## 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...

## 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...

## 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...

## 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...

## 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...

## 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...

## 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....

## 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...

## Theoretical Strength of Solids Griffith I

7 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 1921). 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. 8 In this section, we shall develop an expression for the theoretical strength of perfect crystals (theoretically the strongest form of solid). This...

## 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...

## 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...

## Elliptical hole in a Uniformly Stressed Plate Inglis

24 Next we consider the problem of an elliptical hole in an infinite plate under uniform stress, Fig. 6.2. Adopting the curvilinear coordinate system described in sect. 5.8, we define a and b as the major and minor semi-axes respectively. The elliptical hole is itself defined along a ao, and as we go around the ellipse varies from 0 to 2n. 25 Thus substituting fJ 0 and 3 in Eq. 5.139 and 5.140 we obtain x p o a c cosh a cos 3 c cosh a0 a c cosh a0 x2 p n 2 b c sinh a sin 3 c sinh a0 b c sinh a0...

## 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...

## Plasticityvon Mises Plasticity

51 For J2 plasticity or von Mises plasticity, our stress function is perfectly plastic. Recall perfectly plastic materials have a total modulus of elasticity Et which is equivalent to zero. We will deal now with deviatoric stress and strain for the J2 plasticity stress function. since lt 7 0 in perfect plasticity, the second term drops out and F becomes S 2G e 2G - p 2.71 F 2Gs - m 0 2.73 4. Tangential stress-strain relation deviatoric Now we have the simplified expression Gep 2G 4 2-78 is the...

## 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...

## Energy Transfer in Crack Growth Griffith II

7 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. 8 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. 9 Griffith's main achievement, in providing a basis for the fracture strengths of...

## 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...