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the distinction between the energy involved in creating a new surface , as compared to stretching an already existing one , has been reemphasized many times over the years @xcite . while the former can fairly be taken as a constant per unit area , the latter may have complicated dependence on surface strain . here
we try to find the effect of the latter ( ` surface elasticity ' ) on crack field solutions when this dependence is quadratic akin to its bulk counterpart .
physically , this effect will be important when the solid dimensions are of the order of @xmath2 where @xmath3 and @xmath4 are surface and bulk elasticity moduli respectively . for real materials
this ratio is of the order of nanometers .
since the dimensions of cracks in many materials such as glass range from few angstroms to micrometers , and crack elastic field solutions depend largely on surface tractions , it seems worthwhile to analyze the effect of surface elasticity on crack field solutions . at the outset
, the use of continuum elasticity at such dimensions may be questioned .
some years ago shenoy et al compared the continuum elasticity results with atomistic simulations @xcite and were able to show that bending and torsional behavior of a solid at such dimensions can be accounted for by taking surface elasticity into account . here in similar spirit
we try to formulate the problem of a mode-3 crack for a material having elastic surfaces .
in this section we solve for the elastic field of a mode-3 crack for a solid having an elastic surface .
we follow a perturbative approach with the field in the absence of surface elasticity as the unperturbed solution .
the domain @xmath5 consists of @xmath6 ( see fig 1 ) .
let the loading at the external boundary @xmath7 be of the form @xmath8 .
the specimen ]    in the absence of surface tension and surface elasticity effects , the only non - zero displacement @xmath9 in the z - direction satisfies    @xmath10    supplementing the above equations with the condition of finite displacement at @xmath11 , the anti - symmetric solution ( i.e. the one which contributes to crack - opening ) to above set of eqns . can be easily found as    @xmath12    where @xmath13 .
a general formulation of elastic surfaces was given by gurtin and murdoch @xcite .
following @xcite , in the presence of surface elasticity / tension effects , the equation of equilibrium in the bulk remain unchanged ( i.e. same as eq .
( [ eq : bulk ] ) ) while the boundary conditions at a free surface are changed from @xmath14 to :    @xmath15    here @xmath16 denote surface stress tensor while @xmath17 , @xmath18 and @xmath19 are surface elastic moduli .
if the normal to the surface lies along direction 2 , then latin subscripts @xmath20 range over 1,2,3 while greek subscripts
@xmath21 range over only the directions lying along the surface viz . 1 and 3 . the derivatives such as @xmath22 or @xmath23 employed in the above eqns .
are surface derivatives i.e. projection of usual derivatives along the surface .
note that @xmath24 is linear in surface strain @xmath25 where @xmath26 .
we switch to cartesian coordinates for a while for sake of mathematical convenience . along the crack
faces ( @xmath27 ) , the negative @xmath28 axis coincides with @xmath29 for @xmath30 .
substituting @xmath31 in above equations    @xmath32    taking surface divergence of @xmath24 one obtains @xmath33 where @xmath34 .
the first term arises because the constant part of surface stress in eq .
( [ eq : surfstress ] ) drops abruptly to zero at the crack tip . its effect on stress fields
was solved for many years ago by thomson et al @xcite who showed that such terms lead to @xmath35 singularity in stresses at the crack - tip .
as we are concerned with the effect of surface elasticity and not surface tension in this work and since the overall field can be found as superposition of the fields due to both terms individually , we drop the first term in above eqn . keeping this in mind , the above eqn . in @xmath36
coordinates may be written as @xmath37    eqn .
( [ eq : bulk ] ) in conjunction with  ( [ eq : bc1wel ] ) and  ( [ eq : bc2wel ] ) completes the specification of boundary value problem for our problem .
we were unable to find a straightforward method to solve this set of equations
. we will solve for @xmath38 as a series expansion in terms of the parameter @xmath39 .
we make the assumption @xmath40 .
next write @xmath41 as    @xmath42    where @xmath43 is given by  ( [ eq : w_on1 ] ) . throughout our analysis
we assume that @xmath40 .
since @xmath44 s are independent , each must satisfy the equation @xmath45 in the bulk and the boundary conditions  ( [ eq : bc1wel ] ) ,  ( [ eq : bc2wel ] ) individually .
thus one gets the following set of equations    @xmath46    for all @xmath47 , and @xmath48    where upper sign stands for @xmath49 and lower for @xmath50 . substituting for @xmath43 from  ( [ eq : w_on1 ] ) in  ( [ eq : g_1n ] ) and solving for @xmath51 @xmath52 where @xmath53 is an arbitrary coefficient .
since the loading due to surface elasticity is symmetric ( note the sign change in boundary condition at @xmath54 and @xmath55 ) , all the terms containing @xmath3 should be symmetric in @xmath56 .
therefore @xmath57 .
thus the terms containing @xmath3 do not contribute to the crack opening displacement but the stress field is necessarily altered due to surface elasticity . substituting the expression for @xmath51 in  ( [ eq : g_2n ] ) @xmath58 again by the same argument @xmath59
is seen to be identically equal to zero .
it immediately follows that @xmath60 .
thus the perturbative solution is @xmath61
from equations  ( [ eq : w_tot1 ] ) ,  ( [ eq : pert1 ] ) one observes that the displacement diverges if we include the term @xmath62 . as and
ruina @xcite have shown through their analysis that one needs to be careful while discarding non-@xmath0 singular(and non - singular ) terms for crack problems .
in fact , such terms are always present if one take a more realistic situation where there is a nonlinear zone @xmath63 surrounding the crack as shown in figure 2 . here
we do an analysis similar to @xcite to determine the condition for small - scale yielding in the presence of surface effects .     the specimen ]
assume that the nonelastic zone is centered at the origin @xmath64 and has dimensions smaller than the crack length @xmath65 .
we now consider a completely elastic problem in the domain @xmath66 , @xmath67 being the outermost radius of @xmath63 .
the traction at the inner radius @xmath68 is taken as @xmath69 together with the loading @xmath70 at @xmath71 . the antisymmetric solution to the elastic problem in the absence of surface effects
can now be found as @xmath72 where @xmath13 and @xmath73 .
following the notation of @xcite , the above field can be expressed as a sum of two terms : @xmath74 and @xmath75 where @xmath76 stands for _ _ a__ntisymmetric , _ _ i__nner traction free and @xmath77 for _ _ a__ntisymmetric , _ _
o__uter traction free . for sake of completeness
we write the expressions for these fields :    @xmath78    with @xmath73 , and @xmath79 with @xmath80 , and@xmath81 with @xmath82 , in the presence of surface effects the boundary conditions  ( [ eq : bc1woel ] ) ,  ( [ eq : bc2woel ] ) remain unchanged except that now our domain is @xmath83 .
following exactly the same strategy as before , we obtain the following perturbative solution :    @xmath84    from this , the stress fields are readily obtained ,    @xmath85    and    @xmath86    from above equation the condition for @xmath0 ( @xmath87 term ) field dominance can be determined .
comparing the total stress field due to @xmath0 field to that of next singular @xmath88 term one obtains the following expression for the region in which @xmath0 field dominates :    @xmath89    where @xmath90 it immediately follows from above eqn . that when @xmath91 , the k - dominance region diminishes in its radius by @xmath92 while for the case when @xmath93 the @xmath88 term always dominates the @xmath0 term as it does in the absence of surface elasticity .
to put our analysis in perspective , many articles pertaining to effect of surface stress on crack field solutions have appeared in the past @xcite .
the main difference with our work is that we have considered surface stress @xmath94 as having linear ( and thus surface energy quadratic ) dependence on surface strain @xmath95 instead of being independent of it . as mentioned earlier that a constant surface stress leads to @xmath96 singularity for a mode-3 crack @xcite .
we have shown that the linear term causes a @xmath97 singularity at the crack - tip for @xmath98 .
physically speaking , non-@xmath0 singular terms due to surface elasticity should be dominant only in the region of extent @xmath99 .
indeed we see from eq .
( [ eq : region ] ) that surface elasticity decreases the extent of @xmath0-dominant region by @xmath100 .    though we are not aware of any direct experimental evidence which favors a quadratic dependence of @xmath101 on @xmath95 over a linear one , miller et al @xcite showed that a quadratic dependence was consistent with atomistic simulations for simple problems of elongation , bending and torsion .
note that the variable @xmath94 , which is responsible for size - dependent elastic properties in @xcite , has same significance as @xmath3 in this article under gurtin - murdoch s formalism @xcite .
finally , the green s function obtained by thomson et al @xcite for symmetric dipole acting along a mode-3 crack s surface may also be used for obtaining the results in this article but since in our problem the loading itself is dependent on the displacement field , we found it more expedient to directly solve the equations of elasticity instead of a solving a singular integro - differential equation .
the surface stress can have important effects on the elastic properties of nano - sized structures . here 
 we analyze the effect of surface stress on mode-3 crack displacement and stress - field solutions under the assumption of linear elasticity . 
 we show that surface effects generate non-@xmath0 terms near the crack - tip . 
 we also find the effect of such terms on the conventional small - scale yielding assumption .    
 @xmath1 mechanical engineering department , i.i.t.- kanpur , u.p . - 208016 , india