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Influence of material properties of SIFs determined by frozen stress
PERGAMON
Engineering Fracture Mechanics 61 (1998) 555±568
In¯uence of material properties of SIFs determined by frozen stress C.W. Smith a, *, E.F. Finlayson a, C.T. Liu b a
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA b Phillips Laboratory, Edwards Air Force Base, CA 93524-7680, USA Received 28 April 1998; received in revised form 6 August 1998; accepted 10 August 1998
Abstract In a prior study utilizing two materials for which thermal and mechanical properties of both materials in bimaterial specimens were matched except for modulus, it was found that modulus mismatch had a negligible eect on the stress intensity factor (SIF) level and distribution for cracks in bond lines of stress frozen photoelastic bimaterial specimens. However, bond line residual stresses increased the SIF values. Since it is almost impossible to obtain commercial stress freezing materials with matching critical temperatures (Tc) with dierent moduli, the present study of cracks parallel to and within bond lines utilized such materials with Tc mismatch and compared results with parallel studies with matched Tc values in order to evaluate Tc mismatch eects. Results show that Tc mismatch creates mixed mode conditions in cracks parallel to the bond line with signi®cant increases in the mode I SIF (K1) for cracks very near the bond line but shows little eect for cracks within or away from the bond line. The absence of a modulus mismatch eect is again con®rmed as well. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Stress intensity factor; Bond line cracks; Bimaterial specimens; Frozen stress photoelasticity
1. Introduction Beginning in 1993, the authors undertook an experimental program directed towards an evaluation of the frozen stress photoelastic method for determining the stress intensity factor (SIF) distribution along cracks located near to or within bond lines joining dierent incompressible materials with substantial thickness. The objective was to approximately * Corresponding author. Tel.: 001 540 231 6159; Fax: 001 540 231 4574. 0013-7944/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 8 ) 0 0 0 7 0 - 8
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simulate such distributions for cracks within or near the bond line between rocket motors and their liners. It was expected that there would be residual stress resulting from the bonding procedure. Moreover, it had been shown analytically [1] that when plane strain exists where incompressible materials have been joined, then the interface fracture equations reduce to the classical homogeneous form, thus allowing separation of the mode I and mode II stress intensity factors. It was felt that such would be approximately the case for rocket motor material in a state of generalized plane strain. 2. Experimental procedures In order to separate residual stress eects from modulus mismatch, a three specimen test procedure was employed: (1) an edge cracked homogeneous control specimen; (2) a bonded homogeneous specimen containing an edge crack; and (3) a bimaterial edge cracked specimen. Two types of bimaterial specimens were employed. Their material properties are given in Tables 1±2. Specimens A included homogeneous Araldite specimens and homogeneous bonded Araldite and specimens B included homogeneous PLM4B specimens and homogeneous bonded PLM4B specimens. Ideally, the two materials employed in the bimaterial specimens should have identical properties except for the modulus in order to assess modulus mismatch eects. As indicated in Table 1, it was not possible to fully match all material properties except the modulus. In bimaterial specimens A, the thermal coecients diered by 17% at critical temperature, and in bimaterial specimens B, the critical temperatures diered by about 13%. However the former dierence appeared to have no eect upon the results. Table 1 Material Properties Model A Model A
Tcritical
EHot
fr
Araldite Aral-Alum1
2408F 2408F
2698 psi (18.60 Mpa) 5349 psi (36.88 Mpa)
1.64 psi in (286.9 Pa m) Ð
1 Created by mixing aluminum powder into araldite.Matched thermal coecients at 688F; a = 15.3 10ÿ6 per 8F. At critical temperature (Tc) thermal coecients were 119 10ÿ6220 10ÿ6 per 8F
Table 2 Material Properties Model B Model B
Tcritical
EHot
fr
PLM4B PSM9
1808F 2058F
1879 psi (12.96 Mpa) 6912 psi (47.67 Mpa)
2.40 psi in (420.3 Pa m) 2.97 psi in (520.1 Pa m)
These materials have the same thermal coecients at room temperature (TR) and critical temperature (TC). They were aRT=39 10ÿ6/8F and aCT=90 10ÿ6/8F, respectively. Bonding agent was liquid PSM9
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Since it was crucial that accurate dimensions be achieved along the crack front it was decided to machine in the cracks with a round tipped Buchler saw. The bond line thickness was kept small but ®nite and constant. All of the near tip test data were collected outside the bond line because the bond line material, generally being dierent from the adherends, produced a local disturbance which we considered part of the near tip nonlinear zone. Test specimen dimensions are shown in Fig. 1. All test specimens were ®rst subjected to a no load stress freezing cycle after which stress freezing under load was carried out. Both through the thickness (tt) and thin slices (L,M,R) (Fig. 1) were analyzed photoelastically (Appendix A) using the algorithms in Appendix B for converting optical data into SIF values. All SIF values were normalized by a control value Kc, an experimentally determined SIF value for a homogeneous edge crack in order to account for possible root radius or other experimental eects, and was within experimental error of $5% of the theoretical 2D K1 value. Most of the results are evaluated by thickness averaging for purposes of comparison. All photos used bright ®elds. Reasonable success [2±4] had been achieved in measuring SIF distributions by employing the A material combination for cracks within the bond line for both mode I and mixed mode conditions. The present study was focused primarily on evaluating the eect of critical temperature (Tc) and modulus mismatch in the B material combination for cracks parallel to and within the bond line.
Fig. 1. Specimen dimensions.
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3. Results and discussion By placing the cracks which were outside the bondline in the higher Tc material (PSM9) in the B specimens, it was felt that the Tc eect would be minimized. The idea was that, since the lower Tc material (PLM4B) would still be rubbery when the PSM9 turned glassy, then the PSM9 modulus ratio of E PLM4B Rubbery/E glassy00.5% should negligibly aect the SIF values in the PSM9. Then, when the PLM4B became glassy, it would only slightly stress the PSM9 and this live stress would be removed upon slicing. For specimens containing cracks parallel to the bond line, the no load tests revealed a near tip residual stress ®eld, [Fig. 2(a)] but the fringe order varied little over the data range, indicating that the residual stresses aected mainly the nonsingular part of the data and only slightly the fringe gradient. Thus it had little eect upon the SIF values. None of the homogeneous bonded specimens using the B material combination revealed any increase in the average SIF above the homogeneous specimen value and no shear mode was observed. However for the bimaterial specimens, a small shear mode, indicated by a clockwise rotation of the fringe loops for which the reading angle yR (i.e. y8m) increased as the crack was moved closer to the bond line, was observed. Moreover, for the crack nearest the bond line, [Fig. 2(b), (c)] the value of K1 increased by 30%, as shown in Fig. 3, but no increase was observed for cracks further from the bond line (i.e. h = 0.25 in, [6.35 mm]; 0.50 in. [12.7 mm]) [4]. Fig. 3 shows data for all three slices across the thickness. Each set of data was extrapolated to the origin and the average result is plotted on the ordinate. In order to clarify whether or not this increase was due to modulus mismatch or Tc mismatch, a test was run with the A material combination which duplicated the above noted geometry (i.e. h = 0.125 in [3.18 mm], Fig. 1). The loaded (tt) fringe pattern is shown in Fig. 4. Data used for analysis were taken above the crack within the indicated data zone. The fringe pattern reveals no fringe loop rotation, and hence no detectable mode II and the value of K1 was the same as from a 2D homogeneous specimen analysis, indicating no increase due to the bond line proximity [Fig. 5(a)]. For fringe loops between the crack and the bondline, there is clearly a distortion imposed on the system by the bondline which acts as a sti, loaded boundary. This is interpreted to mean that a stress gradient results which produces a variable sox in the vertical direction. The simplest modi®cation of sox to account for this eect would be to assume sox to vary linearly in the vertical direction after the fashion of Ref. [5]. This means that the Kap vs Zr/a curve will be a second degree curve in the K1 data zone rather than a linear curve as shown in Appendix B. Using such an approach, one obtains a value of K1 which is within 4% of the value taken above the crack [see Fig. 5(b)]. The linear extrapolation was along the slope of the data curve at Zr/a=0.30. These results con®rm that both the shear mode and the increase in the SIF found in the test shown in Fig. 2 were due to Tc mismatch and not to modulus mismatch. A ®nal test was conducted on the B material with the crack within the bond line. Fringe patterns from the bimaterial test are shown in Fig. 6, indicating the data zone. In interpreting these patterns, it should be noted that even for pure mode I (which is observed here) the fringes will not be symmetric with respect to the crack or the bond line due to the dierence in the material fringe values (Tables 1 and 2) for the two materials. The absence of a detectable shear mode observed for both Figs. 4 and 6 agrees with Ref. [1]. Analysis of the fringe data
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Fig. 2. Fringe patterns; B materials; crack parallel to bondline.
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Fig. 3. Normalized SIF results from Fig. 2.
Fig. 4. Fringe pattern; A material; test geometry of Fig. 2.
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Fig. 5. Normalized SIF results from Fig. 4.
showed no signi®cant dierence between the bimaterial and homogeneous bonded test data, again indicating no bimaterial eect on the SIF values. Fig. 7, which is a closeup of the (tt) loaded Fig. 6 photo, reveals several features. In this test, the bond line adhesive was the same material as the upper material PSM9 and so has the same Tc. Nevertheless, at the upper edge of the bond line, some mismatch of the fringes crossing the upper edge of the bond line still occurs near the crack tip suggesting a slight dierence in the material fringe values between the upper plate material and the bond line material, but not as severe as along the bottom of the bond line where the PSM9 adhesive meets the PLM4B. Data analysis from the slices, L, M, R were again averaged since only 2D comparisons are available. Such an analysis for Fig. 6 is given in Fig. 8. Some of the variation in data from slice to slice
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Fig. 6. Fringe pattern for crack in bondline; B materials.
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Fig. 7. Enlargement of Fig. 6(b).
in the B specimens is conjectured to be due to residual bondline stress variation through the thickness. Typical SIF distributions from an A material combination containing signi®cant residual stresses are compared to their homogeneous bonded counterparts in Fig. 9. These results reveal the signi®cant residual stress eects but clearly show that bimaterial eects are not independently signi®cant and that reasonable SIF distributions can be measured by the method employed. For this case, through thickness averaging of the slice results showed that the homogeneous K1 value increased 16% in the bimaterial specimen and 19% in the homogeneous bonded specimen. It is interesting to note that, when the crack is placed in the bond line for the B material combination, no shear mode is detectable which suggests that if one is only interested in problems for cracks within the bond line, then the Tc mismatch does not appear to have a signi®cant eect on K1 for incompressible materials under generalized plane strain. 4. Summary Using a three specimen test procedure, a convenient incompressible bi-material combination (B) was evaluated for predicting SIF levels and distributions in cracks parallel to and within bimaterial specimens. By comparing results with those from a previously used material combination (A) it was shown that Tc mismatch in the B combination induced both a shear
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Fig. 8. Normalized SIF results from Fig. 6 slices.
mode and an elevation in SIF level for cracks very near the bond line. However, for cracks away from the bond line or within the bond line, the B material combination yielded reasonable mode I results. It is important to note the fact that modulus mismatch, shown earlier to have no eect on cracks within the bond line of A materials was also con®rmed here for cracks parallel to the bond line as well as within the bond line. All of the above observations are limited to the indicated mismatch values of Tc and the modulus.
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Fig. 9. Normalized A material SIF distribution.
Acknowledgements The authors wish to acknowledge the support of Phillips Laboratory through Hughes STX Corp. under subcontracts 96-7055-LI467 and the sta and facilities of the Department of Engineering Science and Mechanics at VPI & SU. Appendix A A.0.1. Frozen stress photoelasticity When a transparent model is placed in a circularly polarized monochromatic light ®eld, and loaded, dark fringes will appear which are proportional to the applied load. These fringes are called stress fringes or isochromatics and the magnitude of the maximum in-plane shear stress is a constant along a given fringe. Some transparent materials exhibit mechanical diphase characteristics above a certain temperature, called the critical temperature (Tc). The material, while still perfectly elastic will exhibit a fringe sensitivity of about 20 times the value obtained at room temperature and it's modulus of elasticity will be reduced to about one six hundredth of its room temperature value. By raising the model temperature above Tc, loading, and then cooling slowly to room temperature, the stress fringes associated with Tc will be retained when the material is returned
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to room temperature. Since the material is so much more sensitive to fringe generation above Tc than at room temperature, fringe recovery at room temperature upon unloading is negligible. The model may then be sliced without disturbing the ``frozen in'' fringe pattern and each slice analyzed as a two-dimensional model, but containing the three-dimensional eects. In the use of the method to make measurements near crack tips, due to the need to reduce loads above critical temperature to preclude large focal deformations, and the use of thin slices, few stress fringes are available by standard procedures. To overcome this obstacle, a re®ned polariscope is employed to allow the tandem use of the Post [6] and Tardy [7] methods to increase the number of fringes available locally. In fringe photographs, integral fringes are dark in a dark ®eld and bright in a bright ®eld. Bright ®elds are used through-out this paper for clarity.
Appendix B B.0.1. Mode I algorithm Beginning with the Grith±Irwin Eqs. we may write, for mode I, for the homogeneous case, sij
K1 2pr
1 2
fij y sij
i; j n; z
B:1
where sij are components of stress; K1 is SIF; r, y are measured from the crack tip (Fig. 10); s8ij are non-singular stress components. Then, along y = p/2, after truncating sij tnz max
K1 8pr
1 2
r
KAP 1
8pr2
where t8 = f(s8ij) and is constant over the data range KAP=apparent SIF.
Fig. 10. Mode I near-tip notation.
B:2
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567
Fig. 11. Middle slice data.
(tnz)max=maximum shear stress in the nz plane p 1 KAP K1 8t r 2 1 1 a s 2 2 s pa s pa 1=2 vs where (Fig. 10) a = crack length, and s=remote normal stress i.e. KAP/s(pa) linear.
B:3 p r=a is
Since, from the stress±optic law (tnz)max=nf/2t where n = stress fringe order; f = material fringe value; t = specimen (or slice) thickness and from Eq. (2) KAP=(tnz)max(8pr)12=nf/ 2t(8pr)12, then KAP (through a measure of n) and r becomes the measured quantity from the stress fringe pattern at dierent points in the pattern. Use of a linear variation of sox over the data zone is shown in Fig. 5(b). B.0.2. Mixed mode algorithm The mixed mode algorithm was developed (see Fig. 12(a) and (b)) by requiring that ( ) max 1=2 d tnz lim 8prm K1 ; K2 ; rm ; Ym ; tij 0 rm !0 dY Ym !Y m
Fig. 12. (a) and (b). Determination of y8m.
B:4
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which leads to 2 K2 4 K2 1 cot 2Ym ÿ 0 ÿ ÿÿ : ÿ K1 3 K1 3
B:5
By measuring Y8m which is approximately in the direction of the applied load, K2/K1 can be determined. Then writing the stress optic law as tmax nz
fn K AP 1 ; 2t 8pr2
p one may plot K *AP/s(pa)12 vs r=a as before; locate a linear zone and extrapolate to r = 0 to obtain K . Knowing, K , K2/K1 and Y8,m values of K1 and K2 may be determined since hÿ 2 ÿ 2 i12 ÿ ÿÿ: B:6 K K1 sin Y m 2K2 cos Y m K2 sin Y m Knowing K* and Y8m, K1 and K2 can be determined from Eqs. (5) and (6). Detailspare found in Ref. [8]. The crack tip linear zones in the graphs such as Fig. 11 were between r=a=0.30 and 0.60. References [1] Hutchinson JW, Suo H. Mixed mode cracking in layered materials. In: Advances in applied mechanics. Vol. 29. New York: Academic Press, 1992. p. 63±91. [2] Smith CW, Finlayson EE, Liu CT. Preliminary studies of three dimensional eects on arti®cial cracks at simulated rocket motor liners by the frozen stress method. In: Proc. VIII International Congress on Experimental Mechanics, Society for Experimental Mechanics, Bethel, Ct 1996. p. 55±61. [3] Smith CW, Finlayson EF, Liu CT. A method for evaluating stress intensity distribution for cracks in rocket motor bond lines. Journal of Engineering Fracture Mechanics 1997;58 (12):97±105. [4] Smith CW, Finlayson EF, Liu CT. Bond line residual stress eects on cracks parallel to the bond lines in rubberlike materials. In: Recent advances in solids/structures and application of metallic materials. 369. ASME PVP, 1997. p. 1±10. [5] Smith CW, Weirsma SJ. Stress fringe signatures for propagating cracks. Journal of Engineering Fracture Mechanics 1986;23(1):229±36. [6] Post D. Fringe multiplication in three dimensional photoelasticity. Journal of Strain Analysis 1966;1(5):380±8. [7] Tardy MLN. Methode Practique Examen de Measure de la Birefringence des Verres D'optique. Optics Reviews 1929;8:59±69. [8] Smith CW, Kobayashi AS. Experimental fracture mechanics. In: Kobayashi AS, editors. Handbook on experimental mechanics, 2nd ed. VCH, New York: 1993. p. 905±8.
Engineering Fracture Mechanics 61 (1998) 555±568
In¯uence of material properties of SIFs determined by frozen stress C.W. Smith a, *, E.F. Finlayson a, C.T. Liu b a
Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA b Phillips Laboratory, Edwards Air Force Base, CA 93524-7680, USA Received 28 April 1998; received in revised form 6 August 1998; accepted 10 August 1998
Abstract In a prior study utilizing two materials for which thermal and mechanical properties of both materials in bimaterial specimens were matched except for modulus, it was found that modulus mismatch had a negligible eect on the stress intensity factor (SIF) level and distribution for cracks in bond lines of stress frozen photoelastic bimaterial specimens. However, bond line residual stresses increased the SIF values. Since it is almost impossible to obtain commercial stress freezing materials with matching critical temperatures (Tc) with dierent moduli, the present study of cracks parallel to and within bond lines utilized such materials with Tc mismatch and compared results with parallel studies with matched Tc values in order to evaluate Tc mismatch eects. Results show that Tc mismatch creates mixed mode conditions in cracks parallel to the bond line with signi®cant increases in the mode I SIF (K1) for cracks very near the bond line but shows little eect for cracks within or away from the bond line. The absence of a modulus mismatch eect is again con®rmed as well. # 1998 Elsevier Science Ltd. All rights reserved. Keywords: Stress intensity factor; Bond line cracks; Bimaterial specimens; Frozen stress photoelasticity
1. Introduction Beginning in 1993, the authors undertook an experimental program directed towards an evaluation of the frozen stress photoelastic method for determining the stress intensity factor (SIF) distribution along cracks located near to or within bond lines joining dierent incompressible materials with substantial thickness. The objective was to approximately * Corresponding author. Tel.: 001 540 231 6159; Fax: 001 540 231 4574. 0013-7944/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 9 8 ) 0 0 0 7 0 - 8
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simulate such distributions for cracks within or near the bond line between rocket motors and their liners. It was expected that there would be residual stress resulting from the bonding procedure. Moreover, it had been shown analytically [1] that when plane strain exists where incompressible materials have been joined, then the interface fracture equations reduce to the classical homogeneous form, thus allowing separation of the mode I and mode II stress intensity factors. It was felt that such would be approximately the case for rocket motor material in a state of generalized plane strain. 2. Experimental procedures In order to separate residual stress eects from modulus mismatch, a three specimen test procedure was employed: (1) an edge cracked homogeneous control specimen; (2) a bonded homogeneous specimen containing an edge crack; and (3) a bimaterial edge cracked specimen. Two types of bimaterial specimens were employed. Their material properties are given in Tables 1±2. Specimens A included homogeneous Araldite specimens and homogeneous bonded Araldite and specimens B included homogeneous PLM4B specimens and homogeneous bonded PLM4B specimens. Ideally, the two materials employed in the bimaterial specimens should have identical properties except for the modulus in order to assess modulus mismatch eects. As indicated in Table 1, it was not possible to fully match all material properties except the modulus. In bimaterial specimens A, the thermal coecients diered by 17% at critical temperature, and in bimaterial specimens B, the critical temperatures diered by about 13%. However the former dierence appeared to have no eect upon the results. Table 1 Material Properties Model A Model A
Tcritical
EHot
fr
Araldite Aral-Alum1
2408F 2408F
2698 psi (18.60 Mpa) 5349 psi (36.88 Mpa)
1.64 psi in (286.9 Pa m) Ð
1 Created by mixing aluminum powder into araldite.Matched thermal coecients at 688F; a = 15.3 10ÿ6 per 8F. At critical temperature (Tc) thermal coecients were 119 10ÿ6220 10ÿ6 per 8F
Table 2 Material Properties Model B Model B
Tcritical
EHot
fr
PLM4B PSM9
1808F 2058F
1879 psi (12.96 Mpa) 6912 psi (47.67 Mpa)
2.40 psi in (420.3 Pa m) 2.97 psi in (520.1 Pa m)
These materials have the same thermal coecients at room temperature (TR) and critical temperature (TC). They were aRT=39 10ÿ6/8F and aCT=90 10ÿ6/8F, respectively. Bonding agent was liquid PSM9
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557
Since it was crucial that accurate dimensions be achieved along the crack front it was decided to machine in the cracks with a round tipped Buchler saw. The bond line thickness was kept small but ®nite and constant. All of the near tip test data were collected outside the bond line because the bond line material, generally being dierent from the adherends, produced a local disturbance which we considered part of the near tip nonlinear zone. Test specimen dimensions are shown in Fig. 1. All test specimens were ®rst subjected to a no load stress freezing cycle after which stress freezing under load was carried out. Both through the thickness (tt) and thin slices (L,M,R) (Fig. 1) were analyzed photoelastically (Appendix A) using the algorithms in Appendix B for converting optical data into SIF values. All SIF values were normalized by a control value Kc, an experimentally determined SIF value for a homogeneous edge crack in order to account for possible root radius or other experimental eects, and was within experimental error of $5% of the theoretical 2D K1 value. Most of the results are evaluated by thickness averaging for purposes of comparison. All photos used bright ®elds. Reasonable success [2±4] had been achieved in measuring SIF distributions by employing the A material combination for cracks within the bond line for both mode I and mixed mode conditions. The present study was focused primarily on evaluating the eect of critical temperature (Tc) and modulus mismatch in the B material combination for cracks parallel to and within the bond line.
Fig. 1. Specimen dimensions.
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3. Results and discussion By placing the cracks which were outside the bondline in the higher Tc material (PSM9) in the B specimens, it was felt that the Tc eect would be minimized. The idea was that, since the lower Tc material (PLM4B) would still be rubbery when the PSM9 turned glassy, then the PSM9 modulus ratio of E PLM4B Rubbery/E glassy00.5% should negligibly aect the SIF values in the PSM9. Then, when the PLM4B became glassy, it would only slightly stress the PSM9 and this live stress would be removed upon slicing. For specimens containing cracks parallel to the bond line, the no load tests revealed a near tip residual stress ®eld, [Fig. 2(a)] but the fringe order varied little over the data range, indicating that the residual stresses aected mainly the nonsingular part of the data and only slightly the fringe gradient. Thus it had little eect upon the SIF values. None of the homogeneous bonded specimens using the B material combination revealed any increase in the average SIF above the homogeneous specimen value and no shear mode was observed. However for the bimaterial specimens, a small shear mode, indicated by a clockwise rotation of the fringe loops for which the reading angle yR (i.e. y8m) increased as the crack was moved closer to the bond line, was observed. Moreover, for the crack nearest the bond line, [Fig. 2(b), (c)] the value of K1 increased by 30%, as shown in Fig. 3, but no increase was observed for cracks further from the bond line (i.e. h = 0.25 in, [6.35 mm]; 0.50 in. [12.7 mm]) [4]. Fig. 3 shows data for all three slices across the thickness. Each set of data was extrapolated to the origin and the average result is plotted on the ordinate. In order to clarify whether or not this increase was due to modulus mismatch or Tc mismatch, a test was run with the A material combination which duplicated the above noted geometry (i.e. h = 0.125 in [3.18 mm], Fig. 1). The loaded (tt) fringe pattern is shown in Fig. 4. Data used for analysis were taken above the crack within the indicated data zone. The fringe pattern reveals no fringe loop rotation, and hence no detectable mode II and the value of K1 was the same as from a 2D homogeneous specimen analysis, indicating no increase due to the bond line proximity [Fig. 5(a)]. For fringe loops between the crack and the bondline, there is clearly a distortion imposed on the system by the bondline which acts as a sti, loaded boundary. This is interpreted to mean that a stress gradient results which produces a variable sox in the vertical direction. The simplest modi®cation of sox to account for this eect would be to assume sox to vary linearly in the vertical direction after the fashion of Ref. [5]. This means that the Kap vs Zr/a curve will be a second degree curve in the K1 data zone rather than a linear curve as shown in Appendix B. Using such an approach, one obtains a value of K1 which is within 4% of the value taken above the crack [see Fig. 5(b)]. The linear extrapolation was along the slope of the data curve at Zr/a=0.30. These results con®rm that both the shear mode and the increase in the SIF found in the test shown in Fig. 2 were due to Tc mismatch and not to modulus mismatch. A ®nal test was conducted on the B material with the crack within the bond line. Fringe patterns from the bimaterial test are shown in Fig. 6, indicating the data zone. In interpreting these patterns, it should be noted that even for pure mode I (which is observed here) the fringes will not be symmetric with respect to the crack or the bond line due to the dierence in the material fringe values (Tables 1 and 2) for the two materials. The absence of a detectable shear mode observed for both Figs. 4 and 6 agrees with Ref. [1]. Analysis of the fringe data
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Fig. 2. Fringe patterns; B materials; crack parallel to bondline.
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Fig. 3. Normalized SIF results from Fig. 2.
Fig. 4. Fringe pattern; A material; test geometry of Fig. 2.
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561
Fig. 5. Normalized SIF results from Fig. 4.
showed no signi®cant dierence between the bimaterial and homogeneous bonded test data, again indicating no bimaterial eect on the SIF values. Fig. 7, which is a closeup of the (tt) loaded Fig. 6 photo, reveals several features. In this test, the bond line adhesive was the same material as the upper material PSM9 and so has the same Tc. Nevertheless, at the upper edge of the bond line, some mismatch of the fringes crossing the upper edge of the bond line still occurs near the crack tip suggesting a slight dierence in the material fringe values between the upper plate material and the bond line material, but not as severe as along the bottom of the bond line where the PSM9 adhesive meets the PLM4B. Data analysis from the slices, L, M, R were again averaged since only 2D comparisons are available. Such an analysis for Fig. 6 is given in Fig. 8. Some of the variation in data from slice to slice
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Fig. 6. Fringe pattern for crack in bondline; B materials.
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563
Fig. 7. Enlargement of Fig. 6(b).
in the B specimens is conjectured to be due to residual bondline stress variation through the thickness. Typical SIF distributions from an A material combination containing signi®cant residual stresses are compared to their homogeneous bonded counterparts in Fig. 9. These results reveal the signi®cant residual stress eects but clearly show that bimaterial eects are not independently signi®cant and that reasonable SIF distributions can be measured by the method employed. For this case, through thickness averaging of the slice results showed that the homogeneous K1 value increased 16% in the bimaterial specimen and 19% in the homogeneous bonded specimen. It is interesting to note that, when the crack is placed in the bond line for the B material combination, no shear mode is detectable which suggests that if one is only interested in problems for cracks within the bond line, then the Tc mismatch does not appear to have a signi®cant eect on K1 for incompressible materials under generalized plane strain. 4. Summary Using a three specimen test procedure, a convenient incompressible bi-material combination (B) was evaluated for predicting SIF levels and distributions in cracks parallel to and within bimaterial specimens. By comparing results with those from a previously used material combination (A) it was shown that Tc mismatch in the B combination induced both a shear
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Fig. 8. Normalized SIF results from Fig. 6 slices.
mode and an elevation in SIF level for cracks very near the bond line. However, for cracks away from the bond line or within the bond line, the B material combination yielded reasonable mode I results. It is important to note the fact that modulus mismatch, shown earlier to have no eect on cracks within the bond line of A materials was also con®rmed here for cracks parallel to the bond line as well as within the bond line. All of the above observations are limited to the indicated mismatch values of Tc and the modulus.
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565
Fig. 9. Normalized A material SIF distribution.
Acknowledgements The authors wish to acknowledge the support of Phillips Laboratory through Hughes STX Corp. under subcontracts 96-7055-LI467 and the sta and facilities of the Department of Engineering Science and Mechanics at VPI & SU. Appendix A A.0.1. Frozen stress photoelasticity When a transparent model is placed in a circularly polarized monochromatic light ®eld, and loaded, dark fringes will appear which are proportional to the applied load. These fringes are called stress fringes or isochromatics and the magnitude of the maximum in-plane shear stress is a constant along a given fringe. Some transparent materials exhibit mechanical diphase characteristics above a certain temperature, called the critical temperature (Tc). The material, while still perfectly elastic will exhibit a fringe sensitivity of about 20 times the value obtained at room temperature and it's modulus of elasticity will be reduced to about one six hundredth of its room temperature value. By raising the model temperature above Tc, loading, and then cooling slowly to room temperature, the stress fringes associated with Tc will be retained when the material is returned
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to room temperature. Since the material is so much more sensitive to fringe generation above Tc than at room temperature, fringe recovery at room temperature upon unloading is negligible. The model may then be sliced without disturbing the ``frozen in'' fringe pattern and each slice analyzed as a two-dimensional model, but containing the three-dimensional eects. In the use of the method to make measurements near crack tips, due to the need to reduce loads above critical temperature to preclude large focal deformations, and the use of thin slices, few stress fringes are available by standard procedures. To overcome this obstacle, a re®ned polariscope is employed to allow the tandem use of the Post [6] and Tardy [7] methods to increase the number of fringes available locally. In fringe photographs, integral fringes are dark in a dark ®eld and bright in a bright ®eld. Bright ®elds are used through-out this paper for clarity.
Appendix B B.0.1. Mode I algorithm Beginning with the Grith±Irwin Eqs. we may write, for mode I, for the homogeneous case, sij
K1 2pr
1 2
fij y sij
i; j n; z
B:1
where sij are components of stress; K1 is SIF; r, y are measured from the crack tip (Fig. 10); s8ij are non-singular stress components. Then, along y = p/2, after truncating sij tnz max
K1 8pr
1 2
r
KAP 1
8pr2
where t8 = f(s8ij) and is constant over the data range KAP=apparent SIF.
Fig. 10. Mode I near-tip notation.
B:2
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567
Fig. 11. Middle slice data.
(tnz)max=maximum shear stress in the nz plane p 1 KAP K1 8t r 2 1 1 a s 2 2 s pa s pa 1=2 vs where (Fig. 10) a = crack length, and s=remote normal stress i.e. KAP/s(pa) linear.
B:3 p r=a is
Since, from the stress±optic law (tnz)max=nf/2t where n = stress fringe order; f = material fringe value; t = specimen (or slice) thickness and from Eq. (2) KAP=(tnz)max(8pr)12=nf/ 2t(8pr)12, then KAP (through a measure of n) and r becomes the measured quantity from the stress fringe pattern at dierent points in the pattern. Use of a linear variation of sox over the data zone is shown in Fig. 5(b). B.0.2. Mixed mode algorithm The mixed mode algorithm was developed (see Fig. 12(a) and (b)) by requiring that ( ) max 1=2 d tnz lim 8prm K1 ; K2 ; rm ; Ym ; tij 0 rm !0 dY Ym !Y m
Fig. 12. (a) and (b). Determination of y8m.
B:4
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which leads to 2 K2 4 K2 1 cot 2Ym ÿ 0 ÿ ÿÿ : ÿ K1 3 K1 3
B:5
By measuring Y8m which is approximately in the direction of the applied load, K2/K1 can be determined. Then writing the stress optic law as tmax nz
fn K AP 1 ; 2t 8pr2
p one may plot K *AP/s(pa)12 vs r=a as before; locate a linear zone and extrapolate to r = 0 to obtain K . Knowing, K , K2/K1 and Y8,m values of K1 and K2 may be determined since hÿ 2 ÿ 2 i12 ÿ ÿÿ: B:6 K K1 sin Y m 2K2 cos Y m K2 sin Y m Knowing K* and Y8m, K1 and K2 can be determined from Eqs. (5) and (6). Detailspare found in Ref. [8]. The crack tip linear zones in the graphs such as Fig. 11 were between r=a=0.30 and 0.60. References [1] Hutchinson JW, Suo H. Mixed mode cracking in layered materials. In: Advances in applied mechanics. Vol. 29. New York: Academic Press, 1992. p. 63±91. [2] Smith CW, Finlayson EE, Liu CT. Preliminary studies of three dimensional eects on arti®cial cracks at simulated rocket motor liners by the frozen stress method. In: Proc. VIII International Congress on Experimental Mechanics, Society for Experimental Mechanics, Bethel, Ct 1996. p. 55±61. [3] Smith CW, Finlayson EF, Liu CT. A method for evaluating stress intensity distribution for cracks in rocket motor bond lines. Journal of Engineering Fracture Mechanics 1997;58 (12):97±105. [4] Smith CW, Finlayson EF, Liu CT. Bond line residual stress eects on cracks parallel to the bond lines in rubberlike materials. In: Recent advances in solids/structures and application of metallic materials. 369. ASME PVP, 1997. p. 1±10. [5] Smith CW, Weirsma SJ. Stress fringe signatures for propagating cracks. Journal of Engineering Fracture Mechanics 1986;23(1):229±36. [6] Post D. Fringe multiplication in three dimensional photoelasticity. Journal of Strain Analysis 1966;1(5):380±8. [7] Tardy MLN. Methode Practique Examen de Measure de la Birefringence des Verres D'optique. Optics Reviews 1929;8:59±69. [8] Smith CW, Kobayashi AS. Experimental fracture mechanics. In: Kobayashi AS, editors. Handbook on experimental mechanics, 2nd ed. VCH, New York: 1993. p. 905±8.