- Email: [email protected]
Three-dimensional photoelastic calibration of a chevron-notched short-bar fracture specimen geometry
0013-7944/86 $3.00+.00 Pergamon Journals Ltd.
Engineering Fracture Mechanics Vol. 24, No. 6, pp. 87%387, 1986 Printed in Great Britain.
THREE-DIMENSIONAL PHOTOELASTIC CALIBRATION OF A CHEVRON-NOTCHED SHORT-BAR FRACTURE SPECIMEN GEOMETRY Istituto
GIANNI di Meccanica Applicata Viale Risorgimento
NICOLETTO alle Macchine, Universita 2,40136 Bologna, Italy
di Bologna,
Abstract-Stress intensity distributions have been experimentally evaluated from cracked photoelastic models of the rectangular chevron-notched short-bar fracture specimen. Two methods were applied to SIF determination from near-tip isochromatic fringes in slices removed perpendicularly to the flaw profiles. No assumption on flaw shape was required in the analysis. Comparisons with experimental and numerical compliance calibrations from the literature are included. Differences between 3-D Finite Element and the present photoelastic SIF distributions are discussed. The observed flaw shapes are compared to others found in different materials. Flaw evolution is interpreted in terms of local SIF variations.
1. INTRODUCTION CHEVRON-NOTCHED fracture specimens are gaining widespread acceptance for fracture toughness testing of materials displaying brittle behavior. They are also being considered for inclusion in standard tests by the American Society for Testing and Materials (ASTM) Committee E 24[1]. The interest in chevron-notched specimens originates from distinct advantages : (1) A crack initiates at the chevron tip at an early stage of specimen loading because of a severe stress concentration. This feature circumvents costly precracking procedures prescribed by the ASTM Test Method for Plane Strain Fracture Toughness of Metallic Materials (E 399-78). (2) The Stress Intensity Factor (SIF) passes through a minimum value at a critical crack depth. This critical length depends only on specimen geometry and type of loading for a brittle material. Hence, the minimum SIF can be associated to the maximum load at fracture and only the test load record is required for fracture toughness evaluation from chevron-notched specimens. The test procedure is therefore greatly simplified. (3) The elevated triaxial constraint which is induced at the crack tip by the chevron notch is expected to relax the stringent E 399-78 specimen minimum thickness validity requirements. In Ref. [2] a reduction in thickness by a factor of 2.5 was proposed based on tests of the 6061-T6 aluminum alloy. More recently, in Ref. [3], this factor has been revised to 2.0 on the basis of results on six metals. Although Barker[4] attempted a standardization of short-bar and short-rod geometries, other rod and bar specimen configurations have been studied[5, 61. The necessary calibration of chevron specimens has been mainly obtained experimentally via compliance measurements[4, 51. Recently, in a round robin organized by the ASTM Task Group on Chevron-Notched Specimen Testing, four specimen configurations were analyzed by three-dimensional numerical methods[l]. In the present paper a three-dimensional experimental calibration of a short-bar specimen geometry is reported. Frozen-stress photoelasticity was used in the study of SIF distributions pertaining to natural flaw shapes of different depths that were monotonically grown in epoxy models. The present results are compared to numerical and experimental results from the literature. Actual flaw shapes are interpreted on the basis of effective SIF distributions. Implications of numerical assumptions on SIFs are also discussed.
2. EXPERIMENTS Specimen geometry
The specimen geometry of this study is shown in Fig. 1. It is equivalent to the rectangular shortbar specimen with straight chevron slots originally proposed for standardization by Barker in Ref. [4]. 879
G. NICOLETTO
880
Sez.
A-A
w/B
=
1.4
H/B
=
0.435
a0/w
=
0.31
t/B
=
0.02
c/B
=
0.1
B
=50mm
c 2
Fig. 1. Rectangular
chevron-notched
short bar specimen.
This particular geometry has the following interesting feature: the ratio of the rectangular cross-section (i.e. H/B = 0.435) was determined by Barker as yielding the same compliance derivative of a short-rod specimen of diameter equal to B. Hence, no separate characterization of two complementary specimens is needed. Actually, more recently, the Raju-Newman 3-D Finite Element (FE) analysis identified a difference of 3.8% in compliance calibration[‘l], between the short-bar and the short-rod specimens. This difference is comparable to the expected scatter in the present experimental SIF estimates. Therefore, in the subsequent discussion and assessment of the photoelastic results reference will be made also to short-rod calibrations from the literature. Experimental procedure
Five specimens were machined to the final geometry and dimensions specified in Fig. 1 from stress-free square epoxy bars (50 mm x 50 mm x 200 mm). The experiments began with the insertion of a tiny starter crack at the specimen chevron notch tip with a sharp blade. The specimen was then accommodated in a loading fixture which was inside a programmable oven. The precracked specimen was heated above critical temperature, at which the crack was monotonically propagated to the desired depth by applying a live load. The flaw evolution was visually monitored from the oven viewport. After the load was reduced to stop the crack growth, an adequate dead loading was applied at the beginning of the cooling portion of the thermal cycle. While in the oven and above critical temperature, the specimen rested on its back surface in order to eliminate asymmetrical crack tip loading due to specimen weight. This precaution was suggested from a previous photoelastic calibration of the Compact Tension specimen. At the end of the stress freezing cycle, natural flaw shapes were accurately recorded under x 10 magnification using a profile projector. Up to eight thin slices were removed with a bandsaw from each specimen perpendicularly to the crack profile to determine local SIF estimates from the near-tip isochromatic fringes. Magnification and the Tardy method for fractional fringe order determination were used to retrieve near-tip data points (i.e. r vs N, where I is the fringe distance from the crack tip and N is the isochromatic fringe order) along two directions perpendicular to the crack plane. These data sets were subsequently used in SIF evaluation. SIF extraction from near-tip isochromatic fringes
Photoelasticity has been extensively used in the experimental SIF determination for two- and three-dimensional fracture problems. The extraction of a SIF estimate from the isochromatic fringes involves the assumption on the governing near-tip stress state. Based on different assumptions, various procedures for SIF extraction from the isochromatic fringes, which are locus of points of constant maximum in-plane shear stress, have been proposed. The two most widely used in the case
881
Photoelastic calibration
of static analysis are often referred to as the “slope” method[S], and the “two-parameter” approach[9-111. While in the first method only the classical singular LEFM stress field equations are assumed to dominate the near-tip measurement zone and are combined according to the maximum in-plane shear stress definition, in the second procedure constant stress contributions are added to Irwin’s singular stress tensor equations. In the course of the present study, a computer program, which runs on an HP 85, was developed. It processes two sets of basic experimental data (i.e. r vs N) extracted from each slice along the +y and - y directions of Fig. 1. For each set of data it yields two plots (normalized maximum in-plane shear stress vs fi w r and normalized apparent SIF vs J;lw>. The least-square fit of straight lines through the LEFM-dominated (hence linear) zones yield two SIF estimates from the slope value and from the intercept with the ordinate axis, respectively. For the “slope” method, the following normalized equation is used
in which, from the stress-optic law, the maximum in-plane shear stress rmaxis given by
Tmax
=fLN 2t' ’
wheref” is the material fringe value, N is the isochromatic fringe order and t’ is the slice thickness. For the two-parameter method the following normalized equation holds in the LEFM region[lO], (3) in which Kap = q,,,,(8w)1/2. The procedures for SIF determination with the two methods are shown in the compound diagram of Fig. 2. Although both methods were verified to yield comparable results in most cases, the twoparameter method was usually preferred as more reliable. 3. RESULTS Previous studies have demonstrated that SIFs and actual flaw shapes are directly related [lo, 111. Therefore, care was given in recording the natural flaw shapes obtained during the
40 KAP B@
P
rmax Bw P
Slice lo-
l
Two
Fig. 2. Typical compounded
D-3-2
Parameter
D Slope Method
diagram for SIF determination method.
Method >
by the “slope” and the “two-parameter”
882
G. NICOLETTO
Fig. 3. Monotonic
flaw shapes in epoxy specimens
propagated
above critical temperature
photoelastic experiments. They are shown in Fig. 3. It is pointed out that the thumbnailing of Tests 4-5 was probably enhanced by an attempted propagation out of the chevron notch plane. Flaw shape evolution will be discussed in conjunction with the local variations in SIF distributions. The present three-dimensional analysis yielded Mode I Stress Intensity Factors at discrete points along the crack front for each crack depth. SIF was taken to be
(4)
where F is determined with the two methods outlined in the previous paragraph and the meanings of the variables are evident from Fig. 1. The photoelastic SIF distributions for the five tests are summarized in Fig. 4. From the specimen of Test 1 only three slices could be retrieved because the flaw was narrow. More slices were obtained from the other specimens. In Fig. 4 the SIF estimates are approximately located at the slice centers. Since the specimen had a symmetry axis, the photoelastic results are plotted referring to half specimen thickness. When two separate points are plotted at a specific 2z/b ratio, it signals a definite difference in SIF estimates for two symmetrical slices. As far as the accuracy of the photoelastic method is concerned, previous studies[lO, 1 l] suggest a maximum + 5% experimental scatterband in SIF estimates. Based on inspection of experimental diagrams such as the one of Fig. 2, it is felt that Test 5 photoelastic distribution is the least accurate. The very deep crack of Test 5 is expected to interact with the specimen back surface. Under these conditions the two-parameter method is expected to be less accurate. Additional terms of the non-singular stress field series expansion should be incorporated in the data reduction [ 121. Since most experimental and analytical studies of the literature determined single-valued chevron-notched specimen calibration curves from compliance measurements or calculations, which result in through-the-thickness “averaged” estimates, for the sake of comparison, a characteristic SIF for each crack depth had to be identified from the photoelastic SIF distributions of Fig. 4. These values were introduced, in a diagram F, against relative crack length, in which F, is given by
(5)
883
Photoelastic calibration
30
20 30
o- 2
8
F
0.58
20 30 0 0
3
0.68
4
0.73
5
0.83
20 30‘ ” 20
0
30
7
20 0
I
0
05
,
1
2 z/b
Fig. 4. Photoelastic stress-intensity distributions.
and the subscript c stands for compliance. The characteristic SIF for each test was defined as the value common to most slices. An averaged value was determined from the significant SIFs. Therefore, while notch intersection values are identified for Test 1, internal SIFs are appropriate for the other tests. 4. DISCUSSION The discussion of the results presented in the preceding section will deal initially with the effective SIF distributions and flaw shapes. A discussion of the STF vs crack depth calibration curve is included in a second subsection. The discussion will refer to recent results obtained in a numerical round robin conducted by the ASTM E 24 Task Group on Chevron-Notched Specimens. In this joint effort, two square-bar and two round-rod specimen geometries of different w/B ratios were studied[l], in addition to Barker’s short-bar geometry. The present specimen geometry (i.e. Fig. 1) is different from the ASTM Task Group geometry in several respects. According to Newman’s specimen length definition as the distance from the back surface to the point of load application, the present specimen has w/B = 1.40 instead of 1.45. The a,,/~ ratio is also different from the round robin value (i.e. 0.31 and 0.332, respectively). Finally, the mathematical model assumes the slot height to be zero while the photoelastic specimens have t/B = 0.02. Although this aspect ratio doesn’t comply to Barker’s range for a square-tipped notch (i.e. 0 < t/B < O.Ol), Ref. [4], the influence on SIF estimates is believed negligible because of the elastic behavior of the epoxy material. Efectivejaw shapes and SIF distributions A through-the-thickness cracked specimen configuration defines only theoretically a bidimensional problem. Crack front thumbnailing under fatigue loading, even in relatively thin plates,
884
G. NICOLETTO
clearly indicates that actual three-dimensional conditions occur along the flaw front. This feature involves, however, theoretical complexities that are often neglected in engineering practice. Procedures for equivalent crack length determination from a curved profile in a standard fracture specimen (i.e. CT) are prescribed in ASTM E 399-78. The chevron-notched short-bar specimen is characterized by a through-the-thickness crack front. However, its front evolution appears more complex than for uniform thickness specimens. A series of photographs of the crack front evolution in a polystyrene chevron-notched specimen are summarized in Fig. 5(a)[l3]. It demonstrates retarded growth in the specimen interior at short crack lengths. As the crack propagates, the front gradually straightens out and then thumbnails. This behavior seems consistent with the photoelastic flaw shapes presented in Fig. 3. However, in order to assess the influence of the Poisson’s ratio on flaw shape (i.e. the epoxy material above critical temperature is nearly incompressible), displacement-controlled crack propagation tests in epoxy specimens at room temperature (i.e. v = 0.35) were performed. On the specimen fracture surfaces ripples were detectable and associated to flaw propagation. They are shown in Fig. 5(b). Again the flaw evolution signaled a development from nearly straight to thumbnailed shape as the back surface was approached. From this experimental evidence it was concluded that the photoelastic modeling of the real boundary conditions in the present fracture problem is effective. Similar conclusions were reached in Refs. [IO, 111. Conversely, numerical methods of analysis assume a mathematical description of the flaw shape (i.e. circular, elliptical, rectilinear). Hence, in the ASTM numerical round robin[l], straight crack fronts normal the specimen longitudinal axis were considered. The analyses found nearly constant SIF distributions in the interior but severe gradients at the notch-crack intersection. Notably, the higher the number of Finite Element layers in the mathematical model, the more severe the SIF gradient in the vicinity of the notch[7]. Presence of SIF gradients signal non-uniform crack propagation to be expected along the flaw front. Upward concave SIF distributions resulted from both the Finite Element method and the Boundary Integral Equation method analyses, these distributions being independent from the crack depth. Besides an inherent inaccuracy in the use of the numerical SIF estimates as stated by Ingraffea[l3], no explanation of the peculiar flaw evolution previously described (i.e. increased thumbnailing with crack length) results from the numerical analyses. Tn Fig. 4 the SIF distributions obtained by frozen-stress photoelasticity clearly signal the presence of SIF gradients. The observed SIF concavity inversion with crack depth (i.e. from upward concave to downward concave) agrees with the observed change in flaw front curvature during propagation [i.e. Fig. 5(a) and (b)]. In Fig. 6 the Finite Element estimates for a short-rod specimen (i.e. w/B = 1.45) are compared to the appropriate frozen-stress test results of the present study. Significantly, besides the difference in SIF gradient evolution already commented on, the closest agreement between numerical and
Fig. 5. Flaw shapes in (a) a polystyrene
short rod[l3], and (b) in an epoxy rectangular temperature.
short bar at room
885
Photoelastic calibration 3D Finite
Element
[7] a/w I
Test I 1
20
-I
30 0.73 0.68
0 (
0
6
05
i
7
2 z/b
Fig. 6. Comparison of 3-D Finite Element results for the short rod[7], with frozen-stress photoelastic SIF distributions.
expe~mental SIFs occurs in the vicinity of minimum SIF depths (i.e. a/w = 0.55) at which natural Aaws (i.e. Fig. 5) approximate the rectilinear configuration. Specimen calibration curve The calibration curve of a fracture specimen gives the relationship between SIF and crack length only [i.e. eqn (2)]. Previous experimental calibrations have mainly studied the short-rod specimen and resorted to the compliance method. Barker[4], measured load-point displacements on fused quartz using a laserinterferometric technique. More recently Shannon et al., as reported in Ref. [l], characterized experimentally the short-rod specimen, analyzing crack opening displacements in aluminum specimens. In the ASTM task group numerical round robin, reported in Ref. [l], the information required from the analyses were, besides the SIF distributions along the flaws, SIF vs crack length relationships obtained from plane stress compliance. The analysis methods of the short-rod specimen included the Finite Element method[7, 131, and the Boundary Integral Equation method[l3]. The results of these numerical and experimental compliance studies for the short-rod fall in the shaded area of Fig. 7. Relatively small scatter is evidenced in view of the different techniques and materials used. The calibration curve for the rectangular short bar of the present investigation is also introduced in the same figure. Inspection of Fig. 7 reveals a similar trend although shifted to slightly lower values. The agreement, however, is believed satisfactory. In fact, a lower FC vs a/w curve for the present specimen was to be expected based on two considerations. First, Raju-Newman’s results[7] demonstrate an imperfect equivalence of the short rod and of the rectangular bar, lower SIFs pertaining to the last geometry. Second, the w/B ratio of the photoelastic specimens was 4% lower
886
G. NICOLETTO 40
Fc 30
Short Rod 20
B
w/6=1.45; adw=.3
Exp. & Num.Complianc
tect. Short Bar w/B=1.4; a,/w=.3 -*-0
.2
0
Fig. 7. Comparison
Photoelasticity .4
.6
a/w
.8
1
of experimental and numerical compliance SlFs for the short rod from the literature with photoelastic SIFs for the rectangular short bar.
than for the specimen considered in the ASTM round robin (i.e. w/B = 1.45). Both these factors tend to reduce the SIFs. The preceding comparison used the results relative to the short-rod specimen assuming approximate compliance equivalence between short rod and rectangular short bar. A hybrid experimental-numerical technique was used by Sanford-Chona[l2] in the determination of the geometric shape function of a thin 2-D through-cracked photoelastic specimen having the characteristic w/H ratio of the rectangular short bar. Their procedure employed data from an extended zone around the crack tip and yielded 2-D SIFs that were subsequently converted estimates for a 3-D chevronnotched analogue applying a compliance equivalence approach proposed by Munz et aZ.[5]. The SIF vs relative crack depth curves from the Sanford-Chona study and results of the present photoelastic analysis are included in Fig. 8. The correlation is good in the minimum SIF region.
80 RECTANGULAR A
Numerical
. 2D local
SHORT comptiance[
collocation
+ 3D equivalent 60 n
BAR
tion b2] Frozen stress
71 [12]
calibrg photoela
sticity
40
20
0.75
a/w
1
Fig. 8. Comparison of 2-D through-the-thickness cracked and 3-D short bar calibration by the local collocation photoelastic method[l2], with numerical compliance results[7], and with the present photoelastic estimates.
Photoelastic calibration
887
However, steeper SIF gradients than for the present results are signaled by the local collocation method at short and very long crack length. For high a/w ratios when a back surface effect could be significant it is conceivable that the local collocation method of Ref. [ 121 could be more accurate than the two-parameter approach. The difference is less justified at short crack lengths. In Fig. 8 a numerical calibration curve for the rectangular short-rod specimen taken from Ref. [7] is also introduced. From the limited number of points of the FE analysis a good agreement with frozen stress results at short crack lengths and with the local collocation method for long crack length is identified. Nonetheless, the three different calibration curves summarized in Fig. 8 yield minimum SIFs in the 0.54.6 a/w range. 5. CONCLUSIONS The present paper has reported on an experimental calibration of the rectangular chevronnotched short-bar specimen[4]. Frozen stress photoelasticity was used in studying a series of cracked epoxy models. A comparison of two procedures for SIF determination from near-tip isochromatic fringes procedures was performed. SIF distributions along the crack front were obtained and correlated to the relative flaw shapes. Natural flaws in epoxy specimens displayed crack-length-dependent front curvatures. These findings were corroborated by observations in different materials. Flaw departures from a straight through-the-thickness shape were interpreted in terms of local photoelastic SIF variations. Upward concave SIF distributions were observed for short cracks. Subsequent evolution to downward concave SIF distributions was found for longer crack lengths. The rectilinear flaw shape, commonly assumed in analytical studies, is found to produce severe gradients at the crack-notch intersection, which cannot explain natural flaw evolution. Finally, comparisons with experimental and numerical compliance calibrations are found to be satisfactory, especially in view of the inevitable variation in geometry, material and method of analysis. REFERENCES Jr., A review of chevron-notched fracture specimens. Chevron-Notched Specimen : Test&q and Stress Analysis, ASTM STP 855 (Edited by J. H. Underwood, S. W. Freiman and F. I. Baratta), pp. 5-3 1.American Society for Testing and Materials, Philadelphia (1984). 121 L. M. Barker, Theory for determining K,, from small non-LEFM specimens, supported by experiments on aluminum. Int. J. Fracture 15, 511-536 (1979). 131 L. M. Barker, Specimen size effects in short-rod fracture toughness measurements. Chevron-Notched Specimen : Tesfin,g and Stress Analysis, ASTM STP 855 (Edited by J. H. Underwood, S. W. Freiman and F. I. Baratta), pp. 117-133. American Society for Testing and Materials, Philadelphia (1984). 141 L. M. Barker, Compliance calibration of a family of short rod and short bar fracture toughness specimens. Engng Fracture Mech. 17,289-312 (1983). 151 D. Munz, R. T. Bubsey and J. E. Srawley, Compliance and stress intensity coefficients for short bar specimens with chevron Notches. Znt. J. Fracture 16, 359-374 (1980). [61 R. T. Bubsey, D. Munz, W. S. Pierce and J. L. Shannon, Compliance calibration of the short rod chevron-notch specimen for fracture toughness testing of brittle materials. Int. J. Fracture 18, 125-133 (1982). finite element analysis of chevron-notched fracture specimens. [71 I. S. Raju and J. C. Newman, Jr., Three-dimensional Chevron-Notched Specimen : Testing and Stress Analysis, ASTM STP 855 (Edited by J. H. Underwood, S. W. Freiman and F. I. Baratta), pp. 32-48. American Society for Testing and Materials, Philadelphia (1984). determination of stress intensity factors for single and interacting cracks and [81 Y. Phang and C. Ruiz, Photoelastic comparison with calculated results. Part I : two dimensional problems. J. Strain Analysis 19,23-34 (1984). cracked body problems. J. Optical Engng 191 C. W. Smith, Use of optical methods in stress analysis of three-dimensional X,69&703 (1982). distributions for corner cracks emanating from open holes in plates of finite width. Theor. [lOI G. Nicoletto, Stress-intensity Appl. Fracture Mech. 3,63-70 (1985). characterization of cracks at straight attachment lugs. Engng Fracture Mech. 22, 829%338 1111 G. Nicoletto, Experimental (1985). calibration of the short-bar chevron notched specimen. Chevron-Notched [I21 R. J. Sanford and R. Chona, Photoelastic Specimen : Testing and Sfress Analysis, ASTM STP 855 (Edited by J. H. Underwood, S. W. Freiman and F. I. Baratta), pp. 81-97. American Society for Testing and Materials. Philadelphia (1984). finite and boundary 1131 A. R. Ingraffea, R. Perucchio, T.-Y. Han, W. H. Gerstle and Y.-P. Huang, Three-dimensional element calibration of the short-rod specimen. Chevron-Notched Specimen : Testing and Stress Analysis, ASTM STP 855 (Edited by J. H. Undenvood, S. W. Freiman and F. I. Baratta), pp. 4949. American Society for Testing and Materials, Philadelphia (1984).
111 J. C. Newman,
(Received 5 August 1985)
Engineering Fracture Mechanics Vol. 24, No. 6, pp. 87%387, 1986 Printed in Great Britain.
THREE-DIMENSIONAL PHOTOELASTIC CALIBRATION OF A CHEVRON-NOTCHED SHORT-BAR FRACTURE SPECIMEN GEOMETRY Istituto
GIANNI di Meccanica Applicata Viale Risorgimento
NICOLETTO alle Macchine, Universita 2,40136 Bologna, Italy
di Bologna,
Abstract-Stress intensity distributions have been experimentally evaluated from cracked photoelastic models of the rectangular chevron-notched short-bar fracture specimen. Two methods were applied to SIF determination from near-tip isochromatic fringes in slices removed perpendicularly to the flaw profiles. No assumption on flaw shape was required in the analysis. Comparisons with experimental and numerical compliance calibrations from the literature are included. Differences between 3-D Finite Element and the present photoelastic SIF distributions are discussed. The observed flaw shapes are compared to others found in different materials. Flaw evolution is interpreted in terms of local SIF variations.
1. INTRODUCTION CHEVRON-NOTCHED fracture specimens are gaining widespread acceptance for fracture toughness testing of materials displaying brittle behavior. They are also being considered for inclusion in standard tests by the American Society for Testing and Materials (ASTM) Committee E 24[1]. The interest in chevron-notched specimens originates from distinct advantages : (1) A crack initiates at the chevron tip at an early stage of specimen loading because of a severe stress concentration. This feature circumvents costly precracking procedures prescribed by the ASTM Test Method for Plane Strain Fracture Toughness of Metallic Materials (E 399-78). (2) The Stress Intensity Factor (SIF) passes through a minimum value at a critical crack depth. This critical length depends only on specimen geometry and type of loading for a brittle material. Hence, the minimum SIF can be associated to the maximum load at fracture and only the test load record is required for fracture toughness evaluation from chevron-notched specimens. The test procedure is therefore greatly simplified. (3) The elevated triaxial constraint which is induced at the crack tip by the chevron notch is expected to relax the stringent E 399-78 specimen minimum thickness validity requirements. In Ref. [2] a reduction in thickness by a factor of 2.5 was proposed based on tests of the 6061-T6 aluminum alloy. More recently, in Ref. [3], this factor has been revised to 2.0 on the basis of results on six metals. Although Barker[4] attempted a standardization of short-bar and short-rod geometries, other rod and bar specimen configurations have been studied[5, 61. The necessary calibration of chevron specimens has been mainly obtained experimentally via compliance measurements[4, 51. Recently, in a round robin organized by the ASTM Task Group on Chevron-Notched Specimen Testing, four specimen configurations were analyzed by three-dimensional numerical methods[l]. In the present paper a three-dimensional experimental calibration of a short-bar specimen geometry is reported. Frozen-stress photoelasticity was used in the study of SIF distributions pertaining to natural flaw shapes of different depths that were monotonically grown in epoxy models. The present results are compared to numerical and experimental results from the literature. Actual flaw shapes are interpreted on the basis of effective SIF distributions. Implications of numerical assumptions on SIFs are also discussed.
2. EXPERIMENTS Specimen geometry
The specimen geometry of this study is shown in Fig. 1. It is equivalent to the rectangular shortbar specimen with straight chevron slots originally proposed for standardization by Barker in Ref. [4]. 879
G. NICOLETTO
880
Sez.
A-A
w/B
=
1.4
H/B
=
0.435
a0/w
=
0.31
t/B
=
0.02
c/B
=
0.1
B
=50mm
c 2
Fig. 1. Rectangular
chevron-notched
short bar specimen.
This particular geometry has the following interesting feature: the ratio of the rectangular cross-section (i.e. H/B = 0.435) was determined by Barker as yielding the same compliance derivative of a short-rod specimen of diameter equal to B. Hence, no separate characterization of two complementary specimens is needed. Actually, more recently, the Raju-Newman 3-D Finite Element (FE) analysis identified a difference of 3.8% in compliance calibration[‘l], between the short-bar and the short-rod specimens. This difference is comparable to the expected scatter in the present experimental SIF estimates. Therefore, in the subsequent discussion and assessment of the photoelastic results reference will be made also to short-rod calibrations from the literature. Experimental procedure
Five specimens were machined to the final geometry and dimensions specified in Fig. 1 from stress-free square epoxy bars (50 mm x 50 mm x 200 mm). The experiments began with the insertion of a tiny starter crack at the specimen chevron notch tip with a sharp blade. The specimen was then accommodated in a loading fixture which was inside a programmable oven. The precracked specimen was heated above critical temperature, at which the crack was monotonically propagated to the desired depth by applying a live load. The flaw evolution was visually monitored from the oven viewport. After the load was reduced to stop the crack growth, an adequate dead loading was applied at the beginning of the cooling portion of the thermal cycle. While in the oven and above critical temperature, the specimen rested on its back surface in order to eliminate asymmetrical crack tip loading due to specimen weight. This precaution was suggested from a previous photoelastic calibration of the Compact Tension specimen. At the end of the stress freezing cycle, natural flaw shapes were accurately recorded under x 10 magnification using a profile projector. Up to eight thin slices were removed with a bandsaw from each specimen perpendicularly to the crack profile to determine local SIF estimates from the near-tip isochromatic fringes. Magnification and the Tardy method for fractional fringe order determination were used to retrieve near-tip data points (i.e. r vs N, where I is the fringe distance from the crack tip and N is the isochromatic fringe order) along two directions perpendicular to the crack plane. These data sets were subsequently used in SIF evaluation. SIF extraction from near-tip isochromatic fringes
Photoelasticity has been extensively used in the experimental SIF determination for two- and three-dimensional fracture problems. The extraction of a SIF estimate from the isochromatic fringes involves the assumption on the governing near-tip stress state. Based on different assumptions, various procedures for SIF extraction from the isochromatic fringes, which are locus of points of constant maximum in-plane shear stress, have been proposed. The two most widely used in the case
881
Photoelastic calibration
of static analysis are often referred to as the “slope” method[S], and the “two-parameter” approach[9-111. While in the first method only the classical singular LEFM stress field equations are assumed to dominate the near-tip measurement zone and are combined according to the maximum in-plane shear stress definition, in the second procedure constant stress contributions are added to Irwin’s singular stress tensor equations. In the course of the present study, a computer program, which runs on an HP 85, was developed. It processes two sets of basic experimental data (i.e. r vs N) extracted from each slice along the +y and - y directions of Fig. 1. For each set of data it yields two plots (normalized maximum in-plane shear stress vs fi w r and normalized apparent SIF vs J;lw>. The least-square fit of straight lines through the LEFM-dominated (hence linear) zones yield two SIF estimates from the slope value and from the intercept with the ordinate axis, respectively. For the “slope” method, the following normalized equation is used
in which, from the stress-optic law, the maximum in-plane shear stress rmaxis given by
Tmax
=fLN 2t' ’
wheref” is the material fringe value, N is the isochromatic fringe order and t’ is the slice thickness. For the two-parameter method the following normalized equation holds in the LEFM region[lO], (3) in which Kap = q,,,,(8w)1/2. The procedures for SIF determination with the two methods are shown in the compound diagram of Fig. 2. Although both methods were verified to yield comparable results in most cases, the twoparameter method was usually preferred as more reliable. 3. RESULTS Previous studies have demonstrated that SIFs and actual flaw shapes are directly related [lo, 111. Therefore, care was given in recording the natural flaw shapes obtained during the
40 KAP B@
P
rmax Bw P
Slice lo-
l
Two
Fig. 2. Typical compounded
D-3-2
Parameter
D Slope Method
diagram for SIF determination method.
Method >
by the “slope” and the “two-parameter”
882
G. NICOLETTO
Fig. 3. Monotonic
flaw shapes in epoxy specimens
propagated
above critical temperature
photoelastic experiments. They are shown in Fig. 3. It is pointed out that the thumbnailing of Tests 4-5 was probably enhanced by an attempted propagation out of the chevron notch plane. Flaw shape evolution will be discussed in conjunction with the local variations in SIF distributions. The present three-dimensional analysis yielded Mode I Stress Intensity Factors at discrete points along the crack front for each crack depth. SIF was taken to be
(4)
where F is determined with the two methods outlined in the previous paragraph and the meanings of the variables are evident from Fig. 1. The photoelastic SIF distributions for the five tests are summarized in Fig. 4. From the specimen of Test 1 only three slices could be retrieved because the flaw was narrow. More slices were obtained from the other specimens. In Fig. 4 the SIF estimates are approximately located at the slice centers. Since the specimen had a symmetry axis, the photoelastic results are plotted referring to half specimen thickness. When two separate points are plotted at a specific 2z/b ratio, it signals a definite difference in SIF estimates for two symmetrical slices. As far as the accuracy of the photoelastic method is concerned, previous studies[lO, 1 l] suggest a maximum + 5% experimental scatterband in SIF estimates. Based on inspection of experimental diagrams such as the one of Fig. 2, it is felt that Test 5 photoelastic distribution is the least accurate. The very deep crack of Test 5 is expected to interact with the specimen back surface. Under these conditions the two-parameter method is expected to be less accurate. Additional terms of the non-singular stress field series expansion should be incorporated in the data reduction [ 121. Since most experimental and analytical studies of the literature determined single-valued chevron-notched specimen calibration curves from compliance measurements or calculations, which result in through-the-thickness “averaged” estimates, for the sake of comparison, a characteristic SIF for each crack depth had to be identified from the photoelastic SIF distributions of Fig. 4. These values were introduced, in a diagram F, against relative crack length, in which F, is given by
(5)
883
Photoelastic calibration
30
20 30
o- 2
8
F
0.58
20 30 0 0
3
0.68
4
0.73
5
0.83
20 30‘ ” 20
0
30
7
20 0
I
0
05
,
1
2 z/b
Fig. 4. Photoelastic stress-intensity distributions.
and the subscript c stands for compliance. The characteristic SIF for each test was defined as the value common to most slices. An averaged value was determined from the significant SIFs. Therefore, while notch intersection values are identified for Test 1, internal SIFs are appropriate for the other tests. 4. DISCUSSION The discussion of the results presented in the preceding section will deal initially with the effective SIF distributions and flaw shapes. A discussion of the STF vs crack depth calibration curve is included in a second subsection. The discussion will refer to recent results obtained in a numerical round robin conducted by the ASTM E 24 Task Group on Chevron-Notched Specimens. In this joint effort, two square-bar and two round-rod specimen geometries of different w/B ratios were studied[l], in addition to Barker’s short-bar geometry. The present specimen geometry (i.e. Fig. 1) is different from the ASTM Task Group geometry in several respects. According to Newman’s specimen length definition as the distance from the back surface to the point of load application, the present specimen has w/B = 1.40 instead of 1.45. The a,,/~ ratio is also different from the round robin value (i.e. 0.31 and 0.332, respectively). Finally, the mathematical model assumes the slot height to be zero while the photoelastic specimens have t/B = 0.02. Although this aspect ratio doesn’t comply to Barker’s range for a square-tipped notch (i.e. 0 < t/B < O.Ol), Ref. [4], the influence on SIF estimates is believed negligible because of the elastic behavior of the epoxy material. Efectivejaw shapes and SIF distributions A through-the-thickness cracked specimen configuration defines only theoretically a bidimensional problem. Crack front thumbnailing under fatigue loading, even in relatively thin plates,
884
G. NICOLETTO
clearly indicates that actual three-dimensional conditions occur along the flaw front. This feature involves, however, theoretical complexities that are often neglected in engineering practice. Procedures for equivalent crack length determination from a curved profile in a standard fracture specimen (i.e. CT) are prescribed in ASTM E 399-78. The chevron-notched short-bar specimen is characterized by a through-the-thickness crack front. However, its front evolution appears more complex than for uniform thickness specimens. A series of photographs of the crack front evolution in a polystyrene chevron-notched specimen are summarized in Fig. 5(a)[l3]. It demonstrates retarded growth in the specimen interior at short crack lengths. As the crack propagates, the front gradually straightens out and then thumbnails. This behavior seems consistent with the photoelastic flaw shapes presented in Fig. 3. However, in order to assess the influence of the Poisson’s ratio on flaw shape (i.e. the epoxy material above critical temperature is nearly incompressible), displacement-controlled crack propagation tests in epoxy specimens at room temperature (i.e. v = 0.35) were performed. On the specimen fracture surfaces ripples were detectable and associated to flaw propagation. They are shown in Fig. 5(b). Again the flaw evolution signaled a development from nearly straight to thumbnailed shape as the back surface was approached. From this experimental evidence it was concluded that the photoelastic modeling of the real boundary conditions in the present fracture problem is effective. Similar conclusions were reached in Refs. [IO, 111. Conversely, numerical methods of analysis assume a mathematical description of the flaw shape (i.e. circular, elliptical, rectilinear). Hence, in the ASTM numerical round robin[l], straight crack fronts normal the specimen longitudinal axis were considered. The analyses found nearly constant SIF distributions in the interior but severe gradients at the notch-crack intersection. Notably, the higher the number of Finite Element layers in the mathematical model, the more severe the SIF gradient in the vicinity of the notch[7]. Presence of SIF gradients signal non-uniform crack propagation to be expected along the flaw front. Upward concave SIF distributions resulted from both the Finite Element method and the Boundary Integral Equation method analyses, these distributions being independent from the crack depth. Besides an inherent inaccuracy in the use of the numerical SIF estimates as stated by Ingraffea[l3], no explanation of the peculiar flaw evolution previously described (i.e. increased thumbnailing with crack length) results from the numerical analyses. Tn Fig. 4 the SIF distributions obtained by frozen-stress photoelasticity clearly signal the presence of SIF gradients. The observed SIF concavity inversion with crack depth (i.e. from upward concave to downward concave) agrees with the observed change in flaw front curvature during propagation [i.e. Fig. 5(a) and (b)]. In Fig. 6 the Finite Element estimates for a short-rod specimen (i.e. w/B = 1.45) are compared to the appropriate frozen-stress test results of the present study. Significantly, besides the difference in SIF gradient evolution already commented on, the closest agreement between numerical and
Fig. 5. Flaw shapes in (a) a polystyrene
short rod[l3], and (b) in an epoxy rectangular temperature.
short bar at room
885
Photoelastic calibration 3D Finite
Element
[7] a/w I
Test I 1
20
-I
30 0.73 0.68
0 (
0
6
05
i
7
2 z/b
Fig. 6. Comparison of 3-D Finite Element results for the short rod[7], with frozen-stress photoelastic SIF distributions.
expe~mental SIFs occurs in the vicinity of minimum SIF depths (i.e. a/w = 0.55) at which natural Aaws (i.e. Fig. 5) approximate the rectilinear configuration. Specimen calibration curve The calibration curve of a fracture specimen gives the relationship between SIF and crack length only [i.e. eqn (2)]. Previous experimental calibrations have mainly studied the short-rod specimen and resorted to the compliance method. Barker[4], measured load-point displacements on fused quartz using a laserinterferometric technique. More recently Shannon et al., as reported in Ref. [l], characterized experimentally the short-rod specimen, analyzing crack opening displacements in aluminum specimens. In the ASTM task group numerical round robin, reported in Ref. [l], the information required from the analyses were, besides the SIF distributions along the flaws, SIF vs crack length relationships obtained from plane stress compliance. The analysis methods of the short-rod specimen included the Finite Element method[7, 131, and the Boundary Integral Equation method[l3]. The results of these numerical and experimental compliance studies for the short-rod fall in the shaded area of Fig. 7. Relatively small scatter is evidenced in view of the different techniques and materials used. The calibration curve for the rectangular short bar of the present investigation is also introduced in the same figure. Inspection of Fig. 7 reveals a similar trend although shifted to slightly lower values. The agreement, however, is believed satisfactory. In fact, a lower FC vs a/w curve for the present specimen was to be expected based on two considerations. First, Raju-Newman’s results[7] demonstrate an imperfect equivalence of the short rod and of the rectangular bar, lower SIFs pertaining to the last geometry. Second, the w/B ratio of the photoelastic specimens was 4% lower
886
G. NICOLETTO 40
Fc 30
Short Rod 20
B
w/6=1.45; adw=.3
Exp. & Num.Complianc
tect. Short Bar w/B=1.4; a,/w=.3 -*-0
.2
0
Fig. 7. Comparison
Photoelasticity .4
.6
a/w
.8
1
of experimental and numerical compliance SlFs for the short rod from the literature with photoelastic SIFs for the rectangular short bar.
than for the specimen considered in the ASTM round robin (i.e. w/B = 1.45). Both these factors tend to reduce the SIFs. The preceding comparison used the results relative to the short-rod specimen assuming approximate compliance equivalence between short rod and rectangular short bar. A hybrid experimental-numerical technique was used by Sanford-Chona[l2] in the determination of the geometric shape function of a thin 2-D through-cracked photoelastic specimen having the characteristic w/H ratio of the rectangular short bar. Their procedure employed data from an extended zone around the crack tip and yielded 2-D SIFs that were subsequently converted estimates for a 3-D chevronnotched analogue applying a compliance equivalence approach proposed by Munz et aZ.[5]. The SIF vs relative crack depth curves from the Sanford-Chona study and results of the present photoelastic analysis are included in Fig. 8. The correlation is good in the minimum SIF region.
80 RECTANGULAR A
Numerical
. 2D local
SHORT comptiance[
collocation
+ 3D equivalent 60 n
BAR
tion b2] Frozen stress
71 [12]
calibrg photoela
sticity
40
20
0.75
a/w
1
Fig. 8. Comparison of 2-D through-the-thickness cracked and 3-D short bar calibration by the local collocation photoelastic method[l2], with numerical compliance results[7], and with the present photoelastic estimates.
Photoelastic calibration
887
However, steeper SIF gradients than for the present results are signaled by the local collocation method at short and very long crack length. For high a/w ratios when a back surface effect could be significant it is conceivable that the local collocation method of Ref. [ 121 could be more accurate than the two-parameter approach. The difference is less justified at short crack lengths. In Fig. 8 a numerical calibration curve for the rectangular short-rod specimen taken from Ref. [7] is also introduced. From the limited number of points of the FE analysis a good agreement with frozen stress results at short crack lengths and with the local collocation method for long crack length is identified. Nonetheless, the three different calibration curves summarized in Fig. 8 yield minimum SIFs in the 0.54.6 a/w range. 5. CONCLUSIONS The present paper has reported on an experimental calibration of the rectangular chevronnotched short-bar specimen[4]. Frozen stress photoelasticity was used in studying a series of cracked epoxy models. A comparison of two procedures for SIF determination from near-tip isochromatic fringes procedures was performed. SIF distributions along the crack front were obtained and correlated to the relative flaw shapes. Natural flaws in epoxy specimens displayed crack-length-dependent front curvatures. These findings were corroborated by observations in different materials. Flaw departures from a straight through-the-thickness shape were interpreted in terms of local photoelastic SIF variations. Upward concave SIF distributions were observed for short cracks. Subsequent evolution to downward concave SIF distributions was found for longer crack lengths. The rectilinear flaw shape, commonly assumed in analytical studies, is found to produce severe gradients at the crack-notch intersection, which cannot explain natural flaw evolution. Finally, comparisons with experimental and numerical compliance calibrations are found to be satisfactory, especially in view of the inevitable variation in geometry, material and method of analysis. REFERENCES Jr., A review of chevron-notched fracture specimens. Chevron-Notched Specimen : Test&q and Stress Analysis, ASTM STP 855 (Edited by J. H. Underwood, S. W. Freiman and F. I. Baratta), pp. 5-3 1.American Society for Testing and Materials, Philadelphia (1984). 121 L. M. Barker, Theory for determining K,, from small non-LEFM specimens, supported by experiments on aluminum. Int. J. Fracture 15, 511-536 (1979). 131 L. M. Barker, Specimen size effects in short-rod fracture toughness measurements. Chevron-Notched Specimen : Tesfin,g and Stress Analysis, ASTM STP 855 (Edited by J. H. Underwood, S. W. Freiman and F. I. Baratta), pp. 117-133. American Society for Testing and Materials, Philadelphia (1984). 141 L. M. Barker, Compliance calibration of a family of short rod and short bar fracture toughness specimens. Engng Fracture Mech. 17,289-312 (1983). 151 D. Munz, R. T. Bubsey and J. E. Srawley, Compliance and stress intensity coefficients for short bar specimens with chevron Notches. Znt. J. Fracture 16, 359-374 (1980). [61 R. T. Bubsey, D. Munz, W. S. Pierce and J. L. Shannon, Compliance calibration of the short rod chevron-notch specimen for fracture toughness testing of brittle materials. Int. J. Fracture 18, 125-133 (1982). finite element analysis of chevron-notched fracture specimens. [71 I. S. Raju and J. C. Newman, Jr., Three-dimensional Chevron-Notched Specimen : Testing and Stress Analysis, ASTM STP 855 (Edited by J. H. Underwood, S. W. Freiman and F. I. Baratta), pp. 32-48. American Society for Testing and Materials, Philadelphia (1984). determination of stress intensity factors for single and interacting cracks and [81 Y. Phang and C. Ruiz, Photoelastic comparison with calculated results. Part I : two dimensional problems. J. Strain Analysis 19,23-34 (1984). cracked body problems. J. Optical Engng 191 C. W. Smith, Use of optical methods in stress analysis of three-dimensional X,69&703 (1982). distributions for corner cracks emanating from open holes in plates of finite width. Theor. [lOI G. Nicoletto, Stress-intensity Appl. Fracture Mech. 3,63-70 (1985). characterization of cracks at straight attachment lugs. Engng Fracture Mech. 22, 829%338 1111 G. Nicoletto, Experimental (1985). calibration of the short-bar chevron notched specimen. Chevron-Notched [I21 R. J. Sanford and R. Chona, Photoelastic Specimen : Testing and Sfress Analysis, ASTM STP 855 (Edited by J. H. Underwood, S. W. Freiman and F. I. Baratta), pp. 81-97. American Society for Testing and Materials. Philadelphia (1984). finite and boundary 1131 A. R. Ingraffea, R. Perucchio, T.-Y. Han, W. H. Gerstle and Y.-P. Huang, Three-dimensional element calibration of the short-rod specimen. Chevron-Notched Specimen : Testing and Stress Analysis, ASTM STP 855 (Edited by J. H. Undenvood, S. W. Freiman and F. I. Baratta), pp. 4949. American Society for Testing and Materials, Philadelphia (1984).
111 J. C. Newman,
(Received 5 August 1985)