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1984 Application of photoelasticity to determine mode one and mode two orthotropic stress intensity factors Masoud Mojtahed Iowa State University
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Mojtahed, Masoud
APPLICATION OF PHOTOELASTICITY TO DETERMINE MODE ONE AND MODE TWO ORTHOTROPIC STRESS INTENSITY FACTORS
Iowa State University PH.D. 1984
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University Microfilms International
Application of photoelasticity to determine mode one and mode two orthotropic stress intensity factors
Masoud Mojtahed
A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY
Department: Engineering Science and Mechanics Major: Engineering Mechanics
Approved: Members of the Committee:
Signature was redacted for privacy. Signature was redacted for privacy.
Signature was redacted for privacy.
Signature was redacted for privacy.
Iowa State University Ames, Iowa
1984 ii
TABLE OF CONTENTS
Page
ABSTRACT viii INTRODUCTION 1 LITERATURE REVIEW 7 Orthotropic Photoelasticity Works 18 Orthotropic Fracture Mechanics Works 31 Finite Elements Works 34 ORTHOTROPIC PHOTOELASTICITY 38 FRACTURE MECHANICS OF ORTHOTROPIC MATERIALS 46 Photo-Orthotropic Fracture Mechanics 59 METHOD OF ANALYSIS 61 Newton-Raphson Method 62 Method of Least Squares 63 Finite Element Method 66 Half-Fringe Photoelasticity 67 Eye Com System 69 HFP-Eye Com Theory 72 Object of Study 75 EXPERIMENTAL PROCEDURE 76 Loading Apparatus 77 Calibration of Optical and Mechanical Properties 83 Experimental Models 95 Preparation of Finite Element Models 98 iii
Page
Data Collection Method 107 Test Sequence 108 RESULTS AND DISCUSSION 113 Compact Tension Specimens 113 The Strip Specimen 125 CONCLUSION 133 ACKNOWLEDGMENTS 135 BIBLIOGRAPHY 136 APPENDIX A 142a APPENDIX B 149 iv
LIST OF FIGURES
Page
Figure 1. Polar coordinates of a fringe loop 8 Figure 2. Normalized stress intensity factor vs. crack length to one-half width 11 Figure 3. General problem geometry and notation 12 Figure 4. Estimating SIF value by extrapolation method 13 Figure 5. Crack tip geometries used in notch crack comparison 16 Figure 6. Normalized vs. (r/a)^/^ 17 Figure 7. Stress ratio ogg/ag as a function of 0 22 Figure 8. Location of photoelastic solutions 25 Figure 9. Normalized stress (a /c) along x-axis for line A 26 Figure 10. Arcan et al. model 29 Figure 11. Plot of normalized mode one apparent stress intensity factor vs. normalized distance from the crack tip 36 Figure 12. Stress notation 40 Figure 13. Mohr circle for birefringence 41 Figure 14. Unidirectional fiber-reinforced composite under uni-axial loading 43 Figure 15. Stress intensity modes 47 Figure 16a. A crack under mode 1 loading 48 Figure 16b. A crack under mode 2 loading 48 V
Page
Figure 17. Crack coordinate system 51 Figure 18. A typical plot of light intensity vs. fringe order 69 Figure 19. HFP system 70 Figure 20. The loading apparatus 78 Figure 21. The load cell dimensions 79 Figure 22. The load cell 81 Figure 23. Load cell calibration 82 Figure 24. The tension calibration model 84 Figure 25. f^i calibrations 85 Figure 26. f22 calibrations 86 Figure 27. f45 calibrations 87 Figure 28. Cutout circular specimen 89 Figure 29. f^y calibration using Arcan's model 90 Figure 30. Strong axis mechanical properties 91 Figure 31. Weak axis mechanical properties 92 Figure 32. 45® mechanical properties 93 Figure 33. Compact tension specimen 96 Figure 34. The strip model dimensions 97 Figure 35. Finite element model, a/w=0.4 99 Figure 36. Crack tip elements 100 Figure 37. Finite element model for mesh size = 0.02", a/w = 0.4 102 Figure 38. Crack tip elements for mesh size = 0.02" 103 VI
page
Figure 39. Stress comparison for two mesh sizes 104 Figure 40. Finite element model of the strip specimen 105 Figure 41. Crack tip elements for the strip model 106 Figure 42. Data collection points 108 Figure 43. Experimental set up 110 Figure 44. Typical fringe pattern 111 Figure 45. Normalized Kj vs. r/a for 6=165, -165 114 Figure 46. Data line locations 116 Figure 47 Kj, crack aligned with weak axis 118 Figure 48. Kji, crack aligned with weak axis 119
Figure 49. Oqx/ crack aligned with weak axis 120 Figure 50. Kj, crack aligned with strong axis 122 Figure 51. Kji, crack aligned with strong axis 123 Figure 52. cToxf crack aligned with strong axis 124 Figure 53. Kj and Kjj, crack aligned with strong axis 128
Figure 54. OQJJ, crack aligned with strong axis 129 Figure 55. Kj and Kjj, crack aligned with weak axis 130
Figure 56. OQJJ, crack aligned with weak axis 131 Figure 57. Kj vs. R for the plexiglas model 151 vii
LIST OF TABLES
Page
Table 1. Elastic constants of titanium and graphite 2 Table 2. Elastic constants of Type 308 austenitic stainless steel weld metal 2 Table 3. Typical laminae elastic constants 4 Table 4. Optical constants 89 Table 5. Mechanical constants 95 Table 6. The strip model results 132 Table 7. Comparison of methods 150 viii
ABSTRACT
With the advancement of technology and application of orthotropic materials in modern industry, the development of an experimental technique to analyze and determine the stress intensity factors in orthotropic materials was needed. To address this necessity, a new experimental method of combining orthotropic photoelasticity and orthotropic linear elastic fracture mechanics laws are developed. A new set of equations are obtained by combining orthotropic photoelasticity laws and orthotropic fracture mechanics formulas. These equations along with half-fringe photoelasticity techniques are used to deter mine the stress intensity factors. To model orthotropic materials, unidirectional fiberglas/epoxy composites, made at IITRI are used. Compact tension specimens and a strip with a slanted edge crack are employed to determine mode one and mode two stress intensity factors. The optical and mechani cal properties of the materials are determined by the use of tension specimens. A microcomputer is employed to collect and analyze the experimental data, and a finite element solution is used to verify the experimental results. Solid SAP, a finite element program capable of analyzing orthotropic materials is utilized for the finite element solution. The experimental results show that photoelasticity can be used effectively in the determination of mode one stress intensity factors. However, problems arise from the low sensitivity and ix low toughness in the case where the cracks are parallel with the fibers in unidirectional composites. Further studies at higher loads, which are difficult because of the low toughness and crack tip plasticity effects, need to be undertaken. In the case of mixed mode stresses, photoelas- ticity is not as effective in determining stress intensity factors, as in the pure mode one case. 1
INTRODUCTION
The importance of linear elastic fracture mechanics is well-documented [1-4]. Although most of the work in this area is performed on isotropic materials, the significance of anisotropic material in modern industry has motivated investigators to establish the theory of anisotropic frac ture mechanics and furthermore to investigate its fracture criteria. Metal material antisotropy can result from rolling or extrusion and can also occur in centrifugal castings. Table 1 indicates two metals that show orthotropic behavior [5]. Titanium is mildly orthotropic with modulus ratio of E^/E^ = 1.38. Graphite shows a very large modulus ratio of 26.7. Erdogan and Delale 15] have argued effectively that these two materials can be treated as being especially ortho tropic. Their results show that the stress intensity factor for cracks in shells can be strongly influenced by the ortho tropic nature of the materials. Another area of concern is the orthotropic properties of weldments. Austenitic stainless steel shows a preferred subgrain and crystallographic orientation that gives an ortho tropic behavior [6-8]. During weld metal solidifications preferred orientation in the substructure of the welds is obtained [9,10]. This leads to the orthotropic behavior of the weldments. Table 2 shows the elastic constant results 2
Table 1. Elastic constants of titanium and graphite [5]
Tiraniuia Graphite
Z, N/m^/psi) 1.04(10)^1(15.1(10)6) 1.034(10)(1.5(10) N/3?(psi) 1. 43(10) (20.8(10) G) 2.76(10)11(4(10)"^) 0.20 0.0075 •^2 0.27 0.2000 GT2 N/af(psi) 4.68(10)1°(5.78(10)^) 2.76(10)1°(4.0(10)^) n/mfcpsi) 4.93(10)1°(7.15 (10)®) 2.57(10)1° (3.73 (10)®)
Table 2. Elastic constants of Type 30 8 austenitic stainless steel weld metal [11]
^11 " ^22 N/m?(psi) 1.42(10)11(20.6(10)^) 1.04(10)11(15.1(10)6) -33 N/m^(psi) 2 ^44 = ^55 iN/m (psi) 8.2(10)1°(11.9(10)6)
^66 N/m^(psi) 5.7(10)10(8.3(10)6) ^12 0.24 ^13 0.53 ^31 0.39 3 obtained by Dewey et al. [6], from ultrasonic measurements and tensile tests on Type 308 austenitic stainless steel weld metal. This particular weld showed a modulus ratio of 1.4. The orthotropic behavior is controlled by the welding process such as, the number of weld passes, the temperature gradients, the weld metal composition, etc. The stress analysis of flaws in weldments has typically assumed the weld metal to be isotropic and homogeneous [11-13]. It is evident now that more attention must be placed on the orthotropic aspect of the weldment analysis. The use of composite materials in the aerospace and automotive industry is growing rapidly. The high strength/ weight ratio, low cost and availability are the important factors of this rapid growth. In the fibrous composites, the orthotropic behavior is well documented [14, 15]. Filamentary composites are usually made up of a number of plies or laminae. A lamina consists of a layer of fibers all aligned in one direction and surrounded by a matrix. The matrix can be either epoxy or metal, and fibers range from glass to metals. Table 3 shows typical laminae characteris tics for three different fiber/matrix combinations [14]. This indicates that Young's modulus ratio of 40 can be easily obtained. The laminae are stacked with various orientations to obtain a composite with the desired stiffness and strength Table 3. Typical laminae elastic constants [14] Glass/Epoxy Boron/Epoxy Graphite/Epoxy
Ej^ N/m^(psi) 5.4(10)10 (7.8(10)^) 2.1(10)^^ (30(10)^) 2.1(10)^° (30(10)^)
Eg N/m^(psi) 1.8(10)1° (2.6(10)6) 2.1(10)(3(10)^) 5.2(10)^ (0.75(10)^) 0.25 0.3 0.25 G^2 N/m^(psi) 9(10)9 (1.3(10)6) 6.9(10)^ (1(10)^) 2.6(10)^ (0.375(10)G) 5 characteristics. The stress analysis of these layered composites now becomes very difficult, especially if voids, inclusions, or cracks are found. In fact, the analysis of these flaws is quite important in the determination of the flaw severity. The problem is no less important in the case of flaws in weldments or flaws in rolled, extruded, and centrif ugally.cast metals. In general, the philosophy has been to replace or repair a part if a flaw is found. This leads to higher costs and increased down time. Retirement for cause (RFC) programs are now being instituted in the aerospace industry. When a flaw is located, the potential threat of failure is analyzed and only those flaws showing a significant failure potential are eliminated. Thus, it is important that flaws, particularly cracks, be analyzed with respect to their orientation, shape and incident stress field. This research addresses the ground work that needs to be done in order to analyze the stress intensity factors in orthotropic experimental models. The experimental models are needed for confirmation of finite element and analytical methods. An orthotropic photoelastic material will be used as a modeling material for studying orthotropic crack behavior. The difficulty involved in producing mechanically and optical ly uniform orthotropic composites exhibiting good optical sensitivity has hampered researchers in applying photo- 6 elasticity to orthotropic materials. The photoelastic sensitivity is especially burdensome since fringe order data are needed at a large number of points surrounding a crack tip. The development of fairly uniform and sensitive orthotropic photoelastic material at Illinois Institute of Technology Research Institute, which will be referred to as IITRI I16J and the use of digital image analysis system were necessary for characterization of a crack in an ortho tropic material. In this investigation, orthotropic LEFM theory and the orthotropic photoelastic optic-law are combined with the application of half fringe photoelasticity, HFP [17] to col lect the experimental data. A standard compact tension specimen and a strip with a 45° edge crack made of an ortho tropic, transparent photoelastic material were used as models. The experimental results are then compared to a set of finite element results. 7
LITERATURE REVIEW
In recent years, photoelasticity has been used extensively in fracture mechanics. Stress intensity factors have been determined by analyzing the fringe pattern of loaded or stress frozen models. Almost all of the research done in this field is concentrated on isotropic materials. With the advances in technology and applications of computers and microcomputers, scientists have improved the fracture-photoelasticity method. The least squares method was introduced by Sanford and Dally [18] in 1979. This method has several advantages with respect to the standard method, for example, 1) by increasing the number of data points, the experimental errors can be reduced extensively, and 2) a variety of mathematical models may be used in analyzing stress intensity factors. Sanford and Dally analyzed the isochromatic pattern of the near crack tip of their models by using a modified near-field solution for the central crack problem. They developed a nonlinear equation to relate and which are mode one and mode two stress intensity factors, and which is the far field stress along the crack to the fringe order N and fringe position r and 9 (Figure 1). They combined Newton- Raphson iterative method with a least-squares method to obtain and A detailed explanation of Newton- Raphson and least squares methods will be presented in the 8
Figure 1. Polar coordinates of a fringe loop
Method of Analysis chapter of this thesis. Daniel et al. [16], at IITRI, developed a new transparent orthotropic material. This material which is made of glass- epoxy, provides fairly uniform orthotropic birefringent material. This material has low birefringence capability, creeps and delaminates at high level loads. Burger and Voloshin [17], at Iowa State University, de veloped a new method that helps solve the sensitivity problem of orthotropic birefringent materials. This method, which is 9 called "Half-Fringe-Photoelasticity" (HFP), consists of a digital image analysis system with high optical resolution. With HFP, the light intensity of a data point on a specimen is measured and converted to fringe order. Voloshin and Burger 119] analyzed a loaded beam made of epoxy-fiberglass composites with the digital analysis system- They investigated beams loaded under pure bending. The fibers of the two beams were parallel and perpendicular to the longi tudinal axis of the beam. The fringe order across the beam was measured by HFP and Tardy-Compensation methods. The average difference between these two approaches was within 3.0%. Voloshin and Burger also ran two more experiments. In one experiment, they applied a concentrated load on a semi- infinite orthotropic plate, and in the second experiment, they tested a circular disk with a load normal to the direction of the fibers. The contour maps of the constant fringe order were then plotted. Miskioglo et al. I20J combined Sanford-Dally least squares method with Burger and Voloshin's half-fringe photo- elasticity method to determine stress intensity factors. He used double edge cracked tension specimens made of PSM-1 for his experiment. The values he obtained for stress in tensity factors were within 4.0% of the existing theoretical and numerical values. Kar [21] employed the method of Miskioglo et al. to 10 determine stress intensity factors in glass. Kar used tension specimens with single edge cracks, models with 45° edge cracks and double edge cracks. Twenty data points were collected along five radial lines. He limited the collection area to +140° and -140° with respect to the X axis (Figure 1). Kar used the collected data to determine and (mode 1, mode 2 and the far- field stress along the crack). These results then were compared with available numerical and experimental results. The average and values were within 5.2% for single edge cracks, 1.0% for double edge cracks, and 12.0% for 45° edge cracks of the compared values. Figure 2 shows normalized vs. a/b (crack length to one-half width) ratio for single edge specimens. Smith [22] introduced the application of the stress freezing method to determine the stress intensity factors. He stated that the apparent stress intensity factor is a function of I and o for mode one loading. This relation- ship for 0=0 (Figure 3) is given by the following equa tion [22]:
(1)
where 1/2 K T (8nr) = the apparent stress intensity AP max factor 11
l.f-
i.r
Theoretical Exper inental
1.0 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Crack length to one-half width ratio (a/b)
Figure 2. Normalized stress intensity factor vs. crack lenqth to one—half width C2lJ 12
Nf/2t stress fringe order stress fringe value thickness of a thin slice of the material taken mutually orthogonal to the crack surface and the crack border (X, Y plane. Figure 3) a load parameter such as remote stress or internal pressure j = x,y = contribution of regular stresses to the fringe order in the measurement zone and produces tilting and change in the ec centricity of the fringe loop as r in creases from zero
Figure 3. General problem geometry and notation. [22] 13
In the above equation it is obvious that ^AP ^ 1/2 q(na) linearly varies with (r/a) in the area dominated by the stress singularity. As seen from Equation 1, by extrapolating the linear curve across a nonlinear zone near the crack tip to r/a = 0, the apparent SIF and the actual SIF coincide- This type of extrapolation will also be used in part of this research. This method is illustrated for a crack in the reentrant nozzle corner of a boiling water reactor in Figure 4. A model of a reactor made of stress freezing material and loaded with internal pressure was used to obtain the data. Normal- izèd apparent was plotted against (r/a)1/2 ' . The linear
m VESSEL
m
O TEST DATA ( a . 45* )
• DATA USED IN determination
10
01 02 03 04 05
SQUARE ROOT OF NORMALIZED DISTANCE FROM CRACK TIP
Figure 4. Estimating SIF value by extrapolation method [22] 14 section of the curve is then extrapolated to (r/a) 1/2 =0 to determine the actual K^. The extrapolation method also can be applied to determine and the presence of modes 2 and 3 or the combina tion of modes 1, 2 and 3. Smith [22] proposed the applica tion of the stress freezing method to photo-orthotropic elasticity in the final part of his paper. It is proposed that the matrix be replaced with a stress freezing material in composites to determine SIF along a crack tip after it pene trates into the resin rich region. The problem which arises here is that the thickness of this region is of the order of only 0.025 mm which is smaller than the range of most crack tip photoelastic analysis. The alternative to this method is to bound the homogeneous orthotropic layers with stress freezing resin. The thickness of this interface must be chosen to be of the order of the magnitude appropriate for the photoelastic resolution of the system. Another disad vantage of this method is the requirement of matching or controlling the thermal coefficients of expansion of the laminate layers and the photoelastic material for thermal stress simulation. At the end. Smith [22] states that those restrictions imposed on isotropic stress freezing also exist for orthotropic stress freezing analysis and should be con sidered. These restrictions are: 15
1. The model material is elastic and incompressible. 2. The model geometry and starter crack should be geometrically similar to those of prototype. 3. Small scale yield conditions must prevail in the prototype. 4. The fatigue loading of the prototype should be tension-tension. One of the important tasks in determining SIF by photoelasticity is to determine an appropriate region in front of the crack tip for data collection. Schroedl and Smith [23] investigated the near field and far field effect in determining SIF by photoelasticity. A technique called the "Taylor Correction Method" was used to evaluate notch effects in the determination of SIF. Four different crack tip shapes used in the investigation were: elliptical, rectangular, triangular and blunted triangular notches (Figure 5). The SIF results were compared with the solution for a line crack with a sharp crack tip. In order to deter mine an appropriate zone for the data collections, they used slots terminating in a 30° V-notch of 0.025 mm root radius, rectangular slots with 0.15 mm width and natural cracks. Figure 6 shows the apparent normalized SIF vs. square root of normalized distance from the crack tip. Three separate zones can be distinguished in this figure. Two nonlinear zones, I and III, show the near and far field effects. Zone II, which behaves like a "line crack", was found to be the 16
1 f î ^ î T '
Q/w «I
« 2W I 1 1 M 1
C 2lb
2lb
Figure 5. Crack tip geometries used in notch crack comparison [23] 17
lb 1.8
o TEST DATA « DATA USED IN KTSCM DETERMINATION a/w«0 .128 â(Tro)^-14.90 kPa-m"2 o —122^.1,41 K;,* WS'T'"'"® m-rra) VEE - NOTCH
0.2 0.4 0.6 0.8 SQ. ROOT OF NORMALIZED DIST. 1/2 FROM CRACK TIP (r/a)
Figure 6. Normalized K^p vs.(r/a)^/^ [233 best region to collect the experimental data. Determination of the linear zone for data collection in this research will also be considered. Schroedl and Smith [23] found that the results for a 30° V-notch were identical to those of natural cracks. The results for the rectangular notch were 10% higher for the same region. They recommended that, although 18 the best results can be obtained by using natural notches and gathering data outside r/a = 10 p/a, rectangular cracks also can be used to simulate line cracks. For the latter case, a minimum distance where the data can be collected should be determined. In the case of blunt rectangular notches, besides the distance from the crack tip require ment, the ratio of b/a ^ 0.1 should also be considered. In the evaluation of the boundary effects they found that the singular zone shrinks abruptly while the crack approaches the free edge of the model, for this case is mainly controlled by interaction among the free boundary and the singular stress field.
Orthotropic Photoelasticity Works
In 1968, Pih and Knight 124], pioneered a new approach to the experimental stress analysis of orthotropic materials. They introduced the application of photoelasticity to the analysis of fiber-reinforced composite materials. First, a technique called "vacuum impregnation" was developed to produce glass-like fiber reinforced composites. In this method, they matched modified "E" fiberglass and epoxy resin indices of refraction to produce a water clear composite material. In the second step, they formulated the stress- optic law for orthotropic materials. The proposed law was 19 then used to predict isochromatic fringe values and isoclinic angles. Theoretical and experimental results for isochromatics and isoclinics for several fiber orientations with respect to the principal stress direction were plotted. They found that the isoclinic fringes do not represent the principal stress directions. Although the experimental results showed good agreement with their proposed optical law in the case of uni directional loading, the results for the case of biaxial loading were scattered and inconclusive. They concluded that a further study was required to modify the proposed ortho- tropic optical law for future applications. Sampson 125] proposed a new optic law based on an appli cation of Mohr circle to the birefringent components in the state of plane stress. He expanded the isotropic optic law and postulated that three optical material constants are re quired to explain the birefringence behavior of orthotropic photoelastic materials. Sampson's law provided equations to predict isochromatic fringe orders and isoclinic angles in the loaded models. Sampson performed several experiments to veri fy the accuracy of the proposed optical law. He also com pared his results with those of Pih and Knight [24]. The results based on his theory showed a good agreement with both his and Pih and Knight's experiments. He also verified the results of Pih and Knight that the isoclinic fringes do not represent principal stress direction in photo-orthotropic 20 materials. This method will be explained in detail in the Orthotropic Optic Law chapter. Dally and Prabhakaran [26], at the Illinois Institute of Technology, developed a new method to produce orthotropic birefringent material. This material which employs fiber glass cloth and polyester, showed much better homogeneity than those made by Sampson. It produced a very good con tinuous fringe pattern under loading. They also developed a set of equations by the stress partitioning method to esti mate the three orthotropic optical constants. Their experi mental and theoretical results verified the accuracy of Sampson's stress optic law. Prabhakaran and Dally [27] later made an effort to em ploy photo-orthotropic elasticity to determine its appli cability and limitations. Two different subjects were con sidered in their project. The first one was to determine the radial stresses of a semi-infinite plate under a concen trated load. A plate was loaded with the load normal to the fibers and fringes along different radial lines were used to determine the radial stress. The results then were plotted against r (radial distance from the load) and compared with the analytical solution- The difference between the analyti cal and experimental results for radial distance of one inch and different radial angles varied from one to eleven percent. The second goal was to determine the tangential stresses 21 of an elliptical hole in a wide plate under uniaxial loading. Bidirectional orthotropic material was used to make the model. The distribution of tangential stress around the hole was determined by using the isochromatic fringes and plotted against the positional angle with respect to the horizontal axis (Figure 7). These results were compared only with the analytical results for isotropic material since there is no analytical solution for bidirectional orthotropic material cases. They concluded that there are several limitations in the application of the photo-orthotropic material to the stress analysis. The obstacles that Prabhakaran and Dally identified were: 1. Failure to achieve a good fringe resolution can be attributed to a lack of homogeneity in the materials. 2. Higher level of fringes cannot be achieved because of occurrence of debonding during the application of high loads. 3. Errors are caused by the misalignment of fibers in the procedure of making the material or making the models. Prabhakaran 128] also investigated the fringe pattern
along the inner and outer radii of a ring under diametrical loading. He used uniaxial stress optic law (a^ = Nf^/h), to determine the tangential stress of the boundaries of the ring The comparison of the results obtained for three different fiber orientations with respect to the load direction was 22
ee
\^Isotropic
Orthotropic
6 degrees
Figure 7. Stress ratio as a function of 0 [27]
performed to examine the effect of the orthotropic be havior of the ring. Pipes and Dally 129] employed the material provided by Dally and Prabhakaran for further investigation of photo- orthotropic material. They observed that residual bire 23 fringence exists in the material and the residual isoclinics coincide with the axis of the material's symmetry. They also verified that the isoclinic fringes do not represent either principal stress or principal strain directions. Prabhakaran [30], by introducing the strain optic law for photo-orthotropic materials in 1974, opened a new chapter to the application of these materials in research and develop ment of orthotropic materials. He used the concept of Mohr's Circle of birefringence based on Sampson's optic law and developed equations for strain fringe values and their rela tion with principal strains. A circular disk under diammetri- cal compression loading was used to verify the accuracy of the proposed law. The comparison showed a good agreement be tween the theoretical and experimental strain fringe values. In 1974, Cernosek [31] investigated the photo-ortho tropic law of Sampson, Dally and Prabhakaran. He verified the accuracy of both laws and reported that the stress optic constants and optical isoclinic parameters can be predicted even in the presence of residual birefringence in unloaded materials. Prabhakaran [32], in his further investigation of photoelastic composite materials, verified that the param eter of optical isoclinic represents neither principal stress nor principal strain directions. He also confirmed that Sampson's law correctly predicts isoclinic angles. 24
Mittal [33] explored the effect of residual birefringence in orthotropic materials. The results showed that presence of residual fringes mostly affect the isoclinic angles and the true N (isochromatic fringe order) is in error of + (+ residual fringe order). Knight and Pih [34], in 1976, investigated the applica tion of shear difference method in orthotropic photoelasticity. They developed a set of equations for application of this method to orthotropic materials. A strip with a central hole made of a photo-orthotropic material under uniaxial loading was employed to check the accuracy of the developed equations. The stress distribution along three directions. A, B and C, were determined (Figure 8). Normalized stress distributions were then compared with a finite element solution. A good agreement was observed between the two results. Figure 9 shows the plot of the normalized longitudinal stress along direction A. Agarwal and Chaturvedi [35], in 1978, developed a high quality orthotropic material made of polyester and fiber glass. Both unidirectional and bidirectional sheets of this material were produced and tested under uniaxial loading. The results were measured and compared with Sampson's theo retical results. They found a good agreement between the theoretical and experimental results. Voloshin [36] combined orthotropic elasticity equations 25
t
Fiber Axis
Along Hoi* Boundary
Figure 8. Location of photoelastic solutions [34] 26
i • Photo*io*iic Results V - Frnir* EUnwnt
V. t
I f *
(o) Af 0 D#y##.
V
(b) At 30 0«9f«et.
(c) At 90 Dtgr«««
Figure 9. Normalized stress (o /o) along x-axis for line A [34] ^ 27 with photoelastic experimental data to perform stress analysis of orthotropic materials. The finite difference method, which provided an over deterministic solution, was used to solve the equations. A balanced laminated coupon (fiber direction at + 30°) under tension was used in the experiment. The stress distribution was calculated by the proposed method and compared with published and finite element solutions. He concluded that his method provided a better solution, especially near the boundary, compared to other methods for stress analysis of orthotropic materials. Determination of individual principal stresses in ortho tropic materials has been one of the important subjects. Prabhakaran [37], in his paper, reviewed the available de veloped methods and proposed a new procedure based on combining transmission and reflection techniques. He showed that the fringe patterns obtained by these methods can be used to determine either principal stresses or strains indi vidually. Simplification with accuracy of orthotropic photoelastic laws was the main concern of Chaturvedi's recent work [38]. He simplified the strain optic law to predict the isoclinic angle and isochromatic fringe values. The comparison of results obtained by this theory and some published results for a circular disk under diametrical compression showed that the 28 approximate theory is valid only for the cases where the material anisotropy is close to unity. In 19 82, Prabhakaran [39] employed photo-orthotropic elasticity to determine the maximum in-plane shearing stress of a bolted joint and recommended application of this method in the future studies. Chandrashekhara and Jacob [40] investigated the appli cation of numerical methods for separation of stresses in photo-orthotropic elasticity. Photoelasticity was used to determine the principal stresses at the boundaries. The finite difference method was employed to solve orthotropic elastic ity equations and the defined stress parameter was determined at any mesh point. The stress parameters were then combined with photoelastic data obtained at the mesh points to deter mine the individual stresses at each mesh point. They per formed two different tests to examine the accuracy of the proposed numerical method. In the first case, a half ortho- tropic plate under a concentrated load was investigated. The computed stress parameters were compared with the exact theory solution and the error was found to be less than 0.1%. For the second case, an orthotropic square plate under symmetric partially distributed load was considered. For the two loading conditions, load parallel and load perpendicular to the fibers, the stress distribution along the horizontal axis passing through the center of the square was calculated. The comparison 29 of the results with the theoretical results showed a good agreement. Therefore, it was concluded that this method can be used for stress separation in orthotropic materials. Arcan et al. [41] designed a plane circular speciman with antisymmetric cutouts. They proposed that the model can be used to produce a state of uniform stress in the gauge section of the model which is the shaded area in Figure 10.
«
Figure 10. Arcan et al, model [41] 30
When the specimen is loaded in the p^ direction, the gauge section is in the state of pure shear and the principal stresses are directed at + 45° with respect to AB. They recommended the application of this model for determination of the shear modulus of elasticity and f^.^, the shear optical constant. Marloff and Gabrielse 142] extensively analyzed Arcan's model and confirmed that the specimen can be used to deter mine the material constants in the state of pure shear loading. Arcan's model will be used in this experiment to deter- mine and Existence of residual stresses in ortho-photoelastic materials is. an unavoidable problem in the employment of these materials. Hyer and Liu [43] modified the stress- optic law to account for the residual fringe value of the materials. Material provided by Daniel et al. [16] was em ployed in their experiments. Besides the application of the tension specimens, they employed four point bending specimen to determine the optical properties of the material. They observed variation of the value of residual stresses in 0, 45 and 90 degree models. They could not find an explanation for this particular behavior. 31
Orthotropic Fracture Mechanics Works
The study of the combination of a stress field and a crack which would cause the crack to grow, opened a new field in mechanics which is called fracture mechanics. Up until the early 1960s, the area of study was concentrated on linear elastic isotropic materials. With the advancement of technology and application of orthotropic material in the modern industry, establishment of a theory for analysis of fracture criterion in orthotropic materials becomes of interest. Irwin 144], in 1962, adapted Westergaard's stress func tion I45J to orthotropic materials. Wu [46] later used the modified equations and developed a series of equations to interpret the stress distribution in the vicinity of a crack in special orthotropic materials. Wu also used Lekhnitskii's stress functions and developed another set of equations. He proved that these two sets were identical equations. In order to verify the validity of these theoretical equations and investigate the existence of the following empirical fracture equation:
= 1 (2) ^Ic ^2c where and a-^e the mixed mode stress intensity factors 32 at fracture and and ^2c the critical stress intensity- factors for simple tension and pure shear stress, respective ly. He designed a special gripping device to apply simple tension, pure shear and tension-shear combination on plates of 1/16" thick balsa wood. The cracks were aligned with the grain direction. The experimental results proved the accuracy of the equations and he found that m and n exponen tials are 1 and 2, respectively. Wu 147], in 1964, made further investigations of the proposed orthotropic fracture mechanics law. Unidirectional fiber-reinforced composite material was employed for this investigation. He found that the roots of the auxiliary equation of the partial differential equation for the stress function are purely imaginary for the scotch-ply plates used in the experiments. He modified the proposed equation given in I46J to account for purely imaginary roots and ran the same kind of experiments with cracks along the fibers. They showed: 1. K^c 3.nd K2g are the material constants for symmetric and skew-symmetric loading and independent of the crack length. 2. The crack propagates in a straight line along the fibers and colinear with the initial crack. 3. The empirical equation presented in reference [46] is also valid for this case and can formulate a general fracture law for orthotropic materials. 4. Intermittent crack propagation can be observed in fiber reinforced materials, especially under skew- symmetric loadings. 33
Sih et al. [48], in 1965, developed a general equation for stress components and displacement in anisotropic material by using a complex variable approach. They simplified the general case of anisotropic material to a simpler case of an orthotropic material with the crack direction aligned with one of the principal directions of the material. A stress func tion was proposed and equations similar to those of Wu [46] for components of stress and displacement were developed. Delale et al. [49], investigated the problem of an in clined crack in an orthotropic strip. In this study, cracks were always parallel to one of the principal axis of the material. Several cases with cracks at different angles with respect to the orthogonal axis of the strip were investigated. These cracks consisted of internal or edge cracks. Analytical equations to explain and stress intensity factors were developed and numerical method was used to solve for and K2. Boron-epoxy composite was used as an example material and tension and bending were considered as loading conditions. They concluded that for the cases where is negative, the problem would turn to a crack contact problem and the solutions were not valid. These solutions though can be used for com bined loading conditions. QUO fang [50J used standard compact tension specimens to investigate the fracture of reinforced composite materials. Unidirectional and cross-ply fiber reinforced models with 34 cracks at 0, 45 and 90 degrees with respect to the fiber direction were investigated in this study. Time-load and time-strain diagrams for each case were developed and com pared with similar curves for a plexiglass models - By obtain ing the critical loads, critical stress intensity factors were calculated. The experiment revealed that the crack extension in fiber reinforced composites is collinear with the fiber orientation. Fiber delamination was observed under im pact loading, but its mechanism was not clearly understood by the author. It was observed that glass fiber composites are notch insensitive and become fully blunt at high strain rates. Goufang questioned the validity of fracture toughness as a material property and recommended further investigation of this subject for composite materials. Jurf and Pipes [51] introduced a crack in the gauge sections of Arc an et al's. [41] specimen and studied the interlaminar fracture of composites. They found that for the case of PQ loading (Figure 10). the shear stress distribu tions in the gauge section is not uniform.
Finite Element Works
Application of finite element method, FEM to LEFM (linear elastic fracture machines) in recent years has increased rapidly and has become one of the important tools for stress and displacement analysis. 35
Chan et al. [52], investigated the application of F EM to LEFM in 1979. This study showed the importance of the mesh size at the vicinity of a crack in the determination of stress intensity factors. They found that at the immediate vicinity of cracks, because of the existence of a singularity, finite element results are in error. In order to improve the finite element results, they recommended that the apparent stress intensity factor for a constant (polar angle) at several r (radial distances) from the crack tip be determined by using the available relationship between stress or dis placement and stress intensity factor. The curve of apparent stress intensity factor versus r then can be drawn and extrapo lated to r = 0 to determine the real stress intensity factor. Figure 11 shows a typical curve for a compact tension speci men . In this figure, at 8 = 0 was used for several nodal points. The accuracy of up to 3.5 percent of the exact value for Kj was achieved. It was also observed that the finite element always provided the lower bound of stress intensity factor values. Watwood 153] investigated stress intensity factors of center slotted panels under tension with the finite element method. He applied Chan's method, employing and 9 = 45°. The predicted value for was compared with the exact value and the error was about 10 percent. He recommended that instead of refining the elements, in order to save the 36
CL
0-05 0-10 r W
Fi-gure 11. Plot of normalized mode one apparent stress intensity factor vs. normalized distance from the crack tip [52] 37 computer cost, the strain energy method can be used. Parhizgar et al. [54], by using the combination of the finite element method and compact tension specimen, investi gated the validity of LEFM to orthotropic materials. Uni directional scotch-ply composites were used in this experi ment- The stress components at the vicinity of the crack were determined by the finite element method and used to determine the stress intensity factor for various crack lengths. Parhizgar used components of elements along 9 = 0 to determine the apparent stress intensity factors. The curve of apparent K versus r (radial distance from the crack) was extrapolated to r = 0 to estimate the actual stress intensity factors. The experimental results revealed that the concept of LEFM is valid for the case of a = 0, (fibers along the crack), and conditionally valid for a different than zero. It was also noticed that the fracture toughness for each case of a is independent of the crack length and almost constant. Therefore, the concept of the fracture toughness as a material constant was verified. Finally, he developed an empirical formula to relate (fracture tough ness of any fiber direction), to (fracture toughness for a = 0). 38
ORTHOTROPIC PHOTOELASTICITY
Photoelasticity is a fundamental method in stress analysis. Although this powerful method has been practiced for more than half a century, the application of this method to orthotropic materials was introduced in the late 1960s. Pih and Knight [24], by producing an orthotropic transparent photoelastic material and introducing an orthotropic optic law, pioneered the application of photoelasticity to ortho tropic materials. Later Sampson [25] applied the concept of the Mohr's circle to the optical components of an ortho tropic birefringence materials and proposed a new orthotropic optic law. It was just last year that Hyer and Liu [43] modified Sampson's ortho-stress optic law to account for unavoidable residual birefringence in these materials. Photoelasticity theory is based on change in the behavior of photoelastic materials under loadings. Maxwell [55] dis covered that the index of refraction of a photoelastic model observed under a polarized light changes when it is subjected to a system of loads. The photoelastic model behaves as a retarder and its optical properties change in such a way that the optical axes of indices coincide with its principal stress directions. Stress optic law can be formulated in the fol lowing form [56 , p. 411] 39 where: A = relative retardation of the light components A = wave length of the emerging light h = model thickness c = relative stress optic coefficient which is a material constant = principal stresses of the loaded model
The practical form of the above equation is
°l-°2 = ^ or N = where n = ^ = fringe order
f = y = the material fringe value
N = ^ = fringe order per unit thickness
For an element in the state of plane stress, the equilibrium of forces provides a relationship between the three stress compo nents in x-y direction and the principal stresses as follows:
cos 2 8 + #2 sin 2 8 (5)
sin 2 8 + Og cos 2 8 (6)
^1~^2 ^xy = 26 (7)
Now if we combine the stress optic law (Equation 4), and Mohr circle stress law, the stress optic law can be formulated 40
X (fiber)
Figure 12. Stress notation [25] as [24, p. 211];
(8) N f' +
It can be seen that there are three birefringence components which are associated with the three stress components. These components of birefringence which are vectorically combined can be defined as ;
(9)
"y = Cy/f (10)
"xy = ^xy/^ (11)
Equation (8) can be extended to orthotropic materials. In Figure 13, an element under stress and made of a uni directional fiber reinforced composite material with its 41
Figure 13. Mohr circle for birefringence [25] fiber direction positioned at angle 0, with respect to the principal stress direction, is considered. In this case, the three components of birefringence are defined as: Nx = Cx/fx (12)
Ny = Cy/fy (13)
Kxy = ?xy/fxy
By substituting N^, and of orthotropic material in Equation (8), the orthotropic stress-optic law for the state of plane stress can be postulated as :
N = (15) X y xy The graphical representation of Mohr circle of birefringence is shown in Figure 13. In the following, the relationship among the components of birefringence which shows the analogy to those of stress components are given [25, p. 211]: 42
= (N^rNy)- + (2N^)^ (16)
«I'Hz = i I
N -N = N COS 26' X y 2NI_ sin 28' = (19) N
Where N = observed birefringence = principal birefringence components 0' = the angle between the major principal component of birefringence and the x-axis
The optical angle 0' and the physical principal angle 0 are not the same and will coincide only if f^ = fy=f^ (iso tropic materials). By substituting N^, and in Equations (18) and (19) , the shearing stress and the net normal stress can be obtained:
N sin 26' (20)
0% °v Y" ~ £ = N cos 20' (21) X y
The stress-optic law in terms of principal normal stresses can be obtained by substituting Equations (5), (6), and (7) into Equation (8): 43
N = *^1 {I (cos 2 6 + ^2— sin 2 0) 2 1 2 ? f 9 (sin 0 + — cos 20) X-, +
[|^(1 - —)sin 28]2}l/2 (22) ^xy *1
Equation (22) is important in determining . Equation (22) for the state of uniaxial stress (a^ = 0) and for a fiber direction at an arbitrary angle 0 with respect to the load direction (Figure 14) becomes: f fa = f [ (cos^ 9 - — sin^ 0)^ + sin 20] (23) y V
Figure 14. Unidirectional fiber-reinforced composite under uniaxial loading 44 for the case of 0 = 45°, the equation can be written as;
(24)
It is obvious that f^^ is not f^ for uniaxial loading and in order to determine f EqEquation (24) should be used. It was mentioned before that the residual stress is an unavoidable problem in fiber reinforced composite materials. The cause of this residual stress is mostly due to the fab rication process. Because the thermoelastic properties of fibers"and the matrix are different during the curing state of production, the residual stresses arise in the material. Residual stress can also occur during the machining process. The residual stresses in orthotropic birefringent materials cause the residual birefringence and introduce errors in the experimental results. Hyer and Liu [43] employed the concept that birefringent materials have dielectric and magnetic prop erties and modified the stress-optic law of orthotropic mater ials to account for the residual birefringence. The proposed law is given below as:
where = the total fringe value = the residual fringe value 8^ = the residual isoclinic angle 45
It can be seen that in the above equation, when o = a = T =0, the total observed fringe order is the X y xy ' = residual fringe value N^. It can also be observed that in the absence of residual birefringence (N^ = 0q = 0), Equa tion (25) reduces to Equation (8). Equation (26) was derived to determine the total residual isoclinic angle:
sin(20j^) tan(20,j) = — (26)
where
0^ = the total isoclinic angle. When the residual birefringence is zero, Equation (26) reduces to:
tan(S^) = (27)
which is the same as the optical isoclinic 0' and independent of the stress level. The above equation can be obtained also by dividing Equation (20) by Equation (21). 46
FRACTURE MECHANICS OF ORTHOTROPIC MATERIALS
Fracture mechanics is the study of the behavior of a stressed or loaded member containing a crack or a flow. The stress field around the crack tip of a loaded, linear- elastic body, may be determined in terms of applied loads or stresses and the geometry of the cracked body. In order to analyze the stress field near the crack tip of elastic materials, it is important to define the type of crack tip deformation. The crack tip displacement can be divided into three modes as shown in Figure 15. 1. Mode I: The opening mode, is characterized by local displacement in which the crack surfaces move per pendicular to each other in opposite directions. The displacements are symmetric with respect to the x-y and x-z planes. 2. Mode II: The edge-sliding or shear mode, is associated with a displacement in which the crack surfaces slide over each other in a direction per pendicular to the line of the crack tip. The dis placements are symraetric with respect to the x-y plane and skew-symmetric with respect to the x-z plane. 3. Mode III: The tearing mode, is related to the move ment in which the two crack surfaces slide over each other parallel to the crack tip. The displacement is skew-symmetric,with respect to the x-y and x-z planes. In general, any crack tip stress field or deformation can be described by one, or the combination of, these three modes. There are two basic theories available to describe the stress field around the crack tip of a linear elastic body. 47
Figure 15. Stress intensity modes [56]
The first one is based on the stress analysis approach, in which the crack-tip stress field can be characterized by a single parameter K, the stress intensity factor. X is pro portional to the magnitude of applied stress and to the crack length, a. Two modes are shown in Figures (15a) and (16b), where an infinite cracked plate under normal and in-plane 48
Figure 16a. A crack under Mode 1 loading
Figure 16b. A crack under Mode 2 loading 49 shear are considered, respectively. For the case where the crack is under a symmetric loading, only the first mode is in effect and the mode one stress intensity is defined as [57, p. 33]:
= a/ira (28)
and
^11 °
For the case where the crack is under skew-symmetric loading, only the second mode is in effect and the mode two
stress intensity factor is given by [57, p. 34]:
= x/rra (29)
and Ki = = 0 In general, the stress field in front of the crack tip can be given in the following form [4, p. 34]:
K = f(g) -a-/a (30)
where f(g) is a parameter that depends on the specimen and crack geometry. The second approach is based on the Griffin-Irwin fracture theory in which the stress field can be described in terms of G, the energy release rate that is related to the change in the elastic-strain energy when the crack 50 extends. Since in this analysis the crack tip behavior was assumed elastic, K and G are directly related to each other and either one can be used to describe the crack tip stress field of a loaded member. Irwin [44] and Wu[46] argued that the linear elastic fracture mechanics method, in addition to homogeneous iso tropic materials, can be used for homogeneous orthotropic materials. A material with three orthogonal planes of symmetry is called an orthotropic material. A special ortho tropic material with two orthogonal axes of symmetry can be used to simplify the adoption of LEFM to orthotropic materials. Let's consider an infinite plate with a through-the- thickness crack in the x-direction. Assume this plate is made of a special orthotropic material with two orthogonal planes of symmetry and E^>Ey. The assumption of the crack in x-direction is very practical, because in many orthotropic materials such as wood and fiber reinforced composites, the flows are commonly along the fibers and also grow in the fiber direction. Figure 17 shows the described plane. The stress-strain relationships for two-dimensional plane strain are [58, p. 595]
(31)
(32) 51
Figure 17. Crack coordinate system
where
bii = l/E^ (34)
"12 = (35) 52
b,, = 1/Ey (36)
^66 = 1/G=y (37)
The compatibility equation of the strain field should be satisfied and is the same as the one for two-dimensional isotropic materials.
9^E 3^E
If we substitute Equations (31-33) into Equation (13), the compatibility equation for our orthotropic material becomes [56, p. 697]: 9 2 2 2 3^0% 3 c, 3 a
^66 3x3y ~ ° (^9)
The Airy stress function U, in the absence of body forces, satisfies equilibrium and compatibility and the stresses can be represented by [43, p. 4]:
a = (40) * 3y^ 53
Using Equations (40-42), Equation (39) becomes:
4 + ° (43) 3x 22 3x 3y 3y^ ^22 where
(2^12 + b66'/b22 = ^ I(E Y /2G„) xy' " V yx'1
bll/b22 = =y/2x
This problem is reduced to the solution of the fourth order homogeneous partial differential equation whose auxiliary equation is [ 58, p. 697] ;
+ 2[(b^2 + ^66^/^22^^^ + (bii/bgg) = 0 (44)
The roots are either complex or purely imaginary and in con jugate pairs 158/ p. 69]
= 2 + ig = a - is (45) Sg = -a + ig Sg = -a - ig where
28^ = (b22/bii)l/2 + (2b^2 + (46) 2a2 = (b22/bii,l/2 - + b55)/2b^^ 54
The general solution of Equation (44) is [46, p. 5]:
U = Z^(Ç^) + ZgCSi) ^ ZgfGz) + 24(52) (47) where = X + S^y
ïj_ = X + S^y
S2 = * + SzY I, = X + SaY
Wu [46] employed Irwin's version of Westergaard's stress function for special orthotropic materials and developed the stress equations to describe the stress field in front of the crack tip. The stresses in terms of mode 1 stress in tensity factors are [58, p. 700]:
1 a + X (2iTr) (cos 0 + sin 0)^^
1 B/a (cos 8 + Sg sin (cos 0T S^sin 8)^/^
1 •j } (48) (cos 8 + Sg sin 8)^^^ 55
Ky 1 a = TT? {Re[ ^ (2irr) ^ (cos 0 + S, sin 8) 1/2
] (cos 8 + Sg sin
+ f I*»!- (cos 0 + sin 8)^-/2
]} (49) (cos 8 + Sg sin 0)^^^
'xy = (*"^(cos 8 t Si sin 8)
ïTj]} (50) (cos 8 + Sg sin 8)
When only the mode II deformation exists, the stress distribu tions are given as [58/ P- 70]:
a = (- »el ^ (2nr)^/^ (cos 0 + sin 8)^^^
] (cos 8 + Sg sin 8)
+ g Iml- (cos 0 + sin 8)^' ^
J} (51) (cos 8 + Sg sin 0)^'^ 56
(2wr)^ ^(cos 0 + S, sin 6)^^^
]} (52) (cos 0 + S2 sin 0)^^^
{i Re[ ^ (2irr)^^^ ^ (cos 0 + S^ sin 0)^^^
+ 172^ (cos 0 + S_ sin 0)
~ 2a t Ï72 (cos 0 + sin 0)^/^
]} (53) (cos 0 + Sg sin 0)^^^
Wu found that for balsa wood and glass-fiber reinforced plastics that the roots of Equation (44) are purely imaginary. This is also the case in this study. The four roots are
Si = iBi S = -iB, ^ (54) ^2 " ^^2 57
Wu modified and superimposed Equations (48-53) to develop the crack tip stress distribution for these special types of orthotropic materials. The combined equations which repre sent the combination of symmetric and skew-symmetric modes are [58, p. 702]
Kj 3^62 ^2 cosfYg/Z) 6^ COS(Y^/2)
^ 3^^ sin(Yi/2) 82^ sin(Y2/2) ^772 ^^172—>-ox
(55) K , cos(y./2) B, cos(y-/2)
'y °71^ ^' ^775 ^T7r->
KT- , sin(y_/2) sin(Y,/2)
ê^Ô2 sin(Y-j_/2) sinfYg/Z) { wo TTo } ^ (2nr)l/2 ^2"^1 p^V2
K , 6- cos(Y?/2) B, cos(y-,/2) ,XX Z _X X "L /c"7 \ (2.r)l/2 G2-B1 p,!/: " p^l/Z wnere
Pj = (cos^ 9 + Bj sin^ 0) 58 and
tan Yj = $j tan 0 with j = 1,2. The term of Equation (55) was not present in Wu's formulation, and it was added to describe the effects of a far field stress applied parallel to the crack. It should be again noted that the x axis is a principal material axis and is aligned with the crack. As Corten (58) points out, stress intensity factors and are present in a man ner very much like the isotropic case but the stress, strains and displacement distributions are much more complex and they involve the elastic constants. In order to observe the dif ference of isotropic and orthotropic stress distribution, mode I stresses are given here [57, p. 32]
a (58) X (27rr)
a (59)
T (60) (2nr) 59
Photo-Orthotropic Fracture Mechanics
In the isotropic case, the photoelastic law and fracture mechanics equations are combined and no mechanical material properties are required to obtain the stress intensity factors. For the special plane orthotropic material, the material prop erties are needed plus the stress-optic law is much more complicated. To establish photo-orthotropic fracture mechanics law, by substituting Equations (55-57) in Equation 25, we obtain
+ cos (29j^)} 2
+ { + sin(292^}2}l/2 (61) ^xy where
C 1 ^ (2nr)l/2(g2-g^)
S>2 cosfYg/Z) COS(Y^/2) C2 — 62^2 I-
2 sin(Y^/2) GG sin(Y2/2) 60
$2 cos(Y^/2) cosfYg/Z) c. =
sin(Y2/2) sin(y^/2)
sin(Y-,/2) sin(Y„/2) ^6 " ^1^2^ 1/2 I/T"^ Pi . Pz
Equation (58) can be abbreviated to the following form
N = g[Kj,Kjj, r, 0 ,6^, 33 / f ^ 6 j^] (62)
For a given model ' f and 8^ can be considered constants. Thus, (59) can be rewritten as another func tion
N = g^CKi, Kjj., o^x, r, 0) (63)
The fringe order for a model is a function of the location of the point of interest, the stress intensity factors and the far field stress a ox 61
METHOD OF ANALYSIS
Equation (60) can be solved in two different ways:
1. Kj,, and are assumed to be known, therefore. Equation (63) can be used to determine the geometry of fringe loops of a given order. This study was performed by Zachary and Mojtahed [59] for orthotropic materials. 2. The geometry of the fringe loop is known and Equation (63) can be used to determine and a^^,. In order to determine the three unknowns, isochromatic fringe data collected at three arbitrary points can be used. These collected data (N^, 8^, Yj) with j = 1 to 3 can then be plugged into Equation (63) to produce three equations with three unknowns.
g.(Kj, Kji, = 0 (64) where j = 1, 2 and 3. The above equations are involved in a series of compli cated relationships and the closed solution is tedious. With the help of a microcomputer, these equations can be easily solved using a numerical method. The Newton-Raphson method was used in this research to solve Equation (64). 62
Newton-Raphson Method
Newton-Raphson method is an iterative method for solving differentiable equations. Using this method, initial esti mates of Kj, and are made and substituted in Equa tion (6 3). Since the initial estimates are in error, the equality of the equations can not be achieved. To correct the estimates, a series of iterative equations based on a first order Taylor series expansion of g^ are written in [19, p. 7].
3g. ox where the subscript i indicates the i^^ iteration step. In Equation (65), AK^, and are the corrections to the previous estimates. The corrections in the iteration process can be determined to give =0, and there fore, Equations (65) became
To solve for the correction terms, AK^, AK^^/ and Aa^^, Equation (66) is applied to Equation (64) and the result in matrix notation is : 63
-i-l 3gi 391 391 AK, 3Ki 3Kii 392 agg 392 AK (67) II 3Ki 3Kii 393 393 393 Aa.ox 3Kii
The corrected values of ^, and for the i+1 iteration are
(Ki)i^l = + iKj,
"'ll'i+l ^ '^11' (68)
'°Ox'i+l = <"ox'i + AKiI-
The iteration process ends when the correction terras are all less than allowable error values.
Method of Least Squares
By application of the least squares method to the solu tion of Equations (54), we will be able to use several arbi trary points instead of only three. This will sharply reduce the experimental errors. The least squares approach is simi lar to the Newton-Raphson method but this approach uses several 64 arbitrary points (in this case 20 points) to obtain a better fit of function g to the data points. When the iteration scheme provided by Equation (66) is applied to k equations of the form given by Equation (61), an overdetermined set of linear equations in terms of AK^, AK^^/ and Ao^^ of the following form is obtained [19, p. 9]
3GI 39L 39I
CD 3KII H AK, 92 92 ^2 AK 3KI 3KII 3*OX II (69) Aa ox
9K 9K 9K 9K, 3K 3a. II ox —I 1 Equation (69) in the shortened form is:
[g] = [a][AK] . (70)
To determine [AK], both sides of Equation (70) from the left are multiplied by [a] T , as follows
[a]^[g] = [a]^[a] [AK] (71) 65 or
[AK] = [C]"^[d] (72) where d = [a]^ [g] and [cj = [a]^[a]
The A's obtained from Equation (72) are used to correct the initial estimates of K^, and Miskioglu et al.'s least sauares program [20] which is the Fortran version of Sanford and Daily's program [l8j is employed to determine Kj, Kn, and Oox- This program which was written for isotropic materials was modified for orthotropic materials and used under the file name of SICRES. A second program, SRTEST, which was again adapted for orthotropic materials, was used to check the accuracy of SICRES. SRTEST generates 20 data points (r^, S^, N^) for provided values of and The generated data points were then used in SICRES to determine K^, and cr^^. The values obtained for ,
KIIT-T , and a ox from program^ SICRES were identical to those fed to SRTEST, and this proved that program SICRES runs accurate ly. It is useful to mention that in order to run program SICRES, an error value small enough for the accuracy required in this research was given to the program. The iteration process of this program is continued until the error values 66 go below the assigned value. The best estimates of K's and which best fit function g to the k (20) data points are finally obtained at the end of iteration process. This procedure was used for all experimental data collected through out this research. These two programs are listed in Appendix A.
Finite Element Method
The finite element method is a numerical technique which is used in this research to verify the accuracy of the experimental results. This method, with the employment of fast digital computers, has advanced and become one of the common methods used in structural analysis. Employing this method, structural parts are divided into small but finite elements. These elements are considered interconnected at joints which are called nodes or nodal points and super imposed onto a coordinate grid system. The nodal displace ments of each element are related to the nodal forces by the stiffness matrix. Finally, the displacement of the nodal points of each element is related to the external forces of the structure by a matrix which is composed of the matrices of all elements of the structure. Once the dis placement field is determined, the strain-displacement and stress-strain relationships can be used, respectively, to 67 determine the strains and stresses. Solid SAP is a static analysis program used for analyzing two and three-dimensional solid structures [60]. Solid SAP is capable of handling structural parts with orthotropic be havior and is used in this project. This program is written to analyze complex structures with linear elastic behavior.
Half-Fringe Photoelasticity
Burger and Voloshin [17] at Iowa State University com bined photoelasticity with a digital image analysis system to establish the method of half-fringe photoelasticity. This method provided the opportunity to use photoelastic materials with high material fringe value. Since the photoelastic material used in this research possesses high fringe values, this method was found suitable for the collection of ortho- tropic photoelastic data. In order to explain this method, let's go back to the principal of photoelasticity. In a circular polariscope, the intensity of transmitted light emerging from the analyzer for a dark field arrange ment is [56, p. 422]
I = K sin^ I (73) where I = light intensity K = a constant 68
A = relative angular retardation when A/2 = nir, where n = 1,2,3...,1 = 0 and no light is transmitted. This will cause a loci of points which are called the full order fringe patterns, N = ^ = n = 1,2,3... . For a light field set-up the light intensity is [56, p. 424]
I = K cos^ Y (74) and extinction occurs when
I = for n = 0,1,2,3... therefore, the observed fringe pattern, N = j = -^+n = -^, "I" ... are of j order and called the half order fringe pat terns. Half-fringe photoelasticity employs an image analysis system which can measure the light intensity. This system is capable of distinguishing 256 gray levels of a half fringe pattern. In other words, this system can presently only operate in the intervals of 0 to ^ to tt , etc. Figure (18) shows a typical plot of light intensity (I) versus fringe order (N). In this figure, a typical inter val used in a photoelastic analysis is shaded. With HFP, the light intensity interval of a loaded model that is to be investigated, is digitized and converted to fringe orders. 69
I K sin
A/2
0 1/2 1 3/2 2 N
Figure 18. A typical plot of light intensity vs. fringe order [17]
Eye Com System
VThen HFP is used, the light intensity of an interval is digitized and converted to fringe orders by an optical- computer system which is called "Eye Com". The Eye Com, Picture Digitizer and Display system pro vides both input and output function for the processing of an image [61]. An Eye Com system employed in half-fringe photo- elasticity set-up is shown in Figure 19. When a photoelastic model is analyzed by an Eye Com system, the Eye Com Scanner is the source of the image data that is to be processed and analyzed. The scanner. N \ \ HIGH-RESOLUTION DIGITIZER RADIATION EYE COM PHOTOELASTIC 640 X 480 X 8 BITS SOURCE SCANNER MODEL ) / Z-REGISTER \ /
1-POLARIZER 2-QUARTER-WAVE 3-ANALYZER REAL-TIME KEYB OARD DIGITIZER 640 X 480 X 8 BIT
r INTERFACE DISPLAY LSI-1 CONTROLLER MONITOR PRINTER COMPUTER REFRESH MEMORY 640 X 480 X 8 BIT 256 GRAY LEVELS DOUBLE DENSITY HARD COPY DUAL FLOPPY DISK UNIT
Figure 19. HFP system [19] 71 through a special vidicon television camera tube ^ accurately senses and scans the model. The produced image, which serves as the input image to the system, is then processed by analog circuits in real time and can be seen on the Scanner Display immediately. Thus, the Scanner Display shows the image just as it will be digitized. The scanned image is divided into 480 lines and each line consists of 640 elements which are called pixels. Each of the 640 x 480 pixels can be randomly accessed by the High-Resolution Digitizer. The digitizer is capable of digitizing each pixel with 256 gray levels of resolution. The whole image can be digitized in 21 seconds and with the application of special controls, the digitizing time can be reduced to 2.4 seconds. When a pixel is digitized, its posi tion (X, Y), and its brightness (2) are registered by the digi tizer cursor and saved in the computers' memory for future use. For high-speed image analysis, an optional Real-Time digi tizer can be used. By using this option, the entire picture (640 X 480 pixels) is digitized in 1/30 of a second. The data are collected at such a high rate that the data can not be transferred to the computer all at once. In order to solve this problem, the data are stored in the refresh memory of the system, where it can be used later by the computer. The Eye Com system is connected to an LSI-11 computer through an interface. Through the computer, one can communi 72 cate with the system, run programs, save and print the col lected information.
HFP-Eye Com Theory
When the Eye Com is used in HFP analysis, the light intensity of the digitized image is given by [gl/ p. 19]
Z = (75) where Z = value of the digital output I = light intensity K = a proportionality constant Y = slope of the vidicon sensitivity curve
For the maximum brightness. Equation (75) becomes
"max = KImIx
Dividing Equation (75) by Equation (76), gives
1 = (g^)^ (77) max max
'For the liqht intensity of a dark field set-up of a circu lar polariscope. Equation (73) can be rewritten as [17, p.
3]
I = IQ sin^(N7r) (78) 73 where Ig = a reference light intensity N = fringe order For maximum light intensity. Equation (78) becomes
Dividing Equation (78) by Equation (79), gives
I _ sin^ (N ) (80) ^max sin^(N IT) max Now, by combining Equations (77) and (80), the relationship between the digital grayness level of the light and fringe order can be established as 1 2 , Z = sinWT) (81) max sin
For the maximum workable fringe in HFP, N = i. Equation 81 becomes 1 Y 2 (-2^—)2 = sin^(NTT) (82) max or, 2 Y - = [sin (Nu)] (83) Z max Finally, the fringe order from Equation (82) is _1_ ' N = isin"^[(2^—)2Y] (84) max When is measured at N = 1/2 and y is known from 74 the calibration procedure, the fringe orders of an interval can be determined precisely. Calibration of y can be done easily by employing the Tardy compensation method. Taking the logarithm of Equation (83) gives
log —) = Y log [sin^ (NIT) ] (85) max Equation (85) is of a straight line form
y = yx + B (86) where B = log A = constant S should be zero but due to experimental errors, will exist as a small number. A will be nearly equal to 1. Equation (83) can be rewritten as
^7 = A[sin^2 (NTT) ]^ (87) max and, N = I sin~^[(^^^)^] (88) max Equation (88) is used in a program under the file name of CALI [20] to determine y and B. Using this program for a dark field polariscope, the Tardy compensation can be per formed by measuring the light intenstiy of an unloaded model. This can be done by registering the gray level of a uniform field in the model for each setting of the analyzer while it 75 is rotated between 0 and w/2. For each gray level, the equivalent fringe order is provided by the operator. CALI fits the best line to the data points, and calculates y and B.
Object of Study
The primary objective of this study is to utilize orthotropic photoelasticity in the investigation of mode I and mode II crack tip stress intensity factors. Since flaws and cracks are mainly oriented along one of principal direc tions of orthotropic materials, appropriate models with crack along their weak and strong axes under mode I and mode II loading are analyzed. Finite element solutions are used to verify the experimental results. 76
EXPERIMENTAL PROCEDURE
The implementation of the described research proposal requires a series of preparations. A good transparent ortho- tropic material, thin enough for use in plane stress photo- elasticity, and with the least number of flaws for stress in tensity analysis are requirements. Experiments have shown that it is impossible to obtain a good fringe pattern with more than
one fringe without delamination or creep occurring. TO overcome these obstacles, a good transparent orthotropic material must be produced and a system should be adopted to measure less than one fringe precisely. The material difficulty was solved by obtaining a 13 in. X 13 in. sheet of fiberglass/epoxy from NASA. This material that was produced at IITRI [16] is the best on the market. The
material is a unidirectional glass/epoxy laminae and is 0.09 to 0.10 inch thick. In the production process, care was taken to match the indices of refraction of fibers and the resin. The unidirectional composite was produced by impreg nating 37 plies of unidirectional fibers in Maraset 658/558 epxoy resin, using special techniques and equipment. The second problem was addressed by using half-fringe photoelasticity. When this method is used, fringes of up to one-half can be measured precisely. Since only one 13 in. x 13 in. sheet of the above material 77 was available to run the entire experiment including the material's calibration, care was taken in choosing the experi mental and calibration models.
Loading Apparatus
Another major concern of this project was to find a loading frame and an appropriate load cell for it to suit the half-fringe photoelasticity set up with the minimum cost. The load frame was designed and built at the Iowa State Uni versity Research Institute Machine Shop. This frame, shown in Figure 20, is suitable for the educational polariscope made by Photoelastic Inc. and used in the HFP set up. The frame has two sets of arms of different lengths which enable it to be used for models of different dimensions.
The load cell design To measure the applied load, a load cell capable of measuring loads up to 100 pounds and suitable for the loading frame was designed and built. Aluminum alloy 6061-T6 with = 40,00 0 psi and thickness of 1/2 inch was used for the element of the transducer. The dimensions of the load cell are shown in Figure 21. A factor of safety of two was used in the design. In the design of a transducer, choosing strain gages of proper size and characteristics constitutes an important Figure 20. The loading apparatus 79
il/2">Hl/2">- -1 1/4" B» 1/2" 1/2" -1 1/4"- '•py 1—r GAGES I I ^gases^ I I I—I 1 r = 1/4" J izz r = 1/2"
THREADED HOLE 4 1/2"
Figure 21. The load cell dimensions 80 factor to its accuracy. Fatigue life, creep resistance, and the range of working temperature are some of the factors in a design decision. These factors also must be con sidered for the adhesive used in bonding the gages to the element. Considering the above requirements, four EA-06-125TG-350 strain gages and M-Bond 610 adhesive, all products of Micro- Measurements Corporation, are employed here. The gages were glued to the element by the adhesive and cured in an oven according to the cure cycled specified for M-Bond 510. Wires are then soldered to the gages and the transducer was connected to a strain gage indicator to produce a full-bridge circuit (Figure 22). The load cell which is capable of measuring tension and compression loads was finally calibrated under tension and compression loadings. By increasing the load from 0 to 120 pounds, several strain measurements were recorded. A strain-load curve was plotted and least squares method was used to fit the best line to the data points. The slope of 36 y/pound was obtained. With a strain gage indicator capable of reading ly, the load cell can measure loads as low as 0.03 pounds (13 grams). Figure 23 shows a typical calibration plot of the load cell. The accuracy of the load cell was checked by calibrating it at the end of all experiments, again, the same calibration curve was obtained. Figure 22. The load cell 82
5000
c 4000 C
3000
LU S o 2000 SLOPE = 36 v/lb
1000
0 20 40 60 80 100 120 LOAD P, Ib
Fig. 23. Load cell calibration 83
Calibration of Optical and Mechanical Properties
For the application of transparent unidirectional composites in photoelasticity and plane stress analysis, the material's optical and mechanical properties have to be determined. If the x-y system is assumed to coincide with the princi pal material axes of orthotropic material (fiber and normal to the fiber directions), the required optical properties are f^^,
^22' ^12 the material properties are ^22' ^12'
^21' =12- To determine these properties, three standard tension specimens of the shape of Figure 24 with fibers along, perpen dicular, and at 45° with respect to the longitudinal axis were cut and machined from the sheet of material.
Determination of the optical properties For the determination of the optical properties, the load- fringe order plot for each of the three models were developed separately. Figures 25 to 27- Employing the least squares method, slopes of these lines are determined and used in c h f = —^ to estimate f,,, f_- and f.^. Equation (23) for 0 = a N 11 22 45 45°, reduces to [25, p. 254] LONGITUDINAL TRANSVERSE GAGE GAGE
1/2 1/2
1 1/2 7 1/2
Fig. 24. The tension calibration model 85
140
120
100
a. Q O< —I
W = 833 Ib/fr-in
0.0 .2 0.3 0.4 FRINGE ORDER, N
Fig. 25. fii calibrations 60 —
452 Ib/fr-in.
-0. 0.2 0.3 0.4 FRINGE ORDER, N
Fig. 26. ±22 calibrations 87
100
a.
384 Ib/fr-in.
0.0 0.1 0.2 0.3 0.4 0.5 FRINGE ORDER, N
Fig. 27. £45 calibrations 88 and this equation was used to determine f^2• Residual fringes varying from 0.07 to 0.18 were observed in all models during the entire period of experimental process. The time varying residual fringes were not the same for these models. The cause of these differences is not clear to the author and also to Hyer and Liu [43]. The optical proper ties are tabulated in Table 4 and compared with those ob tained by Daniel et al. [16] and Hyer and Liu [43]. Arcan's circular specimen [41] was also used to deter mine f^2' ^ cutout circular model with the gage length of 0.46 inch (Figure 28) was loaded and the load-fringe order curve for the state of pure shear loading was developed (-Figure 29). The slope of the straight line is determined by least squares method, and used in the following relationship [26]
to determine f±2-
^^12^ fl2 = (90)
where h is the thickness of the model. The value of f~ 416 Ib/fr.in. was obtained by this method.
Determination of the mechanical properties The mechanical properties were determined by attaching four strain gages (one longitudinal and one transverse on each side) to the three tension specimens (Figure 24). Stress- strain diagrams were developed (Figures 30 to 32), and the 89
Table 4. Optical constants Optical ^ Experimental Daniel's Hyer's constants values values values
833 834 849 452 458 385
384 398 - 392 406 319
^All in-lb/fr.in.
Figure 28. Cutout circular specimen 90
CL O c o—J
ÛN = 416 Ib/fr-in.
0.0 0.10 0.20 0.30 0.40 0.50 0.60 FRINGE ORDER, N
Fig. 29. fjjy calibration using Arcan's model 91
200
180 O P vs £l A Ey VS EL 160
140
120
100
80
60 = 5.355 X 106 Psi
40 V12 = fx ~ 0.30
20
0 100 200 300 400 500 600 700
LONGITUDINAL STRAIN E^, pin./in.
30. Strong axis mechanical properties 92
L O P VS EL A e-j- VS E|_
1.732 X 10^ Psi WtG[
100
80 5
60 ^ < DC H- (/)
LU OO ÛC LU to> z <ÛC
200 400 600 800 1000
LONGITUDINAL STRAIN , pin./in.
Fig. 31. Weak axis mechanical properties 93
800 .
400 LU
200 <
200 400 600 800 1000 1200 LONGITUDINAL STRAIN yin./in.
Fig. 32. 45° mechanical properties 94 mechanical constants '^21' ^45' "^12' ^21' ^45 *^12 were obtained from their corresponding diagrams. The fol lowing equation was used in the determination of G^2 [14, p. 66].
(_î i ^ ^45 ^11 ^22 ^11
The experimental mechanical constants and their corresponding values obtained by Daniel and Hyer are given in Table 5. The cutout circular specimen was also used to deter mine G^2' installing strain gages at 45° with respect to the load direction. The value determined for G^^ is 1.0 74 X 10^ psi, which is too large in comparison with the reference values. This inflated value may have been caused by a small gage section, nonlinear stress distribution in the gage section or strain gage positioning and its size. The optical and mechanical constants obtained from tension specimen are in good agreement with the experimental results of references 16 and 43 and the expected values using the analytical equations of Sampson 125], and Pih and Knight [24] Therefore, these values are used for all calculations in this research. 95
Table 5. Mechanical constants Material's Experimental Daniel' s Hyer's constants values values values
^1 5.355 5.540 5.400 %2 1.732 1.670 1.780
^45 1.840 1.540 - 0.660 0.520 0.570 ^2 ^12 0.300 0.300 0.300 0.094 0 .080 0.090 ^21 0.365 0 .400 45
and G in (10)^ psi.
Experimental Models
Standard compact tension specimens of Figure 33 are employed in the experimental determination of mode 1 stress intensity factors along the strong and the weak axes of the orthotropic material. The advantage of using compact tension specimens can be summarized in the following: 1. These models are small and of a rectangular shape, therefore, a minimum amount of material is used for the experimental investigation. 2. Only one specimen is needed in determining stress intensity factors for each fiber direction. Several crack sizes can be introduced into each model, thus, material and model preparation time are held to a minimum. 96
1.25 W
0.275 W
B = THICKNESS W = 2 INCH
Figure 33. Compact tension specimen
3. Since the model is small and symmetric, with respect to the crack axis, a great deal of computer costs and grid preparation time can be saved. A strip with fibers at a 45° angle with respect to the longitudinal axis is used for the determination of mixed mode stress intensity factors (Figure 34). The crack is aligned with the strong axis. •4"-
Ô7r r 0.5" 45°
W "1 vo 0.75" -J FIBER DIRECTION _1
0.75"
Fig. 34. The strip model dimensions 98
Preparation of Finite Element Models
Finite element model of compact tension specimens Development of finite element models for all the speci mens used in the experiments is required for the verification of the experimental results. Displacements of compact tension specimens, where the fibers are either along or perpendicular to the crack, is symmetric with respect to the crack axis. Therefore, the finite element model consists of only one-half of the specimen and this must also be used to complete the modeling process. An orthotropic finite element model having a near crack field mesh size of 0.01" was designed to determine the near crack tip stresses. The computer plot for a half specimen finite element model with a/w = 0.4 is shown in Figure 35. The mesh with quadrilateral elements, has 60 4 nodes and 545 elements. An enlarged plot of crack tip elements is shown in Figure 36. To account for the symmetrical boundary conditions, all of the nodes between point A and B (crack tip) of Figure 35 were fixed in the direction normal to the crack. Point A was also fixed from moving in the direction parallel to the crack and this eliminated the rigid body motion of the model. A dummy load of the magnitude of one hundred pounds was applied to the model and was analyzed by the solid SAP I— I X CJ
7Î\
rrn D Qlir (J m
Fiqure 35. Finite element model, a/w=0.4 (.)
CJ
lO cj
Figure 36. Crack tip elements 101 finite element program. In order to check the accuracy of this model, a second model having a near crack field mesh size of 0.02 inch for a/w = 0.4 was developed. The computer plot of the mesh of this model and its crack tip elements are shown in Figures 37 and 38, respectively. This model has 416 nodes and 367 ele ments. The same dummy load and boundary conditions were also applied in this case. The plot of stress versus the radial distance (r) on 0 = 45° line for both mesh sizes were developed (Figure 39). The plots show little difference (less than 1%) in the results. Finite element models with the crack tip mesh size of 0.01" for the crack sizes a/w = 0.5, 0.6, and 0.7 were also developed and analyzed by solid SAP under the same load and boundary conditions.
Finite element model of the strip specimen The crack of the strip specimen is oriented at 45° angle with respect to the longitudinal axis and along the fibers. This case does not provide a symmetrical displacement with respect to the crack axis, therefore, a full finite element model is required for the stress study. A finite element mesh with 1524 nodes and 1443 elements is developed for the strip. Figures 4 0 and 41 show the finite element mesh for the strip m d
(D (o to Xd"^ ΠI >-
CO d"* (T) I—I X CE I >-
a o -0. 05 0. 02 0. 09 0. 22 0. 29 0. 35 X-AXIS Figure 38. Crack tip elements for mesh size = 0.02" 104
3200 a/w 0.4 0 450
• MESH SIZE 0.01", y 2800 — = O MESH SIZE 0.02", ay • MESH SIZE 0.01",
• MESH SIZE 0.02", Of VS R 2400 — 1 A = A MESH SIZE 0.01", •^xy vs R = A MESH SIZE 0.02", fxy vs R
0.0 0.02 0.04 0.06 0.08 0.10 0.12
RADIAL LOCATION r, INCH
Fig. 39. Stress comparison for two mesh sizes Û. 00 1.00 J. 00 y. 00 4.00 b. 00 G. 00 7.00 8.00 X nxi;,
Figure 40. Finite element model of the strip specimen 106
' Ml ! M I
nwTviYYîvrwîvn rr !r X cr (T I c-\J 1> s -L:3 TT Ti I rr A^l ^ l /-\ \
I I I I I 1 I -1 1 1 1 xc:r I I /I
c c 1 I r
X-AX ;
Figure 41. Crack tip elements for the strip model 107 specimen and its crack tip elements. The boundary conditions must be imposed on this model too. In Figure 40, points A and B were fixed in X-direction. Point A was also fixed in y- direction to prevent any rigid body motion. To check the accuracy and effects of the boundary conditions, a model with 181 nodes and 144 elements was developed. Solid sap program was run and the boundary condition effects on crack-tip stresses were of a negligible size. Special care was taken in the development of all models to keep the mesh band width small to save computer time.
Data Collection Method
A program with the file name of COLF, developed at Iowa State University [20] was used to collect the experimental data. When the upper and lower radial angles lower and upper r/a (radial distance/crack length) ratios with respect to the X-axis are given to the program, it auto matically collects 20 data points (N, r, and 9) along five radial lines (Figure 42) and saves them on a floppy disk. Each data point is the average light intensity of nine pixels surrounding the cursor. In order to minimize effects of the random noise caused by electronics, sixteen consecutive pictures of the same photoelastic field are averaged and saved in the refresh memory for data collection. 108
DATA POINTS
TYPICAL
Figure 42. Data collection points
Test Sequence
Two compact tension specimens, one with a crack along the fibers and the second one with a crack perpendicular to the fibers, a strip with fibers and a crack at 45° with respect to its longitudinal axis, were studied with HFP system. A crack was introduced in each of the compact tension models and the crack tips were sharpened by a very thin disk saw (0.0 06") to produce the crack length of a/w = 0.4. Each model was 109 installed in the load frame separately and was viewed in a diffuse field polariscope by an Eye-Corn scanner. Mono chromatic sodium light was used as the light source of the polariscope. The experimental set up consisting of the light source, load frame, strain gage indicator, and Eye-Com system is shown in Figure 43. The following procedure was employed for each compact tension specimen to collect the experimental data and deter mine the stress intensity factors and the far field stress. 1. Program CALI was used to calibrate the system by using the unloaded model. The calibration values Y and B (Equation 88) were saved on a floppy disk for later use. 2. The model was loaded to obtain less than half a fringe pattern at the vicinity of the crack tip and the applied load was recorded from the indi cator. A typical crack tip fringe pattern is shown in Figure 44. 3. Program COLF was employed to collect the experimental data. This program uses the calibration values and collects 20 data points. 4. The upper and lower r/a, and 6 and 9^ was deter mined by using program COLF. 5. The model was loaded and calibrated three separate times and each time 15 sets of 20 data points with arbitrary upper and lower angles were collected. The collected data are saved on a floppy disk under different file names.
6. Program SICRES was used to determine Kj, Kji and Qqx for each 20 data points. These 45 values of Kj, K%% and Oqx were averaged to determine the stress in tensity factors and far field stress (K^, and Figure 43. Experimental set up Ill
Figure 44, Typical fringe pattern 112
7. The experimental values were finally compared with a finite element solution. Solid SAP was used to de termine the stresses for elements around the crack tip. Stresses of 30 elements were used in a program written in SAS (Statistical Analysis System). The values of Kj, Ijj, and Cqx were determined through a multiple regression analysis. 8. The experimental and theoretical values were compared. After satisfactory results were obtained, the crack in the model was elongated and procedure 1 to 8 were repeated for each new crack length. Crack length of a/w = 0.4, 0.5, 0.6, and 0.7 were studied for both compact tension specimens. The strip specimen was only studied for one crack length. A crack of the length of 0.425" along the fibers (.45° with respect to the longitudinal axis) was machined into the speci men using the same procedures used for compact tension speci mens. Steps 1-8 were also used in this case to determine the mixed mode stress intensity factors and the far field stress. It is important to mention that the accuracy of the half-fringe-photoelasticity method was determined before applying it to orthotropic materials. The examination of HFP is given in Appendix B. 113
RESULTS AND DISCUSSION
Compact Tension Specimens
For the determination of stress intensity factors, the data collection zone has to be determined. In order to determine this region, upper and lower r/a (radial distance/ crack length) and 0g and 9^ (Figure 42) were required. To de termine r/a ratios, Schroedl and Smith curve (Figure 6) had to be established. Several sets of data for 6y and 9^ between IIQO, -110° to 165°, -165° were collected. Figure 45 shows the graph of normalized Kj vs. r/a for the model with the crack normal to its fibers and a/w = 0.4. The upper and lower angles were limited to 165° and -165° respectively. Each experimental data point represents the averaged value of Kj obtained from 20 data points on five lines, 4 points per line, collected in the range of r/a + 0.01. The results show that for each of these data a line can be fitted into the data points between r/a = 0.03 - 0.08. In the process of data collection, the data points could be observed on the televi - sion screen. Therefore, any collected data in a region with defects could be detected easily. During this investigation two defects (white spots in Figure 44) were detected in the vicinity of r/a = 0.10 and 0.13. The effect of these de fects can be observed as sharp deviations of the K values in those regions. 114
PHOTOELASTIC FINITE ELEMENT
Q.
CO
0.00 0.03 0.06 0.09 0.12 r/a
Figure 45. Normalized Kj vs. r/a for 9 = 165, -165 115
The accuracy of the above range was investigated by employing a finite element solution. Equation (56) for finite element data along 8=45° line was used to evaluate Kj. Normalized Kj vs. r/a for this case was also plotted on Figure 45. The comparison of these plots show a good agreement. Therefore, it was decided to collect data for r/a =0.03-0.08 and limit 9 to +160°. Upper and lower limits of 0 was chosen in a way to collect data from the whole region. In Figure 46, the locations of the data lines with respect to the crack direction are shown. During the tests, the residual fringe orders ranged from 0.03 to 0.18 over the period of the project. At the time of a test, the up-to-date residual fringe value was used in Equations (62) and (63). The larger residual values were disturbing since the HFP method involved using less than 0.5 fringes. Thus, the fringe order due to load was limited by the residual fringe order. It is felt that better results can be obtained with larger loads which reduces the effects of the residual fringe orders. The results for the case where the fibers are normal to the crack longitudinal axis are presented in Figures 47, 48 and 4 9. Both the finite element and the photoelastic results are plotted versus the nondimensional crack length. The normalized 116
Figure 46. Data line locations 117
Kj plot. Figure 47 shows a very good agreement between the two methods. At a/w=0.6, a localized flaw near the crack tip affected the results. The load applied to this crack length case was 6 lb., whereas all other cases were run with an 8-lb. load. Since this is a symmetrical case, Kjj should be zero. Figure 48 indicates that both methods give small values for Kii with the photoelastic results being essentially zero.
Figure 49 shows the far field stress Cqx versus crack length. The far field affect must be separated from the residual stress field in the photoelastic analysis. If the values of the residual fringe order is in error or varies from point to point near the tip, then the calculated will also be in error. The finite element data also indicated that changes from point to point near the crack, thus the variation in results in Figure 49 are considered reasonable under the circumstances. It was observed that by limiting the upper and lower data collection angles to 15 0° and -150° the photoelastic results can be improved further. Overall with the fibers normal to the crack, crack aligned with the weak principal axis, the results were very good and gives hope that the photoelasticity can be used in such future studies. The next case has the fibers aligned with the crack. The same procedure was followed to determine the r/a and 9 ranges. 118
O PHOTOELASTIC • FEM
Q.
5ca
O—
0.4 0.5 0.6 0.7 a/w
Fig. 47. Kj, crack aligned with weak axis 119
O PHOTOELASTIC i 20 — • FEM
Cû
0.4 0.5 0.6 0.7 a/w
Fig. 48. Kji, crack aligned with weak axis 120
-20
çr-
-40
o PHOTOELASTIC • FEM \
-60 0.4 0.5 0.6 0.7 a/w
Fig. 49. t^ox crack aligned with weak axis 121
The load had to be reduced to 5 lbs. compared to the 8 lbs. in the above case to assure that the crack would not propagate. Because of low level loading and the direction of fibers with respect to the load, the fringe loops were much smaller and more smeared than the previous case and the Schroedl-Smith curve could not be constructed. But, the experimental data showed that the best result can be obtained in the range of r/a=0.015 to 0.05. It was also found that the best results were obtained when the data was taken between 150° and -150°. The results are given in Figures 50, 51 and 52. These results are not nearly as consistent as in the previous cases. The trend in the Kj versus a/w are the same for the finite element and the photoelastic analysis, however, the photoelastic value is as much as 2 times the finite element result at a/w= 0.4. One effect that has not been completely eliminated is the load level. For instance at a/w=0.7, the photoelastic model gave a normalized Kj value of 72 for a 2-lb. load, but for a 5-lb. load this reduced to 52. Thus, there is an indica tion that the low level of fringe order on top of the residual fringe order is causing errors in the photoelastic results. Further tests are needed to completely analyze this problem and it is anticipated that fringe orders greater than a half fringe will be required to give accurate results. For a/w=0.6, the model broke during the machining process; therefore, no photoelastic result could be obtained. A spare model with an 122
O PHOTOELASTIC • FEM
Q.
CQ
0.4 0.5 0.6 0.7 a/w
Fig. 50. Kj, crack aligned with strong axis 123
20 O PHOTOELASTIC • FEM
0 I
-20 —
0.4 0.5 0.6 0.7 a/w
Fig. 51. Kji, crack aligned with strong axis 124
O PHOTOELASTIC • FEM
Q. 9r- oX o
1 n 0.4 0.5 0.6 0.7 a/w
Fig, 52. Gox, crack, aligned with strong axis 125
initial a/w greater than 0.6 was machined to a/w=0.7 to determine the photoelastic value of Kj for a/w=0.7. Figure 51 again shows that the Kjj values are small. Figure 52 illustrates that the values have large dif ferences between photoelastic and finite element methods. The comments for Kj values are also applicable to this problem.
The Strip Specimen
For the determination of mixed mode stress intensity factors, the strip specimen with a 45° edge crack (Figure 34) was investigated by both photoelasticity and finite element methods. To obtain the experimental data, the same procedure as mentioned for compact tension specimens was used. The obser vation of the fringe pattern on the TV screen showed that the fringes in this case were severely smeared and distorted. More than 200 data sets under different loads, and for various angular and radial positions were collected. These data sets later were analyzed by program SICRES to determine Kj, Kjj and Values of stress intensity factors and far field stress did not converge for any of the collected data sets, thus , Kji and could not be determined for this case. It can be observed that the asymmetrical position of fibers and the crack 126
with respect to the load direction severely distorts the fringe pattern. This phenomenon with less severity has also been observed in the photoelastic study of mixed mode stress intensity factors using isotropic materials.
Smith et al. [62], and Smith and Smith [63] experienced a large scattering among the photoelastic experimental results during the investigation of mixed mode isotropic stress inten sity factors. Errors up to 12% in comparison with the theo retical values of stress intensity factors were reported in the above investigations. Kar [21] also experienced severe scattering and large errors in his investigation of mixed mode stress intensity factors of glass strips with slanted cracks. It can be concluded that in the determination of stress intensity factors, photoelasticity in general and for orthotropic material specifically is currently not an accurate method in mixed mode problems. Other methods should be inves tigated and further development of isotropic and orthotropic photoelastic methods is needed.
Finite element method was also employed for the stirp model. The finite element model under a dummy load of 100 lbs. was analyzed by the Solid SAP program. To determine the stress intensity factors and the far field stress, the finite element stresses for elements along the crack direction were used in equations (55-57). For each element, values of apparent Kj,
Kjj and were determined. The normalized values of Kj, Kjj 127
and were then plotted against dimensionless r/a and extrapolated to r/a=0 to estimate the true values of stress intensity factors and the far field stress. Two different models, one with the crack along the fibers and the other with the crack perpendicular to fibers were investigated.
In Figures 53 and 54 the plots of normalized values of K^,
Kjj and vs. r/a for the model with its crack along fibers is shown. The ratio Kjj/Kj=0.52 and the value of Oqx BW/p=
-4.45 were determined from these graphs. Similar plots for the model with its crack perpendicular to fibers are given
in Figures 55 and 56. KJJ/KJ=0.44 and BW/P=4.7 were obtained for this model. The above results and the numerical results obtained by Delale et al. [49] for a model made of boron-epoxy and E22_/E22=3.09 with a 45° crack along its weak axis are shown in Table 5. The isotropic results for the same a/w obtained from reference [63] are also given in the table.
The table shows a good agreement among the finite element and Delale et al. results. It also confirms their conclusion that in the inclined crack problem beside the material orthotropy, the orientation of the axes of ortho-
tropy may likewise significantly affect the stress intensity
factors. 128
2.0 — O
S
0.8
0.00 0.04 0.08 0.12 0.16 0.20 0.24 a/w
Figure 53. Kj and Kij, crack aligned with strong axis 129
0.0
-1.0
-2.0
3 en X o o
-4.0
-5.0 0.00 0.04 0.08 0.12 0.16 0.20 0.24 r/a
Figure 54. crack aligned with strong axis 130
Q. . t=
S
0.8
0.4
0.0 0.00 0.04 0.08 0.12 0.15 0.20 0.24 r/a
Figure 55. Kj and crack aligned with weak axis 131
5.0
4.0 a. g X o o
2.0
0.00 0.04 0.08 0.12 0.16 0.20 0.24 r/a
Figure 56- crack aligned with weak axis 132
Table 6. The strip model results
Model Kji/Kj a^j^BW/P
Crack along the strong axis 0.52 -4.45
Crack along the weak axis 0.44 4.7
Crack along the weak axis [49] 0.41
Isotropic [63] 0.50
The experimental and numerical analysis of the strip
specimen with a slanted edge crack show, where the experi
mental methods fail to provide a solution, a numerical
method can be used as an alternative to determine the
mixed mode stress intensity factors. 133
CONCLUSION
The photoelastic study of unidirectional composite materials proved that this method can be used as a powerful tool in the determination of the mode one stress intenstiy factors, when the crack or flaws are oriented along the weak axis. It also showed that at higher levels of loading a better accuracy can be achieved. The loading level importance was more evident in the case where the crack was parallel to the strong axis. In this case, because of existence of residual stresses and the model's sensitivity to the crack growth, a lower level of load was applied. The results show a large
difference among the Ki and values obtained by the experi
mental and the finite element methods. This difference was
reduced in one of the experiments by just increasing the
applied load.
In the study of the mixed mode stress intensity factors,
the experimental approach failed to provide any solution. The
finite element approach provided a very good solution.
The research brings about the first step of the applica
tion of photoelasticity to the fracture of composite materials.
Although the material used in this project was the best
available in the market, it is felt that more study is required
to produce even a better material by reducing the flaws and
the residual stresses in the materials. It is also seen that 134
more investigation of these materials at higher loads is needed to see if the low material sensitivity and residual stresses problems can be overcome.
Since the load was limited to a small amount in the case of load along the weak axis, it can be concluded that in the case of bidirectional photoelastic composite materials this obstacle can be removed. Therefore,this method shows a great promise in the study of fracture in bidirectional composite materials. 135
ACKNOWLEDGMENTS
I would like to express my sincerest appreciation to my major professor. Dr. Loren W. Zachary, for his guidance, patience and encouragement throughout my study.
I would also like to thank Dr. Lowell F. Greimann, in charge of my minor area of study, and the other members of my graduate committee. Professor William F. Riley, Dr.
Frederick M. Graham and Dr. Wallace W. Sanders for their valuable guidance and for serving as my committee members.
Special thanks go to Dr. Harry J. Weiss and the
Engineering Research Institute for their cooperation and financial support.
I very much like to thank Dr. Ibrahim Miskioglu for his computer programming assistance and the use of his valuable programs.
Special acknowledgment is due to Dr. Paul Cooper from
NASA Langley Research Center for providing the composite materials used in this research.
This project was partially supported by the National
Science Foundatio under the Grant Number MEA-811353 5. 136
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APPENDIX A PROGRAM 31GRES LOGICAL*! ANS REAL#4 T(20),R(20),SN(20>,S(3 >,SDM(3) COMMON /VAR/FX,FY,FXY,B1,B2,TH,FR,TR t squares technique applied to composite materials to determine stress intensity factors TYPE INPUT the material Fringe constants FX,FY,FXY' READ(5,K)FX,FY,FXY TYPE INPUT the material constants BETAl & BETA2' READ(5,*)B1,B2 TYPE *,'INPUT Model thicKness ' READ(5,*)TH TYPE INPUT the residual Frinae & residual isoclinic FR,TR' READ (5,#) FR.TR TYPE #,'INPUT Allowable error & Max. no. oF iterations' READ(5,*)ER,MA TYPE INPUT estimates oF K1,K2,S0X' READ(5,*) (SDM(I ) , I == 1 , 3 ) TYPE *,'INPUT the data File number -- 2 digits' READ(5» *)JJ DO 850 I==1,3 G(I ) =SDM(I) OPEN (UNIT= JJ,TYPE== 'OLD' ,ERR= 9G) READ(JJ,1000>N,TINC FORMAT(14,FG. 1) DO 30 I=1 , N READ(JJ,2000)T(I),R(I),GN(I) FORMAT(3F15.5) CLOGE(UNIT=JJ) GO TO GG9 GTOP 'BAD OPEN FILE' CONTINUE CALL GIFC(T,Û,GN,G,ER,E1,N,MA,INO) WRITE(7,70)JJ,TINC WRI TE(G,70)JJ,TINC IF( I NO .r.Q . 0 ) WRI TE ( B, 800 ) WRITE(G,80)E1 DO 20 1=1,3 WRITE(G,50)I,S(I) 20 WRITE(7,50)I,G(I) TYPE TYPE */' Do you want to run the proaratii asain ? ' READ<5rGO)ANS IF(ANS.E0.'Y')G0 TO 100 GO FORMAT(A) 50 FORMAT(' S(',I1,')=',F10.4) 70 FORMAT(//,' FILE NUMBER 13,lOX,'Anale increment =',F6.1/) GO FORMAT(' RELATIVE ERROR =',F10.G,/) GOO FORMAT(' NO CONVERGENCE WITHIN THE SPECIFIED NO OF ITER.',/) STOP END
SUBROUTINE SIFC(T,R,GN,G,ER,E1,N,MA,IN0) ^ REAL T(20),R(20),GN(20),G(3),C(3,3),D(3),X(3),G(20> ^ REAL CO(20),C1(20),C2(20),C3(20),C4(20>,C5(20),CG(20),C7(20) REAL DG(20),SX(20),SY(20),TXY(20),A(20,3),W(3,20) REAL G1(20),02(20),R1(20),R2(20) COMMON /VAR/FX,FY,FXY,81,B2,TH,FR,TR C **** C COMPOGITE MATERIAL C **** INi] : 1 WRÎTE(7,G0)TM,ER,N,MA GO FORMAT ( ' Tlii c k . = ', F5 . 3 , ' Err.= ',F0.7,' N-',I2,' MA-',14) PI:: 4. SATAN ( 1. ) DO 1 I=1,N G1 (I)=ATAN(B1*SIN(T(I))/CQS(T(I))) .TF(T(I ) .LT. rM/2. )G1 (I) =01 (I ) PI IF(T(I ) .GT.i'I/2. )G1 (I ) :|::1 (I ) iPI G2;: , •iVrAi.Ki;J2:iG:iN(T( I) )/i:Gi:{T(I ) ) ) IR(T( [). GT.PI/;'. >02(1)^02') ) IFI IF(T(n .LT. -PI/2. )G2(I)=G2(I) -PI Rid)-GQRKCOGCK I ) ) **2+ G 1 ^GIN ( K I ) )**2) R2(I)^GQRT(CGS(T(I))**2+B2#GIN(T(I))#*2) C0(I)^SQRT(2.*PI*R(I))tt(B2-Bl) C1(I)=1./CO(I) C2(I)=C1(I)*Bl%B2*(B2*C0S(G2(I)/2.)/GQRT(R2(I))-Bi*COG(Dl(I)/2.)/ &GQRKR1( I))) C3(I):C1(I)*(B1**2*SIN(G1(I)/2.)/GQRT(Rl(I))-B2**2*SIN(G2(I)/2.)/ &SQRT(R2(I))) C4(I)=C1( I)*(B2*C0S(G1( I )/2. ) /GQRKRl (I ) )-B1 stCOG ( G2 (I )/2 . ) / &6QRT(R2(I))) C5(I ) =C1 (I > « (S;CN(G2(I ) /2. ) /SQRT(R2(I ) )--GIN(G.t (I )/2. )/SQRT(Rl (I))) CG(I)=C1(I)*B1*B2*(SIN(G1(I)/2.)/GQRT(Rl(I))-GIN(G2(I)/2.)/ &GQRT(R2(I))) C7(I)=C2(I)/(B1*B2) CONTINUE M=1 E1=0 WRI TE(7,70)M FORMAT ( ' *It Iteration No.= ',I3) DO 3 1=1,N GX(I)=C2(I)*G(1)+C3(I)ttG(2)-G(3) GY (I ) •-••C4 (I)*G(1)+CI5 (I)*G(2) TXY SUBROUTINE SUBN(A,B,N) REAL*U A(N,N),B(N,N) DO 40 I=1,N DO 40 J=1,N 40 B(I,J)=A(I,J) RETURN END PROGRAM SRTEST C This proariuii will generate data For the program SICRES C UP to 20 points and store it in the File FTN**.DAT. C The Far Field stress SOX is included. C il * C COMPOSITE MATERIAL C *##* REAL T(20),R(20),SN(20),S(3) TYPE *,'File number -- 2 diaits -- Number oF points' READ(5,*)JJ,N OPEN(UNIT=JJ,TYPE='NEW',ERR=99) WRITE(JJr1000)N 1000 FORMAT(14) TYPE *,' INPUT the material Frinae constants FX,FY,FXY' READ(5,*)FX,FY,FXY TYPE INPUT the material constants BETAl & BETA2' READ(5,%)B1,B2 TYPE 'INPUT Model thicKness (inches)' -J READ(5,*)TH TYPEi^ 7 ' INPUT the residual values FR,TR' READ(5,*)FR,TR TYPE INPUT the values oF K1,K2,S0X' READ(5,*)(S(I),1=1,3) TYPE ' INPUT the anale increment (desrees) ' READ(5,K)TINC C Generate 20 data points For SICRES )]0 1. 1=1,4 RI=0.05%I DO 2 J =•• 1 , 5 TI= -45. I TIN(:»( J-1) TJ^TIn4.HATAN(1.)/180, PI::4. %ATAN( 1 . ) Gl=ATAN(Bl :iSIN(TJ ) /CGS APPENDIX B To determine the accuracy of HFP, a model made of plexiglass material which possesses a high fringe value was investigated. The fringe value, fa=670 lb./ fr.in. was determined by loading a circular disk of plexiglas in a circular polariscope. A standard compact-tension specimen with a/w=0.04 and a thickness of 1/8 inch (Figure 33) was cut from the plexiglas material. The model was loaded with 10 pounds and used in HFP system for data collection. Eighteen sets of 20 data points were collected. These data sets were then analyzed by a program called SIF [20] written for isotropic material to determine Kj, and o^y.. The average values of Kj were then compared with 's determined by using the finite element method and tabulated empirical formulas [4, p. 76]. The solid SAP finite element program was used to determine the stresses in the vicinity of the crack tip. The stresses along the 9 =45® line were then used in equation (59) to determine the nominal Kj's. The graph of Kj vs. r was plotted and extrapolated to r=0 to determine the actual stress inten sity factor (Figure 57). The empirical formula given in equation (92) which is valid for compact tension specimens with a/w between 0.3 and 0.7 and is accurate within 0.25% was used to determine the 150 empirical value of K^. K]. = P/Bwl/2[29.6(a/w)l/2 _ 185.5 (a/w) 3/2 + 6 55 .7 (a/w) - 1017.0(a/w)7/2 + 638.9(a/w)9/2] (89) where P=load, lb, B=thickness of specimen, in, W=width of specimen, in, a=crack length, in. A load of 10 pounds was applied to the model for all calculations. The values of Kj obtained from the above pro cedures are given in Table 7. Table 7. Comparison of methods Method Kj, Psii/in, HFP 430 FEM 420 Empirical 414 The table shows a small difference among the K_ values obtained by these three methods and proves the accuracy of HFP method. 151 420 400 380 —O 360 340 0.00 0.02 0.04 0.06 0.08 0.10 R, in. Figure 57. vs. R for the plexiglas