## Usw

Embedded Elliptical Crack A large number of naturally occurring defects are present as embedded, surface or corner cracks (such as fillet welding) Irwin, (Irwin 1962) proposed Table 7.2 c Factors for Point Load on Edge Crack Figure 7.12 Embedded, Corner, and Surface Cracks the following solution for the elliptical crack, with x a cos 6 and y b sin 6 if b2 Ki (6) sin2 6+ cos2 6 aV b where o is a complete elliptical integral of the second kind An approximation to Eq. 7.17 was given by Cherepanov...

## Additional Design Considerations Leak Before Fail

16 As observed from the preceding example, many pressurized vessels are subject to crack growth if internal flaws are present. Two scenarios may happen, Fig. 7.14 Break-through In this case critical crack configuration is reached before the crack has day-lighted , and there is a sudden and unstable crack growth. Leak Before Fail In this case, crack growth occur, and the crack pierces through the thickness of the vessel before unstable crack growth occurs. This in turn will result in a sudden...

## Analytical Models for Isotropic Solids

12 First four models for the mixed mode crack propagation will be presented, three for isotropic materials, and one for anisotropic ones. Subsequently the actual algorithmic implementation of those models will be presented. 10.1.1 Maximum Circumferential Tensile Stress. 13 Erdogan and Sih (Erdogan, F. and Sih, G.C. 1963) presented the first mixed-mode fracture initiation theory, the maximum circumferential tensile stress theory. It is based on the knowledge of the stress state near the tip of a...

## Basic Equations of Anisotropic Elasticity

119 An alternate form of Eq. 5.104 is where the indices i and j go from 1 to 6. In the most general case this would yield 36 independent constants aij, (Lekhnitskii 1981), however by virtue of symmetry (a a , ) this reduces to 21 ilKil j _ If the the material has one plane of elastic symmetry, then there would 13 independent constants If it has three mutually orthogonal planes of elastic symmetry, then we would say that it is orthogonally anisotropic or orthotropic, and we will have ai6 a26 a36...

## Classification of Failure Kinematics

I We can distinguish three separate modes of failure according to the degree of C-(dis)continuity in failure processes, Fig. 1.1 ( ). Figure 1.1 Kinematics of Continuous and Discontinuous Failure Processes Diffuse failure corresponds to C1 -continuity of motion. We do have a continuity of both displacement (increments) and strains across a potential failure surface. Localized failure corresponds to a C0 continuity of motion. Whereas continuity of displacements is maintained, we do have a...

## Complex Variables

99 As will be shown in the next chapter, we can use Airy stress function with real variables to determine the stress field around a circular hole, however we need to extend Airy stress functions to complex variables in order to analyze 1) stresses around an elliptical hole (Inglis), and stresses at the tip of a crack (Westergaard). 100 First we define the complex number z as where i l. x and x2 are the cartesian coordinates, and r and 6 are the polar coordinates. 101 We further define an...

## Constitutive Equations

The Generalized Hooke's Law can be written as so The (fourth order) tensor of elastic constants Dijkl has 81 (34) components however, due to the symmetry of both a and e, there are at most 36 distinct elastic terms. si For the purpose of writing Hooke's Law, the double indexed system is often replaced by a simple indexed system with a range of six ok Dkm em k,m 1,2,3,4,5,6 (5.105) 82 In terms of Lame's constants, Hooke's Law for an isotropic body is written as 83 In terms of engineering...

## Contents

1 FINITE ELEMENT MODELS FOR for PROGRESSIVE FAILURES 1 1.1 Classification of Failure 2 Continuum Mechanics Based Description of Failure Plasticity 1 2.1.1 Uniaxial Behavior 2.1.2 Idealized Stress-Strain 2.1.3 Hardening 2.2 Review of Continuum 2.2.1.1 Hydrostatic and Deviatoric Stress Tensors 5 2.2.1.2 Geometric Representation of Stress States 6 2.2.2.1 Hydrostatic and Deviatoric Strain Tensors 8 2.3 Yield 2.4 Rate Theory 2.5 J2 Plasticity von Mises 2.5.1 Isotropic Hardening Softening( J2...

## Coverage

Following this brief overview, chapter two will provide the reader with a review of elasticity. In particular we shall revisit the major equations needed to analytically solve simple problems involving elliptical holes or sharp cracks. Those solutions will be presented in detail in chapter three. This chapter, mathematically the most challenging, is an important one to understand the mathematical complexity of solutions of simple crack problem, and to appreciate the value of numerical based...

## Crack at an Interface between Two Dissimilar Materials Williams

77 We shall now consider the problem of a crack at the interface between two dissimilar isotropic materials, (Williams 1959, Zak and Williams 1963). Accordingly, we rewrite Eq. 6.61 as where the subscript i refers to material 1 and 2, Fig. 6.6 Fi (0, A) Ai cos(A 1)0 + Bi cos(A + 1)0 + Ci sin(A 1)0 + Di sin(A + 1)0 (6.88) 79 Boundary conditions for this problem are zero stresses, aoo, on the free edges (at 0 n). Thus from Eq. 6.69-a a0e o n rA-1A(A +1) A1 cos(A 1)n + B1 cos(A + 1)n +Ci sin(A 1)n...

## Crack Stability

Crack stability depends on both the geometry, and on the material resistance. 52 From Eq. 9.6, crack growth is considered unstable when the energy at equilibrium is a maximum, and stable when it is a minimum. Hence, a sufficient condition for crack stability is, (Gtoudos 1993) < 0 unstable fracture > 0 stable fracture 0 neutral equilibrium and the potential energy is n U W. 53 If we consider a line crack in an infinite plate subjected to uniform stress, Fig. 9.7, then the potential energy...

## Curvilinear Coordinates

111 A variable z x1 + ix2 in the cartesian coordinate system can be expressed as p a + i3 where a and 3 are co-ordinates in a curvilinear system. 112 We next seek to solve for xi and x2 in terms of a and 3. The relationship between z and p is given by where c is a constant. 113 Recalling that we substitute those equations into Eq. 5.136 ea cos 3 + iea sin (3 + e a cos ( ie a sin ( cos (3 (ea + e a) +i sin (3 (ea e a) Separating reals from immaginary parts we obtain If we eliminate 3 from those...

## D S

Fracture initiation is assumed to occur when Somin reaches the maximum critical value 29 We note that it is also possible to decompose the strain energy density into two components a volumetric one and a dilational one where the dilational part is given by (ax - ay)2 + (ay - az)2 + (az - ax)2 + 6(Xy + Tyz + TXz) 30 We note that Smin is associated with brittle fracture, and Smax is associated with yielding. Its direction coincides with the direction of maximum distortion while Smin coincides...

## Derivation of CTOD

16 The vertical displacement of a point next to the crack tip due to mode I loading is given by Eq. 6.56-f, Fig. 12.2 Figure 12.2 Estimate of the Crack Tip Opening Displacement, (Anderson 1995) 17 If we substitute 6 we obtain the upper and lower displacements of the crack face, and due to symmetry their sum corresponds to the crack opening displacement. Hence the crack opening is given by

## Due dup or

Where R represents the rate of energy dissipation during stable crack growth. The first part corresponds to plastic deformation, and the second to energy consumed during crack propagation. 58 Back to Eq. 9.52, crack instability will occur when for an infinitesimal crack extension da, the rate of energy released is just equal to surface energy absorbed. Which is Eq. 9.27 as originally derived by Griffith (Griffith 1921). 59 This equation can be rewritten as 60 In general, the critical energy...

## Dw fdue dup dr

17 This equation represents the energy balance during crack growth. It indicates that the work rate supplied to the continuum by the applied external loads is equal to the rate of strain energy (elastic and plastic) plus the energy dissipated during crack propagation. that is the rate of potential energy decrease during crack growth is equal to the rate of energy dissipated in plastic deformation and crack growth. 19 It is very important to observe that the energy scales with a2, whereas...

## Examples of Structural Failures Caused by Fracture

Some well-known, and classical, examples of fracture failures include Mechanical, aeronautical, or marine 1. fracture of train wheels, axles, and rails 2. fracture of the Liberty ships during and after World War II 3. fracture of airplanes, such as the Comet airliners, which exploded in mid-air during the fifties, or more recently fatigue fracture of bulkhead in a Japan Air Line Boeing 747 4. fatigue fractures found in the Grumman buses in New York City, which resulted in the recall of 637 of...

## Extensions to Anisotropic Solids

38 The maximum circumferential tensile stress theory, originally developed for isotropic solids, has been extended to the anisotropic case by Saouma et al. (Saouma, Ayari and Leavell 1987). 39 The fracture toughness, as the elastic modulus, is uniquely defined for an isotropic material. However, for a homogeneous transversely isotropic solid with elastic constants e1,e2, g12, and 12, two values are needed to characterize the brittle behavior of the crack, Kc and K2c, as shown in Fig. 10.6 along...

## Fracture Mechanics vs Strength of Materials

In order to highlight the fundamental differences between strength of materials and fracture mechanics approaches, we consider a simple problem, a cantilevered beam of length L, width B, height H, and subjected to a point load P at its free end, Fig. 4.1 Maximum flexural stress Figure 4.1 Cracked Cantilevered Beam Figure 4.1 Cracked Cantilevered Beam We will seek to determine its safe load-carrying capacity using the two approaches2. 1. Based on classical strength of materials the maximum...

## Fracture Properties of Materials

15 Whereas fracture toughness testing will be the object of a separate chapter, we shall briefly mention the appropriate references from where fracture toughness values can be determined. Metallic Alloys 16 Testing procedures for fracture toughness of metallic alloys are standardized by various codes (see (Anon. n.d.) and (British Standards Institution, BS 5447, London 1977)). An exhaustive tabulation of fracture toughnesses of numerous alloys can be found in (Hudson and Seward 1978) and...

## Kfa

Combining this equation with Eqn. 10.62 gives G (1o 2) ( c 2 + d 2)KK + 2Re(cdK2a2i ) (10.67) 4Note the analogy between this expression and Eq. 10.13 for the SIF at the tip of a kinked crack as developed by Hussain et al. for the Maximum Energy Release Rate criterion. This expression can be reduced further by writting K as (He and Hutchinson 1989), where by 10.59, L is the in-plane length quantity characterizing the specific interface crack problem when a 0. From Equations 10.65, 10.67 and...

## Homogeneous Anisotropic Material Sih and Paris

102 To analyze an anisotropic body with with a crack, we need to derive the two stress functions 1 and 2 in Eq. 5.154 such that they satisfy the boundary conditions of the problem under consideration. 103 For an infinite plate with a central crack in an anisotropic body the derivation for the stress functions was undertaken by Sih, Paris and Irwin, (Sih, Paris and Irwin 1965). This solution is the counterpart or generalization of Westergaard's solutions. S1P2 (cos 9 + S2 sin 9)2 S2P1 (cos 9 + s...

## Info

Figure 8.4 Influence of Atomic Misfit on Ideal Shear Strength Figure 8.4 Influence of Atomic Misfit on Ideal Shear Strength This equation, combined with Eq. 8.9 will finally give which is an approximate expression for the theoretical maximum strength in terms of E. 8.1.2 Shear Strength 14 Similar derivation can be done for shear. What happen if we slide the top row over the bottom one. Again, we can assume that the shear stress is 15 Because we do have very small displacement, we can elliminate...

## Interface Cracks

57 Interface crack propagation is rapidly gaining wide attention. Such crack growth is exceedingly important for composite materials, and can be extended to concrete rock crack interfaces1. 58 A crack once initiated within an interface can either be stable or propagate in an unstable manner. The unstable crack can propagate along the interface or kink into one of the materials. It can also branch out, that is, it can propagate along the interface and then kink. Whether a crack is forced to...

## Kf fd

For a sharp crack and linear elastic systems during crack extension AV 0 and during crack extension and t* t0, thus -An a E dV-- t AuidT (13.27) 7. Back to sharp crack but not necessarily linear elastic systems, we can write dU d ( 1 wdV - t*du*dr) (13.28) For unit thickness dV dxdy and dA dx It is identical to the previous expression but we do not restrict ourselves to go around Av , but around any closed path. This integral was first proposed by Eshelby who had defined a number of contour...

## Kinematic

The Lagrangian finite strain tensor can be written as The Eulerian finite strain tensor can be written as or E* i(uVx + Vxu Vxu-uVx) z v v y du1 du1 du2 du2 du3 du3 dx1 dx2 dx1 dx2 dx1 dx2_ 56 Alternatively these equations may be expanded as 1. We define the engineering shear strain as 2. If the strains are given, then these strain-displacements provide a system of (6) nonlinear partial differential equation in terms of the unknown displacements (3). 3. ik is the Green-Lagrange strain tensor....

## List of Tables

4.1 Column Instability Versus Fracture 5.1 Number of Elastic Constants for Different Materials 27 6.1 Summary of Elasticity Based Problems 7.1 Newman's Solution for Circular Hole in an Infinite Plate subjected to Biaxial Loading, and Internal 7.2 C Factors for Point Load on Edge 7.3 Approximate Fracture Toughness of Common Engineering Materials 12 7.4 Fracture Toughness vs Yield Stress for .45C N Cr Mo Steel 12 10.1 Material Properties and Loads for Different Cases 23 10.2 Analytical and...

## Major Historical Developments in Fracture Mechanics

As with any engineering discipline approached for the first time, it is helpful to put fracture mechanics into perspective by first listing its major developments 1. In 1898, a German Engineer by the name of Kirsch showed that a stress concentration factor of 3 was found to exist around a circular hole in an infinite plate subjected to uniform tensile stresses (Timoshenko and Goodier 1970). 2. While investigating the unexpected failure of naval ships in 1913, Inglis (Inglis 1913) extended the...

## Mixed Mode Crack Propagation

7 Practical engineering cracked structures are subjected to mixed mode loading, thus in general Ki and Kii are both nonzero, yet we usually measure only mode I fracture toughness Kic (Kiic concept is seldom used). Thus, so far the only fracture propagation criterion we have is for mode I only (Ki vs Kic, and Gi vs R). 8 Whereas under pure mode I in homogeneous isotropic material, crack propagation is colinear, in all other cases the propagation will be curvilinear and at an angle 00 with...

## 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) 2. large elastic deformation (jamming) 3. gross plastic deformation (yielding) 4. tensile instability (necking) 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...

## Multiaxial Yield Criteria

31 All the previous models have restricted themselves to 6 0 and have used uniaxial yield criteria, but the size of the plastic zone can be similarly derived from a multi-axial yield criterion. The principal stresses at a point with respect to the crack tip are given by where the stresses were obtained in Eq. 6.56-a, 6.56-b, and 6.56-c 33 With those stress expressions, any yield criteria could be used. Using the Von-Mises criteria, we would obtain and yielding would occur when ae reaches ayld....

## P

Figure 13.7 Experimental Derivation of J Figure 13.8 j Resistance Curve for Ductile Material, (Anderson 1995) Figure 13.8 j Resistance Curve for Ductile Material, (Anderson 1995) propagation occurs. An arbitrary definition of a critical J, Jjc can be specified at the onset of initiation. 13.7 f Load Control versus Displacement Control 47 Crack propagation can occur under either load or displacement control. Whereas in both cases J (and G) is the same, the rate of change of J depends on the...

## S V

Figure 6.5 Plate with Angular Corners Figure 6.5 Plate with Angular Corners 58 Using the method of separation of variables in 1952, Williams (Williams 1952, Williams 1957) proposed the following solution where F(6, A) em(A , and m(A) is yet to be determined, to the bi-harmonic equation, Eq. 5.124 V I dr2 r 8r r2 062 J I Or2 r Or r2 062 J 59 Note that the problem he originally considered was not a crack, but rather a plate under tension with angular corners, Fig. 6.5 making an angle 2y. For y 0...

## Thermodynamics of Crack Growth

13 If we consider a crack in a deformable continuum aubjected to arbitrary loading, then the first law of thermodynamics gives 14 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. 15 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...

## U uV

In which i and v are the shear modulus and Poisson's ratio, respectively, and subscripts 1 and 2 refer to the materials above and below the interface, respectively. Dunders defines an additional mismatch parameter a where E is the plane strain Young's modulus, E E (1 v2).We note that Q, a and vanish when the materials above and below the interface are identical. When 0, Equation 10.53 shows that the stresses oscillate3heavily as the crack tip is approached (r 0). Furthermore, the relative...

## Uniaxial Stress Criteria

12 First we shall examine criteria in which only the uniaxial stress state (ayy normal to thee crack axis) and we shall consider three models of increasing complexities. 1Due to the intrinsically different behavior of concrete compared to metals, estimates of the fracture process zone will be separately discussed. The simplest estimate of the size of a process zone is obtained by equating ay (Eq. 6.56-b) (27rr)t 2 to the yield stress ayld for 0 0, Fig. 11.1 Figure 11.1 First-Order Approximation...

## Yield Criteria

36 In uniaxial stress states, the elastic limit is obtained by a well-defined yield stress point a0. In biaxial or triaxial state of stresses, the elastic limit is defined mathematically by a certain yield criterion which is a function of the stress state aij expressed as For isotropic materials, the stress state can be uniquely defined by either one of the following set of variables those equations represent a surface in the principal stress space, this surface is called the yield surface....

## Derivation

9 We start by exploring the energy of interaction between two adjacent atoms at equilibrium separated by a distance a0, Fig. 8.1. The total energy which must be supplied to separate atom C from C' is where 7 is the surface energy1, and the factor of 2 is due to the fact that upon separation, we have two distinct surfaces. 8.1.1.1 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...

## Kinetics

5.2.1 Force, Traction and Stress Vectors 38 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 pbdVol. 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. 39 The surface force per unit area acting on an element dS is called traction or more accurately stress vector. f tdS if txdS jf tydS kf tzdS 5.37...

## Fundamental Laws of Continuum Mechanics

58 We have thus far studied the stress tensors Cauchy, Piola Kirchoff , and several other tensors which describe strain at a point. In general, those tensors will vary from point to point and represent a tensor field. 59 We have also obtained only one differential equation, that was the compatibility equation. 60 In this chapter, we will derive additional differential equations governing the way stress and deformation vary at a point and with time. They will apply to any continuous medium, and...

## Review of Continuum Mechanics Stress

2.2.1.1 Hydrostatic and Deviatoric Stress Tensors If we let a denote the mean normal stress p o -p i lt 7n 722 f 33 U tr a 2.17 then the stress tensor can be written as the sum of two tensors Hydrostatic stress in which each normal stress is equal to p and the shear stresses are zero. The hydrostatic stress produces volume change without change in shape in an isotropic medium. _ Deviatoric Stress which causes the change in shape. 2.2.1.2 Geometric Representation of Stress States 23 Using the...

## Stress Intensity Factors

12 As shown in the preceding chapter, analytic derivation of the stress intensity factors of even the simplest problem can be quite challenging. This explain the interest developed by some mathematician in solving fracture related problems. Fortunately, a number of simple problems have been solved and their analytic solution is found in stress intensity factor handbooks. The most commonly referenced ones are Tada, Paris and Irwin's Tada et al. 1973 , and Roorke and Cartwright, Rooke and...

## List of Figures

1.1 Kinematics of Continuous and Discontinuous Failure Processes 1 1.2 Discrete-Smeared Crack 2.1 Typical Stress-Strain Curve of an Elastoplastic 2.2 Elastoplastic Rheological Model for Overstress 2.3 Bauschinger Effect on Reversed 2.4 Stress and Strain Increments in Elasto-Plastic 2.5 Stress-Strain diagram for Elastoplasticity 2.6 Haigh-Westergaard Stress Space 2.7 General Yield 2.8 Isotropic 2.9 Kinematic 4.1 Cracked Cantilevered 4.2 Failure Envelope for a Cracked Cantilevered 4.3 Generalized...

## Griffith Theory

17 Around 1920, Griffith was exploring the theoretical strength of solids by performing a series of experiments on glass rods of various diameters. He observed that the tensile strength at of glass decreased with an increase in diameter, and that for a diameter 0 t in., at 500,000 psi furthermore, by extrapolation to zero diameter he obtained a theoretical maximum strength of approximately 1,600,000 psi, and on the other hand for very large diameters the asymptotic values was around 25,000 psi....

## Crack Tip Opening Displacements

7 Within the assumptions and limitations of LEFM we have two valid and equivalent criteria for crack propagation 1 k vs kic which is a local criteria based on the strength of the stress singularity at the tip of the crack and 2 g vs gic or R which is a global criteria based on the amount of energy released or consumed during a unit surface crack's propagations. 8 In many cases it is found that LEFM based criteria is either too conservative and expensive as it does not account for plastification...

## Crack Westergaard

35 Just as both Kolosoff 1910 and Inglis 1913 independently solved the problem of an elliptical hole, there are two classical solutions for the crack problem. The first one was proposed by Westergaard, and the later by Williams. Whereas the first one is simpler to follow, the second has the advantage of being extended to cracks at the interface of two different homogeneous isotropic materials and be applicable for V notches. 36 Let us consider an infinite plate subjected to uniform biaxial...

## Tensors

10 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. 11 We hereby adopt the dyadic notation for tensors as linear vector operators 12 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...

## Lefm Design Examples

7 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. 8 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...

## Plane Strain vs Plane Stress

36 Irrespective of a plate thickness, there is a gradual decrease in size of the plastic zone from the plate surface plane stress to the interior plane strain , Fig. 11.8. 37 The ratio of the plastic zone size to the plate thickness must be much smaller than unity for plane strain to prevail. It has been experimentally shown that this ratio should be less than 0.025. _ Figure 11.8 Plastic Zone Size Across Plate Thickness 38 We also observe that since rp is proportional to the plate thickness...