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Mechanics of (Brittle) Fracture

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Fracture mechanics = Mechanical behavior of Materials containing cracks

• Crack geometry is assumed given (we do not ask where the crack comesfrom!)

• Brittle Fracture: Failure by rapid expansion of a crack. We try to determine

conditions for the formation of ‘critical’ cracks capable of rapid expansion.

Basic idea :

• Introducing or expanding a crack relaxes the stresses in a material andthus reduces the elastic energy (strain energy release).

• Introducing or expanding a crack requires energy to be expendedto create free surfaces (crack faces)due to energy dissipation in case of plastic deformation at the crack tip

to overcome friction between the crack faces (in case of mode-II ormode-III cracks).

• The condition for rapid crack expansion is simply that the strain energyrelease exceeds the energy that needs to be spent for expanding a crack.

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Some basic notations

Three basic fracture modes:

Mode I: Opening (or anti-opening!) under tensile (compressive!) stressMode II: Plane shear (shear in in direction perpendicular to crack front)Mode III: Anti-plane shear (shear in direction parallel to crack front)_

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NB: Compressive ‘Anti-cracks’ exist in materials that densify during failure

Example: Snow, snow slab avalanches,Heierli, Gumbsch + Zaiser, Science 31,(2008) p. 241

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Experimentally: Usually testing is done in one of two simple geometries.Plane stress deformation:

The stress state is completely characterized by 2 principal stresses. Realization:

the in-plane deformation of a thin plate.

Plane strain deformation:

The strain state is completely characterized by 2 principal strains. Realization:a very thick plate with homogeneous loading across thickness

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Estimate the strain energy release upon

introducing a crack of length 2a in a thinsheet (plane strain) of thickness wconsisting of a linearly elastic materialof Young’s modulus E :

Energy is reduced in a region of characteristicdiameter 2a . Prior to introducing the crack, thestrain energy density in this region is

We thus estimate the released strain energy tobe given by

where f is a factor of the order of unity

E W 2

2

σ =

wa E

f 22

2π

σ

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A detailed calculation (Griffith 1920) shows that for an infinite sheet f =2

Assuming a perfectly brittle material (no energy dissipation), the energy neededto create the crack is the surface energy 4wa γ

We see that for short cracks the surface energy increases faster than the strainenergy release. Short cracks can therefore not grow. To obtain the critical cracklength we calculate the change in strain energy upon increase of the cracklength by da and equate this to the strain energy release. With f =2 we obtain

The right hand side is called the critical stress intensity, or fracture toughness.

IcK E a

wdaada E

wdW

==

==

γ π σ

γ π σ

2

42 2

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Generalization by Irwin (1957): If expanding the crack goes along withenergy dissipation (eg. due to plastic deformation at the crack tip) thenwe replace

where G D

is the amount of energy that is dissipated when extending thecrack area by a unit amount. The (plane stress) fracture toughness is then

Irwin’s theory assumes that the material is linearly elastic except in theimmediate vicinity of the crack tip. In the linearly elastic region the stressfield has for an infinite plate the form

In other geometries the stress intensity must be modified by geometry-dependent factors (see e.g. Tada, Paris, Irwin The Stress Analysis of Cracks Handbook . American Society of Mechanical Engineers.)

DGG +=⇒ 22

EGK Ic =

aK f r

K ijij π σ θ

π σ == ,)(

2