In materials modeled by
linear elastic fracture mechanics (LEFM), crack extension occurs when the applied
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 fractu ...
exceeds
, where
is the material's resistance to crack extension.
Conceptually
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. At any moment in time, if
then crack extension is energetically favorable. A complication to this process is that in some materials,
is not a constant value during the crack extension process. A plot of crack growth resistance
versus crack extension
is called a crack growth resistance curve, or R-curve. A plot of energy release rate
versus crack extension
for a particular loading configuration is called the driving force curve. 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
where
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.
R-curves can alternatively be discussed in terms of
stress intensity factors rather than
energy release rates , where the R-curves can be expressed as the
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 ...
(
, sometimes referred to as
) as a function of crack length
.
Types of R-Curves
Flat R-Curves
The simplest case of a material's crack resistance curve would be materials which exhibit a "flat R-curve" (
is constant with respect to
). In materials with flat R-curves, as a crack propagates, the resistance to further crack propagation remains constant and thus, the common
failure criteria of
is largely valid. In these materials, if
increases as a function of
(''which is the case in many loading configurations and crack geometries''), then as soon as the applied
exceeds
the crack will unstably grow to failure without ever halting.
Physically, the independence of
from
is indicative that in these materials the phenomena which are energetically costly during crack propagation do not evolve during crack propagation. This tends to be an accurate model for perfectly
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 such as
ceramics
A ceramic is any of the various hard, brittle, heat-resistant and corrosion-resistant materials made by shaping and then firing an inorganic, nonmetallic material, such as clay, at a high temperature. Common examples are earthenware, porcelain ...
, in which the principal energetic cost of fracture is the development of new
free surfaces on the crack faces. 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.
Rising R-Curves
![Rising R Curve Stability Criterion V2](https://upload.wikimedia.org/wikipedia/commons/7/77/Rising_R_Curve_Stability_Criterion_V2.0.png)
Another category of R-curve that is common in real materials is a "rising R-curve" (
increases as
increases). In materials with rising R-curves, as a crack propagates, the resistance to further crack propagation increases, and it requires a higher and higher applied
in order to achieve each subsequent increment of crack extension
. As such, it can be technically challenging in these materials in practice to define a single value to quantify resistance to fracture (i.e.
or
) as the resistance to fracture rises continuously as any given crack propagates.
Materials with rising R-curves can also more easily exhibit stable crack growth than materials with flat R-curves, even if
strictly increases as a function of
. If at some moment in time a crack exists with initial length
and an applied energy release rate which is infinitesimally exceeding the R-curve at this crack length