The crack opening displacement (COD) parameter was proposed to provide a more physical explanation for crack extension processes. [Wells, 1961; Burdekin & Stone, 1966]  The philosophy was based on a crack tip strain based model of cracking that would allow for the occurrence of elastic-plastic material behavior.  The initial modeling, however, was based on elasticity solutions of crack tip displacements.  Equation 2.2.54 describes the x and y displacements (u and v, respectively) in the crack tip region of an elastic material:

		(2.2.54a)
	(2.2.54b)

where k = 3 - 4n for plane strain and k = (3 - n)/(1 + n) for plane stress, and where G is the shear modulus (G = 0.5E/(1 + n)).  If the angle q  is chosen to be 180° (p), the displacements are those associated with crack sliding (u component) or opening (v component).  Under mode 1 (symmetrical) loading, the case covered by Equation 1.3.54, the sliding displacement term is noted to be identically zero; and all displacement is perpendicular to the crack, i.e. only opening is observed.  Based on Equations 2.2.54 and 2.2.18 and the definition of shear modulus (G), the displacement of the crack relative to its longitudinal axis (x axis) is

(2.2.55)

The relative movement of the crack faces is the COD and it is twice the value obtained by Equation 2.2.55, i.e.

COD = 2v

(2.2.56)

One immediate observation is that COD will vary as a function of position along the crack, and that the COD at the crack tip, i.e. at r = 0, is zero.  In the quasi-elastic-plastic analysis performed by Wells, the crack was allowed to extend to an effective length (a_e), one plastic zone radius larger than the physical crack length (a); the crack opening displacement was then determined at the location of the physical crack tip.  Figure 2.2.15 describes the model used to define the crack tip opening displacement (CTOD).  The Wells modeling approach leads one to

(2.2.57)

which after some simplification gives the CTOD as

(2.2.58)

 

 

Figure 2.2.15.  Description of Model Used to Establish the CTOD Under Elastic Conditions

 

It is immediately seen that the CTOD is directly related to the stress-intensity factor for elastic materials; thus, for elastic materials, fracture criteria based on CTOD are as viable as those based on the stress-intensity factor parameter.  The other relationships developed between K and G or J in this section allow one to directly relate G and J to the CTOD in the elastic case.

In the late 1960’s, Dugdale [1960] conducted an elasticity analysis of a crack problem in which a zone of yielding was postulated to occur in a strip directly ahead of the crack tip.  The material in the strip was assumed to behave in a perfect plastic manner.  The extent of yielding was determined such that the singularity at the imaginary crack tip (see Figure 2.2.16) was canceled due to the balancing of the remote positive stress-intensity factor with the local yielding negative stress-intensity factor.  The Dugdale quasi-elastic-plastic analysis provided an estimate of the relative displacement of the crack surfaces for a center crack (crack length = 2a) in an infinite plate subjected to a remote tensile stress (s) and having a yield strength equal to s_o, the CTOD is

(2.2.59)

at the tip of the physical crack tip (a) and the extent of the plasticity ahead of the crack is

(2.2.60)

 

 

Figure 2.2.16.  Dugdale Type Strip Yield Zone Analysis

For the case of small scale yielding, i.e., when s/s₀ is low, the CTOD and extent of plasticity (w) reduce to

(2.2.61)

and

(2.2.62)

It can first be noted that the extent of the plasticity (w) is only about 20% higher than would be predicted using the Irwin estimate of the plastic zone diameter (2r_y).  The level of CTOD estimated by Equation 2.2.61 also compares favorably with that given by Equation 2.2.58; Equation 2.2.61 gives an estimate that is about 30 percent lower than Equation 2.2.58.  Numerous other studies have shown that the CTOD is related to the stress-intensity factor under conditions of small scale yielding through

(2.2.63)

where the constant a ranges from about 1 to 1.5.  Experimental measurements [Bowles, 1970; Roberson & Tetelman, 1973] have indicated that a is close to 1.0, although there is substantial disagreement about the location where CTOD should be measured.

One difficulty with elastic analyses is that the crack actually remains stationary and thus one must reposition the crack through a quasi-static crack extension so that the CTOD for the actual crack can be assessed.  During loading, cracks in ductile materials tend to extend through a slow tearing mode of cracking prior to reaching the fracture load level.  In these cases, the amount of opening that occurs at the initial crack tip represents one measure of the crack tip strain; but, this parameter depends not only on load, initial crack length and material properties, it also depends on the amount of crack extension from the initial crack tip.  Rice and co-workers [Rice, 1968b; Rice & Tracey, 1973] attempted to provide an alternate choice of locating the position where CTOD would be measured.  They found that when the CTOD was determined for the position shown in Figure 2.2.17, the CTOD and J integral were related (for ideally plastic materials) by

(2.2.64)

For the case of plane stress behavior, d_n is unity and for plane strain behavior, d_n is about 0.78.

 

Figure 2.2.17.  Definition of the Crack Tip Opening Displacement (CTOD)

For strain hardening materials controlled by Equation 2.2.45, Shih and co-workers [Shih & Kumar, 1979; Shih, 1979] have shown that Equation 2.2.64 relates J and CTOD if the constant d_n is replaced with a function that is strongly dependent on the strain hardening exponent and mildly dependent on the ratio s_o/E.  Thus, there is a direct relationship between CTOD and J throughout the region of applicability of the J-Integral and CTOD can likewise be considered a measure of the magnitude of the crack tip stress-strain field.