Crack growth resistance curve
For materials that can be modeled with linear elastic fracture mechanics (LEFM), crack extension occurs when the applied energy release rate
is called a crack growth resistance curve, or R-curve.
R-curves can alternatively be discussed in terms of stress intensity factors
, can be thought of as the energetic gain associated with an additional infinitesimal increment of crack extension, while
can be thought of as the energetic penalty of an additional infinitesimal increment of crack extension.
The nature of the applied driving force curve relative to the material's R-curve determines the stability of a given crack.
The usage of R-curves in fracture analysis is a more complex, but more comprehensive failure criteria compared to the common failure criteria that fracture occurs when
is simply a constant value called the critical energy release rate.
An R-curve based failure analysis takes into account the notion that a material's resistance to fracture is not necessarily constant during crack growth.
The reduction in stress intensity factor due to high deformation can result in a flat R curve), as a crack propagates, the resistance to further crack propagation remains constant.
This tends to be an accurate model for perfectly brittle materials such as ceramics, in which the principal energetic cost of fracture is the development of new free surfaces on the crack faces.
[2] The character of the energetic cost of the creation of new surfaces remains largely unchanged regardless of how long the crack has propagated from its initial length.
As such, it can be technically challenging in these materials in practice to define a single value to quantify resistance to fracture (i.e.
and an applied energy release rate which is infinitesimally exceeding the R-curve at this crack length
then this material would immediately fail if it exhibited flat R-curve behavior.
was gradually increased over time (through increasing the applied force for example) then this would lead to stable crack growth in this material as long as the instantaneous slope of the driving force curve continued to be less than the slope of the crack resistance curve.
This tends to be the case in materials which undergo ductile fracture as it can be observed that the plastic zone at the crack tip increases in size as the crack propagates, indicating that an increasing amount of energy must be dissipated to plastic deformation for the crack to continue to grow.
[3] A rising R-curve can also sometimes be observed in situations where a material's fracture surface becomes significantly rougher as the crack propagates, leading to additional energy dissipation as additional area of free surfaces is generated.
, and instead will asymptotically approach some steady-state value after a finite amount of crack growth.
It is usually not feasible to reach this steady-state condition, as it often requires very long crack extensions before reaching this condition, and thus would require large testing specimen geometries (and thus high applied forces) to observe.
Polycrystalline graphite has been reported to demonstrate falling R-curve behavior after initially exhibiting rising R-curve behavior, which is postulated to be due to the gradual development of a microcracking damage zone in front of the crack tip which eventually dominates after the phenomena leading to the initial rising R-curve behavior reach steady-state.
[5] Size and geometry also plays a role in determining the shape of the R curve.
Ideally, the R curve, as well as other measures of fracture toughness, is a property only of the material and does not depend on the size or shape of the cracked body.
ASTM evolved a standard practice for determining R-curves to accommodate the widespread need for this type of data.
While the materials to which this standard practice can be applied are not restricted by strength, thickness or toughness, the test specimens must be of sufficient size to remain predominantly elastic throughout the test.
The size requirement is to ensure the validity of the linear elastic fracture mechanics calculations.
Specimens of standard proportions are required, but size is variable, adjusted for yield strength and toughness of the material considered.
While the C(W) specimen had gained substantial popularity for collecting KR curve data, many organizations still conduct wide panel, center cracked tension tests to obtain fracture toughness data.
As with the plane-strain fracture toughness standard, ASTM E399, the planar dimensions of the specimens are sized to ensure that nominal elastic conditions are met.
For the M(T) specimen, the width (W) and half crack size (a) must be chosen so that the remaining ligament is below net section yielding at failure.
