Stress Intensity Factor (K)

 

Stress Concentration at Cracks

    Recall the Inglis solution in the previous section describing the maximum elastic stress at the tip of an elliptical notch of depth a and root radius ρ is:

 

    where in traditional engineering design, the stress concentration factor (k_t) is given by:

    Now consider for a sharp crack the radius of the tip is extremely small. The above equation would therefore predicts that σ_max tends to infinity as ρ tends to zero. This suggests that for a very sharp crack, any applied stress will cause infinitely high stresses at the crack tip, regardless of the magnitude of the stress is. In conclusion, the concept of stress concentration factor breaks down as crack tip radius tends to zero.

    Consider instead the product of σ_max at the crack tip and the crack tip radius ρ:

    If substitute for σ_max from the Inglis solution and allow ρ to tend to zero, the above expression reduces to:

 

    which remains finite and is formed from the physical quantities a and σ which define the problem. This is in fact the definition of the stress intensity factor (K).

 

Plasticity and Triaxiality

    One can realise from the above equation that it suggests a material which can respond elastically up to infinitely high stress levels. It is quite obvious that no material possesses such properties and in reality, some amount of plastic response must be present near the crack tip. However, this plasticity must also be somewhat limited, since cracking occurs before general yield. 

    The next step is to try and describe the extent and geometry of a plastic zone which could exist in a stress field described by the Westergaard solutions:

 

    The following equations represent the principal stresses for the same Westergaard field:

 

    While Tresca or Von Mises may be used as the yield criteria for metals and they are both indicative of the presence of shear (Tresca - Max Shear Stress/ Von Mises - Shear Strain Energy), therefore 2 assumptions are made to value the third principal stress component, σ₃.

    (i) Plane stress => σ₃=0
    As the radial distance r decreases, the magnitude of σ₁ and σ₂ increases whereas remains 0. The shear stress component can therefore increase dramatically.

    (ii) Plane strain => ε₃=0 and σ₃=v(σ₁+σ₂)
    σ₃ is linearly related to the sum of (σ₁+σ₂) and therefore is a singular. The shear stress component becomes limited.

 

    In practice, plane stress is a common description of the stress state in thin sheets, whereas plane strain describes the stress state of thick components. (Thick components are usually associated with the terms: Plane strain, High constraint, High triaxiality, Contained yield and High severity.)

 

The Stress Intensity Factor (K) and the Shape Factor (Y)

    Recall the Stress Intensity Factor (K) is written:

 

    In general, the stress intensity factor depends on the geometry of the cracked body (including the crack length) and it is usual to express it as:

 

    where Y is the Shape Factor and is a function of the geometry of the body and crack length. (Note: Here the square root π term is absorbed into the Y factor while some publications and textbooks do leave the term outside the Y.)

    Note: This expression for K is only applicable to an ideal siutaion of an infinte plate containing a centre crack of length 2a. In practice, the presence of finite boundaries and the way which the crack is loaded affects the value of stress intensity factor.

 

Loading Geometry

    A crack can be classified into 3 different modes of loading conditions:

 

The Fracture Criterion - Critical Stress Intensity K_C

    The only parameter distinguishing a crack situation from another is K. Therefore if fracture is controlled by conditions in the vicinity of the crack tip, failure must be associated with the critical value of K - the Critical Stress Intensity - K_C. At fracture:

 

    Note: The Stress Intensity Factor K is a stress field parameter independent of the material, whereas K_C is a material property, the fracture toughness.

 

K_C and the Component Thickness

    Since the thickness (geomerty) of the component plays a major role in the validity of the elastic description, it is important to examine how the value of K_C varies with thickness for a certain material. The following behaviour is observed when K_C is measured as a function of component thickness:

 

    An important feature of the above graph shows, when above a certain thickness B_min, the fracture toughness has a minium value which is no longer dependent on the component thickness. This value K_IC is known as the plane strain fracture toughtness.

    The B_min required to ensure the plane strain conditions necessary for K_IC is given by:

 

 

Relationship between K_C and G_C

    Recall the fracture stress (σ_F) equations derived from previous section:

    re-arranging the equations by substituting K gives the relationship at brittle fracture:

K_C² = EG_C (for plane stress)

K_IC² = EG_IC (for plane strain)

    Hence, the value K_IC can actually be converted to a value of G_IC and vice-versa with the main assumption being that linear elasticity describes the stress/ strain field adequately.
    Note: both G and K are called the fracture toughness of a material.

 

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