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[ ""':[;"'2['u2',- II,' "' [[[1' 2.112 11111 '11" [ ['[.[.1[ [."11' [I[. ' [2'11 ..['121'. , "'2' "'2......'["[lrl1‘[».11 11112 """'["'[I [[[[1. '1 ['11'1"'1""' '1 1"[[II'[[[[""' .11 1[[ 121 [[[...M}. 11' """.1[[.'"'~'L'2" """"'.'.'1"2'".111" 1. "" 12"".222121' '[1["1. 11'11211'11'1'111'" [['[ ['[ ' '"1112'1" 11 [1| [1"“1111'12111 ' .111... ['. [r'|:.‘[1[1"1[:'1 12 «r- - --.—o' —'—.__v~r"—~— '-" -_,,; ' 'ii:[[2' ' [...22..1 [[[“ THEsra I)ate 0-7 639 This is to certify that the thesis entitled MEASUREMENT OF TRANSIENT RESPONSE OF A CENTRAL CRACK TO A TENSILE PULSE presented by Abdurahman Ahmed Sukere has been accepted towards fulfillment of the requirements for Eb. I2. deg-gem Mechanics Major professor OVERDUE FINES: 25¢ per day per item RETURNIN LIBRARY MATERIALS: Place in book return toreno charge from ctrculatton records MEASUREMENT OF TRANSIENT RESPONSE OF A CENTRAL CRACK TO A TENSILE PULSE BY Abdurahman Ahmed Sukere A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1979 ABSTRACT MEASUREMENT OF TRANSIENT RESPONSE OF A CENTRAL CRACK TO A TENSILE PULSE BY Abdurahman Ahmed Sukere The goal of this investigation was to study the behavior of a crack in a stress wave environment. The crack was approximated by electromachined slots in the center of thin aluminum specimens. A tensile impact apparatus has been developed for generating a plane ten- sile pulse with a ramp function profile. The amplitude and rise-time of the pulse were fairly repeatable. Dynamic displacements along the crack surface were measured using a laser interferometric displacement gage. The data were analyzed within the bounds of linear elastic fracture mechanics to estimate the dynamic stress intensity factor. The experimental results for the dynamic crack displace- ments were found to oscillate, a phenomena attributed to the cancellation and reinforcement of the incident waves by the various scattered waves. Analytical and numerical solutions compare favorably with the short time experi- mental results; i.e., with results for a time regime corresponding to the time it takes for the first scattered Abdurahman Ahmed Sukere P wave to travel from a crack tip to the nearest boundary and back. The comparison of the long time finite element solution with experimental is not good. The descrepancy may be attributed to the imperfect experimental wave front, the influence of which could not be isolated for this time regime. ACKNOWLEDGEMENTS I wish to extend a sincere thanks to Professor W.N. Sharpe, Jr., for his guidance and patience throughout the research program and for the experience and motivation provided by his research investigations. To Professor M.A. Medick, my gratitude for his encouragement and suggestions. Sincere thanks are also due to the other members of my guidance committee, Professor G.E. Mase and Professor L.J. Segerlind. I would also like to thank Mr. D. Childs and Mr. R. Jenkins for assistance in constructing the apparatus. ii TABLE OF CONTENTS Chapter I. INTRODUCTION II. EXPERIMENTAL TECHNIQUES 2.1 TENSILE PULSE GENERATING SYSTEM AND INSTRUMENTATION A. Background B. General Description and Principle of Operation C. Tensile Impact Apparatus D. Loading Machine E. Strain Pulse Measuring and Recording System 2.2 SPECIMEN GEOMETRY AND PREPARATION A. Material Properties B. Specimen Geometry C. Specimen Preparation 2.3 DISPLACEMENT MEASUREMENT A. Basics of the Interferometric Displacement Gate B. The IDG for Dynamic Displacement C. Relationship Between Displace- ment and Stress Intensity Factor D. DiSplacement Measuring and Recording System 2.4 PROCEDURES A. Preliminary Checks B. Eccentricity of Loading C. Testing Procedure iii Page 10 10 10 12 14 21 23 28 28 30 30 32 32 37 40 43 44 44 49 50 Page III. ANALYSIS OF CRACK RESPONSE TO LONGITUDINAL PLANE WAVES OF NORMAL INCIDENCE 53 3.1 ANALYTICAL CONSIDERATIONS 53 A. Equations of Elasticity in Two Dimensions 53 B. Formulation of the Problem 58 C. Solution of the Problem 61 3.2 EXPERIMENTAL DATA CONSIDERATIONS 65 A. Strain Time History of the Incident Wave 65 B. Dynamic Separation of Crack Faces 66 C. Dynamic Stress Intensity Factor 68 D. Dynamic Crack Profile 73 IV. RESULTS AND COMPARISON WITH THEORY 75 4.1 EXPERIMENTAL RECORDS 75 4.2 GENERAL CRACK BEHAVIOR 82 4.3 CORRELATION WITH ANALYSIS 93 4.4 COMPARISON WITH FINITE ELEMENT 96 V. SUMMARY AND CONCLUSIONS 102 LIST OF REFERENCES 105 iv LIST OF FIGURES Schematic of the experimental setup Schematic of the tensile impact apparatus Tensile pulse generating system Close-up of projectile and tup assemblies Schematic of hyge shock tester Strain gage potentiometer circuit Stress-strain of type 2219 aluminum Schematic of the specimens Schematic of the IDG General view of the experimental setup Photograph of the displacement measuring instrument Notations used in measuring crack profile Strain signal trigger circuit Strain gage locations on the specimen before and after machining the slot Observed strain reSponse at various locations of the Specimen without the slot Geometry for generalized plane stress Geometry of the problem Experimental records for crack displace- ment at Xl (a/b = 0.54) Results of crack tip opening displace- ment for the trimmed specimen showing the displacement associated with each fringe pattern v Page 15 16 17 19 22 24 29 31 33 38 39 41 45 47 48 54 59 67 69 Local rectangular stress components Experimental records for crack displace- ment at X3 (a/b = 0.167) Experimental records for crack displace- ment at X2 (a/b = 0.167) Experimental records for crack displace- ment at Xl (a/b = 0.167) Variations of displacement at X with time for specimen with a/b = O. 67 Variations of displacement at X with time for specimen with a/b = 0.167 Variations of displacement at X1 with time for specimen with a/b = 0.167 Variation of dynamic stress intensity factor with time for specimen with a/b = 0.167 Variations of dynamic stress intensity factor with time for specimen with a/b = 0.54 Crack profile at three different time steps Comparison of experimental dynamic k-calibration data with theoretical static results Input pulse at location 2; experimental points and ramp function approximation Comparison of analytical and experi- mental results for the dimensionless mode I stress intensity factor Finite element model Comparison of finite element and experi- mental results for the dimensionless mode I stress intensity factor vi Page 70 76 77 78 83 84 85 87 90 91 92 94 95 98 99 Figure Page 4.15 Comparison of finite element and experi- mental results for the normalized dis- placement at the center lOO vii CHAPTER I INTRODUCTION Since the time of Griffith's [1] original idea (1920) that fracture of a brittle material is the result of the growth of inherent minute cracks or flaws, a series of technological problems of braod public interest has stimulated interest in the field of fracture mechanics. The U.S. Liberty ship episodes from 1940 to 1945 provide a vivid example. During that period more than 250 serious failures to ship hulls occurred due to factors associated with brittle fracture. These problems led to the modifica- tion of the Griffith idea, extension of the idea to metals and fatigue fracture, and the introduction of the concept of a critical stress intensity factor as a criterion for crack growth. Research in this area has been greatly accelerated, resulting in a profusion of publications. This is because of fracture mechanic's past successes, ever increasing de- mand and because of the support and impetus provided by various governmental and industrial agencies. For the most part, the physical systems which have been studied are quasi-static. In view of the fact that structures such as pipe lines, gun barrels, submarine hulls 1 and reactor components are subjected to dynamic loads that may be tensile in nature before or after reflection from interfaces, there are many fracture problems which cannot be viewed as being quasi-static and for which the inertia of the material must be taken into account. How— ever, there are relatively few solutions available for the stresses in a cracked elastic solid subjected to dynamic loading. The primary reason for this is the extreme com- plexity of elastodynamic crack problems. A comprehensive review of the literature regarding application of elastodynamics to the determination of stress intensities and to brittle fracture will not be undertaken here. In fact several very good and complete reviews have become available recently, among them those of ErdOgan [2], Achenbach [3], [4], Sih [5] and Freund [6]. For the purpose of pertinence to the present work the literature concerned here relates to the class of elastodynamic problems that deals with a stationary crack subjected to time-dependent loads. The brief discussion that follows, also excludes solutions of idealized prob- lems of cracks subjected to vertical shear (SV waves) waves and horizontal shear (SH waves) waves. The stationary crack solutions which have been obtained can be put in one of two categories, depending on whether the applied load is periodic or transient. In the former category, significant progress has been made by Sih and Loeber [7], who obtained effective solutions involving the interaction of wave length with crack size. Among the solutions of the latter category are the analysis of the scattering phenomenon of plane waves due to the presence of a semi-infinite crack by de Hoop [8], and Nuismer and Achenbach [9]. Mue [10], and Ang [ll] exPlored transient problems in which a semi-infinite crack appears instantaneously in a uniformly stressed elastic medium. Their work was later extended by Baker [12], to the case where the crack pro- pagates at a constant velocity after it has suddenly appeared. Equivalent problems have also been studied by Broberg [l3] and Freund [14]. It is well to note that the mathematical formulation of the problems stated in References [10, ll, 12] are equivalent to the specification of a uniform impact loading on the surfaces of the crack. Furthermore, the running crack solutions of References [9, 12] can be reduced to stationary crack solutions by considering the crack velocity to be zero. Transient problems for a finite length crack were explored by Papadopoulos [15], Thau and Lu [16], Ravera, Embley and Sih [l7], and Chen and Sih [18]. The study in Reference [15] is qualitative in nature in that attention is focused on the closure of crack surfaces rather than obtaining quantitative results which characterizes the elastic fields in the vicinity of the crack tip. Of particular interest to the present study is the work of Thau and Lu [l6]. Thau and Lu used the Wiener- Hopf method to study the diffraction of a plane dilational wave of arbitrary profile and arbitrary angle of incidence by a finite stationary line crack in an infinite medium. They derived explicit expressions for the normal dynamic stress intensity factor, kl(t), and shear dynamic stress intensity factor, k2(t). The numerical results reported, which are exact for two crack transit times, are however only for the case of an incident Heaviside step stress pulse. Presented also in their study is the time history of the separation of crack surfaces at various points along the crack. Among the notable findings is the dynamic over- shoot in k1 and k2 of about 30 percent compared with static values for the same loading. The finite dimensions of the cracked specimen have been taken into account numerically by Chen [19], who used the finite difference method and by Anderson, Aberson and King [20], Glazik [21] and Aoki, et al. [22], who used the finite element method. Again the stress pulse was of the step variety, and References [19] and [20] report a dynamic normal stress intensity factor, k1(t), which is close to three times the static counterpart. A substantial number of experimental studies have been conducted to investigate fracture dynamics phenomena. The studies in this field can be classified into two categories according to the quantity intended for measure— ment. One category deals with a phenomenological in- vestigation where the measured quantities of interest are magnitudes of pulse causing fracture, locations and types of fracture, velocity of running cracks and pulses gen- erated due to sudden fracture. The other is concerned with an elastodynamic analysis where the measured quantities pertain to the transient strain or stress field near the crack tip. The earliest work on the phenomenological type of experimental analysis was carried out by J. Hopkinson [23]. Hopkinson measured the strength of steel wires when they were suddenly stretched by a falling weight. The next significant investigation was carried out by B. Hopkinson [24], who detonated an explosive charge in contact with a metal plate. In this work, B. Hopkinson demonstrated the effect of "spalling" or "scabbing," which occurs when the compressive pulse generated by the explosive is re- flected at the opposite side as a tensile pulse. More recently a very extensive investigation of this nature was carried out by Rinehart and Peterson [25]. Detailed des- criptions of these and similar investigations can be found in a recent survey by Kolsky and Rader [26]. Roberts and Wells [27], Schardin [28], and Clark and Irwin [29] measured crack velocities in various materials. Later Kolsky [30] measured pulses generated by a brittle fracture for the cases of Hertzian, simple tensile and flexural fractures. The existing experimental works on fracture dynamics which are based on elastodynamic analysis are very few. Most of these studies were carried out by the methods of dynamic photoelasticity. Among the early works reporting the transient stress field surrounding a propagating crack for a statically loaded specimen are References [31-33]. In a later work Bradley and Kobayashi [34] made a study to correlate stress intensity factor with crack velocity. In 1969 Sommer and Soltesz [35] determined the dynamic stress intensity factor for an edge crack in a plate of Araldite B subjected to wave motions generated by a travelling air shock wave. Later Smith and Knauss [36] determined the critical stress in- tensity factor resulting from a stress wave loading for edge cracked Homalite 100 plates. The dynamic loads were applied directly to the crack surfaces by electromagnetic means. More recently Costin, Duffy and Freund [37] de- termined the fracture toughness for round bar steel speci- mens with prefatigued circumferential notches loaded to failure by rapidly rising tensile pulses. The wave motions were generated by explosive detonations and the pulses had a rise time of 35 to 40 microseconds at the fracture site. In 1978 Theocaris and Katsmanis [38] examined the behavior of notched plexiglas specimens under stress pulses created from an air-gun impact. In that investigation, plexiglas plates of dimensions 300 x 40 mm and thickness 4 mm, and sawn notches of width 0.2 mm, and varying length (4, 6 and 10 mm) were impacted at one end by a steel sphere of diameter 10 mm. Using the method of caustics with a high- Speed Cranz-Schardin camera, they determined the stress intensity factor and crack velocity of a running crack. Furthermore they made an attempt to correlate stress in- tensity factor and crack velocity and crack velocity and normalized crack length. However to the author's knowledge at present there are no experimental publications dealing with the interaction of a tensile stress pulse with a central crack in a thin strip. Furthermore, the studies of References [37] and [38], which have close relevance with the present investigation, lack an accurate deter- mination of the loading applied to the plates. Thus, the objectives of the present study are: 1. To develop experimental techniques for accomplishing a quantitative study of the elastodynamic stress fields in the vicinity of a stationary crack tip. These include a technique to generate a plane longitudinal tensile stress pulse in a plate with reason- able dimensions and a technique to measure dynamic crack displacement. 2. For a finite plate with a central crack, to study some aspects of the interaction of a step-like tensile stress pulse with a crack that has neighboring boundaries. These in- clude the time history of the separation of crack surfaces and the dynamic normal stress intensity factor. 3. To evaluate existing theory. 4. To provide some basic experimental data which can be used to guide future theoretical work. The practical significances are: 1. To help upgrade fracture mechanics considerations in the design of structures and machine components by accounting for dynamic effects. 2. To help better understand geological failures such as mining operations, earthquakes and the interaction of ground shocks with faults in the earth's lithosphere. The experimental techniques developed for this investigation are discussed in Chapter II. The first section describes the tensile pulse generating system and the instrumentation. The specimen geometry and prepara- tion are described in section two. Also discussed in this chapter are the basics of the interferometric displacement gage (IDG), relationship between crack displacement and stress intensity factor, and the procedures used in the experimental work. In Chapter III the theory of the dynamic response of a central crack to a tensile pulse in a plate is reviewed. The data reduction procedure is also discussed here. The analytical work is based upon a two dimensional theory of elasticity for the case of generalized plane stress. The mode I dynamic stress intensity factor and the dynamic separation of crack surfaces were computed by employing the Duhamel superposition integral in conjunction with short time results, for Heaviside step loading, obtained by Thau and Lu [16]. The experimental work which takes into account the influence of the boundaries neighboring the crack was carried out using the IDG technique. The results for the dynamic crack behavior are discussed in Chapter IV. It is believed that the results presented herein are the first experimental data on the dynamic response of a central crack with neighboring boundaries. Chapter V, which summarizes the findings of this investigation, concludes the thesis. CHAPTER II EXPERIMENTAL TECHNIQUES 2.1 TENSILE PULSE GENERATING SYSTEM AND INSTRUMENTATION A. Background Tensile impact tests have been performed by many investigators, most of the tests being performed to de— termine the effects of loading rate on such prOperties as yield strength, energy absorption, reduction in area and elongation. The earliest work in this field was per- formed by J. Hopkinson [23] and Mason [39], both of whom applied tensile stress pulses to wires by means of a falling tup. Ginns [40] used a spring mechanism type loading machine to apply a sudden load and attempted to measure the stress with a resistance—pressure gage. Brown and Vincent [41], developed a pendulum type impact machine and used piezoelectric crystals to measure stress. Clark [42], Mann [43] and Nadi and Manjoine [44] all used versions of a high Speed rotary impact machine. The impact was applied by two hammers mounted on a heavy fly- wheel which was rotated by a variable speed direct current motor. At a predetermined speed the hammers were allowed, by means of a trigger mechanism, to engage an anvil fastened to the bottom of the specimen. 10 11 Clark and Wood in a paper of 1949 [45] described the construction of a new type of a tensile impact machine in which the load is applied pneumatically and reaches a maximum in times as short as 5 milliseconds. In 1959 Austin and Steidel [46] developed an explosive impact tensile tester. All of the above-mentioned techniques suffered one or more of the following problems: distortion of the pulse due to vibration, non-axial loading and long rise time. The difficulty of axial loading may be adequately overcome, if care is taken, in tests involving compression impact, but when a tensile load has to be applied the problem is more difficult. Various attempts have been made to reduce the effect of non-axial impact. One solution [47] has been to use a hollow cylindrical specimen fixed at one end and impacted at the other by'a bullet which has travelled down its length. The requirement of a Specialized design of Specimen makes this technique undesirable. Harding et a1. [48] developed an apparatus in which a tensile stress (strain) pulse was obtained by reflection of the loading wave which produced a compression impact on a tubular weighbar fastened to the tup, so that the eccentricity of tensile loading is improved to that of a compression impact test. With this loading machine they were able to pro— duce loading pulses as short as 25 microseconds which 12 could have been improved if the impact had been at the tup, since the travel in the weighbar contributes to the deterioration of the rise-time of the compression pulse. Later on, Harding [49] developed a magnetic load- ing machine in which the tup was accelerated away from a coil when a capacitor bank was discharged through the coil. This produced loading pulses with rise-times as short as 5 microseconds. The main problem with this technique is the large transient electrical field associated with the discharge. He allowed the pulse to travel for 60 micro- seconds before it reached the specimen so that the transient would die out and resistance strain gages could be used. More recently, 1977, Costin et al. [37] developed an explosive impact tensile tester which is an adaptation of the Klosky pressure bar. The tensile pulse was pro- duced by detonating an explosive charge on a loading head attached to the end of the bar by a large bolt. Reflec- tion of the compressional wave from the back face of the loading head produced a tensile pulse in the bolt, which was transmitted to the end of the bar. The rise time measured at the incident gage was 20 to 25 microseconds and that at the site of interest was 35 to 40 microseconds. B. General Description and Principle of Operation Because the primary interest of the present in- vestigation was the response of the impingement of plane l3 stress waves on a crack, it was necessary to devise an apparatus that would generate plane waves over a relatively large area representing one edge of the speci- mens. As noted in the preceeding section the tensile pulse generating devices used to date are designed for round bar specimens, which makes the problem of non-axial impact relatively easy. Thus it became apparent that the strain-wave generation devices currently in use were either unsuitable or too expensive a venture to adapt for this investigation. It was then decided to design and con- struct a new type of a tensile impact apparatus that would satisfy the following criteria: 1) Capable of generating plane tensile stress waves that resemble a Heaviside step function, to facilitate comparison with theory. 2) Capable of allowing the input pulse to reach its maxi- mum before reflected waves from the boundary of reason— able size specimens reach the crack, i.e. the rise-time of the pulse should be on the order of 10 microseconds or less. 3) Capable of producing repeatable pulses. 4) Should have provisions that would enable the amplitude of the pulse to be varied. 5) Should have provisions to make specimens accessible for optical measurement. 14 6) Should be inexpensive to build, and 7) Should be easy to operate. The major components of the pulse generating system were the loading machine and the tensile impact apparatus. A schematic representation is shown in Figure 2.1. The loading machine has an attached loading assembly that accelerates a projectile which rides down a launch tube and impacts a tup. During contact, com- pression waves travel away from the impacting interface in both the projectile and the tup. Meanwhile, a tensile wave travels through the specimen away from the tup. Since the projectile is shorter in length in comparison with the tup, when the wave reflected (as tensile) from the end of the projectile reaches the interface, the pro- jectile is thrown back. Thus, the length of the projectile determines the duration of the "square" pulse. Brass was chosen for the material of the impactor and the tup on account of its low stress wave velocity, and their respective length dimensions were chosen to give the necessary pulse duration. A detailed discussion of the construction of apparatus is given in the following two sections. C. Tensile Impact Apparatus The arrangement and dimensions of the tensile impact apparatus are shown in Figures 2.2, 2.3, and 2.4. The apparatus consisted of four main components; a launch tube, a loading assembly, a projectile assembly and a tup. 15 .msuwm HmucoEwuomxw wcu mo caumaocom mmmoomoHHMUmO TH Ddsoufiu wmmu uwmwflua .//////////////////4/. / _l.l-II-'/ \\\\\\\\\ \\\\\\\\\\\\\7 \” Ill-ll...- 8%V\§\\ \\NVm Ill-llll—I ./’////////////////I ./ 92m Em Ewwm memq nameHocH wufisouflu wmmw ofluquoflucwuom _D L U msumummmd uommEH mfiflmcoe .H.N ousmwm umpcflazu cwmouuflz has: Houucoo wmxz ® ® uwumwB xoocm mmxm 16 .msumummmm uommefl mafimcwu on» we oflumEonom .N.N madman >Hnfiommm Q56 coEwome mEmon ewnuwnmom Econ .q acmpfiocH >HnEommm ofiwu00noxm . mo» :Oumwm mc>z mo :ofiwcwuxm nouommefl emf»: [ill/[I’l’lllll I Im. \\\\\\\\\\\\\, & \\\\\\ \\\\\\\\m l1 _ . EU on 50 0.0 fil Eu 0m ED NmH l7 Tensile pulse generating system. Figure 2.3. 18 The launch tube was a commercially available cold- drawn seamless steel tube with an inner diameter of 15.240 i 0.058 cm, an outer diameter of 17.78 cm and a length of 152 cm. This launch tube was supported securely in steel pillow blocks constructed from an "I" beam. The blocks were in turn supported by a 183 cm long "I" beam which was bolted to the floor. Due to lack of a lathe large enough to handle the size of the tube no attempt was made to bore it for a more precise clearance. Three ports, which were later fitted with quartz windows, were cut in the tube to allow optical measurements on the specimen. Additional two ports were cut in to allow access to the inside of the tube. Furthermore, a collar was welded on one end of the launch tube to attach the tup. The loading assembly is an extension of the load- ing machine. This unit consists of two parallel aluminum rods threaded onto aluminum circular plates on either sides, one of which was threaded onto the piston shaft of the loading machine, the other left free to apply the accelerating force on the projectile assembly when in contact. The spacing between the loading assembly and the tup was such that the projectile was in free motion within the launch tube just prior to impact. The projectile assembly (see Figure 2.4), which was supported within the bore of the launch tube, con- sisted of a rectangular C Shaped brass impactor, a 19 .mwznaowmm as... can mHHuoonouQ mo annomoHo .v.~ manage 20 cylindrical aluminum supporting and guiding structure, and a steel end plate which held the impactor and the cylinder together. The impactor and the supporting cylindrical structure were designed with provisions to allow optical measurement on the Specimen. Some lubrication was applied to allow free sliding of the projectile in the bore of the launch tube. The tup (see Figure 2.4) consisted of a two piece rectangular brass with serrations to grip the specimen, two clamping brackets, and a steel end plate that bolts onto the launch tube. The dimensions and materials of the impactor and the tup were chosen partly to permit the recording of strain pulses without interference from re- flections arriving from the ends of the tup or impactor for the test duration. The launch tube was aligned longitudinally by making careful measurements with respect to parts of the loading machine. The diameter of the projectile assembly was chosen to closely fit the bore of the launch tube. The ends of the impactor and the tup, which were to be in contact at impact, were carefully turned in a lathe and finished with emery paper in order to reduce lateral friction. To further assure substantial simultaneity of contact throughout the surface of contact there was the provision for inserting shims between the brass bar and the end of plate of the tup. 21 D. Loading Machine The projectile assembly was accelerated by means of a commercial Hyge Shock Tester, Type HY-3422, manu- factured by Consolidated Electrodynamics Corporation. The tester is shown in Figure 2.3 and a schematic is shown in Figure 2.5. The piston diameter is 7.62 cm and the full stroke is 42.55 cm. The acceleration of the piston is achieved in the following manner: A given set pressure is applied to chamber B by means of compressed nitrogen. This pressure acts over the full area of the piston, pushing it against a ring seal at C. A load pressure is then applied to chamber A, this pressure being restricted to a smaller area of the piston by the ring seal thus causing a smaller force to act on this side of the piston. When the load pressure reaches a magnitude of 4.2 times that of the set pressure, the forces acting on the piston become equal. If additional load pressure is applied, the seal at C is broken and the load pressure expands over the entire face of the piston causing a large acceleration. The acceleration is regulated by a conical pin which regulates the amount of gas supplied to the piston face during the initial part of the acceleration. The acceleration is aided by a hydraulic fluid which occupies approximately one-half of the volume of chamber B. The deceleration is accomplished by a pin which gradually 22 To To load pressure . Piston pressure control control I D A C / / R’ B 7 \ Acceliiation Deceliiation Piston p p shaft Figure 2.5. Schematic of Hyge Shock Tester. 23 closes the area of the escape orifice D through which the hydraulic fluid is forced. The system is controlled from a separate control panel which contains pressure gages and control valves. E. Strain Pulse Measuring and Recording System Resistance strain gages change resistance when they are subjected to a strain. A potentiometer circuit converted the resistance change into a voltage change. Calibration factors then determined the strain from the voltage change. The Specimens were instrumented with foil strain gages type ED-DY-062AA-350, manufactured by Micro-Measure- ments. The distortion of the pulse introduced by the 0.157 cm gage length was considered negligible since the gage length was small in comparison with the rising portion of the pulse. The resistance of this gage as given by the manufacturer was 350 ohms : 0.4 percent and the gage factor was 3.16 i 2.0 percent. The circuit voltage was supplied byaisix volt battery, and the voltage was recorded before each test by means of a voltmeter connected in parallel with the battery. The potentiometer circuit is shown in Figure 2.6. A ballast resistor, Rb' is in series with the strain gage, Rg. For any fixed supply voltage, Eb’ the maximum sensitiv- ity of the circuit will occur when Rb = R9. The coupling capacitor, CC, prevents the passage of direct current to 24 \l/n O J. | 0' -v- o m Circuit supply voltage (DC) [11 U‘ I: Circuit current Ballast resistor 05’ R9: Strain gage CC: Coupling capacitor E0: Output voltage Figure 2.6. Strain gage potentiometer circuit. 25 the recording instrument. Thus, this circuit will only respond to dynamic strains or the dynamic components of combined strains. By Ohm's Law the current in the circuit is I = Eb/(Rb + Rg) (2.1) and the voltage across the gage is E = I R (2.2) Substituting Equation (2.1) into Equation (2.2), gives the voltage across the strain gage as Eg = EbRg/(Rb + Rg) Applying a strain to the gage changes the gage resistance to R + AR . 9 9 The current then becomes I = Eb/(Rb + R9 + ARg) (2.3) and the voltage across the gage becomes E + AB = I(R + AR ) (2.4) 9 9 9 9 Substituting Equation (2.3) into Equation (2.4) gives the new voltage across the gage as E R + AR b( g g) (R + R + AR ) b g 9 E + AB = (2.5) 9 9 Successive algebraic operations on Equation (2.5) give 26 E + AEg EbRg(l + ARg/Rg)/{(Rb + Rg)[1 + Rg/(Rb + Rg)]} ER1+ARR _b9( 9/9) (R +R) 9 9 [1 - (ARg/(Rb + Rg)) 2 2 + (ARg/(Rb + Rg)) - (ARg/Rb + Rg)) +...] =E-l9-Eg-[1+ARR R(R +R)-AR2R/R(R +12)2 RbRg 9b'9 b 9 9b 9 b 9 +AR3Rb/R (R +R)3-...] (2.6) 9 9 b 9 The first term on the right hand Side of Equation (2.6) is Eg which is the DC component of the voltage across R9. The remaining terms are then the dynamic com- ponent of the voltage across the gage. Since the coupling capacitor will only pass the dynamic component of the voltage, we have as a circuit voltage E0 = AEg (2.7) Thus Equations (2.6) and (2.7) give E R 2 bRb g _ E0 [1 ARg/(Rb + R9) 2 (Rb + Rg) 2 2 R R + R -... 2.8 + g/< b g) 1 < ) Typical foil strain gages subjected to elastic strains have the following values: R9 = 120 or 350 AR < 20 9 27 Therefore, the higher order terms in Equation (2.8) can be neglected;this results in an error in E0 of less than one percent. The output voltage for foil gages is then B = EbRbARb 0 2 (Rb + Rg) Two Tektronix preamplifiers were used to amplify (2.9) the signals from the potentiometric circuits. Tektronix Tyep 1A7A plug-in preamplifier was used for the upper trace signal. This is a high gain differential preamplifier with a vertical sensitivity to 10 microvolts per centi- meter and a frequency response from DC to l megahertz. The lower trace signal was amplified with the use of a Tektronix Type lAl dual-trace plug-in preamplifier. Both channels of the Tektronix Type 1A1 plug-in-unit had a vertical sensitivity to .005 volts per centimeter. In order to pick up the low-level strain signals Channel 1 was used as a wide band AC-coupled 10X preamplifier for Channel 2 by connecting Channel 1 and Channel 2 in cascade with coaxial cable meant for this purpose. The resulting output noise level was reduced by inserting a low-pass filter between the Channel 1 signal output and the Channel 2 input. The preamplifiers were mounted in a Tektronix Type 551 dual bean oscilloscope which permitted the Signals from two gage stations to be displayed at the same time with one horizontal sweep of the beams. The sweep speed 28 could be varied from 0.1 microsecond per centimeter to five seconds per centimeter. The input impedance of the oscilloscope was one M0. This value was much greater than the strain gage resistance. Therefore, the effect of the oscilloscope impedance on the output of the potentiometer circuit was negligible. The sc0pe was triggered by means of a delayed trigger signal from the Tektronix Type 555 oscilloscope. A record of the oscilloscope trace was obtained by using a Tektronix camera system type C-12 with Polaroid Land film type 47. Using an open shutter at a setting of f1.9 and an oscilloscope sweep speed of 5 microseconds/ centimeter a clear picture of each trace was recorded. 2.2 SPECIMEN GEOMETRY AND PREPARATION A. Material Properties The specimen material was Type 2219 aluminum 1/8 inch (3.2 mm) thick furnished by NASA-Lewis. The specimen was oriented so that the rolling direction was parallel to the loading direction. The stress-strain curve from an ASTM specimen with the same orientation is shown in Figure 2.7. This curve was obtained using an Instron test machine and foil gages on the specimen. Another specimen was instrumented with foil gages in the longitudinal and transverse directions to obtain Poisson's Ratio. From these data, the elastic properties are determined to be: Stress - MPa 29 400 l 300 q 200 ‘ 100 ‘ I 1 I I I r j 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Strain-Percent Figure 2.7. Stress-strain of type 2219 aluminum. 3O Elastic Modulus 70 1 1G Pa Poisson's Ratio 0.33 i 0.01 B. Specimen Geometry A schematic of the specimen is given in Figure 2.8. The dimensions of the plate Specimen were determined by three considerations. The maximum thickness was governed by the desire to reduce geometric disperson to negligible proportionsiknrthe frequencies contained in the stress pulse. The lateral dimensions are chosen to produce reasonable levels of strain relative to electrical noise and to yield an essentially plane wave front for this type of excitation away from the impact point. Furthermore, the intention to compare results with previous static measurements of crack behaviors forced the selection of Specimen with dimensions 0.32 cm x 7.62 cm x 20.32 cm. C. Specimen Preparation A slot nominally 12.5 mm long was electromachined in the plate Specimen. The thickness of the Slot was measured at five positions along the slot - at the ends (x1 and x5), the middle (x3) and the quarter-points (x2 - x4) as indicated in Figure 2.8. The data on the slot geometry are measured to be tl = .314 mm, tl .307 mm, t| = .313 mm, 1 X2 X3 .320 mm fl II .290 mm, tlx s 3l pk 2a L 2b ssaq ----- ---------------------------------------.P--J!- Clamped portion Basic specimen Narrow speciman 2a = 1.27 cm 2a = 1.27 cm 2b = 7.62 cm 2b = 2.34 cm L = 5.08 cm L = 5.08 cm ——""—— 3“-““-~___ .L.—'T Figure 2.8. Schematic of the specimens. 32 The preparation and maintenance of the specimen surface is critical for generating useable fringe patterns since the patterns are degraded by stray reflections surrounding the indentations. A flat surface was attained on the Specimen by first sanding with 320, 400, and 600 grit metallurgical paper and then polished with 1 micron and finally 0.3 micron alumina paste following standard metallurgical procedures. The reflective indentations were applied to the Specimen surface with a Vicker's hardness tester using the 100 p weight. The crosshairs in the x-y micrometer stage of the microhardness tester allow placement of the indenta- tion in the desired locations to within : 2 microns if care is taken. 2.3 DISPLACEMENT MEASUREMENT A. Basics of the Interferometric Displacement Gages Shallow reflective indentations are pressed into the polished surface of the Specimen on either Side of a crack or slot as shown in Figure 2.9. When coherent light impinges upon the indentations, it is diffracted back at an angle (do) with respect to the incident beam shown shcematically in Figure 2.9. Since the indentations are placed close together, the respective diffracted beams overlap, resulting in interference fringe patterns on either side of the incident laser beam. 33 Laser Left Hand Right Hand Pattern Pattern a c a ‘3 t: Indentations Figure 2.9. Schematic of the IDG. 34 In observing the fringe pattern from a fixed posi- tion at the angle do, fringe movement occurs as the dis— tance (d) between the indentations changes. Application of a tensile load, causing the distance between the in- dentations to increase, results in positive fringe motion towards the incident beam. Conversely, the removal of the tensile load results in negative fringe motion away from the incident beam. The intensity, I, of the fringe pattern is given by [50]. [51] I = I 51“ 8 coszg (2.10) 0 2 B where B = %fl (sin do - sin a) 0 and In these equations, I0 is the maximum intensity at the center of the pattern, W is the width of an indentation side, d is the spacing between indentations, A0 is the wave length of light, and a and do are defined in Figure 2.9. In Equation (2.10) the c052: term is modulated by the more slowly varying sinZB/B2 term since W << d. If one now fixes the observation position at do and monitors the intensity changes as d changes, Equation (2.10) becomes 35 .. 21:9.- I — Iocos (A0 Sln do ) (2.11) The intensity has a minimum whenever sin a0 = (m + g), m = 0, :1, :2, :3,... A0 Using this as a basis one can then write the re- lationship between the indentation spacing and the fringe order shown schematically in Figure 2.7 as d Sln do = mko (2.12) Here m is the fringe order, and a the angle 0 between the incident and reflected beams, thus defining the zeroth fringe order. The relationship between the change of indentation spacing (6d) and the change in fringe order (6m) is given by 6mlo Sin a 0 5d = (2.13) It is this relation that serves as the basis of the IDG. Fringe motion can be caused by rigid body motion as well as relative displacements. When the specimen moves parallel to its surface and along a line between the in- dentations, one fringe pattern moves toward the incident beam, and one moves away. Therefore, averaging the fringe motions eliminates the rigid body motion, and one should calculate the displacement from: 36 A 6m + 6m _ 0 l ‘2 5d * 3373‘ 7 > (”4) This component of rigid-body motion is present in every ordinary system for loading Specimens, so that it is very important that it be averaged out. Other rigid-body motions (e.g., one perpendicular to the specimen surface) are not averaged out and can lead to errors. In carefully aligned testing machine this components of rigid-body motion can be made small, eliminating the need for corrections. Using typical values of A0 = 0.6328 microns (He-Ne laser) and do = 42°, the calibration constant AO/sin a0 is 0.95 microns. In other words, when one complete fringe shift has been observed, the corresponding displacement is about one micron. The gage consists of two reflecting indentations. These are applied with a Viker's harness tester which, with its diamond indentor, permits the accurate applica- tion of high quality pyramidal indentations. These in- dentations are typically 25 microns long on each side and can be placed as close as 25 microns to the edge of a crack. The sensitivity of the displacement measurement is determined by the quantity lo/sin a in Equation (2.14) 0 and by the capability of measuring fractions of a fringe shift. For large displacements the measuring of one-half of a fringe shift is sufficiently sensitive since this refers to a displacement half a micron. 37 A very small 6m can be resolved if one has a record of the absolute intensity variation at a position on the pattern about one-half the distance between a maximum and minimum. B. The IDG for Dynamic Displacement The technique for dynamic interferometric displace- ment measurement described here is essentially similar to that used previously by Sharpe [52]. A schematic of the setup for the dynamic displacement measurement is given in Figure 2.1, and photographs are shown in Figures 2.10 and 2.11. Convenience of the arrangement necessitated the use of a mirror to direct the laser beam onto the indenta- tions; it is important that this mirror not only has a high reflectivity but also be flat in order to preserve the coherence of the laser beam. A dielectric mirror with a reflectivity of 99.7 percent that is flat to 1/10 wave length was used. The three point adjusting feature on the mirror guarantees the easy adjustment of the light field on the indentations. With this technique one simply monitors the fringe shift of each pattern by a photomulitplier tube whose window is covered except for a thin slit with a width less than the width of a fringe in a pattern. As the fringes pass over this slit, the intensity seen by the photo- multipler tube varies and, in the ideal case, the output signal is a sine wave. Adjusting the fringe pattern so 38 .dsuwm Hmucwfifluwmxw may mo 3ofl> Hmuocwu .OH.N engage V 'muw lllllllu. slllllll ‘llllllll ‘lllllll' ll‘lilli. “5 i. 39 Photograph of the displacement measuring instrument. Figure 2.11. 40 that the static signal from the photomultiplier tube is midway between maximum and minimum and observing the posi- tion of the nearest bright or dark fringe relative to the slit, permits one to determine whether the displacement signal corresponds to a relative separation or closing of the indentations. C. Relationship Between Displacement and Stress Intensity Factor The mode I stress intensity factor, k is the 1' linear elastic fracture mechanics parameter which ade- quately characterizes crack behavior near the tip in many materials. With reference to Figure 2.12 the y-direction displacement (Uy) associated with the opening mode crack tip stress field is given by [53] U = £1. y G L _ (%;)2 sin % [2 - 20 - cos2 %] (2.15) where G = shear modulus v = Poisson's ratio r,0 = Polar coordinates referred to the crack tip cl ll 0, for plane strain = v/(l + v) for plane stress For the present case where plane stress conditions pre- vail the crack opening (U3) near the tip may be given as l 2 2 0] 1+0 2 (2.16) N|CD ._ Ll:- U ( fl) Sln 41 Th ‘n Figure 2.12. Notations used in measuring crack profile. 42 Examining Equation (2.16) it is obvious that with the knowledge of elastic constants G and v, and care— ful measurements of r,8, and the vertical separation of crack surfaces near the tip one can determine the mode I stress intensity factor. In fact, such measurements have been made with great success by many researchers. Adams [54] used a photographic technique to measure the relative displacement of identifiable surface markings astride a fatigue crack and only 76 microns apart. His technique, though somewhat laborious, could be used to ob- tain k. Elber [55] developed an accurate clip gage with gage length of 1.27 mm that could be used for k-calibra- tions of larger specimens. Dudderar and Gorman [56] made k measurements in thin PMMA sheet specimens using holo- graphic interferometry tomeasure the displacement per- pendicular to the sheet. Their technique is applicable only to thin transparent models. Sommer [57] and Crosely et a1. [58] measured the crack displacements inside glass models by optical interferometry and used them to deter- mine k. The advantage of interior displacement measure- ments near the crack tip is offset by the transparency requirement of the specimen. Evans and Luxmore [59] used the laser speckle method to measure inplane displacements on the specimen surface. A brittle plastic specimen was used, but the technique works on metals as well. Sharpe and Grandt [60, 61], Macha, Sharpe and Grandt [62], 43 Sharpe [63], [64] used a laser interferometric technique to measure crack surface displacements. This technique which is adapted for the present research program can be used to measure displacements of 0.02 microns at 50 microns from the crack tip. Further details of the technique are given in section B. The hitch to the measurement of the stress in- tensity factor is the requirement of a technique capable of measuring small displacements very near the crack tip. As a rule of thumb the displacement relations given in Equation (2.16) are regarded to be sufficiently accurate within a distance a/lO from the crack tip, where "a" is half the crack length [65]. D. Displacement Measuring and Recording System The photomultiplier tubes used were Amperex Type XP-lll7 with a divider circuit load resistance of 1k and an operating voltage of 1500V. The output from each photo- multiplier tube was fed into a Tektronix Type 53/54 D plug-in preamplifier. These are high gain differential amplifiers with a vertical sensitivity of l mv and a frequency response from .35 to 2MC. The two plug-in preamplifiers were mounted in a Tektronix Type 555 dual beam oscilloscope. The output from each photomultiplier tube was thus recorded simultaneously on separate channels of the oscilloscope. The sweep speed of the oscilloscope could be varied from 44 0.1 microseconds/cm to five seconds/cm. The sweep was triggered by a differentiated signal from a foil strain gage mountedcnithe tup about 0.64 cm from the impacting interface. A schematic of the triggering circuit is shown in Figure 2.13 [66]. A Tektronix Type 0 operational amplifier unit was used in the differentiator circuit. The unit consisted of three parts: a vertical preamplifier and two Operational amplifiers. The vertical amplifier was such that it can be used either as an independent oscilloscope preamplifier or to monitor the output of either of the operational amplifiers. The Operational amplifiers could be used for applications involving integration, differentiation as well as many others. Besides having provisions for select- ing input and feedback impedances, Ci and R from f: several values, the Type 0 unit had an internal circuitry to limit high frequency reSponse. A record of the oscilloscope trace was obtained by using a Tektronix camera system type C-12 with Polaroid Land film, type 47. Using an open shutter at a setting of f1.9 and an oscilloscope sweep rate of 5 microseconds/cm a clear picture of each trace was recorded. 2.4 PROCEDURES A. Preliminary Checks In order to check the applicability of the plane stress assumption, we can consider simple normal impact 45 '——’\/\f/\/\—i [”9 'F 7 JO 4. U I () .|||}_ Figure 2.13. Strain signal trigger circuit. 46 of a free-ended rectangular plate. A schematic of an instrumented Specimen prior to machining the slot is shown in Figure 2.14a. The consiStance of the plane stress assumption is implied by the magnitudes of the response of gages 2 and 12 in Figure 2.15. Also, Figure 2.15b shows that there is little bending; note that the gages are separated by one inch in the longitudinal direction. The signals in Figure 2.15 show a longer rise-time than the later results as they were taken during development of the impact apparatus. Dispersion, induced by finiteness of geometry, will alter the rise time and amplitude of a propagating step wave. The desire to generate a tensile pulse with a short rise time necessitated the use of a flat ended impactor for the present investigation. A flat ended impactor produces a pulse with a short rise time so that the shorter wave length components are more significant and therefore, according to the Pockhammer-Chree theory as discussed by Davies [67] , the dispersive effects are greater. This phenomena poses a problem. Due to the presence of a slot in the specimen one cannot measure the input pulse at the slot. On the other hand one can measure the pulse at a location one inch before the slot and assume dispersion to be negligible. The validity of this assumption is implied by the response of gages 2 and 12, which cover the region of interest, as shown in 47 mcficflsomfi umuwm paw mHOMon :oEflommm on» co mcoflumooH mmmm Samuum m 7IIIIi\IlI)\lllLll!!!)IlllllI))\ll. .u: L +- +- EU vm.m EU SN.H ED vm.m EU h~.H EU N®.m hN.H vm.m vm.m hN.H .uon are M EU 50 .eH.m manage 2.. Ln 48 I fl V U I I I V U P I 0.5 mv Gage 2 Gage 12 0.5 mv Gage ll Gage 9 Figure 2.15. Observed strain response at various locations of the specimen without the slot. 49 Figure 2.14a. Note that even though the change in amplitude and rise time are negligible the oscillations in the pulse are indicative of some dispersive effects. An air-cushion between the impactor and the tup would increase the strain pulse rise-time in the specimen. In order to eliminate the air-cushion, the tensile impact apparatus was designed such that a vacuum could be drawn in the launch tube. Great care was taken in sealing the launch tube. Even then leakage was unavoidable. Tests with slightly reduced launch tube pressures showed no improvement of the strain pulse rise time. The attempt to draw a vacuum in the launch tube was then discontinued. B. Eccentricity of Loading Since the primary objective of the present in— vestigation was to study the response of the interaction of a longitudinal wave of normal incidence with a crack, particular care was taken to reduce the eccentricity in the apparatus. Further provision was made to check the planarity of the pulse during the actual test by observing if gages such as one and three in Figure 2.14b respond simultaneously to the wave front. It was then found that the axiality problem was worse than expected - a few per- cent of all impacts being axial to within two microseconds. Another problem observed during the course of testing was an occasional vertical non axial impact. This, in most cases was a result of bolts loosening in the tup or the projectile assembly. C. 50 Testing Procedure The potentiometric strain gage circuits, the amplifiers, the oscilloscopes, the laser and photomultiplier power supplies were turned on at least one—half hour before a test was performed in order to obtain nearly steady state operating conditions. 2) 3) The procedure adopted in testing was as follows: The oscilloscopes with plug-in preamplifiers were calibrated for amplitude by means of the square-wave calibrator output incorporated in the oscilloscope. The sweep rate of the oscilloscope was calibrated with one, five and fifty microseconds timing marks from a time mark generator. These calibrations were per- formed against a graticule-scale over the face of the cathode ray tube. The test specimen was carefully mated with the tup assembly. The bolts in the clamping brackets were tightened to secure the test specimen which is now sandwiched between the two brass bars of the tup assembly. The entire assembly was then mounted on the end of the launch tube and the projectile assembly pulled back and aligned against the accelerating assembly. The incident laser beam was aligned so that the in- dentations are illuminated at the beginning of load- ing and throughout the test. This is done using the 4) 5) 6) 7) 8) 51 adjusting screws on the mirror directing the beam. The specimen reflects a portion of the incident beam on the laser head. In order to insure the per- pendicularity of the incident beam the reflection on the laser head was used as a guide in adjusting the laser position. The photomultiplier tubes were positioned so that the fringe patterns fall on the windows. This was accomplished using the travelling stages which angularly orient the photomultiplier tubes. Further adjustments were made by rotating the windows of the photomultiplier tubes so that the fringes are parallel to the slits. The oscilloscopes were set on single sweep lockout so that the entire test was recorded on one sweep. This required a sweep rate of five microseconds/cm. The sweep for the fringe motion record was triggered by a signal from a strain gage mounted about a quarter of an inch behind the impacting interface. The type 551 oscilloscope was triggered using a delay triggering signal from the Type 555. The vertical gain adjustments of the amplifiers were set as desired in accordance with the particular test. A set pressure was applied to the forward chamber of the Hyge unit and locked in. The camera shutters were opened and held open until after the Hyge unit fired. 9) 10) ll) 12) 13) 52 The Hyge load pressure was gradually increased until firing. The load pressure was released and the Hyge piston was retracted to be ready for the next test. The potentiometric strain gage circuit voltage was read and recorded. The projectile assembly was pulled back and aligned for the next test. The procedure 3 through 12 was repeated for the three sets of indentations. CHAPTER III ANALYSIS OF CRACK RESPONSE TO LONGITUDINAL PLANE WAVES OF NORMAL INCIDENCE 3.1 ANALYTICAL CONSIDERATIONS A. Equations of Elasticity in Two Dimensions Consider the problem depicted in Figure 3.1. The plate which is assumed isotropic linear elastic has the width 2b, where b >> h. The displacement components are U , U x y’ tively. Since the plate is thin and free from stresses and U2 in the x, y and 2 directions respec- at surface 2 = :h, we assume 0 = 1 = T = 0 (3.1) In the absence of body force, integrating the three-dimensional stress equation of motion across the thickness of the plate, and ignoring transverse inertia as well as the integrations across the thickness of the plate of the transverse shear stresses, give 2 Bo 8T 3 U _l‘. __XY= X 8x + By D 2 at 2 (3.2) 8T 80 3 U xy + x = y -——‘ ——— 0 3x 3y at2 54 -'\:<— Figure 3.1. Geometry for generalized plane stress. 55 In these equations the field variables represent averages across the thickness of the plate, and p is the mass density. A Similar averaging process gives the average field variables for the stress-strain relations and strain-displacement relations as Cij = Aekkéij + Zneij 1:3 = 1:2:3 (3.3) = L 81] 2mid + Ujpl) where A and n are the Lame's constants and 6ij is the Kronecker delta. From Hooke's Law it follows that 633 is related to all + 622 by E =—4_U (34) 33 A + 2U k,k ‘ Substitution of (3.4) into the expression for the stress yields _ ZuA OaB - A + 2p UkykdaB + uwouB + U8.a) (3.5) (118 = 112 which may be expanded as 2 BUX 3U oX — p Cp[§—— + v 3y ] C2 3U aux O'y - O Cp[_13+ \) W] (3.6) 3U SD = 2 —X _Y. Txy p Cslay + 3x I 56 where, C2 = 4U(X + U) P 0(K + 2h) 2 _ _ . , . Cs — u/o v — POisson 3 ratio By substituting Equation (3.6) into Equation (3.2) the displacement equations of motion for a thin plate are reduced to (c2 - c2)vv-E + c2 26 = E (3.7) p s s where U is the displacement vector having two com- ponents, UX and Uy in x and y directions, respec- tively. The symbol V represents the gradient operator in two dimensions. The dot over the displacement U denotes the time derivative, and V2 is the two—dimensional Laplacian Operator. Let 5 be a unit vector normal to the median plane of the plate. By introducing a scalar displacement potential ¢ and a vector displacement potential w = Em and taking 6 = v¢ + VXE (3.8) which may be expanded as (3.9) >4 0) X 2:): a: 57 Equation (3.7) can be reduced to two scalar wave equations G "St II B. (3.10) (3.11) o U) 33 H Bus 0 . 81¢ O o -v-l U) c 4 0 ° 0) S 0 E U 0 W4 0 O 0 I ' ‘ ' U ' ' ‘ - . -l.0 -0.5 0 Figure 4.9. Distance from crack tip x/a Crack profile at three different time steps. 92 3. Q _“ 2d 73 (: >§ m E C \U ' Theory fi Reference E E: [72] H x 1 n ,_q .M 0 1 T 1 I t W 1 1 W 0.0 O 2 0.4 0.6 0.8 Figure 4.10. a/b Comparison of experimental dynamic k— calibration data with theoretical static results. 93 theoretical k-calibration for the elastostatic case is shown in Figure 4.10. The comparatively long rise time for the narrow specimen is likely to influence the magnitude of kl(t)max.° 4.3 CORRELATION WITH ANALYSIS The normalized mode I stress intensity factor, kl(t), for various times after the arrival Of the wave front at the crack, was determined using Equation 3.21. It should be noted that in Equation 3.21, f(0) = 0, there- fore t ~ _ 3f(t') ~ _ . . kl(t) _ Jo at' skl(t t )dt In evaluating the foregoing integral, the experi- mental input pulse at location 2 was approximated by a ramp function time-varying force with a rise-time tR = 4.2 microseconds as shown in Figure 4.11. The step response, Skl(t)' in Reference [16] was approximated by a ninth-order polynomial (least square fit). Also used in the evaluation was Cp = 200,000 in./sec., which was ob- tained from the experiment. The appropriate integration was then carried out by a simple numerical technique. The result of the comparison is Shown in Figure 4.12. The experimental result here represents the re- sponse during the time interval before the arrival of the reflected scattered waves from the nearest boundary. 94 F(t) Time .(microseconds) Figure 4.11. Input pulse at location 2; experi- mental points and ramp function approximation. 95 Analytical o' [Thau & Lu] o' Time (microseconds) Figure 4.12. Comparison Of analytical and experimental results for the dimensionless mode I stress intensity factor. 96 Also note that the theoretical result is shifted to omit the premature experimental response due to disturbance leading the pulse front. As can be seen from Figure 4.12 the analytical result based on the work of Thau and Lu appear to do a good job Of predicting the quantitative behavior of the experimental result. The deviations may be attributed to a number of reasons: 1. The ramp function pulse used in the analytical computa- tion is an approximation Of the actual pulse. 2. The states of stress and deformation are not truly two dimensional as assumed in the analytical com- putations. 3. The existence of a slight curvature to the wave front might add to the discrepancy. 4. Errors may have arisen in the digitizing and curve- fitting process. 4.4 COMPARISON WITH FINITE ELEMENT King et a1. [20], have reported finite element analyses Of similar elastodynamic crack problems. Their analysis was carried out using a singularity element capable of mixed mode deformation and by simulating the incident wave by applying a step function time varying finite element equivalent of uniform stress at one end of the finite element model. 97 Professors King, Anderson, and Shreeves of Georgia Tech provided an analysis Of this experiment. The finite element model they used is depicted in Figure 4.13. The model is a uniform grid model composed of 127 nodes, 200 ramp function time varying strain triangles, and one eight-node crack tip singularity-element. Constraints against horizontal displacement on the left edge Of the model are dictated by symmetry of the problem with respect to a vertical axis through the center Of the crack. A plane wave is induced by uniform loading on the upper edge of the model. In an effort to represent the behavior of the experimental model, the dimensions and the material properties for the finite element model were selected to correspond to that of the experimental model. The experi— mental pulse was approximated by a ramp function time varying force with a rise-time of five microseconds. The details of the finite element analysis may be found in Reference [20]. Figures 4.14 and 4.15 show comparisons of the finite element approximation with experimental results for the normalized mode I stress intensity factor and the nor- malized crack displacement at the center respectively. The finite element approximation and the experimental results show fairly good agreement for the short time results. It must be noted here that the short time experimental results have been corrected for the influence of the disturbance leading the wave front. 98 if) i k“ a._1 crack tip Figure 4.13. Finite element model. 99 Experimental (t) Static A._.’° \ Finite \\ element Figure 4.14. P1 I2 I u 1 1 1 j 10 20 30 Time (microseconds) Comparison Of finite element and experimental results for the dimensionless mode I stress intensity factor. 7d 7 6‘ O r-l X Hm 95‘ \ E I ié E4“ C \ >‘ :3 Normalized Crack Displacement, 100 Experimental Finite Element \ \\\ Static Figure 4.15. 10 20 30 Time (microseconds) Comparison of finite element and experimental results for the normalized displacement at the center. 101 Although the qualitative behavior of the long time results seem to be similar, as evidenced by the curve shapes in Figures 4.14 and 4.15, the experimental results are not strictly comparable. The long—time experimental response is the sum effect Of the "ramp" forcing function and the reflection from the lateral boundaries of the diffracted waves resulting from the impingement on the crack Of the disturbance leading the wave front. This somewhat exaggerates the quantitative values of the long time experimental results. Also causing quantitative deviations may be the following reasons: 1. The ramp function time varying force used in the numerical computations is an approximation of the actual pulse. . 2. The existence Of a slight curvature to the wave front might add to the discrepancy. Even though the quantitative values Of the experi- mental curves may be somewhat exaggerated for reasons mentioned above, the finite element approximations are felt to under estimate the maximum stress intensity factor. CHAPTER V SUMMARY AND CONCLUSIONS An apparatus that generates a plane tensile pulse in a plate with finite dimensions was developed. The pulse had an approximate shape Of a ramp function with a rise—time of about five micro-seconds and a undisturbed flat top of about 30 micro-seconds. This loading apparatus was used in conjunction with the interferometric displace- ment gage (IDG) technique to study the dynamic response Of a central crack in a finite plate subjected to longitudinal waves. The experiments were conducted for two different widths Of the specimens. Three different sites along the faces Of the crack were used to measure the displacements. Displacement-time curves were plotted for each site and points were taken from these curves to plot the crack profile for various time steps. Further- more, the displacement-time result for the crack tip was used to compute the mode I dynamic stress intensity factor, k1(t). The tensile impact apparatus generated a main pulse, which approximates a ramp function, and a precursor pulse. The precursor pulse may have been caused by imper- fections of the impacting interfaces. The amplitude and 102 103 rise-time Of the main pulse were fairly repeatable. Con- sidering the difficulty of constructing such an apparatus, the financial and facility constraints at the time, the loading mechanism has proven to be satisfactory for the present investigation. Furthermore, the new apparatus has advantages over other techniques due to its simplicity, versatility, safety and low cost. The interferometric displacement gage technique was successfully used in measuring dynamic crack displace- ments. Besides being easy to use the technique also proved to have the following definite advantages: satisfactory resolution, ease of specimen preparation and convenience of data recording. The displacement-time and the normalized mode I dynamic stress intensity factor, Rl(t), curves were found to oscillate, a phenomena attributed to the cancellation and reinforcement of the incident waves by various scattered waves. It was also found that the dynamic over- shoot Of El(t) is 170% in contrast to Chen's numerical solution of 175% which corresponds to a Heaviside step function loading of a bar with Poisson's ratio of 0.3 and a/b = 0.24. The analytical computations of k1(t), were carried out by employing the elastodynamic counterpart of the theory Of generalized plane stress and the principle of superposition in time. The experimental results show good 104 agreement with analytical computations based on the results of Thau and Lu. The finite element solutions provided by King et a1. [73] compare favorably with the Short time experi- mental results, i.e., with results for a time regime corresponding to the time it takes for the first scattered longitudinal wave to travel from a crack tip to the nearest boundary and back. The comparison Of the long time results is not good. The descrepancy may be attributed to: the imperfect experimental wave front the influence of which could not be isolated for this time regime, and some ambiguity in fringe interpretation. The dynamic stress intensity factor, in general, depends on the shape of the pulse, the wave length Of the pulse, the crack length, the Specimen geometry, and the Poisson's ratio Of the material. In the present study emphasis was put on a particular ramp function type Of pulse and a particular rectangular geometry of Specimen. The development of an experimental technique for generating pulses Of varying pulse shapes appears to be very desirable. Studies Of the diffraction Of a plane pulse by multiple normal cracks, by a single Oblique crack, or by a combina- tion of oblique and normal cracks appear to be a fruitful area for future investigators. LIST OF REFERENCES 10. LIST OF REFERENCES Griffith, A.A., "The Phenomena of Rupture and Flow in Solids", Phil. Trans. Roy. Soc. Lond. 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