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Unformatted text preview: Fracture Mechanics Brittle fracture Fracture mechanics is used to formulate quantitatively • The degree of Safety of a structure against brittle fracture • The conditions necessary for crack initiation, propagation and arrest • The residual life in a component subjected to dynamic/fatigue loading Fracture mechanics identifies three primary factors that control the susceptibility of a structure to brittle failure. 1. Material Fracture Toughness. Material fracture toughness may be defined as the ability to carry loads or deform plastically in the presence of a notch. It may be described in terms of the critical stress intensity factor, KIc, under a variety of conditions. (These terms and conditions are fully discussed in the following chapters.) 2. Crack Size. Fractures initiate from discontinuities that can vary from extremely small cracks to much larger weld or fatigue cracks. Furthermore, although good fabrication practice and inspection can minimize the size and number of cracks, most complex mechanical components cannot be fabricated without discontinuities of one type or another. 3. Stress Level. For the most part, tensile stresses are necessary for brittle fracture to occur. These stresses are determined by a stress analysis of the particular component. Other factors such as temperature, loading rate, stress concentrations, residual stresses, etc., influence these three primary factors. Fracture at the Atomic level Two atoms or a set of atoms are bonded together by cohesive energy or bond energy. Two atoms (or sets of atoms) are said to be fractured if the bonds between the two atoms (or sets of atoms) are broken by externally applied tensile load Theoretical Cohesive Stress If a tensile force ‘T’ is applied to separate the two atoms, then bond or cohesive energy is (2.1) Where is the equilibrium spacing between two atoms. Idealizing forcedisplacement relation as one half of sine wave (2.2) o x Tdx F φ = x o x C T sin( ) π λ φ = + + x o Bond Energy Cohesive Force λ Equilibrium Distance x o P o t e n t i a l E n e r g y Distance Repulsion Attraction Tension Compression A p p l i e d F o r c e k Bond Energy Distance Theoretical Cohesive Stress (Contd.) Assuming that the origin is defined at and for small displacement relationship is assumed to be linear such that Hence forcedisplacement relationship is given by (2.2) Bond stiffness ‘k’ is given by (2.3) If there are n bonds acing per unit area and assuming as gage length and multiplying eq. 2.3 by n then ‘k’ becomes young’s modulus and beecomes cohesive stress such that (2.4) Or (2.5) If is assumed to be approximately equal to the atomic spacing + + x o Bond Energy Cohesive Force λ Equilibrium Distance x o P o t e n t i a l E n e r g y Distance Repulsion Attraction Tension Compression A p p l i e d F o r c e k Bond Energy Distance o x x x sin( ) π λ π λ C x T T π λ C T k π = λ o x o x C T C σ c o E x λ σ = π c E σ = π λ Theoretical Cohesive Stress (Contd.)Theoretical Cohesive Stress (Contd....
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 Fall '11
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 Stress, Fracture mechanics, crack tip, Theoretical Cohesive Stress

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