Lecture Notes in: FRACTURE MECHANICS. Victor E. Saouma

Transcription

1 Lecture Notes in: FRACTURE MECHANICS Victor E. Saouma Dept. of Civil Environmental and Architectural Engineering University of Colorado, Boulder, CO

2 Contents 1 INTRODUCTION Modes of Failures Examples of Structural Failures Caused by Fracture vs Strength of Materials Major Historical Developments in Coverage PRELIMINARY CONSIDERATIONS 1.1 Tensors Indicial Notation Tensor Operations Rotation of Axes Trace Inverse Tensor Principal Values and Directions of Symmetric Second Order Tensors Kinetics Force, Traction and Stress Vectors Traction on an Arbitrary Plane; Cauchy s Stress Tensor E -1 Stress Vectors Invariants Spherical and Deviatoric Stress Tensors Stress Transformation Polar Coordinates Kinematic Strain Tensors Compatibility Equation Fundamental Laws of Continuum Mechanics Conservation Laws Fluxes Conservation of Mass; Continuity Equation Linear Momentum Principle; Equation of Motion Moment of Momentum Principle Conservation of Energy; First Principle of Thermodynamics Constitutive Equations Transversly Isotropic Case Special D Cases Plane Strain

3 CONTENTS iii Ideal Strength in Terms of Engineering Parameter Shear Strength Griffith Theory Derivation ENERGY TRANSFER in CRACK GROWTH; (Griffith II) Thermodynamics of Crack Growth General Derivation Brittle Material, Griffith s Model Energy Release Rate; Global From Load-Displacement From Compliance Energy Release Rate; Local Crack Stability Effect of Geometry; Π Curve Effect of Material; R Curve Theoretical Basis R vs K Ic Plane Strain Plane Stress MIXED MODE CRACK PROPAGATION Analytical Models for Isotropic Solids Maximum Circumferential Tensile Stress Maximum Energy Release Rate Minimum Strain Energy Density Criteria Observations Empirical Models for Rocks Extensions to Anisotropic Solids Interface Cracks Crack Tip Fields Dimensions of Bimaterial Stress Intensity Factors Interface Fracture Toughness Interface Fracture Toughness when β = Interface Fracture Toughness when β Crack Kinking Analysis Numerical Results from He and Hutchinson Numerical Results Using Merlin Summary II ELASTO PLASTIC FRACTURE MECHANICS 9 8 PLASTIC ZONE SIZES Uniaxial Stress Criteria First-Order Approximation Second-Order Approximation (Irwin) Example

4 CONTENTS v 11.11Dynamic Energy Release Rate Effect of Other Loading Surface Tractions on Crack Surfaces Body Forces Initial Strains Corresponding to Thermal Loading Initial Stresses Corresponding to Pore Pressures Combined Thermal Strains and Pore Pressures Epilogue IV FRACTURE MECHANICS OF CONCRETE 41 1 FRACTURE DETERIORATION ANALYSIS OF CONCRETE Introduction Phenomenological Observations Load, Displacement, and Strain Control Tests Pre/Post-Peak Material Response of Steel and Concrete Localisation of Deformation Experimental Evidence σ-cod Diagram, Hillerborg s Model Theoretical Evidence Static Loading Dynamic Loading Loss of Hyperbolicity Wave Equation for Softening Maerials Conclusion Griffith Criterion and FPZ FRACTURE MECHANICS of CONCRETE Linear Elastic Fracture Models Finite Element Calibration Testing Procedure Data Reduction Nonlinear Fracture Models Hillerborg Characteristic Length MIHASHI data Jenq and Shah Two Parameters Model Size Effect Law, Bažant Carpinteri Brittleness Number Comparison of the Fracture Models Hillerborg Characteristic Length, l ch Bažant Brittleness Number, β Carpinteri Brittleness Number, s Jenq and Shah s Critical Material Length, Q Discussion Model Selection

5 CONTENTS vii V FINITE ELEMENT TECHNIQUES IN FRACTURE MECHANICS SINGULAR ELEMENT Introduction Displacement Extrapolation Quarter Point Singular Elements Review of Isoparametric Finite Elements How to Distort the Element to Model the Singularity Order of Singularity Stress Intensity Factors Extraction Isotropic Case Anisotropic Case Numerical Evaluation Historical Overview Other Singular Elements ENERGY RELEASE BASED METHODS Mode I Only Energy Release Rate Virtual Crack Extension Mixed Mode Cases Two Virtual Crack Extensions Single Virtual Crack Extension, Displacement Decomposition J INTEGRAL BASED METHODS Numerical Evaluation Mixed Mode SIF Evaluation Equivalent Domain Integral (EDI) Method Energy Release Rate J D case D Generalization Extraction of SIF J Components σ and u Decomposition RECIPROCAL WORK INTEGRALS General Formulation Volume Form of the Reciprocal Work Integral Surface Tractions on Crack Surfaces Body Forces Initial Strains Corresponding to Thermal Loading Initial Stresses Corresponding to Pore Pressures Combined Thermal Strains and Pore Pressures Field Equations for Thermo- and Poro-Elasticity

6 List of Figures 1.1 Cracked Cantilevered Beam Failure Envelope for a Cracked Cantilevered Beam Generalized Failure Envelope Column Curve Stress Components on an Infinitesimal Element Stresses as Tensor Components Cauchy s Tetrahedron Flux Through Area ds Equilibrium of Stresses, Cartesian Coordinates Curvilinear Coordinates Transversly Isotropic Material Coordinate Systems for Stress Transformations Circular Hole in an Infinite Plate Elliptical Hole in an Infinite Plate Crack in an Infinite Plate Independent Modes of Crack Displacements Plate with Angular Corners Plate with Angular Corners Middle Tension Panel Single Edge Notch Tension Panel Double Edge Notch Tension Panel Three Point Bend Beam Compact Tension Specimen Approximate Solutions for Two Opposite Short Cracks Radiating from a Circular Hole in an Infinite Pla 4.7 Approximate Solutions for Long Cracks Radiating from a Circular Hole in an Infinite Plate under Tensio 4.8 Radiating Cracks from a Circular Hole in an Infinite Plate under Biaxial Stress Pressurized Hole with Radiating Cracks Two Opposite Point Loads acting on the Surface of an Embedded Crack Two Opposite Point Loads acting on the Surface of an Edge Crack Embedded, Corner, and Surface Cracks Elliptical Crack, and Newman s Solution Growth of Semielliptical surface Flaw into Semicircular Configuration Uniformly Stressed Layer of Atoms Separated by a Energy and Force Binding Two Adjacent Atoms

7 LIST OF FIGURES iii 10.1 Crack Tip Opening Displacement, (Anderson 1995) Estimate of the Crack Tip Opening Displacement, (Anderson 1995) J Integral Definition Around a Crack Closed Contour for Proof of J Path Independence Virtual Crack Extension Definition of J Arbitrary Solid with Internal Inclusion Elastic-Plastic versus Nonlinear Elastic Materials Nonlinear Energy Release Rate, (Anderson 1995) Experimental Derivation of J J Resistance Curve for Ductile Material, (Anderson 1995) J, J R versus Crack Length, (Anderson 1995) J, Around a Circular Path Normalize Ramberg-Osgood Stress-Strain Relation (α =.01) HRR Singularity, (Anderson 1995) Effect of Plasticity on the Crack Tip Stress Fields, (Anderson 1995) Compact tension Specimen Center Cracked Panel Single Edge Notched Specimen Double Edge Notched Specimen Axially Cracked Pressurized Cylinder Circumferentially Cracked Cylinder Dynamic Crack Propagation in a Plane Body, (Kanninen 1984) Test Controls Stress-Strain Curves of Metals and Concrete Caputring Experimentally Localization in Uniaxially Loaded Concrete Specimens Hillerborg s Fictitious Crack Model Concrete Strain Softening Models Strain-Softening Bar Subjected to Uniaxial Load Load Displacement Curve in terms of Element Size Localization of Tensile Strain in Concrete Griffith criterion in NLFM Servo-Controlled Test Setup for Concrete K Ic and G F *Typical Load-CMOD Curve from a Concrete Fracture Test Effective Crack Length Definition Energy Transfer During Infinitesimal Crack Extension Central Crack With Constant Cohesive Stresses Nominal Strength in Terms of Size for a Center Crack Plate with Constant Cohesive Stresses Dugdale s Model Size Effect Law for an Edge Crack with Constant Cohesive Stresses Linear Cohesive Stress Model Energy Transfer During Infinitesimal Crack Extension Size Effect Law for an Edge Crack with Linear Softening and Various Orders of Approximation Three Point Bend Specimen with Linear Cohesive Stresses Size Effect Law Inelastic Buckling

8 LIST OF FIGURES v.4 Interface fracture Failure function Bi-linear softening laws Stiffness degradation in the equivalent uniaxial case Interface element numbering Local coordinate system of the interface element Algorithm for interface constitutive model Definition of inelastic return direction Influence of increment size Shear-tension example Secant relationship Line search method Griffith criterion in NLFM Mixed mode crack propagation Schematics of the direct shear test setup Direct shear test on mortar joint Experimental set-up for the large scale mixed mode test Nonlinear analysis of the mixed mode test Crack propagation in Iosipescu s beam, (Steps 1 & 3) Crack propagation in Iosipescu s beam, (Increment 11 & 39 in Step 6) Multiple crack propagation in Iosipescu s beam (Steps 3,4) Multiple crack propagation in Iosipescu s beam (Step 5) Meshes for crack propagation in Iosipescu s beam (Steps 1,3,4,5) Iosipescu s beam with ICM model Crack paths for Iosipescu s beam Large Iosipescu s beam, h = 50 x 100 mm Crack propagation for anchor bolt pull out test I Crack propagation for anchor bolt pull out test II Crack patterns Load displacement curve for test I Load displacement curve for test II

9 List of Tables 1.1 Column Instability Versus Fracture Instability Number of Elastic Constants for Different Materials Summary of Elasticity Based Problems Analysed Newman s Solution for Circular Hole in an Infinite Plate subjected to Biaxial Loading, and Internal Pres 4. C Factors for Point Load on Edge Crack Approximate Fracture Toughness of Common Engineering Materials Fracture Toughness vs Yield Stress for.45c N i C r M o Steel Material Properties and Loads for Different Cases Analytical and Numerical Results Numerical Results using S-integral without the bimaterial model Comparison of Various Models in LEFM and EPFM Effect of Plasticity on the Crack Tip Stress Field, (Anderson 1995) h-functions for Standard ASTM Compact Tension Specimen, (Kumar, German and Shih 1981) Plane Stress h-functions for a Center-Cracked Panel, (Kumar et al. 1981) h-functions for Single Edge Notched Specimen, (Kumar et al. 1981) h-functions for Double Edge Notched Specimen, (Kumar et al. 1981) h-functions for an Internally Pressurized, Axially Cracked Cylinder, (Kumar et al. 1981) F and V 1 for Internally Pressurized, Axially Cracked Cylinder, (Kumar et al. 1981) h-functions for a Circumferentially Cracked Cylinder in Tension, (Kumar et al. 1981) F, V 1, and V for a Circumferentially Cracked Cylinder in Tension, (Kumar et al. 1981) Strain Energy versus Fracture Energy for uniaxial Concrete Specimen Summary relations for the concrete fracture models When to Use LEFM or NLFM Fracture Models Experimentally Determined Values of Bf t, (Bažant, Z. and Planas, J. 1998) Size Effect Law vs Column Curve Fractal dimension definition Concrete mix design Range and resolution of the profilometer (inches) CHECK Mapped profile spacing, orientation, and resolution for the two specimen sizes investigated Computed fractal dimensions of a straight line with various inclinations

10 Chapter 1 INTRODUCTION In this introductory chapter, we shall start by reviewing the various modes of structural failure and highlight the importance of fracture induced failure and contrast it with the limited coverage given to fracture mechanics in Engineering Education. In the next section we will discuss some examples of well known failures/accidents attributed to cracking. Then, using a simple example we shall compare the failure load predicted from linear elastic fracture mechanics with the one predicted by classical strength of materials. The next section will provide a brief panoramic overview of the major developments in fracture mechanics. Finally, the chapter will conclude with an outline of the lecture notes. 1.1 Modes of Failures The fundamental requirement of any structure is that it should be designed to resist mechanical failure through any (or a combination of) the following modes: 1. Elastic instability (buckling). Large elastic deformation (jamming) 3. Gross plastic deformation (yielding) 4. Tensile instability (necking) 5. Fracture Most of these failure modes are relatively well understood, and proper design procedures have been developed to resist them. However, fractures occurring after earthquakes constitute the major source of structural damage (Duga, Fisher, Buxbam, Rosenfield, Buhr, Honton and McMillan 1983), and are the least well understood. In fact, fracture often has been overlooked as a potential mode of failure at the expense of an overemphasis on strength. Such a simplification is not new, and finds a very similar analogy in the critical load of a column. If column strength is based entirely on a strength criterion, an unsafe design may result as instability (or buckling) is overlooked for slender members. Thus failure curves for columns show a smooth transition in the failure mode from columns based on gross section yielding to columns based on instability. By analogy, a cracked structure can be designed on the sole basis of strength as long as the crack size does not exceed a critical value. Should the crack size exceed this critical value, then

11 Chapter PRELIMINARY CONSIDERATIONS Needs some minor editing! 1 Whereas, ideally, an introductory course in Continuum Mechanics should be taken prior to a fracture mechanics, this is seldom the case. Most often, students have had a graduate course in Advanced Strength of Materials, which can only provide limited background to a solid fracture mechanics course. Accordingly, this preliminary chapter (mostly extracted from the author s lecture notes in Continuum Mechanics) will partially remedy for occasional deficiencies and will be often referenced. 3 It should be noted that most, but not all, of the material in this chapter will be subsequently referenced..1 Tensors 4 We now seek to generalize the concept of a vector by introducing the tensor (T), which essentially exists to operate on vectors v to produce other vectors (or on tensors to produce other tensors!). We designate this operation by T v or simply Tv. 5 We hereby adopt the dyadic notation for tensors as linear vector operators u = T v or u i = T ij v j (.1-a) u = v S where S = T T (.1-b) 6 Whereas a tensor is essentially an operator on vectors (or other tensors), it is also a physical quantity, independent of any particular coordinate system yet specified most conveniently by referring to an appropriate system of coordinates. 7 Tensors frequently arise as physical entities whose components are the coefficients of a linear relationship between vectors.

12 .1 Tensors 3 A fourth order tensor (such as Elastic constants) will have four free indices. 4. Derivatives of tensor with respect to x i is written as, i. For example: Φ x i = Φ,i v i x i = v i,i v i x j = v i,j T i,j x k = T i,j,k (.6) 14 Usefulness of the indicial notation is in presenting systems of equations in compact form. For instance: x i = c ij z j (.7) this simple compacted equation, when expanded would yield: x 1 = c 11 z 1 + c 1 z + c 13 z 3 x = c 1 z 1 + c z + c 3 z 3 (.8-a) x 3 = c 31 z 1 + c 3 z + c 33 z 3 Similarly: A ij = B ip C jq D pq (.9) A 11 = B 11 C 11 D 11 + B 11 C 1 D 1 + B 1 C 11 D 1 + B 1 C 1 D A 1 = B 11 C 11 D 11 + B 11 C 1 D 1 + B 1 C 11 D 1 + B 1 C 1 D A 1 = B 1 C 11 D 11 + B 1 C 1 D 1 + B C 11 D 1 + B C 1 D A = B 1 C 1 D 11 + B 1 C D 1 + B C 1 D 1 + B C D (.10-a) 15 Using indicial notation, we may rewrite the definition of the dot product a b = a i b i (.11) and of the cross product a b = ε pqr a q b r e p (.1) we note that in the second equation, there is one free index p thus there are three equations, there are two repeated (dummy) indices q and r, thus each equation has nine terms..1. Tensor Operations 16 The sum of two (second order) tensors is simply defined as: S ij = T ij + U ij (.13)

13 .1 Tensors Rotation of Axes 3 The rule for changing second order tensor components under rotation of axes goes as follow: u i = a j i u j From Eq.?? = a j i T jqv q From Eq..1-a = a j i T jqa q pv p From Eq.?? (.0) But we also have u i = T ip v p (again from Eq..1-a) in the barred system, equating these two expressions we obtain T ip (a j i aq pt jq )v p = 0 (.1) hence T ip = a j i aq pt jq in Matrix Form [T ] = [A] T [T ][A] T jq = a j i aq pt ip in Matrix Form [T ] = [A][T ][A] T (.) By extension, higher order tensors can be similarly transformed from one coordinate system to another. 4 If we consider the D case, From Eq.?? cos α sin α 0 A = sin α cos α T = T xx T xy 0 T xy T yy T = A T T A = = T xx T xy 0 T xy T yy (.3-a) (.3-b) (.3-c) cos αt xx + sin 1 αt yy + sin αt xy ( sin αt xx + sin αt yy + cos αt xy 0 1 ( sin αt xx + sin αt yy + cos αt xy sin αt xx + cos α(cos αt yy sin αt xy (.3-d) alternatively, using sin α = sin α cos α and cos α = cos α sin α, this last equation can be rewritten as T xx T yy T xy = cos θ sin θ sin θ cos θ sin θ cos θ sin θ cos θ sin θ cos θ cos θ sin θ cos θ sin θ T xx T yy T xy (.4).1.4 Trace 5 The trace of a second-order tensor, denoted tr T is a scalar invariant function of the tensor and is defined as Thus it is equal to the sum of the diagonal elements in a matrix. tr T T ii (.5)

14 . Kinetics 7. Kinetics..1 Force, Traction and Stress Vectors 3 There are two kinds of forces in continuum mechanics body forces: act on the elements of volume or mass inside the body, e.g. gravity, electromagnetic fields. df = ρbdv ol. surface forces: are contact forces acting on the free body at its bounding surface. Those will be defined in terms of force per unit area. 33 The surface force per unit area acting on an element ds is called traction or more accurately stress vector. tds = i t x ds + j t y ds + k t z ds (.34) S S S S Most authors limit the term traction to an actual bounding surface of a body, and use the term stress vector for an imaginary interior surface (even though the state of stress is a tensor and not a vector). 34 The traction vectors on planes perpendicular to the coordinate axes are particularly useful. When the vectors acting at a point on three such mutually perpendicular planes is given, the stress vector at that point on any other arbitrarily inclined plane can be expressed in terms of the first set of tractions. 35 A stress, Fig.1 is a second order cartesian tensor, σ ij where the 1st subscript (i) refers to X 3 σ 33 σ 31 σ 3 σ 3 X 3 σ 11 σ σ σ σ X 1 X X 1 X Figure.1: Stress Components on an Infinitesimal Element the direction of outward facing normal, and the second one (j) to the direction of component force. σ 11 σ 1 σ 13 t 1 σ = σ ij = σ 1 σ σ 3 = t (.35) σ 31 σ 3 σ 33 t 3

15 Chapter 3 ELASTICITY BASED SOLUTIONS FOR CRACK PROBLEMS 3.1 Introduction 1 This chapter will present mathematically rigorous derivations of some simple elasticity problems. All the theoretical basis required to follow those derivations have been covered in the previous chapter. A summary of problems to be investigated is shown in Table Circular Hole, (Kirsch, 1898) Analysing the infinite plate under uniform tension with a circular hole of diameter a, and subjected to a uniform stress σ 0, Fig The peculiarity of this problem is that the far-field boundary conditions are better expressed in cartesian coordinates, whereas the ones around the hole should be written in polar coordinate system. 4 We will solve this problem by replacing the plate with a thick tube subjected to two different set of loads. The first one is a thick cylinder subjected to uniform radial pressure (solution of which is well known from Strength of Materials), the second one is a thick cylinder subjected to both radial and shear stresses which must be compatible with the traction applied on the rectangular plate. Problem Coordinate System Real/Complex Solution Date Circular Hole Polar Real Kirsh 1898 Elliptical Hole Curvilinear Complex Inglis 1913 Crack Cartesian Complex Westergaard 1939 V Notch Polar Complex Willimas 195 Dissimilar Materials Polar Complex Williams 1959 Anisotropic Materials Cartesian Complex Sih 1965 Table 3.1: Summary of Elasticity Based Problems Analysed

16 3. Circular Hole, (Kirsch, 1898) 3 1. Outer boundaries: around an infinitely large circle of radius b inside a plate subjected to uniform stress σ 0, the stresses in polar coordinates are obtained from Strength of Materials [ σrr σ rθ σ rθ σ θθ ] [ cos θ sin θ = sin θ cos θ ] [ σ ] [ ] T cos θ sin θ (3.7) sin θ cos θ yielding (recalling that sin θ = 1/ sin θ, and cos θ = 1/(1 + cos θ)). (σ rr ) r=b = σ 0 cos θ = 1 σ 0(1 + cos θ) (3.8-a) (σ rθ ) r=b = 1 σ 0 sin θ (3.8-b) (σ θθ ) r=b = σ 0 (1 cos θ) (3.8-c) For reasons which will become apparent later, it is more convenient to decompose the state of stress given by Eq. 3.8-a and 3.8-b, into state I and II: (σ rr ) I r=b = 1 σ 0 (3.9-a) (σ rθ ) I r=b = 0 (3.9-b) (σ rr ) II r=b = 1 σ 0 cos θ (3.9-c) (σ rθ ) II r=b = 1 σ 0 sin θ (3.9-d) Where state I corresponds to a thick cylinder with external pressure applied on r = b and of magnitude σ 0 /. Hence, only the last two equations will provide us with boundary conditions.. Around the hole: the stresses should be equal to zero: (σ rr ) r=a = 0 (3.10-a) (σ rθ ) r=a = 0 (3.10-b) 1 Upon substitution in Eq. 3.6 the four boundary conditions (Eq. 3.9-c, 3.9-d, 3.10-a, and 3.10-b) become ( A + 6C b 4 ( A + 6Bb 6C b ( 4 A + 6C + 4D ) b ) D b a 4 + 4D ) a ( A + 6Ba 6C a 4 D ) a = 1 σ 0 (3.11-a) = 1 σ 0 (3.11-b) = 0 (3.11-c) = 0 (3.11-d)

17 3.5 V Notch, (Williams, 195) Thus we can define a n = A n = cos(λ n 1)α B n cos(λ n + 1)α = ω sin(λ n 1)α sin(λ n + 1)α b n = C n = sin(λ n 1)α D n sin(λ n + 1)α = ω cos(λ n 1)α cos(λ n + 1)α these ratios are equal to 1/3 and 1 respectively for α = π and λ = 1/. and F (θ) = [ ( a n sin 3 θ + sin 1 ) ( 1 θ + b n 3 cos 3 θ + cos 1 )] θ (3.77-a) (3.77-b) (3.78) 67 The stresses are obtained by substituting σ rr = [ b n r ( 5 4 cos θ 1 4 σ θθ = [ b n r ( 3 4 cos θ σ rθ = [ b n r ( 1 4 sin θ θ cos 3θ cos 3θ sin ) + a ( n r ) + a ( n r 5 4 sin θ sin θ 3 4 ) + a ( n 1 r 4 cos θ )] 3θ sin )] 3θ sin cos 3θ )] (3.79-a) (3.79-b) (3.79-c) 68 These equations can be further simplified into σ rr = [ b n cos θ ( 1 + sin θ r ) + a ( n r σ θθ = [ b n cos θ ( 1 sin θ ) + a ( n r r σ rθ = [ b n r sin θ cos θ + a n r ( 1 4 cos θ sin θ sin θ 3 4 )] 3θ cos )] 3θ sin )] 3θ sin (3.80-a) (3.80-b) (3.80-c) 69 Finally, it can be shown that the displacements will be given by u = 1 { Re a n r λn [(κ + λ n cos α + cos λ n α) cos λ n θ λ n cos(λ n )θ] µ } b n r ξn [(κ + ξ n cos α cos ξ n α) sin ξ n θ ξ n sin(ξ n )θ] v = 1 { Re a n r λn [(κ λ n cos α cos λ n α) sin λ n θ + λ n sin(λ n )θ] µ } +b n r ξn [(κ ξ n cos α + cos ξ n α) cos ξ n θ + ξ n cos(ξ n )θ] (3.81-a) (3.81-b) 70 This solution can be compared with Westergaard s solution by comparing Equations 3.55-a and 3.56-a with Eq a; Eq b and 3.56-b with Eq b; and Eq c and 3.56-c with Eq c for n = 1. From this we observe that b 1 = K I π a 1 = K (3.8) II π

18 3.6 Crack at an Interface between Two Dissimilar Materials (Williams, 1959) 19 or F 1 (π) = F ( π) = 0 Continuity of σ θθ at the interface, θ = 0 Continuity of σ rθ at θ = 0 along the interface A 1 + B 1 = A + B (3.87) (λ 1)C 1 + (λ + 1)D 1 = (λ 1)C (λ + 1)D (3.88) Continuity of displacements (u r, u θ ) at the interface. Using the polar expression of the displacements u i r = 1 µ i r λ { (λ + 1)F i (θ) + 4(1 α i )[C i sin(λ 1)θ + A i cos(λ 1)θ]}(3.89-a) u i θ = 1 µ i r λ { F i (θ) 4(1 α i )[C i cos(λ 1)θ A i sin(λ 1)θ]} where µ is the shear modulus, and α i we obtain ν i 1+ν i (3.89-b) 1 [ (λ + 1)F 1 (0) + 4A 1 (1 α 1 )] = 1 [ (λ + 1)F (0) + 4A (1 α )](3.90-a) µ 1 µ 1 [ F µ 1 (0) 4C 1 (1 α 1 ) ] = 1 [ F 1 µ (0) 4C (1 α ) ] (3.90-b) Homogeneous Equations 74 Applying those boundary conditions, will lead to 8 homogeneous linear equations (Eq a, 3.85-b, 3.86-b, 3.86-c, 3.87, 3.88, 3.90-a, 3.90-b) in terms of the 8 unknowns A 1, B 1, C 1, D 1, A, B, C and D. 75 A nontrivial solution exists if the determinant of the 8 equations is equal to zero. This determinant 5 is equal to where k = µ 1 µ. [ ] k(1 cot α ) (1 α 1 ) (k 1) λπ + = 0 (3.91) k(1 α ) + (1 α 1 ) 76 For the homogeneous case α 1 = α and k = 1, the previous equation reduces to cot λπ = 0 or sin λπ = 0 thus we recover the same solution as the one of Eq b for a crack in one material: λ = n n = 1,, 3,... (3.9) Note that we exclude negative values of n to ensure finite displacements as the origin is approached, and the lowest eigenvalue controls. 5 The original paper states:... After some algebraic simplification...

19 3.6 Crack at an Interface between Two Dissimilar Materials (Williams, 1959) 1 80 Thus, Eq finally leads to Re(cot λπ) = 0 (3.100-a) Im(cot λπ) = ±β (3.100-b) we thus have two equations with two unknowns Solve for λ 81 Let us solve those two equations. Two sets of solutions are possible: 1. If from 3.99-b tan λ r π = 0 then λ r = n = 0, 1,, 3,... (3.101) and accordingly from Eq b λ j = ± 1 π coth 1 β (3.10). Alternatively, from Eq a cot λ r π = 0 tan λ r π = and 6 : λ r = n + 1 n = 0, 1,, 3,... (3.103-a) λ j = ± 1 π tanh 1 β (3.103-b) = 1 [ ] β + 1 π log (3.103-c) β 1 We note that for this case, λ j 0 as α 1 α and k 1 in β Near Crack Tip Stresses 8 Now that we have solved for λ, we need to derive expressions for the near crack tip stress field. We rewrite Eq as Φ(r) = r} λ+1 {{} F (θ, λ) (3.104) G(r) we note that we no longer have two sets of functions, as the effect of dissimilar materials has been accounted for and is embedded in λ. 83 The stresses will be given by Eq.?? 6 Recall that tanh 1 x = 1 σ rr = 1 Φ r θ + 1 Φ r r = r G(r)F (θ) + r 1 G (r)f (θ) (3.105-a) σ θθ = Φ r = G (r)f (θ) (3.105-b) σ rθ = 1 r Φ θ 1 r log 1+x 1 x Φ r θ = r G(r)F (θ) r 1 G (r)f (θ) (3.105-c)

20 3.6 Crack at an Interface between Two Dissimilar Materials (Williams, 1959) 3 thus, Re {sin [(λ r ± 1) + iλ j ] θ} = sin(λ r ± 1) cos(θ) cosh λ j θ (3.114-a) Re {cos [(λ r ± 1) + iλ j ] θ} = cos(λ r ± 1) cos(θ) cosh λ j θ (3.114-b) 87 Substituting those relations in Eq Re [F (θ)] = cosh λ j θ }{{} f(θ) (3.115-a) [A cos(λ r 1)θ + B cos(λ r + 1)θ + C sin(λ r 1)θ + D sin(λ r + 1)θ] (3.115-b) }{{} g(θ) Re [Φ(r, θ)] = r λr+1 cos(λ j log(r)) cosh λ j θ [A cos(λ r 1)θ + B cos(λ r + 1)θ +C sin(λ r 1)θ + D sin(λ r + 1)θ] (3.115-c) 88 For λ r = 1 g(θ) = A cos θ 3θ + B cos C sin θ 3θ + D sin (3.116) 89 Applying the boundary conditions at θ = ±π, σ θθ = 0, Eq b F (θ) = 0, that is g 1 ( π) = g (π) or C = D = a (3.117) 90 Similarly at θ = ±π, σ rθ = 0. Thus, from Eq c F (θ) = 0, or g 1 ( π) = g (π) or A = 3B = b (3.118) 91 From those two equations we rewrite Eq ( g(θ) = a sin θ ) ( 3θ + sin + b 3 cos θ ) 3θ + cos (3.119) 9 We now determine the derivatives f (θ) = λ j sinh λ j θ (3.10-a) ( 3 g 3θ (θ) = a cos + 1 cos θ ) ( + b 3 3θ sin 3 sin θ ) (3.10-b) 93 Thus, we now can determine F (θ) = f (θ)g(θ) + f(θ)g (θ) (3.11-a) { [ 3 3θ = a cosh λ j θ cos + 1 cos θ ] [ + λ j sinh λ j θ sin 3θ + sin θ ]} { ] ]} +b cosh λ j θ + λ j sinh λ j θ (3.11-b) [ 3 3θ sin 3 sin θ [ cos 3θ + 3 cos θ

21 3.8 Assignment 5 98 where s 1 and s are roots, in general complex, of Eq..17 where s j = α j + iβ j for j = 1,, and the roots of interests are taken such that β j > 0, and p j = a 11 s j + a 1 a 16 s j (3.15) q j = a 1 s j + a s j a 6 (3.16) 99 After appropriate substitution, it can be shown that the cartesian stresses at the tip of the crack for symmetric loading are [ ( )] K I s1 s σ x = s s 1 Re (3.17-a) πr s 1 s (cos θ + s sin θ) 1 (cos θ + s 1 sin θ) 1 [ ( )] K I 1 s σ y = 1 s Re (3.17-b) πr s 1 s (cos θ + s sin θ) 1 (cos θ + s 1 sin θ) 1 [ ( )] K I s1 s σ xy = 1 s 1 Re (3.17-c) πr s 1 s (cos θ + s 1 sin θ) 1 (cos θ + s sin θ) and, for plane skew-symmetric loading: σ x = K [ ( II 1 s Re πr s 1 s (cos θ + s sin θ) 1 σ y = K [ ( II 1 1 Re πr s 1 s (cos θ + s sin θ) 1 σ xy = K [ ( II 1 s 1 Re πr s 1 s (cos θ + s 1 sin θ) 1 s 1 )] (cos θ + s 1 sin θ) 1 )] 1 (cos θ + s 1 sin θ) 1 )] s (cos θ + s sin θ) 1 (3.18-a) (3.18-b) (3.18-c) 101 For in-plane loadings, these stresses can be summed to give the stresses at a distance r and an angle θ from the crack tip. 10 An important observation to be made is that the form of the stress singularity r 1/ is identical to the one found in isotropic solids. 103 It should be noted that contrarily to the isotropic case where both the stress magnitude and its spatial distribution are controlled by the stress intensity factor only, in the anisotropic case they will also depend on the material elastic properties and the orientation of the crack with respect to the principal planes of elastic symmetry (through s 1 and s ). 3.8 Assignment CVEN-6831 FRACTURE MECHANICS

22 3.8 Assignment 7 3. The stress intensity factor of the following problem: x x P B A x 1 P a a is given by: K A = K b = P a + x πa a x P πa a x a + x (3.130) (3.131) Based on those expressions, and results from the previous problem, determine the stress function Φ. 4. Barenblatt s model assumes a linearly varying closing pressure at the tip of a crack, c σy (a - c) σy c Using Mathematica and the expressions of K A and K B from the previous problem, determine an expression for the stress intensity factors for this case. 5. Using Mathematica, program either: (a) Westergaard s solution for a crack subjected to mode I and mode II loading. (b) Williams solution for a crack along dissimilar materials.

23 Chapter 4 LEFM DESIGN EXAMPLES 1 Following the detailed coverage of the derivation of the linear elastic stress field around a crack tip, and the introduction of the concept of a stress intensity factor in the preceding chapter, we now seek to apply those equations to some (pure mode I) practical design problems. First we shall examine how is linear elastic fracture mechanics (LEFM) effectively used in design examples, then we shall give analytical solutions to some simple commonly used test geometries, followed by a tabulation of fracture toughness of commonly used engineering materials. Finally, this chapter will conclude with some simple design/analysis examples. 4.1 Design Philosophy Based on Linear Elastic 3 One of the underlying principles of fracture mechanics is that unstable fracture occurs when the stress intensity factor (SIF) reaches a critical value K Ic, also called fracture toughness. K Ic represents the inherent ability of a material to withstand a given stress field intensity at the tip of a crack and to resist progressive tensile crack extension. 4 Thus a crack will propagate (under pure mode I), whenever the stress intensity factor K I (which characterizes the strength of the singularity for a given problem) reaches a material constant K Ic. Hence, under the assumptions of linear elastic fracture mechanics (LEFM), at the point of incipient crack growth: K Ic = βσ πa (4.1) 5 Thus for the design of a cracked, or potentially cracked, structure, the engineer would have to decide what design variables can be selected, as only, two of these variables can be fixed, and the third must be determined. The design variables are: Material properties: (such as special steel to resist corrosive liquid) K c is fixed. Design stress level: (which may be governed by weight considerations) σ is fixed. Flaw size: 1, a. 1 In most cases, a refers to half the total crack length.

24 4. Stress Intensity Factors 3 Figure 4.: Single Edge Notch Tension Panel Figure 4.3: Double Edge Notch Tension Panel Single Edge Notch Tension Panel (SENT) for L W =, Fig. 4. [ ( ) ( ) a a ( ) a 3 ( ) ] a 4 K I = σ πa W W W W (4.4) } {{ } β We observe that here the β factor for small crack ( a W approximately ) is grater than one and is Double Edge Notch Tension Panel (DENT), Fig. 4.3 K I = [ ( ) ( ) a a ( ) ] a σ πa W W W (4.5) } {{ } β Three Point Bend (TPB), Fig. 4.4 [ K I = 3 a W 1.99 ( ) ( ) ( a W 1 a W a W +.7 ( ) )] a W ( 1 + a ) ( ) 3 W 1 a W P S BW 3 (4.6) Compact Tension Specimen (CTS), Fig. 4.5 used in ASTM E-399 (399 n.d.) Standard Test Method for Plane-Strain Fracture Toughness of Metallic Materials [ ( ) ( ) a a ( ) a 3 ( ) ] a 4 P K I = πa (4.7) W W W W BW } {{ } β }{{} σ We note that this is not exactly the equation found in the ASTM standard, but rather an equivalent one written in the standard form.

25 Chapter 5 THEORETICAL STRENGTH of SOLIDS; (Griffith I) 1 We recall that Griffith s involvement with fracture mechanics started as he was exploring the disparity in strength between glass rods of different sizes, (Griffith 191). As such, he had postulated that this can be explained by the presence of internal flaws (idealized as elliptical) and then used Inglis solution to explain this discrepancy. In this section, we shall develop an expression for the theoretical strength of perfect crystals (theoretically the strongest form of solid). This derivation, (Kelly 1974) is fundamentally different than the one of Griffith as it starts at the atomic level. 5.1 Derivation 3 We start by exploring the energy of interaction between two adjacent atoms at equilibrium separated by a distance a 0, Fig The total energy which must be supplied to separate atom C from C is U 0 = γ (5.1) where γ is the surface energy 1, and the factor of is due to the fact that upon separation, we have two distinct surfaces Tensile Strength Ideal Strength in Terms of Physical Parameters We shall first derive an expression for the ideal strength in terms of physical parameters, and in the next section the strength will be expressed in terms of engineering ones. 1 From watching raindrops and bubbles it is obvious that liquid water has surface tension. When the surface of a liquid is extended (soap bubble, insect walking on liquid) work is done against this tension, and energy is stored in the new surface. When insects walk on water it sinks until the surface energy just balances the decrease in its potential energy. For solids, the chemical bonds are stronger than for liquids, hence the surface energy is stronger. The reason why we do not notice it is that solids are too rigid to be distorted by it. Surface energy γ is expressed in J/m and the surface energies of water, most solids, and diamonds are approximately.077, 1.0, and 5.14 respectively.

26 Chapter 6 ENERGY TRANSFER in CRACK GROWTH; (Griffith II) 1 In the preceding chapters, we have focused on the singular stress field around a crack tip. On this basis, a criteria for crack propagation, based on the strength of the singularity was first developed and then used in practical problems. An alternative to this approach, is one based on energy transfer (or release), which occurs during crack propagation. This dual approach will be developed in this chapter. 3 Griffith s main achievement, in providing a basis for the fracture strengths of bodies containing cracks, was his realization that it was possible to derive a thermodynamic criterion for fracture by considering the total change in energy of a cracked body as the crack length increases, (Griffith 191). 4 Hence, Griffith showed that material fail not because of a maximum stress, but rather because a certain energy criteria was met. 5 Thus, the Griffith model for elastic solids, and the subsequent one by Irwin and Orowan for elastic-plastic solids, show that crack propagation is caused by a transfer of energy transfer from external work and/or strain energy to surface energy. 6 It should be noted that this is a global energy approach, which was developed prior to the one of Westergaard which focused on the stress field surrounding a crack tip. It will be shown later that for linear elastic solids the two approaches are identical. 6.1 Thermodynamics of Crack Growth General Derivation 7 If we consider a crack in a deformable continuum subjected to arbitrary loading, then the first law of thermodynamics gives: The change in energy is proportional to the amount of work performed. Since only the change of energy is involved, any datum can be used as a basis for measure of energy. Hence energy is neither created nor consumed. 8 The first law of thermodynamics states The time-rate of change of the total energy (i.e., sum of the kinetic energy and the internal energy) is equal to the sum of the rate of work done by

27 6.1 Thermodynamics of Crack Growth 3 Figure 6.1: Energy Transfer in a Cracked Plate length a located in an infinite plate subjected to load P. Griffith assumed that it was possible to produce a macroscopical load displacement (P u) curve for two different crack lengths a and a + da. Two different boundary conditions will be considered, and in each one the change in potential energy as the crack extends from a to a + da will be determined: Fixed Grip: (u = u 1 ) loading, an increase in crack length from a to a + da results in a decrease in stored elastic strain energy, U, U = 1 P u 1 1 P 1u 1 (6.6) = 1 (P P 1 ) u 1 (6.7) < 0 (6.8) Furthermore, under fixed grip there is no external work (u = u 1 ), so the decrease in potential energy is the same as the decrease in stored internal strain energy, hence Π Π 1 = W U (6.9) = 1 (P P 1 )u 1 = 1 (P 1 P )u 1 (6.10) Fixed Load: P = P 1 the situation is slightly more complicated. Here there is both external work W = P 1 (u u 1 ) (6.11) and a release of internal strain energy. Thus the net effect is a change in potential energy given by: Π Π 1 = W U (6.1)

28 Chapter 7 MIXED MODE CRACK PROPAGATION 1 Practical engineering cracked structures are subjected to mixed mode loading, thus in general K I and K II are both nonzero, yet we usually measure only mode I fracture toughness K Ic (K IIc concept is seldom used). Thus, so far the only fracture propagation criterion we have is for mode I only (K I vs K Ic, and G I vs R). Whereas under pure mode I in homogeneous isotropic material, crack propagation is collinear, in all other cases the propagation will be curvilinear and at an angle θ 0 with respect to the crack axis. Thus, for the general mixed mode case, we seek to formulate a criterion that will determine: 1. The angle of incipient propagation, θ 0, with respect to the crack axis.. If the stress intensity factors are in such a critical combination as to render the crack locally unstable and force it to propagate. 3 Once again, for pure mode I problems, fracture initiation occurs if: K I K Ic (7.1) 4 The determination of a fracture initiation criterion for an existing crack in mode I and II would require a relationship between K I, K II, and K Ic of the form F (K I, K II, K Ic ) = 0 (7.) and would be analogous to the one between the two principal stresses and a yield stress, Fig. 7.1 F yld (σ 1, σ, σ y ) = 0 (7.3) Such an equation may be the familiar von Mises criterion. 5 In the absence of a widely accepted criterion for mixed mode crack growth, three of the most widely used criterion are discussed below.

29 7.1 Analytical Models for Isotropic Solids 3 Solution of the second equation yields the angle of crack extension θ 0 ( tan θ 0 = 1 K I ± 1 ) KI + 8 (7.8) 4 K II 4 K II 10 For the crack to extend, the maximum circumferential tensile stress, σ θ (from Eq b and 3.56-b) σ θ = K I cos θ ( 0 πr 1 sin θ 0 ) + K ( II 3 πr 4 sin θ sin 3θ ) 0 (7.9) must reach a critical value which is obtained by rearranging the previous equation σ θmax πr = KIc = cos θ 0 [ K I cos θ 0 3 ] K II sin θ 0 (7.10) which can be normalized as K I cos 3 θ 0 K Ic 3 K II cos θ 0 K Ic sin θ 0 = 1 (7.11) 11 This equation can be used to define an equivalent stress intensity factor K eq for mixed mode problems K eq = K I cos 3 θ 0 3 K II cos θ 0 sin θ 0 (7.1) 7.1. Maximum Energy Release Rate 1 In their original model, Erdogan, F. and Sih, G.C. (1963) noted that: If we accept Griffith (energy) theory as the valid criteria which explains crack growth, then the crack will grow in the direction along which the elastic energy release per unit crack extension will be maximum and the crack will start to grow when this energy reaches a critical value (or G = G(δ, θ)). Evaluation of G(δ, θ) poses insurmountable mathematical difficulties. 13 Finding G(δ, θ) will establish for the general mixed mode case the duality which is the basis of fracture mechanics: the equivalence in viewing fracture initiation from either a global energy balance or a local stress intensity point of view. 14 This (insurmountable) problem was solved in 1974, by Hussain et al. (1974). Fundamentally, Hussain et al. (1974) have solved for the stress intensity factor of a major crack with an infinitesimal kink at an angle θ, K I (θ) and K II (θ) in terms of the stress intensity factors of the original crack K I and K II, Fig. 7.: { KI (θ) K II (θ) } ( ) 4 ( 1 θ π = 3 + cos θ 1 + θ π ) θ π { K I cos θ + 3 K II sin θ K II cos θ 1 K I sin θ } (7.13)

30 Part II ELASTO PLASTIC FRACTURE MECHANICS

31 Chapter 8 PLASTIC ZONE SIZES 1 it was shown in chapter?? that, under linear elastic fracture mechanics assumptions, the stress at the crack tip is theoretically infinite. Clearly, all materials have a finite strength, thus there will always be a small plastified zone around the crack tip. If this zone is small compared to the crack size, then our linear elastic assumptions are correct; if not, LEFM is not applicable (thus it would be incorrect to use a K or G criterion) and a nonlinear model must be used. This damaged zone is referred to as a plastic zone for metals, and a fracture process zone for cementitious materials and ceramics. 3 Thus there are two important issues associated with nonlinear fracture: 1. What is the size of the plastic or process zone?. What are the criteria for crack growth? 4 This chapter will answer the first question by focusing on metals 1, whereas the next chapter will develop criterions for crack growth. 5 The evaluation of the plastic zone for plastified materials can be determined through various levels of approximations: 1. Uniaxial stress criteria (a) first order approximation (b) second order approximation (Irwin) (c) Dugdale s model. Multiaxial yield criteria Each one of them will be separately reviewed. 8.1 Uniaxial Stress Criteria 6 First we shall examine criteria in which only the uniaxial stress state (σ yy normal to thee crack axis) and we shall consider three models of increasing complexities. 1 Due to the intrinsically different behavior of concrete compared to metals, estimates of the fracture process zone will be separately discussed.