Crack Paths 2009

Figure 1. (a) Discontinuous displacement field in a 2-Dsolid; (b) Schematisation in a PE.

where B(x) is the compatibility matrix. In the above expression, the crack jump

displacement vector w(x) is unknown, and must be evaluated by taking into account

the material cohesive crack bridging law for the normal and tangential stresses.

V A R I A T I O NF AOLR M U L A TOIFOD INS C O N T I N UROEU. S

The equilibrium problem can be stated through the following weakform [14]:

I W(dn *)cd§2 = [an * bdo + I an * tdr + [(511+ * -5n**)t+dr

(5)

r,

s

Q \ S

Q \ S

for any virtual displacement field on * and corresponding strains and stresses 5s* I VSo‘u’k, 50* I 60 * lVsdu By introducing the FE notation, it can be written as:

J‘[B-66*—[B+®6w*]ijClB-6—[B+

®wHdQI

§2\S

(6)

= ][N-dd*+[H —N+]6w*]bd£2+j[N-66*+[H—N*]5w*]tdl"+j6w*t*dl"

r,

s

Q\S

Since the virtual displacements 5w* and 63* are arbitrary, we can assume

5 W * I0and& * I 0separately in eq. (6) and get the following expressions, after

eliminating the arbitrary displacements & * and the discontinuity vector 5w* :

jB’ CBdQ _ In’ CBVIQ

jN’bdo+jN’tdr

m s

o\s

{ } : Q“

R

(7)

jB’+CBdQ J-B’+CB*dQ

w j[H-N+]bdo+j[H —N*]' tdr

Q \ S

Q\S

Q \ S

l“,

or, in a compactform, after performing a static condensation:

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