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FRACTURE INITIATION BY STRESS WAVE LOADING
J. Duffy, C. Shih, L. Freund, R. Hawley
To cite this version:
J. Duffy, C. Shih, L. Freund, R. Hawley. FRACTURE INITIATION BY STRESS WAVE LOADING.
Journal de Physique Colloques, 1985, 46 (C5), pp.C5-163-C5-169. �10.1051/jphyscol:1985521�. �jpa-
00224751�
JOURNAL DE PHYSIQUE
Colloque C5, suppl6ment a u n08, Tome 46, aoQt 1985 page C5-163
FRACTURE INITIATION
BY
S T R E S S W A V E LOADINGJ. D u f f y , C.F. Shih, L.B. Freund and R.H. Hawley
Brown University, Providence, Rhode Island 02912, U . S . A.
~e'sume'
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On dicrit une mkthode d'essai dkveloppke pour permettre l'&valuation de la tcnacite' KI, ou J,, sous conditions de chargement dynamique et de dkformation plane.La vitesse de la tdnacite rkalise'e damcet essai, avec des aciers structuraux, donne une valeur de K, Cgale 5 2 X 106 MPa a s - ' . On examine la prkcision de cette m6thode d'essai & I'aide d'une ricente analyse par kldments finis.
Abstract
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We describe an experimental technique which has been developed to determine the fracture toughness KIc or JIc under dynamic plane strain loading conditions.Loading rates K, of 2 X 106 M P a m S-' have been achieved in experiments with structural steels. The accuracy of the experimental procedure is discussed in the light of a recent finite element analysis.
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IntroductionTo answer the need for an experimental technique that would provide a value for the dynamic plane strain fracture toughness in structural materials, Costin et a1 [ l ] developed a notched round bar dynamic fracture test, which is an adaptation of the Kolsky pressure bar. In this test a specimen in the shape of a notched round bar with a fatigue induced crack a t the notch root is subjected to a dynamic tensile loading pulse of sufficient magnitude to fracture the specimen during the initial rise of the loading pulse. This technique has the distinct. advantage of achieving extremely high loading rates KI up to 2 X 106 ~ ~ a , K i i s - ' , while providing a load-displacement record which allows a direct evaluation of. the fracture toughness. Thus it is possible to determine the loading rate sensitivity of the parameters usually used to characterize fracture initiation in structural materials.
Recently a critique of the notched round bar fracture initiation test based on a finite element analysis of the specimen during the passage of a tensile pulse was presented by Nakamura et a1 [2].
On the basis of an elastic-plastic analysis which employed the actual measured stress-strain relations appropriate to the dynamic test conditions, they concluded that the experimental method provides an accurate evaluation of the K,, and J,, provided the remaining ligament diameter (after fatigue precracking) is smaller than about 30 percent of the specimen diameter.
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Descri~tion of the Fracture Initiation E x ~ e r i m e n tThe dynamic fracture initiation test consists in loading to failure a pre-cracked notched round bar of 2.54 cm diameter by means of a rapidly rising tensile pulse resulting from an explosive detonation initiated a t one end of the bar to produce a loading rate o f . K, = 2 X 106 MPadiiis-', see figure 1. A circumferentially uniform fatigue crack is grown i n from the root of the notch in a rotating bending apparatus leaving a remaining ligament approximately 7.6 mm in diameter.
Various methods are available to initiate the tensile pulse and to measure the crack mouth opening displacement [3]. In our tests, a charge of duPont Deta Sheet plastic explosive is placed against a mass that is bolted to the end of the specimen. Upon detonation, the mass is pushed axially away from the end of the specimen, thus initiating a tensile loading pulse in the specimen. To avoid
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985521
C5-164 J O U R N A L DE PHYSIQUE
possible damage to the instrumentation, the charge is detonated inside a vacuum chamber. The charge and loading apparatus have been calibrated to ensure that the amplitude of the tensile pulse is sufficiently large so that fracture occurs on the rising slope of the pulse. The tensile pulse is monitored by strain gages as it travels toward the pre-cracked section of the bar. A second set of strain gages mounted one diameter beyond the notch monitors the pulse transmitted through the pre-cracked section a n d thus provides a measure of the average stress a t the fracture site as a function of time i n accordance with the principle of Kolsky's split-Hopkinson pressure bar [4]. The instant of fracture initiation is determined from observations of the transmitted and reflected pulses as recorded on a n oscilloscope.
Crack mouth opening displacement is measured optically by a n adaptation of the moire technique i n which a glass slide is mounted to span the notch in the specimen, while cemented only to the transmitter side of the notch. A grid of 33 Iines/mm is deposited photographically on a polished flat portion on the incident side of the notch with a matching grid on the slide. As the specimen is loaded by the tensile pulse, the crack opens and the two grids slide past each other. Their relative displacement as a function of time is detected from changes i n their interference pattern using a photodiode. The resulting record has a wavelength that varies i n proportion to the velocity of the incident edge of the notch relative to the transmitter edge. The calibration of the crack mouth opening displacement instrumentation depends only upon the grid geometry and f o r the grid of 33 lines/mm, each wavelength on the record is equivalent to 15 microns displacement. Since the observed wave can easily be divided into quarter wavelengths, a crack opening displacement measurement as small as 3.8 microns can be determined from the record. The instrumentation described above provides unambiguous measurements of the average stress a t the fracture site and of crack opening displacement, as functions of time. The load transmitted across the ligament versus the crack mouth opening displacement is recorded over the interval leading to the fracture event.
This information is employed to determine the fracture toughness KIc if fracture initiation takes place under small scale yielding conditions. If fracture is preceded by large scale yielding, JIc can be determined f r o m the load-deflection record.
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Analvsis of DataFor a nominally brittle material the fracture parameter of interest is KIc, the plane strain fracture toughness. For a n externally cracked round bar, the stress intensity factor is given by [7] as
where r is the radius of the uncracked ligament, P is the applied load, D is the diameter of the specimen, a n d F(Zr/D) is a dimensionless size function
.
T h e load P f o r calculating KIc from equation (1) is determined from the load-displacement curve in accordance with ASTM standards using the 5% slope offset procedure. In order to apply linear fracture mechanics concepts, the size at the crack tip plastic zone must be small compared to the nominal dimensions of the specimen. A criterion f o r a valid KIc test requires that the radius of the uncracked ligament satisfywhere ay is the flow stress of the specimen material a t a strain rate comparable to the strain rate near the crack tip during the fracture test.
The size requirement (2) presents a dilemma f o r more ductile materials. Unrealistically large specimen sizes would be required if the K approach is to be followed. In this case a J-integral approach has been adopted. Rice et a1 [6] have shown that the value of J for an externally notched round bar can be determined from a load-displacement curve using
where is the load-point displacement d u e *to the presence of the crack. In the context of a dynamic fracture experiment, i t is not obvious which displacement quantity should serve as the load-point displacement. Since the crack mouth (or notch) opening displacement is readily measured during the experiment, this quantity was taken as equivalent to load-point displacement.
Furthermore, the force P in (3) is taken to be the load transmitted through the ligament. The accuracy of this approximation has been investigated by Nakamura et a1 [2]. Their study showed that the above reinterpretation of (3) resulted i n a slightly conservative estimate of J,, (see following section).
Paris [7] has suggested that a possible criterion f o r a valid JIc test is
r b SO J,, / U,
This size requirement has been adhered to by Costin et a1 and other investigators who have reported data using this test technique [S -101. A more recent review of the size requirements for a valid JIc test has been given by Hutchinson [ l l].
IV
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Finite Element Analvsis of t h e Notched Round Bar SoecimenA critical analysis of the circumferentially notched round bar test f o r the determination of plane strain fracture toughness KIc under dynamic a n d quasi-static loading conditions has been given by Nakamura et a1 [2]. Their dynamic el'astic-plastic finite element simulation of the actual fracture experiment uses the experimental dynamic stress-strain curves developed in [9] f o r a n AISI 1020 steel. Two relative crack depths, a/R, of 0.6 and 0.75, were analyzed; the former crack depth is typical of the test specimens utilized in [S, 91. A time-dependent tensile load which matches the measured incident tensile pulse is applied a t one end of model of the specimen. The choices of loading rates, material properties and geometric configuration were guided by the experiments carried out in [S, 91. The d'ynamic J-integral was calculated throughout the interval of interest using a n energy release rate integral expression adapted f o r the axisymetric configuration [2].
A schematic of the natched round bar of radius R, crack length a and remaining ligament radius r is shown in figure 2. The remotely applied load P, the load point displacement A and the notch opening displacement 6 are also indicated. Under quasi-static conditions, the J-integral for a round bar with a deep external notch is given by Equation (3). In the large scale plasticity regime, A, may be approximated by the notch opening displacement 6. Under these conditions the deep crack formula reduces to
Y Y
Costin et a1 [ l ] evaluated the dynamic J-integral by equation (5) justifying this by pointing out that the pulse length i n this experiment is large compared to the length of the crack o r the ligament.
The finite element mesh i n the r-z plane f o r the notched round bar is shown in figure 3. The location of the strain gages is indicated i n the figure as z, and z,. In the computational studies, the J-integral (Jdc) was calculated from the numerically determined load-displacement behavior for the cracked geometry using the above deep crack formula (5). In addition the J-integral was calculated from a n exact expression (appropriate to a dynamically loaded notched bar) using the field quantities obtained from the finite element analysis. Nakamura et a1 [2] compared the values determined form (5) and from the exact expression for J over the entire interval of interest. Their results are shown i n figure 4. The following conclusions concerning the deep crack formula for a n externally notched round bar can be drawn. In the elastic a n d contained plastic regime, the deep crack formula f o r the J-integral underestimates the actual value by about 20 percent if the relative crack depth is about 0.7. As yielding progresses, the accuracy of the deep crack formula improves and the formula is very accurate a t fully yielded conditions. The computational study suggests that dynamic fracture toughness, JIc,determined from the experimentally recorded P-6 data should be within 20 percent of the actual value [l, 21. Alternatively the fracture toughness expressed by KIc should be within 10 percent. Subsequent fracture toughness tests [ I l l have adhered to the
C5-166 JOURNAL DE PHYSIQUE
recommendation that the relative crack depth be greater than 0.7. The computer simulation of the experiment provided additional information on the full field which is worthy of mention.
An examination of the stress and deformation fields induced by the tensile pulse load reveals that the pulse transmitted through the ligament can be accurately inferred from surface strain measurements taken a t a distance of one bar diameter downstream from the crack plane. These results confirm that the location of the strain gages on the test specimens are appropriate for measuring the transmitted pulse [ l , 21. Zones depicting two levels of effective and hydrostatic stress in the vicinity of the crack plane are shown in figure 5. Upstream from the crack plane the plastic zone is slightly larger. A close inspection of the crack tip region reveals that the near-tip fields are symetrical, i.e., they can be characterized as Mode I.
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ConclusionsAn experimental technique f o r determining the value of the stress intensity factor KIc or JI, at fracture initiation under dynamic loading conditions utilizing the principles of the
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Kolsky (split-Hopkinson) bar has been developed. Loading rates %of 2 X 106 MPayii S-' or more have been attained in experiments o n externally notched .round bars of structural steels. The plane strain fracture toughness KIc or JIc is determined from measurements of the crack mouth opening, 6, and the load, P, transmitted through the ligament. These measurements can be accurately recorded in the experiment. A finite element analysis of the experiment showed that the dynamic fracture toughness can be determined to good accuracy from the experimentally obtained P-6 record, as long as the ration of the crack depth to bar radius exceeds 0.7.V1
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AcknowledaementsThe authors wish to acknowledge support from the Materials Research Laboratory a t Brown University funded by the National Science Foundation and the Office of Naval Research, Grant N00014-78-C-0051.
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References1. Costin, L.S., Duffy, J. and Freund, L.B., "Fracture Initiation Under Stress Wave Loading Conditions," Fast Fracture and Crack Arrest ASTM STP 627, American Society for Testing and Materials, 1977, pp. 301-318.
2. Nakamura, T., Shih, C.F., and Freund, L.B. "Elastic-Plastic Analysis of a Dynamically Loaded Circumferentially Notched Round Bar," To be published in Engineering Fracture Mechanics, 1985.
3. Dormeval, R., Chevallier, J.M., and Stelly, M "Fracture Initiation of Metals at High Loading Rates," Fifth International Conference on Fracture, Cannes, France, 1981.
4. Kolsky, H., "An Investigation of the Mechanical Properties of Materials a t Very High Rates of Loading," Proceedings of the Physical Society, Vol. 62,1949, pp. 676-699.
5. Tada, H., Paris, P.C. and Irwin, G.R., The Stress Analvsis of Crack Handbook, Del Research Corp., Hellertown, Pa., 1973.
6. Rice, J.R., Paris, P.C., Merkle, J.G., "Some Further Results of J-Integral Analysis and Estimates," Proaress in Flaw Growth and Fracture Toughness test in^, ASTM STP 536, 1973, pp. 23 1-245.
7. Paris, P.C., written discussion of J.A. Belgey and
J.C.
Landes, "The J-Integral as a Fracture Criterion," Fracture Toughness, ASTM STP 514, 1973, pp. 1-23.8. Costin, L.S. a n d Duffy, J., "The Effect of Loading Rate and Temperature on the Initiation of Fracture in a Mild, Rate-Sensitive Steel," Journal of Engineering Materials and Technolo~v, Transactions American Society of Mechanical Engineers, Vol. 101,1979, pp. 258-264.
9. Wilson, M.L., Hawley, R.H. and Duffy, J., "The Effect of Loading Rate and Temperature on Fracture Initiation in 1020 Hot Rolled Steel," Engineering Fracture Mechanics, Vol.
13, No. 2, 1980, pp. 371-385.
10 Couque, H. Duffy, J. and Asaro, R.J., "Effects of Prior Austenite and Ferrite Grain Size on Fracture Properties of a Plain Carbon Steel," Brown University, Division of Engineering Report, July, 1984.
11. Hutchinson, J.W., "Fundamentals of the Phenomenological theory of Nonlinear Fracture," J. Appl. Mechanics, 1983, Vol. 50, pp. 1042-1051.
GRIDS FATIGUED
NOTCH
Figure 1
Figure 2
\ I--- 39 cm
--A
I N C I D E N T S T R A I N l TRANSMITTER /,STRAIN GAGE V
A-
/
\
EXPLOSIVE U U
PHCTO-DIODE LIGHT
CHARGE SOURCE
Schematic diagram of apparatus for the dynamic notched round bar fracture initiation experiment.
Schematic cross-section of an externally notched round bar subjected to tensile loading. The dimensions of the bar and the locations f o r measuring the load point displacement, A, and the notch opening displacement 6 are indicated.
JOURNAL DE PHYSIQUE
Figure 3
Figure 4
(a) Axisymetric finite element model of an externally notched round bar with a ratio crack length to bar radius of 0.6.
DEEP CRACK L I M I T
.
, \-.-.-.-.-.-.-
,
---_-
----, .-Ho / R = 0 7 5 DYNAMIC
0 0
0 5 1 .O 1.5 2.0 2 . 5 3.0
P / PO
Ratio, Jd,/J, determined from dynamic analysis plotted against the normalized load for a/R = 0.6 and a/R = 0.75. The deep crack limit is indicated.
Figure 5
DIRECTION O F P U L S E P R O P A G A T i O N
Results o f dynamic analysis a t t = 46 microseconds f o r a / R = 0.6. (a) Zones where the effective stress level exceeds 0.7 times yield stress and at yield stress; the higher level indicated by darker shade. (b) Zones where hydrostatic stress level exceeds 0.7 times yield stress a t yield stress; the higher level indicated by the darker shade.
