![cracktipcoords](https://upload.wikimedia.org/wikipedia/commons/2/2b/Cracktipcoords.svg)
In
fracture mechanics, the stress intensity factor () is used to predict the
stress
Stress may refer to:
Science and medicine
* Stress (biology), an organism's response to a stressor such as an environmental condition
* Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...
state ("stress intensity") near the tip of a
crack or
notch caused by a remote load or
residual stress
In materials science and solid mechanics, residual stresses are Stress (physics), stresses that remain in a solid material after the original cause of the stresses has been removed. Residual stress may be desirable or undesirable. For example, l ...
es.
It is a theoretical construct usually applied to a homogeneous, linear
elastic
Elastic is a word often used to describe or identify certain types of elastomer, elastic used in garments or stretchable fabrics.
Elastic may also refer to:
Alternative name
* Rubber band, ring-shaped band of rubber used to hold objects togeth ...
material and is useful for providing a
failure criterion for
brittle
A material is brittle if, when subjected to stress, it fractures with little elastic deformation and without significant plastic deformation. Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Bre ...
materials, and is a critical technique in the discipline of
damage tolerance
In engineering, damage tolerance is a property of a structure relating to its ability to sustain defects safely until repair can be effected. The approach to engineering design to account for damage tolerance is based on the assumption that flaws ...
. 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
In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
(
) with origin at the crack tip, has the form
:
where is the stress intensity factor (with units of stress × length
1/2) 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 (small ) because
plasticity typically occurs at stresses exceeding the material's
yield strength
In materials science and engineering, the yield point is the point on a stress-strain curve that indicates the limit of elastic behavior and the beginning of plastic behavior. Below the yield point, a material will deform elastically and wi ...
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 (
tensile
In physics, tension is described as the pulling force transmitted axially by the means of a string, a rope, chain, or similar object, or by each end of a rod, truss member, or similar three-dimensional object; tension might also be described as ...
) mode where the crack surfaces move directly apart. Mode II is a sliding (in-plane
shear
Shear may refer to:
Textile production
*Animal shearing, the collection of wool from various species
**Sheep shearing
*The removal of nap during wool cloth production
Science and technology Engineering
*Shear strength (soil), the shear strength ...
) 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
Antiplane shear or antiplane strainW. S. Slaughter, 2002, ''The Linearized Theory of Elasticity'', Birkhauser is a special state of strain in a body. This state of strain is achieved when the displacements in the body are zero in the plane of inte ...
) 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:
[
:
{, class="toccolours collapsible collapsed" style="text-align:left"
!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
In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analys ...
conditions, the strain energy release rate In fracture mechanics, the energy release rate, G, is the rate at which energy is transformed as a material undergoes fracture. Mathematically, the energy release rate is expressed as the decrease in total potential energy per increase in fracture ...
() 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
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied leng ...
and is the Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
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
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
conditions, the equivalent relation is a little more complicated:
:
:
For pure mode III loading,
:
where is the shear modulus
In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain:
:G \ \stackre ...
. 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
The J-integral represents a way to calculate the strain energy release rate, or work ( energy) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov and independently in 1968 ...
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 (load type). 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
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
is referred to as the critical fracture toughness () of the material. has units of stress times the root of a distance (e.g. MN/m3/2). 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 (or fracture toughness) to the stress intensity factors for the three modes. This failure criterion is written as
:
where is the fracture toughness, for plane strain
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
and for plane stress
In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analys ...
. The critical stress intensity factor for ''plane stress'' is often written as .
Examples
Infinite plate: Uniform uniaxial stress
{,
, - align = "left" valign = "top"
,
The stress intensity factor for an assumed straight crack of length perpendicular to the loading direction, in an infinite plane, having a uniform stress field is
:
,
Penny-shaped crack in an infinite domain
{,
, - align = "left" valign = "top"
,
The stress intensity factor at the tip of a penny-shaped crack of radius in an infinite domain under uniaxial tension is
:
,
Finite plate: Uniform uniaxial stress
{,
, - align = "left" valign = "top"
,
If the crack is located centrally in a finite plate of width and height , an approximate relation for the stress intensity factor is [
:
If the crack is not located centrally along the width, i.e., , the stress intensity factor at location A can be approximated by the series expansion][Isida, M., 1966, ''Stress intensity factors for the tension of an eccentrically cracked strip'', Transactions of the ASME Applied Mechanics Section, v. 88, p.94.]
:
where the factors can be found from fits to stress intensity curves[ for various values of . A similar (but not identical) expression can be found for tip B of the crack. Alternative expressions for the stress intensity factors at A and B are ]
:
where
:
with
:
In the above expressions is the distance from the center of the crack to the boundary closest to point A. Note that when the above expressions do ''not'' simplify into the approximate expression for a centered crack.
,
Edge crack in a plate under uniaxial stress
{,
, - align = "left" valign = "top"
,
For a plate having dimensions containing an unconstrained edge crack of length , if the dimensions
of the plate are such that and , the stress intensity factor at the
crack tip under a uniaxial stress is
:
For the situation where and , the stress intensity factor can be approximated
by
:
,
Infinite plate: Slanted crack in a biaxial stress field
{,
, - align = "left" valign = "top"
,
For a slanted crack of length in a biaxial stress field with stress in the -direction and in the -direction, the stress intensity factors are [
:
where is the angle made by the crack with the -axis.
, ]
Crack in a plate under point in-plane force
{,
, - align = "left" valign = "top"
,
Consider a plate with dimensions containing a crack of length . A point force with components and is applied at the point () of the plate.
For the situation where the plate is large compared to the size of the crack and the location of the force is relatively close to the crack, i.e., , , , , the plate can be considered infinite. In that case, for the stress intensity factors for at crack tip B () are
:
where
:
with , , for plane strain
In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimal ...
, for plane stress
In continuum mechanics, a material is said to be under plane stress if the stress vector is zero across a particular plane. When that situation occurs over an entire element of a structure, as is often the case for thin plates, the stress analys ...
, and is the Poisson's ratio
In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
.
The stress intensity factors for at tip B are
:
The stress intensity factors at the tip A () can be determined from the above relations. For the load at location ,
:
Similarly for the load ,
:
,
Loaded crack in a plate
{,
, - align = "left" valign = "top"
,
If the crack is loaded by a point force located at and , the stress intensity factors at point B are[
:
If the force is distributed uniformly between , then the stress intensity factor at tip B is
:
:
:
, ]
Compact tension specimen
{,
, - align = "left" valign = "top"
,
The stress intensity factor at the crack tip of a compact tension specimen
A compact tension specimen (CT) is a type of standard notched specimen in accordance with ASTMASTM E647-00 ''Standard Test Method for Measurement of Fatigue Crack Growth Rates''. ASTM International, 2000. and ISOISO 7539-6 ''Corrosion of metals an ...
is
:
where is the applied load, is the thickness of the specimen, is the crack length, and
is the width of the specimen.
,
Single-edge notch-bending specimen
{,
, - align = "left" valign = "top"
,
The stress intensity factor at the crack tip of a single-edge notch-bending specimen is[
:
where is the applied load, is the thickness of the specimen, is the crack length, and
is the width of the specimen.
, ]
See also
* Fracture mechanics
* Fracture toughness
In materials science, fracture toughness is the critical stress intensity factor of a sharp crack where propagation of the crack suddenly becomes rapid and unlimited. A component's thickness affects the constraint conditions at the tip of a c ...
* Strain energy release rate In fracture mechanics, the energy release rate, G, is the rate at which energy is transformed as a material undergoes fracture. Mathematically, the energy release rate is expressed as the decrease in total potential energy per increase in fracture ...
* J-integral
The J-integral represents a way to calculate the strain energy release rate, or work ( energy) per unit fracture surface area, in a material. The theoretical concept of J-integral was developed in 1967 by G. P. Cherepanov and independently in 1968 ...
* Material failure theory
Material failure theory is an interdisciplinary field of materials science and solid mechanics which attempts to predict the conditions under which solid materials fail under the action of external loads. The failure of a material is usuall ...
* Paris' law
Paris' law (also known as the Paris–Erdogan equation) is a crack growth equation that gives the rate of growth of a fatigue (material), fatigue crack. The stress intensity factor K characterises the load around a crack tip and the rate of crac ...
References
{{reflist, 2
External links
Kathiresan, K. ; Hsu, T. M. ; Brussat, T. R., 1984, Advanced Life Analysis Methods. Volume 2. Crack Growth Analysis Methods for Attachment Lugs
o
www.fracturemechanics.org
by Bob McGinty
Fracture mechanics