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Review on size effects and their interpretation

1. Size effects due to limitations by the internal length scale

The most prominent size effect caused by internal obstacles to dislocation motion is the so-called

Hall Petch effect [2-4]. Here the limiting length scale is gien by the grain size! " # " o$ %d &'.()

*here " is the actual strength) "ois the friction stress for infinite grain size) % is a constant) and d the

grain size. "oand % depend on the condition of the considered crystal [4) (].+uring deformation) the glide of dislocations is stopped at grain boundaries) *hich are considered

as impenetrable obstacles. This leads to a pile-up of dislocations) creating a bac%-stress on the

acting dislocation source. Therefore) the e,ternal stress reuired for further deformation increases.

or a certain increment in global strain) eual numbers of dislocations hae to be stored in

indiidual grains. /ut for smaller grain size) the spacing bet*een indiidual dislocations in the

created pile-up gets reduced) thus e,erting a stronger bac%-stress on the acting source [0].

Therefore) the measured flo* stress increases *ith decreasing grain size.

1imilar size effects *ere obsered for particles and other obstacles hindering dislocation motion. n

this case the aerage obstacle spacing limits the stress necessary to bo* out the moing dislocation

to bypass the hard obstacle in terms of an 3ro*an mechanism []. 5 comparatie reie* on these

size effects *as gien by 5rzt [6].

2. Size effects as a consequence of non-uniform deformation

n the last t*enty years seeral e,perimental obserations presented pronounced size effects in the

presence of strain gradients caused by non-uniform deformation. lec% et al. [7] obsered increased

torsional resistance of thin *ires) *hile no effect on the tensile properties *as obsered. 1t8l%en

and 9ans [:'] reported increased bending strengths *ith reduced foil thic%ness) and seeral

authors obsered increased nanohardness *ith reduced indentation depth [::-:;]. [:4]!

*here b is the /urgers ector) and A is the shear in the slip system.

This uantity enters the classical Taylor relation!

*here B is the flo* stress) C a numerical constant in the order of '.() > the shear modulus) and @1

the density of statistically stored dislocations. /ased on this idea) seeral formulations of this strain

gradient theory *ere deeloped to describe the influence of strain gradients to the e,perimentally

obsered size effects under non-homogenous loading [7) :;) :() :0]. >enerally) the influence of a

strain gradient D present in the deformation field on the flo* stress " can be e,pressed by the

follo*ing euation [:;]

*here "'is the flo* stress in the absence of a gradient and l 'is a characteristic material length scale.

3. Size effects due to geometrical limitations of the eternal length scale

The characterization of mechanical properties in small dimensions is a maEor topic *hen

considering the ongoing trend in miniaturization and the need for proper material characteristics for

engineering applications in this regime.

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diamond punch [:]. This method remoes most of the constraints present in other techniues and

is able to probe mechanical properties on the micrometer and sub-micrometer scale under nominally

unia,ial loading and therefore in the absence of strain gradients.

Therefore) it *as uite une,pected that these micro-compression specimens e,hibited a strong

geometrical size effect in terms of an increased flo* stress *ith reduced sample dimensions. This

obseration has dra*n considerable interest in the scientific community) and seeral groups

deeloped similar methods and reported comparable results [:-24].Fecently) seeral models *ere formulated to e,plain the obsered size effect. 3ne assumption made

to predict the high obsered flo* stresses compared to bul% single crystals is a lac% in dislocation

multiplication eents during deformation) resulting in a dislocation free test structure and

correspondingly a high flo* stress necessary for dislocation nucleation. The theory goerning this

aspect is termed dislocation staration theory [2(]. 3n the other hand) cutting of pre-e,isting

dislocations during / machining can reduce the aerage dislocation length present in the micro-

compression specimen compared to the bul% crystal it *as fabricated from and introduce single-

ended dislocation sources. The distribution of this arm lengths gies rise to truncation hardening

[20]. urthermore) due to the limited number of dislocations in these small structures) a statistical

aspect enters the scene along *ith the uestion of correlation lengths in these miniaturized specimen

[20-26].Guite some effort *as put on modeling this %ind of e,periments *ith arious methods) on the one

hand to e,amine the influence of misalignment and specimen geometry on the determined data [27-

;:]) on the other hand to identify the mechanisms goerning deformation in miniaturized

compression samples [2) ;2-;(].

5t the moment) no clear mechanism e,plaining the compression size effect is identified. hile on

the one hand in-situ compression tests in a transmission electron microscope =T9I? sho* no

dislocation storage for specimen *ith diameters belo* ;'' nm) in-situ *hite beam Jaue diffraction

during micro-compression [;0] as *ell as post compression inestigation using electron bac%scatter

diffraction =9/1+? [;] present distinct crystal rotations due to the storage of dislocations for

samples ranging from 2 Km to 6 Km in diameter. urther uestions rise by the fabrication method

itself) as the ion damage during machining might influence the determined material properties [;6].

/ei et al. [;7] performed micro-compression e,periments on *his%er-li%e structures and reported

size-independent flo* stresses in the order of the theoretical shear stress) *hich *as not obsered

for / fabricated specimen. urthermore) they found changes of the indentation behaior due to

/ milling [4'].