Fracture mechanics design, Lecture notes of Machine Design

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Prepared by: Dr. Gagandeep Bhardwaj, Assistant Professor, MED Email: gagandeep.med@thapar.edu Contact No. 8954388548

FRACTURE MECHANICS

• The concept of fracture mechanics begins with the assumption that all components contain microscopic cracks.
• In case of ductile materials , there is stress concentration in the vicinity of a crack. When the localized stress near the crack reaches the yield point, there is plastic deformation, resulting in redistribution of stresses. Therefore, the effect of crack is not serious in case of components made of ductile materials.
• The effect of crack is much more serious in case of components made of brittle materials due to their inability of plastic deformation.
• When a component containing a small microscopic crack is subjected to an external force, there is an almost instantaneous propagation of the crack leading to sudden and total failure.
• Fracture failure occurs at a stress level which is well below the yield point of the material. Therefore, failure due to propagation of crack in components made of brittle materials is catastrophic.

FRACTURE MECHANICS

FRACTURE MECHANICS

The fracture toughness is the critical value of stress intensity at which crack extension occurs. The fracture toughness is denoted by KI.

• There is a basic difference between stress intensity factor K 0 and fracture toughness KI , although they have the same units.
• The stress intensity factor K 0 represents the stress level at the tip of the crack in the machine part.
• On the other hand, fracture toughness KI is the highest stress intensity that the part can withstand without fracture at the crack.

STRESS CONCENTRATION

STRESS CONCENTRATION

Stress concentration is defined as the localization of high stresses due to the irregularities present in the component and due to abrupt change in the cross section. In order to consider the effect of stress concentration and find out localized stresses, a factor called stress concentration factor is used. It is denoted by Kt and defined as, min min sec t Highest valueof actual stressnear discontinuity K No al stressobtained byelementaryequations for imum cross tion  where σ 0 and τo are stresses determined by elementary equations and σmax. and τmax. are localized stresses at the discontinuities. The subscript t denotes the ‘ theoretical ’ stress concentration factor. The magnitude of stress concentration factor depends upon the geometry of the component.

CAUSES OF STRESS CONCENTRATION

• Variation in Properties of Materials : There is variation in material properties from one end to other.
  • Load Application :: These forces act either at a point or over a small area on the component. Since the area is small, the pressure at these points is excessive. This results in stress concentration. cavities in welding Internal cracks Air hole in steel component Foreign inclusions

STRESS CONCENTRATION FACTOR

The stress concentration factors are determined by two methods, viz., the mathematical method based on the theory of elasticity and experimental methods like photo-elasticity. For simple geometric shapes, the stress concentration factors are determined by photo-elasticity. The charts for stress concentration factors for different geometric shapes and conditions of loading were originally developed by RE Peterson. The chart for the stress concentration factor for a rectangular plate with a hole loaded in tension or compression is shown in Figure. Stress Concentration Factor (Rectangular Plate with Transverse Hole in Tension or Compression)

05/02/2019 Dr. Gagandeep Bhardwaj, AP MED, TIET 11

STRESS CONCENTRATION FACTOR

Stress Concentration Factor (Flat Plate with Shoulder Fillet in Tension or Compression) Stress Concentration Factor (Round Shaft with Shoulder Fillet in Tension)

The stress concentration factor for some simple geometric shapes using the Theory of elasticity. The theoretical stress concentration factor at the edge of hole is given by:

STRESS CONCENTRATION FACTOR

As b approaches zero , the ellipse becomes sharper and sharper. A very sharp crack is indicated and the stress at the edge of the crack becomes very large. The ellipse becomes a circle when ( a = b ).

DESIGNER GUIDELINES

Ductile Materials Under Static Load : When the stress in the vicinity of the discontinuity reaches the yield point, there is plastic deformation, resulting in a redistribution of stresses. This plastic deformation or yielding is local and restricted to a very small area in the component. There is no perceptible damage to the part as a whole. Ductile Materials Under Fluctuating Load: When the load is fluctuating, the stress at the discontinuities may exceed the endurance limit and in that case, the component may fail by fatigue. Therefore, endurance limit of the components made of ductile material is greatly reduced due to stress concentration. This accounts for the use of stress concentration factors for ductile components. Brittle Materials : The effect of stress concentration is more severe in case of brittle materials, due to their inability of plastic deformation. Brittle materials do not yield locally and there is no readjustment of stresses at the discontinuities. Once the local stress at the discontinuity reaches the fracture strength, a crack is formed. Therefore, stress concentration factors are used for components made of brittle materials subjected to both static load as well as fluctuating load. Whereas in case of ductile materials, stress concentration factor is used for components subjected to fluctuating load.

REDUCTION OF STRESS CONCENTRATION

Methods of reducing stress concentration in cylindrical members with holes.

Problem 1 : A flat plate subjected to a tensile force of 5 kN is shown in Figure. The plate material is grey cast iron FG 200 and the factor of safety is 2. 5. Determine the thickness of the plate.

NUMERICAL PROBLEM

NUMERICAL PROBLEM

NUMERICAL PROBLEM

Problem 2 : A non-rotating shaft supporting a load of 2. 5 kN is shown in Figure. The shaft is made of brittle material, with an ultimate tensile strength of 300 N/mm 2

. The factor of safety is 3. Determine the dimensions of the shaft.