MIXED MODE I – II – III FRACTURE CRITERION AND ITS APPLICATION TO CEMENT MORTAR ZHONG KUI (B. Eng., Tongji U.; M. Eng., NUS) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2002 ACKNOWLEDGEMENTS The author wishes to express his most sincere gratitude to his Supervisor, Associate Professor Lo Kwang Wei for his invaluable, patient and constructive guidance throughout the research period. His in-depth knowledge and expertise in the field of Fracture Mechanics had been a great contribution to this work. The author is furthermore indebted to Research Fellow, Dr T Tamilselvan, and all technicians and colleagues in the Geotechnical Laboratory and Concrete and Structural Engineering Laboratory for their kind co-operation and assistance in the research work. Also, the author would like to take this opportunity to express his deep thanks to his family for their endless support, encouragement and love. Last but not least, the author gratefully acknowledges the financial assistance offered by the National University of Singapore through grants from university research project RP 960643 and a research scholarship. ii CONTENTS TITLE PAGE i ACKNOWLEDGEMENTS ii CONTENTS iii SUMMARY vii NOMENCLATURE ix LIST OF FIGURES xiii LIST OF TABLES xvii CHAPTER INTRODUCTION 1.1 Literature Review 1.1.1 Conventional Fracture Mechanics and Mixed Mode Fracture Criterion 1.1.1.1 The Pioneering Work of Griffith and Irwin iii 1.1.1.2 Traditional Linear Elastic Fracture Mechanics 1.1.1.3 Existing Mixed Mode Fracture Criteria 1.1.2 Laboratory Experiments of Fracture under Mixed Mode Loading 1.1.2.1 Traditional Fracture Test Specimens under Mixed Mode Loading 1.1.2.2 Fracture Tests under Mixed Mode I–II Loading 10 1.1.2.3 Fracture Tests under Mixed Mode I–III Loading 13 1.1.2.4 Fracture Tests under Mixed Mode I–II–III Loading 15 1.1.3 Fracture in Cement Mortar 20 1.1.4 The Unified Model 20 1.2 Scope and Objective of Research CHAPTER A GENERALIZED MODEL FOR FRACTURE 22 24 2.1 Crack Closure Analysis in θ0 Plane under Pure Mode I or II Loading 24 2.2 Crack Closure Analysis in θ0 Plane under Pure Mode III Loading 31 2.2.1 Stress and Displacement Fields in Near Field of Crack Tip 31 2.2.2 Closure Analysis in θ0 Plane 40 2.3 Closure Analysis of Generalized θ Plane 43 2.4 Unified Three-Dimensional Mixed Mode I–II–III Fracture Criterion 48 iv CHAPTER EVALUATION OF STRESS INTENSITY FACTORS BY FINITE ELEMENT ANALYSIS 3.1 Finite Element Modelling and Evaluation of Stress Intensity Factors 52 52 3.1.1 Two-Dimensional Modelling 54 3.1.2 Three-Dimensional Modelling 63 CHAPTER EXPERIMENTAL VERIFICATION OF FRACTURE CRITERION 68 4.1 Test Material and Set-up 68 4.1.1 Test Material 668 4.1.2 Test Set-up 4.2 Pure Modes I and II Fracture Testing 75 78 4.2.1 Geometry of Specimen 78 4.2.2 Laboratory Set-up and Test Procedure 80 4.2.3 Determination of Stress Intensity Factors by Finite Element Analysis 4.2.4 Pure Modes I and II Fracture Toughness 4.3 Mixed Mode I–III Fracture Testing 80 88 91 4.3.1 Geometry of Specimen and Loading Fixture 91 4.3.2 Laboratory Set-up and Test Procedure 95 v 4.3.3 Determination of Stress Intensity Factors by Finite Element Analysis 4.3.4 Comparison of Analytical and Experimental Results 4.4 Mixed Modes I–II–III Fracture Testing 99 108 114 4.4.1 Geometry of Specimen 114 4.4.2 Laboratory Set-up and Test Procedure 120 4.4.3 Determination of Stress Intensity Factors by Finite Element Analysis 4.4.4 Comparison of Analytical and Experimental Results CHAPTER CONCLUSIONS AND RECOMMENDATIONS 120 123 135 5.1 Conclusions 135 5.2 Recommendations for Future Work 137 REFERENCES APPENDIX 139 NUMERICAL AND EXPERIMENTAL RESULTS 156 A.1 Pure Modes I and II Fracture Testing 156 A.2 Mixed Mode I-III Fracture Testing 157 A.3 Mixed Mode I-II-III Fracture Testing 158 vi SUMMARY In the linear elastic fracture mechanics (LEFM) of brittle and slightly ductile materials, there are, in general, three modes of deformation near a crack tip. The combination of modes I and II deformation has constituted the major portion of investigation in traditional fracture mechanics. To study crack behaviour under pure mode III loading, the closure analysis of an infinitesimal crack extension has been carried out, and relationship between the stress intensity factor and energy release rate established. After that, the closure analysis of a small crack extension along the generalized θ plane, under mixed mode I–II–III loading, has been conducted. On the basis of the unified model (Lo et al., 1996a), which specifies the in-plane crack behaviour under mixed mode I-II loading, a three-dimensional mixed mode I–II–III fracture criterion has been derived herein, by adopting the simple conversion of pure to mixed mode loading energy, in the ratio of their corresponding fracture energies (Lo et al., 2001). In verifying the proposed fracture criterion, two experimental methods have been designed using cement mortar specimens, in order to conduct three-dimensional mixed mode fracture tests, as well as the corresponding pure mode fracture tests. In the first case, a traditional compact tension specimen has been adopted, which was subjected to mixed mode I–III loading, as well as the respective pure mode loading, through a loading fixture. An advantage of the method adopted is that the full range of vii mixed mode I–III loading combinations may be readily applied to the same specimen configuration. As to the second case, a beam specimen has been tested under mixed mode I–II–III loading. A groove, which was rotated vertically as well as horizontally with respect to the beam section, leaving a truncated V-shaped throat segment, was formed in the specimen. Each test specimen had different orientations of the groove, so as to provide differing mixed mode loading conditions at the crack front. In doing so, a crack was initiated and hence guided to extend along the V-shaped throat segment as a mixed mode fracture. The stress intensity factors of the respective modes of deformation were evaluated numerically by finite element analysis. As a result of the numerical analyses and measurements made in laboratory tests, reasonable agreement was obtained between the proposed 3-D fracture criterion and the experimental results obtained for its verification. The essential findings of the proposed fracture criterion and experimental results have been published in an invited keynote lecture at the Third International Conference on Micro Materials (MicroMat 2000) (Lo, et al., 2000), the 10th International Conference on Fracture (10 ICF) (Zhong et al., 2001) and an international refereed journal (Lo et al., 2002). These publications collectively constitute the contents of this thesis, where they are dealt with in greater detail. viii NOMENCLATURE α δa εxx, εyy, εzz γxz, γyz loading angle small crack extension normal strain components shear strain components λ positive integer µ shear modulus ν Poisson’s ratio θ direction of plane of interest with respect to existing crack plane θ0 θ0C θC σθθ τθz, τxz, τyz τrθ self-similar direction, that is θ = crack extension in self-similar direction general direction of crack extension in-plane circumferential stress out-of-plane shear stress referred to parent crack tip in-plane shear stress referred to parent crack tip ix Aλ, Bλ constants a0 pre-crack length D depth of beam specimen E Young’s modulus FC fracture load measured in mixed mode fracture test FIC fracture load measured in pure mode I fracture test FIIC fracture load measured in pure mode II fracture test FIIIC fracture load measured in pure mode III fracture test f (θ) function of θ G rate of energy release Gθ mixed mode rate of energy release rate along θ plane Gq strain energy release rate for non-elastic materials GI mode I rate of energy release GII mode II rate of energy release GIII mode III rate of energy release GC critical rate of energy release GIC mode I critical rate of energy release GIIC mode II critical rate of energy release x CHAPTER EVALUATION OF STRESS INTENSITY FACTORS BY FINITE ELEMENT ANALYSIS There are three basic methods to evaluate the stress intensity factor, namely the classical, numerical (analytical) and photoelastic (experimental) methods. The classical method is only applicable to simple boundary value problems, while the photoelastic method is a complicated procedure. The numerical method, on the other hand, is a comparatively practical and effective, especially in determining the stress intensity factors of test specimens. Among the numerical methods, the finite element (FE) method is probably the most widely used. Hence, in the present study, the stress intensity factors have been evaluated using the FE method. The application of the method to LEFM will be outlined in the following discussion. 3.1 Finite Element Modelling and Evaluation of Stress Intensity Factors The finite element method was first introduced by Turner et al. (1956) as a method of solving the mechanical behaviour of an elastic continuum, using traditional structural analysis. A number of standard texts are available (Cheung and Yeo, 1979; 52 Zienkiewicz and Taylor, 1989; Bathe, 1996), which describe, in detail, the various formulations, solution routines and element types of the FE method. In the method, the continuum, with its infinite degrees of freedom, is discretized into a finite number of elements of finite size, with various geometrical shapes such as triangles, quadrilaterals and hexahedra, which are interconnected only at their nodal points. A function is chosen to define the state of displacement within each element uniquely, in terms of the nodal displacements. The stiffness relationships between the nodal displacements and corresponding nodal forces may be derived by means of energy theorems, such as the principle of virtual work and minimum total potential energy. In this way, an infinite degree-of-freedom problem may be addressed as an equivalent finite degree-of-freedom assembly, which is amenable to computer solution. The application of the FE method to determine the crack-tip stress and displacement fields has been extensively used and seen rapid progress, due to its versatility, such as in the analysis of complicated geometries, treatment of three-dimensional problems, and use of elastic-plastic elements to include crack tip plasticity. In using the FE method to calculate the stress intensity factor, one of the main departures from routine application of the method is the need to simulate the square-root strain singularities at the crack tip. Clearly, any attempt to model this singularity with conventional isoparametric elements, which allow linear or quadratic variation of stress across the element, would only give reasonable results if an extremely fine mesh were used in the crack tip region. To overcome this problem, many special crack tip elements have been devised (Byskov 1970; Tracey 1971; Walsh 1971; Barsoum 1976; Whiteman and Akin, 1978). Among these, the one proposed by 53 Barsoum is the most commonly adopted, and has, indeed, been used in the present study. Its application to two- and three-dimensional FE modelling, as well as the subsequent evaluation of corresponding stress intensity factors, will be dealt with in the following discussion. 3.1.1 Two-Dimensional Modelling The eight-noded, isoparametric, quadratic quadrilateral element is generally used in the two-dimensional modelling of the in-plane crack problem. In order to model the square-root singularity of stress and strain at the crack tip, the triangular element of Figure 3.1 is adopted, which is formed by collapsing side 1-4 of the quadrilateral, and moving the two mid-side nodes, and 7, to the quarter points shown in the same figure. Following the notation of Zienkiewicz and Taylor (1989), the geometry of an 8-noded plane isoparametric element would be mapped into the normalized square space (ξ, η), (-1 ≥ ξ ≥ 1, -1 ≥ η ≥ 1) through the following transformations, x = ∑ N i ( ξ, η) xi (3.1a) i =1 and y = ∑ N i ( ξ, η) yi , (3.1b) i =1 54 8-noded quadratic quadrilateral element y,η l /2 x,ξ l /2 l /2 l /2 (a) before collapse y η ξ l /2 x l /2 quarter-point l /4 3l1 /4 (b) after collapse Figure 3.1 Formation of triangular quarter-point crack tip element from eight-noded quadratic quadrilateral element 55 where Ni is the shape functions corresponding to the node i, whose coordinates would be (xi, yi) in the x-y system and (ξi, ηi) in the transformed ξ-η system. Since the shape functions evaluated along the line 1-2 of Figure 3.1 would be given as N = − ξ(1 − ξ) , (3.2a) ξ(1 + ξ) (3.2b) N = (1 − ξ2 ) , (3.2c) 1 x = − ξ(1 − ξ) x1 + ξ(1 + ξ) x + (1 − ξ ) x5 . 2 (3.3) N2 = and equation (3.1a) would be obtained as Choosing x1 = 0, x2 = l1, x5 = l1/4, then x= l ξ(1 + ξ)l1 + (1 − ξ ) , (3.4) therefore, 56  x . ξ =  − +  l   (3.5) Considering only the displacements points 1, and 5, the displacement uxx along the line 1-2 would be given by 1 u xx = − (1 − ξ)u xx1 + (1 + ξ)u xx + (1 − ξ )u xx . 2 (3.6) And writing it in the terms of x, it would be expressed as  x 1 x  x 1 x  x  x  2 −  −1+  2 4 u xx . + + − u xx = −  − + u u   xx xx  l 2 l1   l1   l1   l1  l1   (3.7) The strain in the x-direction would thus be  1 4 1 4 4 − u xx . + u xx +  εx = −  − u xx1 +  −  xl1 l1   xl1 l1   xl1 l1  (3.8) Therefore, the strain singularity along the line 1-2 would be 1/√r, which would be the required singularity for the elastic analysis. The stress intensity factors in the self-similar direction of the existing crack of a boundary value problem, KI0 and KII0, may be evaluated from the corresponding displacement field obtained by FE analysis. Under pure mode I loading, the 57 expressions for displacements in the near field of a crack tip are given as uθθ = (1 + ν) r  3θ θ  K I0 [sin − ( 2κ + 1) sin ]  2E 2π  2  (3.9) u rr = (1 + ν) r  θ 3θ  K I0 [( 2κ - 1) cos − cos ] ;  2E 2π  2  (3.10) and for pure mode II loading, they would be uθθ = (1 + ν) r  3θ θ  K [ cos − + ( κ ) cos ] II0 2E 2π  2  (3.11) u rr = (1 + ν) r  3θ θ  K II0 [3 sin − (2κ - 1) sin ]  2E 2π  2 (3.12) and (Irwin, 1958). In considering node of Figure 3.2, where r = r1 = l/4 and θ = -π, the displacements along the x and y directions would be given by 58 y, uyy crack tip crack face θ r x, uxx l/4 l Figure 3.2 Two-dimensional quarter-point elements around crack tip 59 u xx1 = −urr ( r1 ,− π) = − (1 + ν )(κ + 1) K II0 E r1 2π (3.13) u yy1 = −uθθ ( r1 ,− π) = − (1 + ν )(κ + 1) K I0 E r1 , 2π (3.14) and respectively. The negative signs in equations (3.13) and (3.14) take into account the fact that the direction of the Cartesian system is in opposite sense to the polar system, when θ = ±π. Similarly, for node 2, where r = r2 = l/4 and θ = π, the displacements along the x and y directions may be expressed as u xx = (1 + ν)( κ + 1) K II0 E r2 2π (3.15) u yy = (1 + ν)( κ + 1) K I0 E r2 , 2π (3.16) and respectively. In order to obtain a reasonably accurate stress intensity from the displacement 60 method, a relatively fine mesh would be required in the vicinity of the crack region. Moreover, the size of the crack tip element would have to be much smaller than the overall dimension of the specimen (Chan et al., 1970). In doing so, it would be valid to assume that, in the near field of the crack tip, the K value would vary linearly with √r, as depicted in Figure 3.3. Therefore, for the mode I stress intensity factor, interim values of K, which correspond to nodes and (with radial distances of r = r1 = r2 = l/4), would, in view of equations (3.14) and (3.16), respectively, be obtained as K ( r = r1 ) = (u yy − u yy1 ) E 2π . 2(1 + ν )(κ + 1) r1 (3.17) Similarly, the other interim value of K, which corresponds to nodes and (with radial distances of r = r3 = r4 = l), would be given by K ( r = r3 ) = (u yy − u yy ) E 2π , 2(1 + ν )(κ + 1) r3 (3.18) where uyy3 and uyy4 are the displacements, along the y direction, of nodes and 4, respectively. Based on equations (3.17) and (3.18), the value of the mode I stress intensity factor, KI0, may be taken to be K, at r = 0, by linear extrapolation between K(r = r1) and K(r = r3) with respect to √r1 = √(l/4) and √r3 = √l, respectively, that is to say, 61 K K(r3) K(r1 ) K(r = 0) r1 r3 r Figure 3.3 Determination of stress intensity factor at crack tip by 62 K I0 = 2π E [(4u yy − u yy ) − (4u yy1 − u yy )]. 2(1 + ν)( κ + 1) l (3.19) Similarly, by considering nodes to 4, and using equations (3.13) and (3.15), the mode II stress intensity factor KII0 may be extrapolated as K II0 = 2π E [(4u xx − u xx ) − (4u xx1 − u xx )], (1 + ν)( κ + 1) l (3.20) where uxx1, uxx2, uxx3 and uxx4 are the displacements along the x direction of nodes to 4, respectively. 3.1.2 Three-Dimensional Modelling For three-dimensional modelling, the 20-noded, isoparametric quadratic brick element is generally used, as shown in Figure 3.4. It has been shown that the square-root singularity of stress and strain, at the crack tip, may be simulated by using the triangular prismatic element of the figure, which is generated by collapsing face 3487 of the cubical element, and moving the mid-side nodes 9, 11, 17 and 19 to their quarter points. It has been found (Barsoum, 1976) that the required singularity would develop along the crack front. The variation in the ζ direction would be quadratic, thus allowing a piecewise, continuous change from one element to the next along the crack front, which would, in turn, permit an accurate evaluation of the stresses from one element to the next that lies along the crack front (Figure 3.5). 63 20-noded quadratic brick element 19 15 11 η 14 12 ζ 20 18 16 17 l1 l3 10 ξ l3 13 l2 l1 l2 (a) before collapse crack front y 20 15 12 η 11 17 16 14 10 quarter-point ζ 19 l3 18 l3 x l1 ξ 3l1 l2 13 l2 (b) after collapse Figure 3.4 Formation of triangular prismatic quarter-point crack 64 65 l crack face z, uzz y, uyy crack tip Configuration at plane A crack front Figure 3.5 Three-dimensional quarter-point elements around crack tip Three-dimensional modelling of crack front crack face plane A (plane of intersection of adjacent elements) x, uxx θ The modes I and II stress intensity factors along the self-similar direction, KI0 and KII0, may be evaluated from the nodal displacements obtained from FE analysis, given by equations (3.19) and (3.20), respectively. The mode III stress intensity factor KIII0 may be obtained in a similar way. The out-of-plane displacement field in the near field of a crack tip, which is subjected to mode III loading, has been derived in the foregoing discussion of §2.2.1, in terms of equation (2.53). By considering nodes to of Figure 3.5, the following two interim expressions of K, corresponding to node pairs 1-2 and 3-4, with radial distances of r = r1 = r2 = l/4 and r = r3 = r4 = l, may be determined as K ( r = r1 ) = E 2π (u zz − u zz1 ) 8(1 + ν) r1 (3.21) K ( r = r3 ) = E 2π (u zz − u zz ) , 8(1 + ν) r3 (3.22) and where uzz1, uzz2, uzz3 and uzz4 are the respective displacements of nodes to 4, along the z direction. By applying the same premise, as in preceding §3.1.1, that K would vary linearly with √r in the near field of the crack tip, the mode III stress intensity factor, KIII0, may be evaluated, as K at r = 0, by linearly extrapolating from K(r1) and K(r3) 66 where √r1 = √(l/4) and √r3 = √l, respectively. The result would then be K III0 = E 2π [(4u zz − u zz ) − (4u zz1 − u zz )]. 8(1 + ν) l (3.23) By applying equations (3.11), (3.12) and (3.15) to each intersection plane of adjacent elements at the crack front, as shown in Figure 3.5, the stress intensity factors, KI0, KII0 and KIII0, along the crack front may respectively be evaluated. Since the distribution of the stress intensity factors along the crack front of the inner layers are relatively uniform and the contribution to the overall stress intensity factor, from the end values is relatively insignificant, the value at mid-section of the specimen may be employed with reasonable accuracy. 67 [...]... to the extended crack tip K III mode III stress intensity factor referred to the extended crack tip KIθ unified pure mode I stress intensity factor KIIθ unified pure mode II stress intensity factor KIIIθ unified pure mode III stress intensity factor KI0 mode I stress intensity factor with respect to θ0 plane KII0 mode II stress intensity factor with respect to θ0 plane KIII0 mode III stress intensity... beam specimen of mixed mode I II III fracture test 11 6 Figure 4. 31 Formation of groove and pre-crack by three steel plates 11 8 Figure 4.32 Mould of mixed mode I II III fracture test specimen 11 9 Figure 4.33 Loading cases used in mixed mode I II III fracture tests 12 1 Figure 4.34 Finite element model for beam group A of mixed mode I II III fracture test 12 2 Figure 4.35 Sectional view showing details of...GIIIC mode III critical rate of energy release GIθ mode I rate of energy release with respect to θ plane GIIθ mode II rate of energy release with respect to θ plane GIIIθ mode III rate of energy release with respect to θ plane KI mode I stress intensity factor KII mode II stress intensity factor K I mode I stress intensity factor referred to the extended crack tip K II mode II stress intensity factor... Failure of mixed mode I II III fracture test specimens Figure 4.40 Comparison of fracture criterion with mixed mode I II III fracture test results 13 2 - 13 3 13 4 xvi LIST OF TABLES Table 4 .1 Angles of inclination of beam groups 11 7 Table A .1 Numerical and experimental results of pure modes I and II fracture tests 15 6 Table 4.2 Numerical and experimental results of mixed mode I- III fracture tests 15 9... model of mixed mode I – III loading (α=45°) 10 5 Figure 4.26 Distributions of stress intensity factors across throat of specimen (t/T = 0.5) 10 7 Figure 4.27 KI0 and KIII0 with degree of grooving 10 9 Figure 4.28 Failure surface of mixed mode I III fracture test specimen 11 0 Figure 4.23 Figure 4.24 xv Figure 4.29 Comparison of unified fracture criterion with mixed mode I III fracture test results 11 3 Figure... specimen 92 Figure 4 .16 Specimen mould for mixed mode I III fracture test 93 Figure 4 .17 Schematic diagram of mixed mode I III test components 96 Figure 4 .18 Geometrical configuration of partial loading fixture and coupling 97 Mixed mode I III fracture test arrangement (angle of loading of α=45°) 98 Figure 4 .19 Figure 4.20 Finite element model of mixed mode I III fracture test specimen 10 0 Figure 4. 21. .. data from mixed mode I II III loading, as shown in Figure 1. 6 Modes I and II crack tip loading conditions were created in the specimen by subjecting it to an axial load, while mode III crack tip conditions were produced by applying a torsional load Although the specimen does not cover every KI : KII : KIII ratio, it was reported that any KI : KIII ratio could be covered, and, together with the results... loading, which is referred to as mixed mode loading 1 y x z (a) Mode I (a) mode I (b) Mode II (b) mode II (c) Mode III (c) mode III Figure 1. 1 Three modes of loading 2 Traditionally, the main focus of research of so-called linear elastic fracture mechanics (LEFM), with which the present work is concerned, has been the in-plane problem, that is the pure modes I and II, and mixed mode I – II, loading In... under torsion (Pook, 19 85b) All the specimens of Figure 1. 2 are meant for mixed mode I – II fracture testing, and so are specimens (a) to (d) of Figure 1. 3 Specimen (e) of Figure 1. 3 is aimed at studying crack growth behaviour under mixed mode II – III loading, while specimens (f) and (g) are designed to produce mixed mode I – III loading 1. 1.2.2 Fracture Tests under Mixed Mode I – II Loading Considerable... Figure 4 .10 Experimental set-up of pure mode II fracture test 84 Figure 4 .11 Finite element model of beam specimen 86 Figure 4 .12 K-calibrations of stress intensity factors 87 Figure 4 .13 Typical load-stroke displacement curves for pure mode I and II fracture 89 Figure 3.4 xiv Figure 4 .14 Failure of modes I and II fracture test specimens 90 Figure 4 .15 Geometrical configuration of mixed mode I – III . under Mixed Mode Loading 9 1. 1.2.2 Fracture Tests under Mixed Mode I II Loading 10 1. 1.2.3 Fracture Tests under Mixed Mode I III Loading 13 1. 1.2.4 Fracture Tests under Mixed Mode I II III Loading. K II0 mode II stress intensity factor with respect to θ 0 plane K III0 mode III stress intensity factor with respect to θ 0 plane K IC mode I fracture toughness K IIC mode II fracture. steel plates 11 8 Figure 4.32 Mould of mixed mode I II III fracture test specimen 11 9 Figure 4.33 Loading cases used in mixed mode I II III fracture tests 12 1 Figure 4.34 Finite element model for