The displacements during unloading are shown in Fig. 2. During unload- ing, the reversed plastic zone stress is increased to twice the yield stress used during loading, accounting for an elastic stress range equal to the difference of tensile and compressive yield stresses.
The crack surface displacements at the minimum applied load, shown
'Hsu, T. M. and Lassiter, L. W., "Effects of Compressive Overloads on Fatigue Crack Growth," AIAA Paper No. 74-365, AIAA/ASME/SAE Structures, Structural Dynamics and Materials Conference, Las Vegas, Nev., 17-19 April 1974.
DILL AND SAFF ON CRACK GROWTH 143
ELASTIC DISPLACEMENTS DUE TO REMOTE LOADING, SpEMOTE
ELASTIC DISPLACEMENTS DUE TO PLASTIC ZONE YIELD STRESS, 6.
FIG. 1—Elastic crack surface displacement at maximum load.
in Fig. 3, are found by subtracting the displacements occurring during un- loading from those at maximum load. The crack surface remains wedged open by the stress distributions within the plastic zone even when the min- imum load is zero, if permanent plastic deformations caused by prior crack growth are not considered. The center portion of Fig. 3 shows results of this wedging effect. However, when the permanent plastic deformations are con- sidered, the crack surfaces will interfere upon unloading. The interference is visualized as though the surfaces were allowed to pass through each other.
The potential interference of either surface is the displacement of that surface past the centerline of the crack.
In modeling the residual deformation and surface interference, the plastic deformation at the crack tip at minimum load is considered to be equal to the crack opening displacement (COD) at that load. This crack opening dis- placement for tensile minimum loads is approximated as
CODATmin = U {K\^ - 1/2 KK^) j lEfo (1)
NET DISPLACEMENT AT MAXIMUM
NET DISPLACEMENT DUE TO UNLOADtNG, 6^K
PLASTIC ZONE NET DISPLACEMENT AT
MINIMUM LOAD, 5
CODK„,M = C O D K „ ^ X - A C O D
FIG. 2—Elastic crack surface displacement at minimum load.
PLASTIC ZONE
where
a = 1 for plane stress,
a = (1 - i;')/2 for plane strain, K = stress intensity,
^K = stress intensity range, E = Young's modulus, and /o = yield stress.
This deformation exists just behind the crack tip as well since material adjacent to the crack tip, both ahead and behind, is compressed to this value on unloading. Correlation with constant-amplitude closure stress intensity
DILL AND SAFF ON CRACK GROWTH 145 data from Elber" shows that a closer approximation to the permanent plastic deformation is
8rôidual = a (/C'max " 0.4 AK') I 2Efo (2)
where 5 = crack surface displacement.
The potential interference is the difference between the permanent plastic deformation and the minimum displacement of the elastic surface as shown in Fig. 3.
The potential interference acts as a wedge behind the crack tip, creating a stress intensity at the crack tip. To determine the stresses behind the tip caused by this wedge, a simple contact stress model of closure was developed.
This model, symbolized in Fig. 4, uses twenty-five constant stress elements to idealize the wedge. Bueckner's weight function approach^ was used to develop an influence coefficient matrix for the displacement-stress rela- tionship between elements. The analysis is iterative so that a solution is determined wherein the maximum contact stress is limited to the yield stress and there is no tensile contact stress.
The stresses determined from interference are used to compute the contact stress intensity occurring at zero load, the effective minimum stress inten- sity, and the effective stress intensity range.
PERMANENT PLASTIC DEFORMATION, 6p
^ I
PHYSICAL CRACK TIP
NET DISPLACEMENT AT MINIMUM LOAD, S K M I N
*WEDGE = 5 p - * K M | N POTENTIAL INTERFERENCE, SyvEDGE
FIG. 3—Potential interference at minimum load under constant-amplitude cycling.
"Elber, W. in Damage Tolerance in Aircraft Structures, ASTM STP 486, American Society for Testing and Materials, 1971, pp. 230-242.
'Bueckner, H. P., Zeitschrift Fuer Angewandte Mathematik und Mechanik, Vol. 50, 1970, pp. 529-546.
LOCATION OF DISPLACEMENT COMPUTATION AX/u-
1.4 1.3
X/"TOTAL = '"'.5
REFERENCE LENGTH
• 25 X 25 INFLUENCE COEFFICIENT MATRIX DERIVED THROUGH BUECKNER'S WEIGHT FUNCTION APPROACH
• MAXIMUM STRESS LIMITED TO YIELD
• NO TENSILE STRESSES
• DISPLACEMENTS AND STRESS INTENSITY COMPUTED FOR INPUT VALUES OF K ^ A X . ^APP. E. AND fo
FIG. 4—Contact stress model of closure.
Analysis of COD With Compressive Loads
The Westergaard-Dugdale analysis, including the computation of
CODK min through use of Eq 2, is applicable for minimum loads equal to or greater than zero. However, when the minimum load is less than zero, a portion of the crack surface is in contact, even when permanent plastic de- formations caused by crack growth are not considered. In order to compute CODjc„j„ for compression loads, a necessary step in determining the per- manent plastic deformation left in the wake of the growing crack, a separate analysis was performed. Ordinarily, the computation of displacements and contact stresses is made with the origin of reference for the contact stress model placed at the physical tip. However, for the analysis of CODK^i„ for compression loads, the reference origin of the model was placed at the end of the forward plastic zone. Figure 5 depicts the displacements input to the computation. The wedge is composed of the sum of the residual displace- ments in the forward plastic zone caused by loading and the elastic displace- ments due to applying the compressive load. This wedge is depicted in the lower portion of Fig. 5, and the resulting displacement and stress distribution is depicted in the bottom of Fig. 5.
The results of this analysis procedure are summarized in Fig. 6. Normal- ized crack surface displacements are shown for both positive and negative loads. The analyses of positive minimum loads were performed to verify the analysis procedure by comparison to previous Dugdale-Westergaard analysis results. Figure 7 summarizes the COD at minimum load, for various R ratios. {R = ratio of minimum to maximum applied load.) It can be seen that there is a rapid decrease in COD for positive R ratios, as the minimum load is reduced. However, for negative R ratios, the crack surface contact
DILL AND SAFF ON CRACK GROWTH 147 PLASTIC ZONE
AT K M A X -
DISPLACEMENTS DUE T O K M A X , 5 M A X -
ELASTIC DISPLACEMENTS DUE TO REMOTE LOADING, K|V||N WITHOUT CRACK SURFACE CONTACT, SMIN"
IHHIIMHHMHIHIHt"
COMPRESSION- TENSION-uJ
FIG. 5—Computation of crack surface displacements under compression loading.
prevents this decrease in COD from continuing, and, even at /? = -3.0, the COD is greater than half the COD at /? = 0.
With the effects of compression loads upon residual plastic deformation accounted for, analysis of contact stress intensity, including the effects of compression, could be performed in the same manner as under tensile loading.
Crack Growth Following Compressive High Loads
An excellent test program was conducted by Hsu and Lassiter on Ti-6A1- 4V beta annealed titanium and 7050-T73 aluminum plate materials and was reported in the source cited in Footnote 3. Two types of crack geometries shown in Fig. 8 were utilized: center-through-the-thickness crack, and
through-the-thickness crack emanating from one side of a 0.636-cm hole.
An electrodischarge machine was used to introduce the initial notch. The specimens were then precracked with constant-amplitude fatigue cycling until the initial crack lengths were 1.12 cm for the center-crack panels and 0.17 cm for the open-hole panels. The four basic constant-amplitude load spectra are shown in Fig. 9. Each block of loading consisted of 1 overload cycle and 49 cycles of constant-amplitude stress at 137 MN/m^forTi-6Al-4V specimens and 68.95 MN/m^ for 7050-T73 specimens.
CRACK SURFACE DISPLACEMENT
C O D M A X
1.5 1.0 0.5 0 - 0 . 5 - 1 . 0 DISTANCE FROM CRACKTIP/PLASTIC ZONE SIZE
FIG. 6—Crack surface displacements at minimum load.
The high load ratios, defined as the ratio of the tensile stress high load and the constant-amplitude stress load in the Type A spectrum were 1.0, 1.2, 1.5, and 2.0. The high load ratios in the Type B spectrum were 1.2 and 1.5 with i? = -1.0 for the tension-compression high load cycle, where R is the applied stress ratio defined as Ommlamix- The sequence of tension-compression cycles used in Type C loading was fully reversed from Type B. The com- pressive high load ratios applied in the Type D spectrum were -1.2 and -1.5.