Crack Paths 2012

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Similar (non-homogeneous) equations can be found for the higher-order terms Bes des e ergy s lutions, there eX st singular sol tions r_1/2\I/ of problem (5) - In order to formu ate a fracture criterio the first pairs of this poWe - aW solutions,

UjO,1(y/) 3: T1/2@2,1(€0),

Vfiw') 3: 71_1/2\D2,1(§0):

j I 1, 2, 3

have to be normalized in a mechanical reliable sense. Dueto [8], the energy solutions

can be chosen to

1 §lUi1l(—y1):evil/2e?”

%[U§,1l(—y1): Cr1/2e1, %[U§)71](_y1): CT1/2e2

Where (g1) :: u(y1, +0) — u(y1, —0) is the jump over the crack and C a material

constant. Using this so-called strain basis of power-lawsolutions, the first SIF is

related directly to opening of the crack, the second to sliding of the crack surfaces in

the y’-plane and the third to out-of-plane sliding of the crack surfaces. The singular

solutions can be chosen in such a Way, that

/ N°<1/2><1>2,1<¢>-\12.<¢>

— @3480) -NO<—1/2>@3.<@>d@

: a, m‘ : 1,2,3.

Where N0(A) :: N0(

Also in three dimensions SIFs can be calcu

lated using singular Weight functions. There exist solutions Qj of the homogeneous

equations (1) With singular asymptotic behavior at the crack front:

@(Hw)I HEN/21W)+ - ~ I H($)1°_1/2‘I’§,1(§0)+-~,

ly'l

—> 0,

and for smooth functions H(s) the following integral representation holds [3, 6]:

/ p(9c)-Qj(H;9c)dS: /H($)K,-,1(s)ds,

j :1,2,3.

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a m z i

r

G R I F F I TEH ’ N E RC RG IYT E R I O N

As previously discussed, crack propagation can be predicted using the energy crite

rion [9, 10]. For this it is necessary to calculate the change of potential energy A U

for small crack elongations. Let E(t) be the elongated crack With new crack front

P(t) ;: {96(8) + t(h(s)cos(19(s))n(s)

_ h(s)sin(i9(s))b(s))

:9c(s) e r}

Here, t Z O is a time-like parameter and 0 g th(s) << 1 is the length of the crack

shoot in the y’-plane to direction 19(5). W eassume, that h and '19 are smoothfunctions

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