal plasticity, twinning, martensitic
transformation, coherent precipitation, epitaxial growth, thermal dilatation, magnetostric-
tion. In most situations the stress free deformation is restricted to a given volume
V. This remark has been the starting point of the development of the so called inclusion model due to Eshelby [1].
Although Eshelby’s analysis applies to inclusions of any shape some specific problems arise
when the inclusion has a polyhedral shape, quite a common situation.
The long range stress field, i.e. the stress field at distance r large compared with the size D of the inclusion, is not changed by the occurrence of wedges or vertices. The local stress field however may be drastically changed resulting in stress concentrations which may
provoke plastic deformation or fracture.
The purpose of this paper is to study this point specifically and to give some applications of
the results to physical problems of interest.
Nonlinear elastic effects, anisotropy of the elastic constants, inhomogeneity of the moduli
certainly influence the exact expression of the results, however, they should not change the general conclusions. Therefore we restrict ourselves to the case of linear elasticity and assume
that the elastic medium we consider is characterized everywhere by its shear modulus IL and Poisson’s ratio v.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500180250500
In this paper we shall make use thoroughly of Krôner’s [2] and Kondo’s [3] analysis. Since
there exist very good accounts of this theory [4, 5] we only sketch the general results and notations we use. On the other hand we shall make an extensive use of different kinds of
8 -functions, very useful accounts of which are given in [6-8]. The definitions and properties
relevant to this paper are given in the appendix.
We assume the plastic distorsion tensor pP to be non zero inside a volume V and 0 elsewhere. Then 13P reads
B is a constant second rank tensor.
The plastic strain field eP is defined as the symmetric part of 13P. Then
where E is the symmetric part of B.
’
The incompatibility TI is the symmetric tensor
(êikm is 1, - 1, 0 following ikm is an even or odd permutation of 123 and is zero when two indices are equal, Einstein convention for the summation of dummy indices is used).
Let x’ be a symmetrical second order tensor such that
is satisfied, together with suitable boundary conditions.
Then the stress function x is defined by
and the stress tensor is calculated as
2. One dimensional problems.
2.1 GENERAL RESULTS.
-The case where the stress free strain is restricted to a half space bounded by a plane P has been treated extensively [9-13] ; we just quote a few results.
Equation (3) reads
n is the normal to the plane, pointing outside from the region where E differs from 0.
8 p is a dipolar distribution defined in appendix 1. Let the plane P be the plane
xl Ox2, E differing from 0 for positive X3. Then 6p is simply 5’ (X3)’ the derivative of the Dirac
5 distribution.
From formulae (7-9) one easily obtains the non zero components of the stress field
sgn X3 equals 1 when X3 is positive and - 1 when x3 is negative.
Therefore, either F is 0 and there is no stress field, the wall is said to be a Nye wall, or F
differs from 0 and the stress field is homogeneous in each half space.
Conversely if the strain tensor E is given, we choose the principal axes of E as the axes of
coordinates. Let Ei be the principal values of E. Equation (8) shows that the condition for the existence of Nye walls is that at least one Ei be zero, the two others being of opposite signs.
Assume this is the case and let El be zero, then the normal n to the possible Nye walls is
defined by
If besides the dilation E2 + E3 is zero, then the possible Nye walls are perpendicular.
2.2 DISCRETE DISLOCATION DISTRIBUTION.
-In the case of plastic deformation one has discrete dislocation lines with discrete Burgers vectors. The dislocation density tensor
a is defined as
When 13P is restricted to a half plane, this formula reads
Then the previous formulae apply to the average value of the true (discrete) dislocation
density. The Nye walls are the so-called low angle grain boundaries. If the distribution of dislocation is periodic, it has been shown [11, 12] that besides the stress field calculated by
formulae (10) there exists a fluctuating stress field decreasing exponentially with the distance
from the wall.
Therefore the continuum approach is valid at distances from the boundary, larger than the period f of the dislocation distribution.
We will come back to this problem in section 5.
3. Two dimensional problems.
3.1 GENERAL INTRODUCTION. - Let us consider a prismatic closed surface (Fig. 1). The edges are parallel to Oxi, they intersect the plane xl Ox2 at O102 03 04. This situation is
general enough for our purpose. We study the situation where the stress free strain is constant, non zero, inside the prism, and 0 outside.
Obviously the plastic incompatibility will be non zero on the faces of the prism (we shall call
it surface incompatibility) and on the edges (we shall call it edge incompatibility).
Our first step will be to calculate the incompatibility of a wedge (Fig. 2). This problem has already been treated along similar lines (10, 14) and general expression for the stress has been
obtained. But problems connected with boundary conditions remain and the analysis of the
incompatibility deserves more attention.
Fig. 1. Fig. 2. ~x’-2
Fig. 1.
-Intersection of a prism parallel to Ox, with the plane Oxz X3. The axes OXI Xz X3 are not
indicated. The 0« are the intersection of the edges with the plane X3 Ox2. O102
=LI, 1 oa M ) = (u«)112 oa M
=xa.
Fig. 2.
-The intersection of the wedge with the plane xi Ox2. The axis OX3:t are obtained by rotating Ox’± 2 of + 2013 .
°They are not represented.
Therefore we consider the situation where the stress free strain E is constant, non zero, in a half space bounded by a wedge, whose edge is along Oxl. We shall name P± the two half planes bounding the wedge with the convention that the rotation 0 of P+ toward P- is positive when the former sweeps the volume where E is non zero. n± are the normals to
P± pointing from the region where E is non zero (Fig. 2).
Then, from equations (A.15) we write
Therefore the incompatibility of the wedge is the sum of a line incompatibility on the edge
and a surface incompatibility on each half plane constituting the wedge.
The incompatibility does not vanish unless E does. Note that in that case there may be dislocations on the face and even on the edge of the wedge. For this to occur, the distorsion tensor has to be antisymmetric, i.e. a pure rotation.
Before looking at the case of the prism let us study the problem of the edge common to 3 planes.
3.2 EDGE COMMON TO 3 PLANES.
-Consider now the case of an edge common to three half planes P1, P2, P3 dividing the space into three regions (1), (2), (3). Let Ei be the stress free
strain in each region (Fig. 3). Nothing is changed if we superimpose a stress free strain constant in the whole space. Therefore we may assume that one of the E‘, Let it be
E3, is zero. Let us look for a situation of zero stress. From what we have seen in section 2, one
necessary condition is that the edge, let it be Oxl, is the principal axis of the strain tensors
E1, E2 with 0 as principal value. Let us take axis Ox2 in P3. Then the only non zero
components of E’(i =1, 2 ) are E22, E33, E23. We define the angles 0 and 0 2 as shown in
figure 3 and require that the incompatibility is zero on the half planes and on the edge.
Fig. 3.
-The triple edge.
From equations (14), (15), (16), (A.13) we obtain
With a little algebra equations (17) read :
where E22 is arbitrary and À is obtained from (17c).
We notice that :
e if there are no other restrictions on E1, E2, all the components of these tensors, exept for 1, are completely determined. In other words, the condition that the configuration be stress
free imposes 5 conditions on each strain tensor E1, E2 ;
e if there are restrictions on E1, E2 there is no solution in general. This is the case for
example when the stress free deformation has no dilatation (E1k is equal to zero) ;
e increasing the number of walls arriving to the edge increases the number of degrees of freedom, i.e. the possibility of stress free accomodation.
3.3 CASE OF THE PRISM.
-Coming back to the problem of the prism, we may now calculate
the total incompatibility % as the sum of an edge term le and a surface term lqs. Both are obtained by summing the contribution of the edges 0 (a) } and the faces
a (Fig. 1). From formulae (16) and (A.15) :
where the xk are the coordinates of 0", the n" the normal to the face a. It must be noticed that
The surface term is the sum of the contributions of all the faces. On each face there is a
dipolar distribution of incompability which has an extension La (Fig. 1) and which we denote cP (S). Then the surface term is read
F" is obtained from formula (8).
The calculation of the stress field is done with the help of formulae (3-6). The details of the calculations are given in the appendix. Of course the whole results are tedious and lengthy.
But some general remarks can be made which lead to important simplifications. Let us first analyze the edge term which is the sum of contributions from each edge. For a given edge
there are two terms :
e a singular term, characterized by the tensor S, which depends on the logarithm of the
distance from the edge ;
e a regular term which depends only on the angle ( 9 ") and is characterized by the tensor
R.
From equation (20) we can deduce the following :
e at distances large compared with the size of the prism, the stress field decreases as
r- 2 .
,e close to an edge a the stress field can read to a good approximation
where (Ri§) is the value of R/j averaged on the angle 0 ". llo depends in principle on the
detailed shape of the prism but may, to a very good approximation, be taken as the square of the distance to the closest edge.
The same analysis can be carried out for the face terms. For a given face there are three terms :
e a singular term characterized by the tensor 1 which will depend upon the logarithm of the
distance from the edges ;
. an angular term characterized by the tensor C ;
. a regular term which depends only on the angles 6 a, (J a + 1 characterized by the tensor T.
If we combine the different faces we find that at distances large compared with the size of the prism, the stress field decreases as r- 2.
Close to an edge a the stress field can read to a good approximation as :
’5’ij, Tij, Cij are obtained by taking the contribution of the faces adjacent to a, i.e. by calculating the contribution of a wedge (Fig. 2) assuming the faces have a limited extension.
Dij is a constant tensor which takes into account the contribution of Cf3 and Tf3 of the faces /3 which are not adjacent to the edge a. They are not negligible compared to
Ti . and Cij. Uo is a constant which, in principle, depends on the components Ei, . It’s value can
be taken as the square distance to the closest edge. If one needs to know exactly the stress
field one has to calculate exactly these terms, i.e. take into account the contribution of the far
edges and faces.
It must be noticed that the stress field we have calculated this way does not satisfy the boundary conditions on the surface normal to the OX1 direction. This situation is also met in other problems like that of the edge dislocation for example.
Finally we give the singular part of the stress field close to an edge (Fig. 4) as :
It must be clear that formulae (24-25) do not represent the total stress but only that part of the
stress which is the most important close to the edge.
Fig. 4.
-The model wedge.
4. Three dimensional case.
In that case the calculations are obviously tedious and heavy. But looking at equations (3, 4, 6), it is simple to see how the results obtained in section 3 can be transposed :
. the long range stress field will decrease as r- 3 ;
. the angle r will be replaced by a solid angle Q ;
. the logarithmic divergence close to the edge will still be present but, close to a vertex one
has to add 3 logarithmic terms, one for each edge. The result is that at a vertex, the stress varies like the logarithm of the distance to the vertex.
Therefore, the results we have obtained in section 3 describe the situation correctly.
5. Général discussion.
The previous analysis shows that edges and possibly vertices can be a source of large internal
stresses. This will be studied systematically in a following paper but some examples may be
given which show how important the effect may be. Since we are interested only in orders of
magnitude we shall assume that the prism is a rectangular one which simplifies formulae (25).
a) For crystals which present anisotropy of thermal dilatation coefficients the local stress field a at an edge upon cooling is of the order of :
where Da is the difference between the dilation coefficients, T is the variation of temperature and D the size of the grain. For a cooling of about 1 000 K and Da of the order of
10- 6, and a grain size of 0.1 mm there follows a stress of 10- 3 w at 10- 3 À from the edge. This
is large compared to the yield stress of most metals and comparable to the fracture stress. For
highly anisotropic crystals like aU, or Zn, Da may be larger than 20 x 10-6 [15]. This means
that either cracks or large local plastic deformation occur even during slow cooling of such crystals.
b) Coherent precipitates
-when they are for example parallelepipedic
-will exert the
same kind of stress. From formulae (25), this local stress is minimized when the precipitate is along the principal axis of the strain transformation. From what is said in section 3.3
(Formulae (24)), the stress is minimized if the precipitate is very flat or has the form of a
needle. Similar effects occur if the precipitate has elastic moduli differing from those of the matrix when the metal is under stress.
c) The hydrostatic tension is 2 w [(1 + v)/7T(l- v )] Ez3 Ln (U/Uo) ; then during high temperature deformation the edges are places where vacancies can be generated or absorbed.
In the latter case this may be an explanation for the appearence of cracks at triple edges of grain boundaries.
d) The effects for low temperature plastic deformation are more intricate. There is
certainly an effect, analogue to what has been sketched in b), due to the average elastic stress combined with the anisotropy of elastic constants. However (Fig. 5) plastic deformation may have a tendency to avoid edges, which explains why effects of edges of grain boundaries are not observed [14]. But the localisation of plastic deformation along grain boundaries will create local concentrations of the kind we have studied here. This requires more discussion
however since the plastic deformation results from the glide of discrete dislocations.
e) Let us conclude with a remark on situations like Lüders bands or shear bands in which
macroscopic shear is bounded by non crystallographic planes. From equation (9) this requires cooperative glide of four glide systems at least unless there is a strong frictional stress to
compensate for the average stress calculated in formula 10. This is indeed the case for the observed bands.
As pointed out in the introduction the results obtained in this paper differ from those obtained by Eshelby [1] in that they reveal the logarithmic singularity of the stress. This singularity results from the existence of the edges. It is due to the fact that the planes are of
finite extension and change orientation at each edge. This mostly modifies the local stress close to the edges. For example the stress at infinity is the same as that calculated by Eshelby.
The other terms are not fundamentally different.
Fig. 5.
-Schematic representation of a triple edge E at the meeting of 3 grain boundaries Bl B2 B3. In order to be effective the glide has a tendency to avoid the region inside the triangle A1, A2, A3, so that one sees no effect at the node E. But still Ai A2 A3 are edges in the sense used in this paper.
Appendix.
A.l Général formulae for 6 functions [6, 7, 8].
Let us define for any functions f and g :
Then for any well behaved function ip let us define 6, :
where V is a closed volume or a half space.
Then the derivatives 8 ", k of 6, are defined by
Let S be the surface bounding the volume V. Then
where
Where n is the normal pointing outside from the surface, 5s is a surface distribution of density
one.
The second derivative is calculated in the same way
if S is an infinite plane P
a p is a surface distribution. 6 ’ p a dipole surface distribution.
If the plane is xl OX2
A.2 Case of a wedge (Fig. 2).
Let P± be the half planes bounding the wedge, as defined in the text. Their intersection with the plane x20x3 defines axes Ox’----. 2 We orient Ox2 + positively starting from 0 and Ox2 - when going towards 0 as indicated in figure 2. We define Ox3 ± to complete the orthogonal reference frame.
The calculation of the first derivative gives :
Where 6 ± is a constant surface distribution on the half planes p± . Since n’ is 0,
Sv, e is zero. The same is true for Sv, ke.
Let a± be defined by
Then coordinate transformation between OX1 X2 X3, OX1 x2 ± x3 ± is given by
From which ’6,, kf is easily calculated as :
The non zero 0 ke are :
are dipole distributions on the half planes P’ .
A.3 Calculation of the stress contribution of the edges.
Let
It is well known that
Then let
Then the stress potential for the edges is
To calculate the stress field Qe we only need the second order derivatives of g" which are easily calculated as
Then, by applying formula (6) the stress field is calculated as
The value of the constant Uo is irrelevant since E Si7 is zero from equation (20).
a
