Stress intensity factor

The stress intensity factor, K {displaystyle K} , is used in fracture mechanics to predict the stress state ('stress intensity') near the tip of a crack or notch caused by a remote load or residual stresses. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials, and is a critical technique in the discipline of damage tolerance. The concept can also be applied to materials that exhibit small-scale yielding at a crack tip.The stress intensity factor for an assumed straight crack of length 2 a {displaystyle 2a} perpendicular to the loading direction, in an infinite plane, having a uniform stress field σ {displaystyle sigma } is The stress intensity factor at the tip of a penny-shaped crack of radius a {displaystyle a} in an infinite domain under uniaxial tension σ {displaystyle sigma } is If the crack is located centrally in a finite plate of width 2 b {displaystyle 2b} and height 2 h {displaystyle 2h} , an approximate relation for the stress intensity factor is For a plate having dimensions 2 h × b {displaystyle 2h imes b} containing an unconstrained edge crack of length a {displaystyle a} , if the dimensionsof the plate are such that h / b ≥ 0.5 {displaystyle h/bgeq 0.5} and a / b ≤ 0.6 {displaystyle a/bleq 0.6} , the stress intensity factor at the crack tip under an uniaxial stress σ {displaystyle sigma } is For a slanted crack of length 2 a {displaystyle 2a} in a biaxial stress field with stress σ {displaystyle sigma } in the y {displaystyle y} -direction and α σ {displaystyle alpha sigma } in the x {displaystyle x} -direction, the stress intensity factors are Consider a plate with dimensions 2 h × 2 b {displaystyle 2h imes 2b} containing a crack of length 2 a {displaystyle 2a} . A point force with components F x {displaystyle F_{x}} and F y {displaystyle F_{y}} is applied at the point ( x , y {displaystyle x,y} ) of the plate.If the crack is loaded by a point force F y {displaystyle F_{y}} located at y = 0 {displaystyle y=0} and − a < x < a {displaystyle -a<x<a} , the stress intensity factors at point B areThe stress intensity factor at the crack tip of a compact tension specimen isThe stress intensity factor at the crack tip of a single edge notch bending specimen is The stress intensity factor, K {displaystyle K} , is used in fracture mechanics to predict the stress state ('stress intensity') near the tip of a crack or notch caused by a remote load or residual stresses. It is a theoretical construct usually applied to a homogeneous, linear elastic material and is useful for providing a failure criterion for brittle materials, and is a critical technique in the discipline of damage tolerance. The concept can also be applied to materials that exhibit small-scale yielding at a crack tip. The magnitude of K {displaystyle K} depends on sample geometry, the size and location of the crack or notch, and the magnitude and the modal distribution of loads on the material. Linear elastic theory predicts that the stress distribution ( σ i j {displaystyle sigma _{ij}} ) near the crack tip, in polar coordinates ( r , θ {displaystyle r, heta } ) with origin at the crack tip, has the form where K {displaystyle K} is the stress intensity factor (with units of stress × {displaystyle imes } length1/2) and f i j {displaystyle f_{ij}} is a dimensionless quantity that varies with the load and geometry. This relation breaks down very close to the tip (small r {displaystyle r} ) because as r {displaystyle r} goes to 0, the stress σ i j {displaystyle sigma _{ij}} goes to ∞ {displaystyle infty } resulting in a stress singularity, which can be avoided by representing a crack as a round tipped notch. Plastic distortion typically occurs at stresses exceeding the material's yield strength and the linear elastic solution is no longer applicable close to the crack tip. However, if the crack-tip plastic zone is small, it can be assumed that the stress distribution near the crack is still given by the above relation. Three linearly independent cracking modes are used in fracture mechanics. These load types are categorized as Mode I, II, or III as shown in the figure. Mode I, shown to the left, is an opening (tensile) mode where the crack surfaces move directly apart. Mode II is a sliding (in-plane shear) mode where the crack surfaces slide over one another in a direction perpendicular to the leading edge of the crack. Mode III is a tearing (antiplane shear) mode where the crack surfaces move relative to one another and parallel to the leading edge of the crack. Mode I is the most common load type encountered in engineering design. Different subscripts are used to designate the stress intensity factor for the three different modes. The stress intensity factor for mode I is designated K I {displaystyle K_{ m {I}}} and applied to the crack opening mode. The mode II stress intensity factor, K I I {displaystyle K_{ m {II}}} , applies to the crack sliding mode and the mode III stress intensity factor, K I I I {displaystyle K_{ m {III}}} , applies to the tearing mode. These factors are formally defined as: K I = lim r → 0 2 π r σ y y ( r , 0 ) K I I = lim r → 0 2 π r σ y x ( r , 0 ) K I I I = lim r → 0 2 π r σ y z ( r , 0 ) . {displaystyle {egin{aligned}K_{ m {I}}&=lim _{r ightarrow 0}{sqrt {2pi r}},sigma _{yy}(r,0)\K_{ m {II}}&=lim _{r ightarrow 0}{sqrt {2pi r}},sigma _{yx}(r,0)\K_{ m {III}}&=lim _{r ightarrow 0}{sqrt {2pi r}},sigma _{yz}(r,0),.end{aligned}}} The strain energy release rate ( G {displaystyle G} ) for a crack under mode I loading is related to the stress intensity factor by where E {displaystyle E} is the Young's modulus and ν {displaystyle u } is the Poisson's ratio of the material. The material is assumed to be an isotropic, homogeneous, and linear elastic. Plane strain has been assumed and the crack has been assumed to extend along the direction of the initial crack. For plane stress conditions, the equivalent relation is