In materials modeled by linear elastic fracture mechanics (LEFM), crack extension occurs when the energy release rate G equals R, the material's resistance to crack extension. A plot of R versus crack extension is called a crack resistance curve, or R curve. The corresponding plot of energy release rate versus crack extension is called a driving force curve.
Contents
Types of R-curve
Generally, materials display either a flat R curve or a rising R curve. For a flat R curve, material resistance is constant with respect to crack extension. When the resistance curve is flat, it exhibits a critical value of energy release rate Gc. A material with a rising R curve, however, cannot be uniquely characterized with a single toughness value. A flawed structure fails when the driving force curve is tangent to the R curve, but this point of tangency depends on the shape of the driving force curve, which depends on the configuration of the structure. Materials with rising R curves are generally characterized by the value of G at the beginning of crack growth, but the precise moment of initiation is unclear. The R curve for an ideally brittle material is flat because the surface energy is a fixed property. Nonlinear behavior, like ductile fracture, can result in a rising R curve as the plastic zone at crack tip increases in size with extension. Falling R curves occur when a metal fails by cleavage. Cleavage propagation is unstable and very high strain rates are found near the crack tip. This suppresses plastic deformation.
Effect of size and shape
Size and geometry also plays a role in determining the shape of the R curve. A crack in a thin sheet tends to produce a steeper R curve than a crack in a thick plate because there is a low degree of stress triaxiality at the crack tip in the thin sheet while the material near the tip of the crack in the thick plate may be in plane strain. The R curve can also change at free boundaries in the structure. Thus, a wide plate may exhibit a somewhat different crack growth resistance behavior than a narrow plate of the same material. 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. Much of fracture mechanics is predicated on the assumption that fracture toughness is a material property.
Testing
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.
ASTM Standard E561 covers the determination of R-curves using a middle cracked tension panel [M(T)], compact tension [C(T)], and crack-line-wedge-loaded [C(W)] specimens. 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.