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...

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...

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...

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...

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...

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...

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....

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...

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

Elliptic Hole Infinite Plate

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

Stress Intensity Hole

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...

Crack Tip Opening Displacements

Crack Tip Opening Displacement

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...

Plane Strain vs Plane Stress

Stress And Mechanics

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...