Stress intensity factor


The stress intensity factor,, is used in fracture mechanics to predict the stress state 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 depends on specimen geometry, the size and location of the crack or notch, and the magnitude and the distribution of loads on the material. It can be written as:
where is a specimen geometry dependent function of the crack length,, and the specimen width,, and is the applied stress.
Linear elastic theory predicts that the stress distribution near the crack tip, in polar coordinates with origin at the crack tip, has the form
where is the stress intensity factor and is a dimensionless quantity that varies with the load and geometry. Theoretically, as goes to 0, the stress goes to resulting in a stress singularity. Practically however, this relation breaks down very close to the tip because plasticity typically occurs at stresses exceeding the material's yield strength and the linear elastic solution is no longer applicable. Nonetheless, if the crack-tip plastic zone is small in comparison to the crack length, the asymptotic stress distribution near the crack tip is still applicable.

Stress intensity factors for various modes

In 1957, G. Irwin found that the stresses around a crack could be expressed in terms of a scaling factor called the stress intensity factor. He found that a crack subjected to any arbitrary loading could be resolved into three types of linearly independent cracking modes. These load types are categorized as Mode I, II, or III as shown in the figure. Mode I is an opening mode where the crack surfaces move directly apart. Mode II is a sliding 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 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 and applied to the crack opening mode. The mode II stress intensity factor,, applies to the crack sliding mode and the mode III stress intensity factor,, applies to the tearing mode. These factors are formally defined as:
Equations for stress and displacement fields

The mode I stress field expressed in terms of is
and
The displacements are
Where, for plane stress conditions
and for plane strain
For mode II
and
And finally, for mode III
with.

Relationship to energy release rate and J-integral

In plane stress conditions, the strain energy release rate for a crack under pure mode I, or pure mode II loading is related to the stress intensity factor by:
where is the Young's modulus and is the Poisson's ratio of the material. The material is assumed to be an isotropic, homogeneous, and linear elastic. The crack has been assumed to extend along the direction of the initial crack
For plane strain conditions, the equivalent relation is a little more complicated:
For pure mode III loading,
where is the shear modulus. For general loading in plane strain, the linear combination holds:
A similar relation is obtained for plane stress by adding the contributions for the three modes.
The above relations can also be used to connect the J-integral to the stress intensity factor because

Critical stress intensity factor

The stress intensity factor,, is a parameter that amplifies the magnitude of the applied stress that includes the geometrical parameter . Stress intensity in any mode situation is directly proportional to the applied load on the material. If a very sharp crack, or a V-notch can be made in a material, the minimum value of can be empirically determined, which is the critical value of stress intensity required to propagate the crack. This critical value determined for mode I loading in plane strain is referred to as the critical fracture toughness of the material. has units of stress times the root of a distance. The units of imply that the fracture stress of the material must be reached over some critical distance in order for to be reached and crack propagation to occur. The Mode I critical stress intensity factor,, is the most often used engineering design parameter in fracture mechanics and hence must be understood if we are to design fracture tolerant materials used in bridges, buildings, aircraft, or even bells.
Polishing cannot detect a crack. Typically, if a crack can be seen it is very close to the critical stress state predicted by the stress intensity factor.

G–criterion

The G-criterion is a fracture criterion that relates the critical stress intensity factor to the stress intensity factors for the three modes. This failure criterion is written as
where is the fracture toughness, for plane strain and for plane stress. The critical stress intensity factor for plane stress is often written as.

Examples

Infinite plate: Uniform uniaxial stress

Penny-shaped crack in an infinite domain

Finite plate: Uniform uniaxial stress

Edge crack in a plate under uniaxial stress

Infinite plate: Slanted crack in a biaxial stress field

Crack in a plate under point in-plane force

Loaded crack in a plate

Stress intensity factors for fracture toughness tests

Compact tension specimen

Single edge notch bending specimen

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