AN EXPERIMENTAL STUDY ON THE ELASTIC-PLASTIC FRACTURE
IN A DUCTILE MATERIAL UNDER MIXED MODE LOADING
Perng-Fei Luo and Ching-Hsiung Wang
Department of Mechanical Engineering
Chung Hua University
No. 707, Sec. 2, Wu Fu Rd., Shin Chu 30012, Taiwan, R.O.C.
ABSTRACT
A stereo vision is used to measure the crack tip parameters, such as J integral, plastic mixity, and elastic mixity, of mixed mode
fracture specimens, and to study the applicability of the Shih’s plane strain solution to the mixed mode crack tip fields. The
fracture specimen used in the paper is a compact tension shear (CTS) specimen made of 2024-O aluminum. To conduct mixed
mode fracture experiments at the loading angles of 75° and 45°, a special loading device is used. The in-plane strain and stress
fields near the mixed mode crack tip are determined using the deformation field measured by the stereo vision. Then the J
integral along rectangular contours surrounding the mixed mode crack tip can be evaluated. It can be seen that the computed J
integral values approach constant after r/h > 0.5. In the paper the in-plane strains determined experimentally at several points
near the crack tip are compared with the values calculated using Shih’s plane strain solution. It is found that the measured
values follow the trends of the Shih’s plane strain solution. The elastic mixity evaluated using the measured crack tip stress
fields are close to those obtained from analytical solution. However the evaluated plastic mixity deviates from the analytical
solution.
Introduction
For cracks in elastic-plastic material, strain hardening plasticity solutions have been presented by Hutchinson [1, 2] and Rice
and Rosengren [3] to characterize the behavior at the crack tip in the plastic zone. They are referred as HRR solution and are
restricted to problems in which the stress distribution is either symmetric (Mode I) or antisymmetric (Mode II) with respect to the
crack tip. For a mixed-mode crack problem, Shih [4] used the J-integral [5] which is path independent and defined the plastic
mixity to modify the HRR solution and obtained the asymptotic crack-tip stress and strain fields around the crack tip. For plane
strain mixed-mode fields, Shih [6] presented tables of the plane strain dimensionless functions for several mixities ranging from
Mode I to Mode II.
Pawliska et al. [7] used finite element analysis to determine the stresses around the mixed-mode crack tip at several load
angles. The numerical results were then compared with the values obtained using the Tables of Shih [6]. Aoki et al. used finite
element to compute the J-integral values (JI and JII) around the crack tip of a compact tension shear specimen made of
A5083-O aluminum alloy. The elastic-plastic analysis for the strains at a specific point near the crack tip under the mixed-mode
loading was also conducted. The calculated results were in good agreement with the experimental data obtained from the strain
gage.
Studies on the elastic-plastic fracture behavior for a ductile material under mixed-mode loading are very few, especially for the
determination of the whole crack-tip deformation field. In the paper a stereo vision [9, 10] was used to measure the out-of-plane
and in-plane displacement fields in the vicinity of a crack tip subjected to a mixed-mode loading. The in-plane displacement field
was then used to study whether the strain field can be characterized by the modified HRR solution. In addition, the crack tip
parameters, such as J integral, plastic mixity, and elastic mixity, of a mixed mode fracture specimen were also investigated in
the paper.
Theoretical Background
For a mixed-mode crack in elastic-plastic material, the asymptotic crack-tip strain field around the crack tip can be written as [4,
6]
n
⎡
⎤ n+1
J
ε ij = αε 0 ⎢
εij (θ , n, M p )
⎥
p
⎣ ασ 0ε 0 In( n, M ) r ⎦
(1)
where σ0 is the yield stress, ε0 (= σ0/E) the yield strain, α the strain hardening coefficient, n the strain hardening exponent, r, θ
the polar coordinates at the crack tip, Mp the plastic mixity defined as
Mp =
2
π
tan −1 lim
r →0
σ θθ ( r,0)
,
σ rθ ( r,0)
(2)
and J is the line integral surrounding the crack tip defined by
J = ∫ Γ Wdy − σ ij ⋅ n j
∂ui
ds ,
∂x
(3)
where nj is a normal vector perpendicular to the contour Γ enclosing the crack tip, ui is the displacement vector on the contour,
p
p
ds is the differential arc length along Γ, and W is the strain energy density. It is noted that M = 1 for mode I and M =0 for mode
II. In is a dimensionless constant and εij is a dimensionless function of angular position. A detailed tabulation of In and εij is
given by Shih [6].
In a region called far field whose size is larger than the plastic zone but small compared to the crack length, the asymptotic
elastic solution dominates. Then the J integral can be related to the stress intensity factors KI and KII by
J=
K I 2 + K II 2
π
e
In the far field, Shin [4] defined the elastic mixity M to characterize the relative strengths of KI and KII, which is given by
Me =
=
2
π
2
π
tan −1 lim
r →∞
tan
−1
σ θθ ( r ,0)
σ rθ ( r ,0)
(4)
KI
K II
For a central crack in a large plate subjected to a remote uniaxial stress oriented at an angle β with respect to the axis of the
crack, the ratio of KI and KII is related to β by KI /KII = tanβ. Hence equation (4) can be rewritten as
M e = (2 π ) β .
(5)
Material Properties and Specimen Geometries
The fracture specimen used in the experiment was made of 2024-O aluminum. A uniaxial stress-strain tension test was
performed for AL 2024-O, and then a Ramberg-Osgood relation was used to fit the stress-strain curve, i.e.,
n
⎛σ ⎞
ε
σ
=
+α ⎜ ⎟ .
ε0 σ 0
⎝ σ0 ⎠
(6)
The material properties obtained from the test are shown as follows:
Yield stress, σ0
Elastic Modulus, E
Strain hardening exponent, n
Strain hardening coefficient, α
Poisson's ratio, ν
120 MPa
72 GPa
4.325
1.234
0.333
Figure 1 shows the schematic diagram of the CTS specimen mounted on a special loading device. The CTS specimen with W =
90 mm was fabricated from an 8 mm thick plate of AL 2024-O. A chevron notch was machined in the notch tip. A fatigue
precrack was produced by cyclically loading the notched specimen with a sinusoidal wave form using standard ASTM
procedures. A random pattern was produced on the surface using black and white spray paint. The precrack of the CTS
specimen is about 6 mm and the a/W ratio is 0.5. In the experiment, a loading device designed by Rechard [11] was used so
that both the normal and shear loads can be applied to the CTS specimen. Hence, the investigation on the mixed-mode crack
tip deformation fields can be conducted at some specific loading angle.
Figure 1 CTS specimen with a special loading device
Experiments
The stereo vision was used to measure the three-dimensional displacement components surrounding the crack tip of a CTS
specimen. The experimental setup for the fracture experiment at the loading angle of 75° is shown in Figure 2. The stereo vision
is formed by two Sony XC-77 CCD cameras, one Matrox frame grabber and one Pentium III PC. The region of interest is about
25 × 22 mm. For the loading angle of 45°, the JAI CCD cameras were used and the region of interest is about 34 × 25 mm.
After calibrating the stereo system, an approximate error of 2 µm for each displacement component was obtained. A
displacement loading was applied to the CTS specimen with an Instron material test machine. The images around the crack tip
were taken at several loading stages and stored for later analysis. Combined with digital image correlation [12-14], both the
in-plane and out-of-plane displacement fields of the interested area were determined using the stereo vision [9, 10].
Figure 2 Experimental setup (Loading angle = 75°)
Determination of J integral
Based on the plane stress assumption, equation (3) can be used to evaluate the J integral along contours surrounding the crack
tip. First, the experimental in-plane displacement field was smoothed [15] to obtain the strain field. Using deformation theory, the
strains were separated into elastic strain and plastic strain and the stresses were obtained. Then the J integral was computed
along several rectangular paths around the crack tip. Figure 3 shows the rectangular integration contour, along which the values
of J integral were obtained.
Figure 3 Integration contour for the J integral
Results and Discussions
By using the smoothing technique [15], the in-plane displacement components u and v were smoothed to obtain the strains.
Figure 4 shows the experimentally determined εyy strain distribution around the crack tip. Figure 5 presents the J integral
against distance measured from the crack tip to the path of the line integral. It is well known that the three-dimensional effects
[16-18] usually prevent the data lying inside a circle of radius r = 0.5 h from being analyzable using two-dimensional analyses.
Hence, it can be seen in Fig. 5 that the J integral value only shows path independence for r/h > 0.5.
ε
y (mm)
-4
-2
0
2
yy
4
strain field(%)
6
8
10 12 14 16 18 20
8
8
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
-6
-8
-8
-4
-2
0
2
4
6
8
10 12 14 16 18 20
x (mm)
Figure 4 Contour plot of εyy strain field for the CTS specimen at 23,400 N applied load (Loading angle of 45°)
J (KPa-m)
90
45
0
0
5
10
15
20
Distance from crack tip (mm)
Fig. 5 distribution of J integral at 23,400 N applied load (Loading angle of 45°)
0.035
Experiment data
εxx
εyy
εxy
Plane strain solution
εxx
εyy
εxy
0.028
ε ij
0.021
0.014
0.007
0.000
-0.007
0
10
20
30
Load (kN)
Figure 6 Comparison of the experimental and numerical results at the loading angle of 75°
0.010
Experiment data
xx
ε
εyy
εxy
Plane strain solution
εxx
εyy
εxy
ε ij
0.006
0.002
-0.002
0
10
20
30
Load (kN)
Figure 7 Comparison of the experimental and numerical results at the loading angle of 45°
The in-plane strains determined experimentally at several points near the crack tip are compared with the values calculated
using Shih’s plane strain solution. Figures 6 and 7 show the comparison of the experiment data and the calculated values at the
loading angles of 75° and 45°, respectively. It is found that the measured values follow the trends of the Shih’s plane strain
solution. It is also noted that the location of the points indicated in Figs. 6 and 7 is beyond the region (r/h > 0.5) where the three
dimensional effects exist.
From the Tables of Shih, Mp = 0.835 for the loading angle of 75° and Mp = 0.5 for the loading angle of 45°. By equation (5), Me =
0.833 for the loading angle of 75° and Me = 0.5 for the loading angle of 45°. The evaluation of ( 2 π ) tan −1 σ yy σ xy along the
radial line of θ = 0 is presented in Figures 8 and 9. The values of elastic mixity evaluated using the measured crack tip stress
field are close to those obtained from analytical solution. However the evaluated plastic mixity deviates from the analytical
solution due to the three-dimensional effects.
1.60
Experiment data
σxy
yy
2
_ tan-1σ
___
e
M = 0.833
p
M = 0.835
π
0.80
0.00
0
4
8
12
16
x (mm)
Figure 8 Values of
( 2 π ) tan
−1
along the radial line of θ = 0 at 23,000 N applied load (loading angle of 75°)
σ yy σ xy
1.60
σxy
yy
2
_ tan-1σ
___
Experiment data
π
0.80
e
p
M = 0.5
M = 0.5
0.00
0
4
8
12
16
x (mm)
Figure 9 Values of
( 2 π ) tan
−1
σ yy σ xy
along the radial line of θ = 0 at 23,400 N applied load (loading angle of 45°)
Conclusions
The three-dimensional deformation field near the crack tip of a CTS specimen at the loading angles of 75° and 45° was
determined using the stereo vision. The in-plane displacement field was smoothed to obtain the in-plane strain field. Using the
deformation theory of plasticity and the plane stress assumption, the in-plane stress field was determined. The J integral values
were then evaluated and show path independence for r/h > 0.5. The in-plane strains determined experimentally at several
points, ahead of crack tip and located beyond the region (r/h > 0.5) where the three dimensional effects exist, are compared
with the values calculated using Shih’s plane strain solution. Results indicate that the measured values follow the trends of the
Shih’s plane strain solution. The values of elastic mixity evaluated using the measured far field stress data are close to those
obtained from analytical solution. The plastic mixity determined using the near field stress data deviates from the analytical
solution due to the three-dimensional effects.
Acknowledgements
The authors wish to thank the National Science Council in Taiwan for the support of the project through grants NSC
89-2212-E-216-017.
References
1.
Hutchinson, J. W., “Singular Behavior at the End of a Tensile Crack in a Hardening Material,” Journal of the Mechanics
and Physics of Solids, 16, 13-31 (1968).
2.
Hutchinson, J.W., “Plastic Stress and Strain Fields at a Crack Tip,” Journal of the Mechanics and Physics of Solids,
337-347 (1968).
3.
Rice, J. R. and Roscngren, G. F., “Plane Strain Deformation near a Crack Tip in a Power Law Harding Material,” Journal of
the Mechanics and Physics of Solids, 16, 1-13 (1968).
4.
Shih, C. F., “Small-Scale Yielding Analysis of Mixed Mode Plane-Strain Crack Problems,” Fracture Analysis, ASTM
STP560, American Society for Testing and Materials, 187-210 (1974).
5.
Rice, J. R., “Path Independent Integral and Approximate Analysis of strain Concentration by Notches and Cracks,” ASME
Journal of Applied Mechanics, 35, 368-379 (1968).
6.
Symington, M., Shih, C.F. and Ortiz, M., “Tables of Plane Strain Mixed-Mode Plastic Crack Tip Fields,” Material Research
Laboratory, Brown University, Rhode Island (1988).
7.
Pawliska, P., Richard, H. A. and Kenning, J., “On the Applicability of the HRR Theory to CTS Specimens under
Mixed-Mode Loading Conditions,” International Journal of Fracture, 47, R43-R47 (1991).
8.
Aoki, S., Kishimoto, K., Yoshida, T., Sakata, M. and Richard, H.A., “Elastic-Plastic Fracture Behavior of an Aluminum Alloy
under Mixed Mode Loading,” Journal of Mechanics and Physics of Solids, 2, 195-213 (1990).
9.
Luo, P. F. Chao, Y. J. Sutton M. A. and Peters W. H., “Accurate Measurement of Three-Dimensional Displacement in
Deformable Bodies using Computer Vision,” Experimental Mechanics, 33 (2), 123-132 (1993).
10.
Luo, P. F., Chao, Y.J. and Sutton, M.A., “Application of Stereo Vision to Three Dimensional Deformation Analyses in
Fracture Experiments,” Optical Engineering, 33 (3), 981-990 (1994).
11.
Richard, A., “Loading Device for the Creation of Mixed Mode in Fracture Mechanics,” International Journal of Fracture,
(22), R55-58 (1983).
12.
Peter, W. H., and Ranson, W.F., “Digital Imaging Techniques in Experimental Stress Analysis,” Optical Engineering, 21 (3),
427-432 (1990).
13.
Sutton, M. A., Wolters, W. J., Peters, W. H., Ranson, W. F. and McNeill, S. R., “Determination of Displacements Using an
Improved Digital Correlation Method,” Image and Vision Computing, 1 (3), 133-139 (1983).
14.
Chu, T. C., Ranson, W. F., Sutton, M. A. and Peters, W. H., “Application of Digital-image-correlation Techniques to
Experimental Mechanics,” Experimental Mechanics, 25 (3), 232-244 (1985).
15.
Sutton, M. A., Turner, J. L., Bruck, H. A. and Chae, T., “Full Field Representation of Discretely Sampled Surface
Deformation for Displacement and Strain Analysis,” Experimental Mechanics, 31 (2), 168-177 (1991).
16.
Rosakis, A. J. and Ravi-Chandar, K., “On crack-tip stress state: An Experimental Evaluation of Three-dimensional
Effects,” International Journal of Solids Structures, 22(2), 121-134 (1986).
17.
Narasimhan, R. and Rosakis, A. J., “Three dimensional effects near a crack tip in a ductile three-point bend specimen –
Part I: A numerical investigation,” Journal of Applied Mechanics, 57, 607-617 (1990).
18.
Zehnder, A.T. and Rosakis, A.J., “Three dimensional effects near a crack tip in a ductile three point bend specimen – Part
I: An experimental investigation using interferometry and caustics,” Journal of Applied Mechanics, 57, 618-626 (1990).