Fracture Mechanics - Calculations

• Theory for calculations of severity of crack.

• FEM – J integrals for plane problems.

• Special Finite Elements + video of crack process

• FEM – J integrals for 3-D problems.

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Conclusion of continuum exercise

• At sharp edges there is a stress singularity

• The stresses go to infinity

• The strain energy density is limited

• The magnitude of the stress singularity determines the strain energy density or the stress intensity factor K

• Stress intensity factors should not be confused with stress concentration factors.

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For a very small

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Stress Intensity Factor for a small crack with a length of a:

The crack singularity - polar coordinates

Stress state around crack (Continuum Mechanics)

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Mode I - Opening mode

Stresses depend on inverse of squareroot of r - singularity

Singularity dominates at the crack tip - C11 depends on external load

Symmetric solution.

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Mode II - Shearing mode

Stresses depend on inverse of squareroot of r - singularity

Singularity dominates at the crack tip - C21 depends on external load

Antimetric solution.

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Reformulation of parameters

The D1 is the case with out-of-plane shear (torsion mode).

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How to find the singularity?

Structure with crack is analysed by means of superposition

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Structure without crack + stresses on crack

Crack is closed and stresses will be finite.

The stresses on the crack is easily calculated (FEM – hand calculation)

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Calculation of singularity

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Singular part - Infinite stresses

The infinite stress state around the crack can be found by

Analytically solution for a crack in an infinite plate subjected to

- 2 opposite point forces (normal or shear direction)

Integration of analytically solution

In this way many analytical or semianalytical solutions have been found.

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J - integral

The J – integral is a very effective way of calculating the energy associated with the singularity.

• Numerical stable.

• Independent of material (can be plasticity).

• Easily programmed in a finite element context .

• Basically postprocessing of a Finite Element model with the crack modelled.

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Idea of J - integral

J Integral around a closed curve without singularities can be shown to be 0.

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Definition of J-integral

J-integral is a contour integral of the strain energy (w) – the stresses timesdisplacement gradients.

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Reformulation of solution in Cartesian coordinates

The solution on the boundaries are simple.

Depends on plane stress and plane strain. (kappa value)

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Physical interpretation

• Formulate the polar solution from Mode I in cartesian coordinates

• Insert the Mode I solution

• Calculation of strain energy, stresse and displacement gradients are relative easy.

• Integration around a contour gives

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Mode II and Mode III

• Solutions for Mode II in-plane shearing can be solved in a similar way

• Solutions for Mode III out-of-plane is a little more complicated

• Results depend on plane stress/plane strain (beta)

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Crack driving force

The crack driving force should for stable cracks be less than the material parameter fracture thoughness

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J-integral in Ansys Plane Problem

By Søren Heide Lambertsen, Ph.D. student, Department of Civil Engineering, Aalborg University

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Special elements - Improved convergence, accuracy

• Isoparametric elements with midside node in the quarter point contain strains with inverse squareroot singularity.

• Should only be used near crack tips.

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Triangular, isoparametric elements around crack

Source: Cook et al., ”Concepts and Applications of Finite Element Analysis”,Wiley.

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3-D effects

Plane case: Difference between plane stress and plane strain.

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Video of crack proces in fiberreinforced concrete

• High speed video.

• Pattern recognition to identify different positions.

• Calculation of strains.

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