The R-Curve measures crack resistance to tearing fracture for situations where the material thickness employed within a structure is below the requirement for plane-strain fracture toughness conditions.  The R-curve describes the extent of crack movement from an initial starting condition as a function of the level of applied stress-intensity factor (K) and as such represents a complete history of quasi-static crack growth up until fracture occurs.  It has been shown for several materials that the R-curve for a given thickness is independent of crack size and structural geometry [McCabe, 1973].

For the detailed reasons stated in Section 4 on Residual Strength, the R-curve is not as easily employed in design as abrupt fracture criteria.  Early work on aerospace materials with thicknesses below that required for K_Ic was directed at obtaining the limits on the R-curve, i.e. on obtaining K_ONSET, associated with the K conditions at the start of crack movement, and K_c, associated with the K conditions at the moment of instability.  After it was realized that the plane-stress fracture toughness (K_c) was a function of crack size and structural geometry as well as thickness, attention was focused on obtaining the complete history of the tearing fracture.

ASTM evolved a standard practice for determining the R-curve 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 utilized in tests must be of sufficient size to remain predominantly elastic throughout the duration of the test.  The reason for 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, to be adjusted for yield strength and toughness of the material considered.

The ASTM Standard E561 covers the determination of R-curves using middle cracked tension panel [M(T)], compact tension [C(T)], and crack-line-wedge-loaded [C(W)] specimens.  The compact tension and middle cracked tension panel geometries are illustrated in Figure 7.2.1.  A schematic illustrating the loading arrangement for the crack-line-wedge-loaded specimen is provided in Figure 7.2.4.  The crack-line-wedge-loaded configuration and loading conditions are such that, as the crack grows, the stress-intensity decreases under fixed-displacement conditions.  Such an arrangement facilitates collecting the complete R-curve using one specimen since the crack growth remains stable under decreasing K conditions.  Load control conditions ensure that the stress-intensity factor will increase as the crack grows.  This arrangement results in limiting the K_R versus crack extension (Da) data to a level associated with the fracture of the test specimen.

Figure 7.2.4.  Crack-Line-Loaded Specimen with Displacement-Controlled Wedge Loading [ASTM 2001]

While the C(W) specimen had gained substantial popularity for collecting K_R 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.  It is recommended in ASTM E561 that the M(T) specimen be sized so that the dimensions can be referenced to the plane stress plastic zone size (r_y).

where the specimen sizes are chosen on the basis of the maximum stress-intensity factor expected in the test.  Table 7.2.2 provides a list of minimum recommended M(T) sizes for assumed K_max -to-yield strength ratios.

Table 7.2.2.  ASTM E561-98 Recommended M(T) Dimensions

* Panels wider than 12 in. will require multiple pin grips and the

length requirement is relaxed to 1.5W

It should be noted that the initial crack length is sized to be W/3 to minimize the potential for net section yielding prior to a stress-intensity factor controlled fracture.  Based on data collected from a number of aluminum panels with different widths, it appears that there is a tendency for the calculated fracture toughness K_c to increase with increasing panel width, as shown in Table 7.2.3.  While it is difficult to generalize the observation based on these results to all materials, such data indicates that it is possible to develop conservative predictions of the plane-stress fracture toughness by using sub-size specimens.

Table 7.2.3.  Room Temperature Plane-Stress Fracture Toughness Values for Several

Aluminum Alloys Presented as a Function of Thickness and Width

*Width requirements meet ASTM E 561 requirements. 
+First number represents number of Kapp calculations, the second represents Kc    [ASTM 2001]

Another test condition important to consider during R-curve (or plane-stress fracture toughness) testing is the amount of buckling restraint that should be built into the test fixtures.  Most tests are conducted either with no buckling restraint or with extensive fixturing that tends to maintain inplane loading by preventing buckling.  With tests conducted with limited buckling restraint, the spurious stress distributions created when buckling occurs (at the specimen edges or in the crack tip region) can lead to mechanical driving factors that either enhance or degrade the calculated levels of applied stress-intensity factor.  The ASTM E561 method places restrictions on the amount of buckling exhibited during the R-curve test.

The data collected during an R-curve test includes load and crack size readings.  The stress-intensity factor associated with a given increment of crack size, i.e. K_R, is calculated using the stress-intensity factor formula for the specimen, the applied force (P), and a plasticity enhanced crack size.  The plasticity enhanced crack length is referred to as the effective crack (a_eff) and is calculated by adding the plane stress plastic zone radius (r_y), per Equation 7.2.2, to the current physical crack, i.e.

where a_o is the initial crack length and Da is the increment of crack movement.

Visual and non-visual methods of measuring crack size are available for collecting the data.  Within ASTM E561, the details associated with making crack length measurements based on compliance (force-displacement) methods are fully described. In fact, for those situations where extensive crack tip plasticity can occur, the compliance methods are recommended since these methods yield an estimate of crack length that already accounts for a plasticity correction.

ASTM E561 recommends that the R-curve be presented using an effective crack increment (Da_eff = Da + r_y) so that the instability predictions can be directly made from the plots.  Thus, the R-curve is a plot of K_R = K(a_eff, P) versus Da_eff.  The test engineer must describe how Da_eff and a_eff were calculated so that structural engineers using the data have a full report of the behavior.