when stress is applied to a crack introduced in ductile materials , dislocations are activated and propagate from the crack tip .
the stress concentration at the crack tip is accommodated by dislocation activities such as dislocation glide and multiplication , which lead to an increase in fracture toughness .
fracture toughness increases with temperature , and the fracture mode changes from brittle to ductile , which is the so - called brittle - to - ductile transition ( bdt ) .
silicon single crystals have been used as model crystals to understand the mechanism behind the bdt because they show an abrupt increase in the fracture toughness at a certain temperature and because large dislocation - free silicon single crystals are available .
john   pointed out that the increase in the fracture toughness at the bdt is controlled by dislocation glide in silicon .
hirsch et al .   suggested that the abrupt increase in the fracture toughness at the bdt in silicon could be explained by the assumption that the value of the critical stress intensity factor for dislocation emission at the crack tip should be close to that of the fracture toughness of silicon . a gradual increase in fracture toughness ,
in contrast , could be due to the low critical stress intensity factor for dislocation emission .
figure 1(a ) shows a contour map of the stress component normal to the crack plane , yy , around a crack tip , with a pair of pure edge dislocations in front of the crack tip without any stress applied .
the contour map was obtained by calculating the stress field based on the elastic theory derived by thomson , with the dislocation configuration as shown in fig .
two dislocations are located 45 from the crack line and 100 m away from the crack tip .
the extra half - planes of these dislocations were both at the crack side with respect to their slip planes , as shown in fig .
1(b ) , which represents the situation after two pure edge dislocations have been emitted from the crack tip .
this indicates that some extra tensile stress is necessary for the crack to propagate , cancelling the compressive stress concentration due to the dislocations .
this is the origin of the increase in the fracture toughness from the viewpoint of crack tip shielding due to dislocations [ 2,58 ] . 

1.(a ) contour map of yy around a crack tip with two pure edge dislocations without external stress .
( b ) schematic representation of the edge dislocations used for the calculation of the stress field in ( a ) . ( a ) contour map of yy around a crack tip with two pure edge dislocations without external stress .
( b ) schematic representation of the edge dislocations used for the calculation of the stress field in ( a ) .
the first macroscopic experimental verification of the shielding effect due to dislocations was reported using photoelasticity in ionic crystals .
it was shown that the residual stress field around a crack after deformation was actually a compressive stress field .
in addition , higashida et al . directly observed crack tip dislocations in mgo and silicon single crystals after deformation .
the burgers vectors , including their signs , were determined in this study using high - voltage electron microscopy [ 912 ] .
they pointed out that the local stress intensity factor due to dislocations calculated from the dislocation configuration is negative , which indicated that the dislocations observed were of the shielding type . a recent development in high - resolution transmission electron microscopy ( hrtem ) makes it possible to measure strain at the atomic scale .
htch   suggested geometric phase analysis ( gpa ) which enables the calculation of the strain distribution from hrtem images .
gpa utilizes the crystal periodicity information that is extracted by performing fourier transformations of hrtem images .
it is possible to measure the elastic displacement of atoms from a non - deformed area by calculating the change in phase of the lattice image .
once the displacement of atoms is known , the elastic strain field due to the displacement of atoms can be calculated . in this study , the local strain in front of a crack tip with dislocations in single crystal silicon was measured using gpa software ( hrem research ) , which is a plug - in for digitalmicrograph ( gatan ) , to clarify the existence of the shielding effect of dislocations at the atomic scale .
the sample used in this study was a ( 110 ) single crystal silicon wafer with a thickness of 0.63 mm .
figure 2 shows an indent on the sample surface introduced using a micro vickers indenter ( mvk - e3 , mitutoyo ) with an applied load of 0.5 n and a dwell time of 7 s. cracks propagated from the corners of the indent along both the < 100 > and < 110 > directions .
the { 100 } cracks that propagated along the < 110 > directions were shorter than the { 110 } cracks that propagated along the < 100 > directions .
that is , fracture toughness on the { 100 } planes was higher than that on the { 110 } planes . because { 110 } is the second cleavage plane of silicon , it is essential to elucidate the strain around { 110 } crack tips with dislocations .
therefore , one of the { 110 } crack tips was observed in this study . 

the tem sample used in this study was prepared using focused ion beam ( fib , quanta 3d 200i , fei ) .
first , carbon was deposited on the top surface to protect the crack from damage by ion beams during the machining process .
the area around the deposited region was milled with an ion beam at an operating voltage of 30 kv .
the protective carbon layer was removed and the sample was thinned by an ion beam with an operating voltage of 30 kv , which was gradually lowered to 16 , 8 , 5 and 2 kv .
the thinning at smaller voltages was performed to remove layers damaged during the higher voltage thinning process .
the sample was thinned until the maximum thickness of the sample was < 200 nm . figure 3 shows a hrtem sample after fib machining , wherein a crack is extending from the left edge to the centre .
the crack is located in a very small area that is about 480 nm away from the thinnest edge .
the sample was then subjected to post - fib cleaning using a nanomill system ( model 1040 , fischione ) .
this cleaning was conducted with an operating voltage of 900 v for 5 min at a tilting angle 10 to clean the top and bottom of the sample .
it was followed by milling with 500 v for 10 min on each side to further reduce damage .
the final sample thickness around the crack tip was 65 nm , which was measured during tem observation using the electron energy loss spectroscopy ( eels ) log - ratio technique . 
 fig .
is given by the intensity of the zero - loss electron , io , and the total electron intensity , it , in an energy loss spectrum as follows : 
 ( 1)t=lnitio 
 and 
 ( 2)=106feoemln2eoem , 
 where  is the mean free path of electrons for inelastic scattering , f is the realistic correction factor , eo is the incident energy of electrons , em is the average energy loss of electrons and  is the collection angle .
the foil sample was observed with an aberration corrected hrtem ( jem - arm200f , jeol ) at an operating voltage of 200 kv at the ultramicroscopy research center , kyushu university .
figure 4(a ) shows a low - magnification tem image indicating the position of a crack extending from the edge of the foil sample . to ascertain the exact position of the crack tip at the atomic scale
it is also necessary to use high - angle annular dark field scanning transmission electron microscopy ( haadf - stem ) to determine the exact position of the crack tip .
figure 4(b ) shows a z - contrast haadf - stem image in which the dark area is the crack , which is clearly distinguishable from the matrix .
figure 4(c ) shows an enlarged haadf - stem image of the area indicated by an arrow in fig .
4(b ) at the atomic scale , indicating that the crack is observed nearly edge - on . the crack plane is ( 011 ) and the crack is opened 2 nm wide . 

( b ) haadf - stem images of a crack tip with ( c ) increased magnification .
( b ) haadf - stem images of a crack tip with ( c ) increased magnification .
figure 5(a ) shows an original hrtem image of the crack tip at the atomic scale in the same area as that in fig .
the distorted area in front of the crack tip indicates the existence of dislocations in that region . to determine the character of those dislocations around the crack tip , the original image was filtered to clarify lattice planes .
first , the original hrtem image was transformed into the frequency domain using the fourier fast transformation ( fft ) , as shown in fig .
the background , or low frequency noise , in the centre spot of the fft image was filtered out .
5(d ) was then converted back to image space by performing the inverse fft , resulting in the filtered hrtem image shown in fig .
5(b ) it could be seen that there are four dislocations around the crack tip : d1 , d2 , d3 and d4 .
there are two dislocations in the distorted region in front of the crack tip and two other dislocations on the left - hand and right - hand sides of the crack .
red and green lines in the figure represent the extra half - planes of each dislocation .
the dislocations denoted as d1 and d3 lie on the slip plane of ( 111 ) while those denoted as d2 and d4 lie on the slip plane of ( 111 ) .
the burgers vectors of the dislocations were also determined by using the finish - to - start - right - hand ( fs / rh ) convention .
each burgers circuit coloured in yellow in front of the crack tip contains one dislocation .
these burgers circuits were then transferred to undistorted regions at the right - hand and left - hand sides above the crack tip , forming open circuits . because the image is the projection on the ( 011 ) plane , the exact directions of their burgers vectors can not be determined perfectly .
dislocations d1 and d3 have the same possible pair of burgers vectors , and so do dislocations d2 and d4 .
the burgers vectors of d1d4 are defined to be b1 to b4 . in each case , the edge component of the burgers vectors are the same for d1 , d3 ; and d2 , d4 ; which are a/6[211 ] and a/6[211 ] , respectively . here , we focus on the edge component of the burgers vector , which induces mode i shielding at the crack tip . to clarify the dislocation shielding field at the atomic scale
table 1.possible burgers vectors for b1 , b2 , b3 and b4b1 , b3b2 , b4first possibilitya/2[110]=a/6[211]+a/6[121]a/2=a/6[211]+a/6second possibilitya/2[101]=a/6[211]+a/6[112]a/2[110]=a/6[211]+a/6[121 ] 
 fig .
( b ) filtered image obtained from ( d ) , thompson 's tetrahedron is also shown .
possible burgers vectors for b1 , b2 , b3 and b4 ( a ) original hrtem image at the crack tip .
( b ) filtered image obtained from ( d ) , thompson 's tetrahedron is also shown .
the blue areas indicate compressive strain fields while the red areas indicate tensile strain fields .
there are four butterfly - like points around the crack tip where the blue and red colours discontinuously changed .
it should be noted here that yy is negative around the crack tip , that is , the stress intensity at the crack tip is compressive , which is caused by those dislocations in front of the crack tip . 

6.strain map of yy in front of the crack tip showing intense strain at the crack tip and at the dislocations : ( a ) obtained by gpa and ( b ) calculated based on elastic theory .
the strain intensity at the crack tip is compressive , which indicates the existence of the shielding effect at the atomic scale .
the spatial scale is the same between ( a ) and ( b ) while the colour scale of ( b ) is not exactly the same as that of ( a ) .
strain map of yy in front of the crack tip showing intense strain at the crack tip and at the dislocations : ( a ) obtained by gpa and ( b ) calculated based on elastic theory .
the strain intensity at the crack tip is compressive , which indicates the existence of the shielding effect at the atomic scale .
the spatial scale is the same between ( a ) and ( b ) while the colour scale of ( b ) is not exactly the same as that of ( a ) . to verify the experimental results , the crack tip strain field with the dislocations
the strain components in two - dimensions at the crack tip with dislocations , but without external stress , are given by the following equations , where the crack plane lies along the x - axis and the crack tip is at the origin of the complex field : 
 ( 3)xx=1e[xx(yy+zz ) ] , 
 ( 4)yy=1e[yy(xx+zz ) ] , 
 ( 5)xy=12xy , 
 ( 6)xx+yy=2[(z)+(z) ] , 
 ( 7)yyixy=++(zz) 
 ( 8)=jaj1zjjz1/2 + 1 + 1zjjz1/21+jaj(jj)2(zj)2jz1/2+zj1/22 , 
 ( 9)=jaj1zjjz1/2 + 1 + 1zjjz1/21jaj(jj)2(zj)2jz1/2+zj1/2 + 2 , 
 ( 10)aj=bj2i(+1 ) , 
 ( 11)=(3)(1+),for the plane stress case 
 where z is a complex variable for the position relative to the crack tip in the form of z = x+iy , j is the position of the jth dislocation in the form of j = cx+idy ,  is the shear modulus ( 64 gpa for silicon ) , bj is the burgers vector of the jth dislocation in the form of bj = fx+igy and  is the poisson 's ratio ( 0.278 ) .
the strain field around the crack tip was calculated with four dislocations and without applied stress .
the positions and the burgers vectors are those of the experimentally observed dislocations shown in fig .
5 . figure 6(b ) shows the calculated strain map of yy in the same spatial scale as that in fig .
6(a ) . figure 6(b ) also shows compressive strain fields at the crack tip and at the dislocation cores . the calculated strain map is in good agreement with the experimental result shown in fig .
it is stressed that these results show experimental proof of the dislocation shielding effect observed at the atomic scale in front of the crack tip .
the measured strain field around the crack tip was compressive owing to the dislocation shielding effect .
the dislocations create an extra barrier in front of the crack tip , making it more difficult for the crack to propagate , which increase the fracture toughness at the bdt .
the authors thank jsps kakenhi ( # 24686079 , # 15h04147 ) and kazato research foundation for partly supporting this research . funding to pay the open access publication charges for the article was provided by jsps kakenhi .
 the dislocation shielding field at a crack tip was experimentally proven at the atomic scale by measuring the local strain in front of the crack tip using high - resolution transmission electron microscopy ( hrtem ) and geometric phase analysis ( gpa ) . 
 single crystalline ( 110 ) silicon wafers were employed . 
 cracks were introduced using a vickers indenter at room temperature . 
 the crack tip region was observed using hrtem followed by strain measurements using gpa . 
 the measured strain field at the crack tip was compressive owing to dislocation shielding , which is in good agreement with the strain field calculated from elastic theory . 