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8/12/2019 Lecture 2a Fracture - Theory
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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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8/12/2019 Lecture 2a Fracture - Theory
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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 energydensity or the stress intensity factor K
Stress intensity factors should not be confused with stress concentrationfactors.
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Stress Intensity Factor for a small crack with a length of a:
8/12/2019 Lecture 2a Fracture - Theory
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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 - singularitySingularity 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 associatedwith 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 crackmodelled.
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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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Crack driving force
The crack driving force should for stable cracks be less than the materialparameter fracture thoughness
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J-integral in Ansys Plane Problem
By Sren 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 containstrains 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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