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Fracture Mechanics - Calculations

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Fracture Mechanics - Calculations 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. 1 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. 2 Stress Intensity Factor for a small crack with a length of a: For a very small 3 The crack singularity - polar coordinates Stress state around crack (Continuum Mechanics) 4 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. 5 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. 6 Reformulation of parameters The D1 is the case with out-of-plane shear (torsion mode). 7 How to find the singularity? Structure with crack is analysed by means of superposition 8 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) 9 Calculation of singularity 10 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. 11 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. 12 Idea of J - integral J Integral around a closed curve without singularities can be shown to be 0. 13 Definition of J-integral J-integral is a contour integral of the strain energy (w) – the stresses times displacement gradients. 14 Reformulation of solution in Cartesian coordinates The solution on the boundaries are simple. Depends on plane stress and plane strain. (kappa value) 15 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 16 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) 17 Crack driving force The crack driving force should for stable cracks be less than the material parameter fracture thoughness 18 J-integral in Ansys Plane Problem By Søren Heide Lambertsen, Ph.D. student, Department of Civil Engineering, Aalborg University 19 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. 20 Triangular, isoparametric elements around crack Source: Cook et al., ”Concepts and Applications of Finite Element Analysis”, Wiley. 21 3-D effects Plane case: Difference between plane stress and plane strain. 22 Video of crack proces in fiberreinforced concrete • High speed video. • Pattern recognition to identify different positions. • Calculation of strains. 23.
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