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The influence of loading direction on the shear stress distribution around a slender notch in a uniform tensile field
Engineering
Fructurr
Mechanics.
1972, Vol. 4, pp.
17I-174.
Pergamon Press.
Printed in Great Britain
THE INFLUENCE OF LOADING DIRECTION ON THE SHEAR STRESS DISTRIBUTION AROUND A SLENDER NOTCH IN A UNIFORM TENSILE FIELD H. MD. HASSEEM National Aeronautical Laboratory, Bangalore, India
KURSHID AFFIMIWALA Department of Aeronautics, Indian Institute of Technology. Kharagpur, India Abstract-A
detailed two dimensional photoelastic investigation of some typical star shaped models showed that the shear stress at the tip region of a slender notch in a uniform tensile field attains a maximum when the loading direction makes an angle of 67P with respect to the notch axis. When subjected to higher static loads along this critical direction, the notch was found to open out radially in the direction along which the maximum shear stress was noticed.
INTRODUCTION IN FRACTURE mechanics the intensification of stresses near the tip region of a crack has received utmost attention, since it is at that point the crack growth takes place. For fatigue studies it is obvious that the shear stress is the driving force behind the movements of dislocations. The available results based on theoretical as well as experimental investigations are often built upon the assumption that the critical conditions are achieved when the loading direction is normal to the crack axis, and this may not occur consistently in practice. The pertinent question to be answered is ‘How does the shear stress distribution around the tip of a crack in a sheet loaded in tension get affected by changing the angle between the crack and the loading direction from 90” down to lower values?’ Schijve [ 1I considered this case and theoretically calculated the shear stress distribution for different values of /3 where /3 is the angle between the loading direction and the crack (Fig. 1). He concludes that “there is a more pronounced effect of p on
Fig. 1. Crack in a sheet loaded in arbitrary direction. 171
172
H.
MD. HASSEEM
and
KURSHID
AFFIMIWALA
the distribution of shear stress (7) on slip planes passing through the tip of the crack. Also if p is decreasing from 90” to 45” there is a tendency for an increasing concentration of r in line with the crack. Other circumstances being favorable a crack with an oblique orientation (p f 90’) may persist in growing in the oblique direction.” To take full advantage of this statement and also to find out the critical value of /3 and the corresponding shear stress distribution, a plausible physical model is required. But to the best of the authors’ knowledge, no experimental investigations were so far made on these lines. Hence, a detailed two dimensional study was initiated using typical star shaped models and the results are reported herein. EXPERIMENTAL Photoelastic
DETAILS
model
The photoelastic models were made out of quarter inch thick araldite sheet (Araldite CY 230 with 10 per cent by weight of Hardener HY 951). Figure 2 shows one of the model configurations. This particular geometry was selected to facilitate application of load in a number of directions. Three models with identical notch configurations were made so that load can be applied at p values of O”, 229,45”, 526”, 60”, 67$‘, 75” and 90”. The central notch in each specimen was made in a jig boring machine fitted with a special engraving tool giving partial thickness cuts. The root radius of the notch configuration was maintained at the lowest practical value of 3”2in., the width fs in. and the chordal distance between the tips being gin. Pure tensile load was applied through clamped end plates in a similar fashion to the actual fatigue specimen. To eliminate machining stresses the models were annealed at 80°C the heating and cooling rates being S”C/hr with a soaking time of 2 hr. 24in.
-
II in.
8 P + kin.
Fig. 2. Photoelastic
specimen.
dia.
hole_.+
8
(cl
p=45*
(df p=22 k’ Fig. 3. Isoclinic fringe patterns.
(b) f3=67t+’
(c) f3-45O Fig. 4@-d).
(d)
p=22’/2’
fsxhromatic fringe patterns (darkfield).
(g) p =45O
(h) /?=22’/2”
Fig. 4(e-h). ~s~~~romatic fringe patterns (Iight field).
Infiuence of loading direction on shear stress distribution
I73
EXPERIMENTAL PROCEDURE each model was lightly loaded in a plane polariscope and the isoclinic parameters at a number of points around the notch tip were noted for different values of /3. Figure 3 shows some typical isoclinic patterns. The models were then subjected to a uniform tensile stress of 467 psi and the isochromatic fringe number at each of these points was noted by setting the circular polariscope with respect to the corresponding isoclinic parameter (Fig. 4). The fringes were magnified through proper lens arrangement for accurate measurements. Tardy’s compensation technique was employed for measuring fractionat fringe orders. Readings were taken as near as possible to the end of the notch. While conducting the experiments, the loading direction was always kept vertical and the models were suitably rotated to achieve different values of p, After each unloading the models were annealed to remove any residual stresses. First
RESULTS AND DISCUSSION The variation of shear stress at the notch tip with respect to p is plotted in Fig. 5. From this it is quite clear that the shear stress attains a maximum when the loading direction is at an angle of 67$’ with respect to the notch axis. This result gives the critical value of p as 679, whiie according to Schijve the absolute maximum shear stress occurs when /3 varies from 90” to 45”, which is a rather generalized form of statement. /
pooL_L--’ 0
Loadtrig Fig. 5.
20 angle
40
wtn respect
! 60
467 Iblin’
i 80
90
to notch axis,
/3”
Variation of shear stress at the notch tip with respect to loading direction.
For this critical case the variation of shear stress along different radial lines at the notch tip is plotted in Fig. 6. It can be noticed from this plot that the maximum shear stress occurs along an axis which makes an angle of 45” with respect to the notch axis. Now to study the nature of the opening mode at the critical value of 0, a simple plastic (fully annealed) specimen with a central notch oriented at /3 = 67P was mounted
174
H. MD. HASSEEM
0 8 =90” 0.8=15” B=O”
600
O--
and KURSHID
0.125
Radial
C-25
distance
467
0375
from
AFFIMIWALA
notch
lb/in2
0.50
tip,
in.
Fig. 6. Variation of shear stress along different radial lines for critical loading f@= 67:“).
in an Instron testing machine and subjected to a linearly varying load (0.5 mmlmin) till failure. The failed parts when reassembled clearly indicated the opening mode to lie at an angle of 45” (with respect to notch axis) along which maximum shear stress was noticed (Fig. 6). CONCLUSION It is believed that these results may find some useful application in studies relating to fracture mechanics. Accordingly suitable fatigue tests are now being planned for further investigation. Acknowledgements-The authors are grateful to Mr. K. N. Raju for suggesting this problem. Thanks are due to Dr. S. A. Hussainy and Mr. S. Radhakrishna for their help and support during various stages of this investigation.
REFERENCE [l] J. Schijve, Analysis of the fatigue phenomenon
in aluminium alloys. NLR-TRM
(Received 5 December 1970)
2122,
p. 54, April (1964).
Fructurr
Mechanics.
1972, Vol. 4, pp.
17I-174.
Pergamon Press.
Printed in Great Britain
THE INFLUENCE OF LOADING DIRECTION ON THE SHEAR STRESS DISTRIBUTION AROUND A SLENDER NOTCH IN A UNIFORM TENSILE FIELD H. MD. HASSEEM National Aeronautical Laboratory, Bangalore, India
KURSHID AFFIMIWALA Department of Aeronautics, Indian Institute of Technology. Kharagpur, India Abstract-A
detailed two dimensional photoelastic investigation of some typical star shaped models showed that the shear stress at the tip region of a slender notch in a uniform tensile field attains a maximum when the loading direction makes an angle of 67P with respect to the notch axis. When subjected to higher static loads along this critical direction, the notch was found to open out radially in the direction along which the maximum shear stress was noticed.
INTRODUCTION IN FRACTURE mechanics the intensification of stresses near the tip region of a crack has received utmost attention, since it is at that point the crack growth takes place. For fatigue studies it is obvious that the shear stress is the driving force behind the movements of dislocations. The available results based on theoretical as well as experimental investigations are often built upon the assumption that the critical conditions are achieved when the loading direction is normal to the crack axis, and this may not occur consistently in practice. The pertinent question to be answered is ‘How does the shear stress distribution around the tip of a crack in a sheet loaded in tension get affected by changing the angle between the crack and the loading direction from 90” down to lower values?’ Schijve [ 1I considered this case and theoretically calculated the shear stress distribution for different values of /3 where /3 is the angle between the loading direction and the crack (Fig. 1). He concludes that “there is a more pronounced effect of p on
Fig. 1. Crack in a sheet loaded in arbitrary direction. 171
172
H.
MD. HASSEEM
and
KURSHID
AFFIMIWALA
the distribution of shear stress (7) on slip planes passing through the tip of the crack. Also if p is decreasing from 90” to 45” there is a tendency for an increasing concentration of r in line with the crack. Other circumstances being favorable a crack with an oblique orientation (p f 90’) may persist in growing in the oblique direction.” To take full advantage of this statement and also to find out the critical value of /3 and the corresponding shear stress distribution, a plausible physical model is required. But to the best of the authors’ knowledge, no experimental investigations were so far made on these lines. Hence, a detailed two dimensional study was initiated using typical star shaped models and the results are reported herein. EXPERIMENTAL Photoelastic
DETAILS
model
The photoelastic models were made out of quarter inch thick araldite sheet (Araldite CY 230 with 10 per cent by weight of Hardener HY 951). Figure 2 shows one of the model configurations. This particular geometry was selected to facilitate application of load in a number of directions. Three models with identical notch configurations were made so that load can be applied at p values of O”, 229,45”, 526”, 60”, 67$‘, 75” and 90”. The central notch in each specimen was made in a jig boring machine fitted with a special engraving tool giving partial thickness cuts. The root radius of the notch configuration was maintained at the lowest practical value of 3”2in., the width fs in. and the chordal distance between the tips being gin. Pure tensile load was applied through clamped end plates in a similar fashion to the actual fatigue specimen. To eliminate machining stresses the models were annealed at 80°C the heating and cooling rates being S”C/hr with a soaking time of 2 hr. 24in.
-
II in.
8 P + kin.
Fig. 2. Photoelastic
specimen.
dia.
hole_.+
8
(cl
p=45*
(df p=22 k’ Fig. 3. Isoclinic fringe patterns.
(b) f3=67t+’
(c) f3-45O Fig. 4@-d).
(d)
p=22’/2’
fsxhromatic fringe patterns (darkfield).
(g) p =45O
(h) /?=22’/2”
Fig. 4(e-h). ~s~~~romatic fringe patterns (Iight field).
Infiuence of loading direction on shear stress distribution
I73
EXPERIMENTAL PROCEDURE each model was lightly loaded in a plane polariscope and the isoclinic parameters at a number of points around the notch tip were noted for different values of /3. Figure 3 shows some typical isoclinic patterns. The models were then subjected to a uniform tensile stress of 467 psi and the isochromatic fringe number at each of these points was noted by setting the circular polariscope with respect to the corresponding isoclinic parameter (Fig. 4). The fringes were magnified through proper lens arrangement for accurate measurements. Tardy’s compensation technique was employed for measuring fractionat fringe orders. Readings were taken as near as possible to the end of the notch. While conducting the experiments, the loading direction was always kept vertical and the models were suitably rotated to achieve different values of p, After each unloading the models were annealed to remove any residual stresses. First
RESULTS AND DISCUSSION The variation of shear stress at the notch tip with respect to p is plotted in Fig. 5. From this it is quite clear that the shear stress attains a maximum when the loading direction is at an angle of 67$’ with respect to the notch axis. This result gives the critical value of p as 679, whiie according to Schijve the absolute maximum shear stress occurs when /3 varies from 90” to 45”, which is a rather generalized form of statement. /
pooL_L--’ 0
Loadtrig Fig. 5.
20 angle
40
wtn respect
! 60
467 Iblin’
i 80
90
to notch axis,
/3”
Variation of shear stress at the notch tip with respect to loading direction.
For this critical case the variation of shear stress along different radial lines at the notch tip is plotted in Fig. 6. It can be noticed from this plot that the maximum shear stress occurs along an axis which makes an angle of 45” with respect to the notch axis. Now to study the nature of the opening mode at the critical value of 0, a simple plastic (fully annealed) specimen with a central notch oriented at /3 = 67P was mounted
174
H. MD. HASSEEM
0 8 =90” 0.8=15” B=O”
600
O--
and KURSHID
0.125
Radial
C-25
distance
467
0375
from
AFFIMIWALA
notch
lb/in2
0.50
tip,
in.
Fig. 6. Variation of shear stress along different radial lines for critical loading f@= 67:“).
in an Instron testing machine and subjected to a linearly varying load (0.5 mmlmin) till failure. The failed parts when reassembled clearly indicated the opening mode to lie at an angle of 45” (with respect to notch axis) along which maximum shear stress was noticed (Fig. 6). CONCLUSION It is believed that these results may find some useful application in studies relating to fracture mechanics. Accordingly suitable fatigue tests are now being planned for further investigation. Acknowledgements-The authors are grateful to Mr. K. N. Raju for suggesting this problem. Thanks are due to Dr. S. A. Hussainy and Mr. S. Radhakrishna for their help and support during various stages of this investigation.
REFERENCE [l] J. Schijve, Analysis of the fatigue phenomenon
in aluminium alloys. NLR-TRM
(Received 5 December 1970)
2122,
p. 54, April (1964).