First Edition, 2012 ISBN 978-81-323-0884-3 © All rights reserved. Published by: Academic Studio 4735/22 Prakashdeep Bldg, Ansari Road, Darya Ganj, Delhi - 110002 Email: [email protected] Table of Contents Chapter 1 - Fracture Mechanics Chapter 2 - Fatigue (Material) Chapter 3 - Fractography Chapter 4 - Focal Mechanism Chapter 5 - Fracture Toughness Chapter 6 - Fracture Chapter 7 - Structural Fracture Mechanics Chapter 8 - Peridynamics Chapter 9 - Bending Chapter 10 - Euler–Bernoulli Beam Equation Chapter 11 - Stress (Mechanics) Chapter 12 - Shear Stress Chapter 13 - Timoshenko Beam Theory Chapter 1 Fracture Mechanics Fracture mechanics is the field of mechanics concerned with the study of the propagation of cracks in materials. It uses methods of analytical solid mechanics to calculate the driving force on a crack and those of experimental solid mechanics to characterize the material's resistance to fracture. In modern materials science, fracture mechanics is an important tool in improving the mechanical performance of materials and components. It applies the physics of stress and strain, in particular the theories of elasticity and plasticity, to the microscopic crystallographic defects found in real materials in order to predict the macroscopic mechanical failure of bodies. Fractography is widely used with fracture mechanics to understand the causes of failures and also verify the theoretical failure predictions with real life failures. An edge crack (flaw) of length a in a material. Linear elastic fracture mechanics Griffith's criterion Fracture mechanics was developed during World War I by English aeronautical engineer, A. A. Griffith, to explain the failure of brittle materials. Griffith's work was motivated by two contradictory facts: • The stress needed to fracture bulk glass is around 100 MPa (15,000 psi). • The theoretical stress needed for breaking atomic bonds is approximately 10,000 MPa (1,500,000 psi). A theory was needed to reconcile these conflicting observations. Also, experiments on glass fibers that Griffith himself conducted suggested that the fracture stress increases as the fiber diameter decreases. Hence the uniaxial tensile strength, which had been used extensively to predict material failure before Griffith, could not be a specimen- independent material property. Griffith suggested that the low fracture strength observed in experiments, as well as the size-dependence of strength, was due to the presence of microscopic flaws in the bulk material. To verify the flaw hypothesis, Griffith introduced an artificial flaw in his experimental specimens. The artificial flaw was in the form of a surface crack which was much larger than other flaws in a specimen. The experiments showed that the product of the square root of the flaw length (a) and the stress at fracture (σ) was nearly constant, which is f expressed by the equation: An explanation of this relation in terms of linear elasticity theory is problematic. Linear elasticity theory predicts that stress (and hence the strain) at the tip of a sharp flaw in a linear elastic material is infinite. To avoid that problem, Griffith developed a thermodynamic approach to explain the relation that he observed. The growth of a crack requires the creation of two new surfaces and hence an increase in the surface energy. Griffith found an expression for the constant C in terms of the surface energy of the crack by solving the elasticity problem of a finite crack in an elastic plate. Briefly, the approach was: • Compute the potential energy stored in a perfect specimen under an uniaxial tensile load. • Fix the boundary so that the applied load does no work and then introduce a crack into the specimen. The crack relaxes the stress and hence reduces the elastic energy near the crack faces. On the other hand, the crack increases the total surface energy of the specimen. • Compute the change in the free energy (surface energy − elastic energy) as a function of the crack length. Failure occurs when the free energy attains a peak value at a critical crack length, beyond which the free energy decreases by increasing the crack length, i.e. by causing fracture. Using this procedure, Griffith found that where E is the Young's modulus of the material and γ is the surface energy density of the material. Assuming E = 62 GPa and γ = 1 J/m2 gives excellent agreement of Griffith's predicted fracture stress with experimental results for glass. Irwin's modification The plastic zone around a crack tip in a ductile material. Griffith's work was largely ignored by the engineering community until the early 1950s. The reasons for this appear to be (a) in the actual structural materials the level of energy needed to cause fracture is orders of magnitude higher than the corresponding surface energy, and (b) in structural materials there are always some inelastic deformations around the crack front that would make the assumption of linear elastic medium with infinite stresses at the crack tip highly unrealistic. F. Erdogan (2000) Griffith's theory provides excellent agreement with experimental data for brittle materials such as glass. For ductile materials such as steel, though the relation still holds, the surface energy (γ) predicted by Griffith's theory is usually unrealistically high. A group working under G. R. Irwin at the U.S. Naval Research Laboratory (NRL) during World War II realized that plasticity must play a significant role in the fracture of ductile materials. In ductile materials (and even in materials that appear to be brittle), a plastic zone develops at the tip of the crack. As the applied load increases, the plastic zone increases in size until the crack grows and the material behind the crack tip unloads. The plastic loading and unloading cycle near the crack tip leads to the dissipation of energy as heat. Hence, a dissipative term has to be added to the energy balance relation devised by Griffith for brittle materials. In physical terms, additional energy is needed for crack growth in ductile materials when compared to brittle materials. Irwin's strategy was to partition the energy into two parts: • the stored elastic strain energy which is released as a crack grows. This is the thermodynamic driving force for fracture. • the dissipated energy which includes plastic dissipation and the surface energy (and any other dissipative forces that may be at work). The dissipated energy provides the thermodynamic resistance to fracture. Then the total energy dissipated is G = 2γ + G p where γ is the surface energy and G is the plastic dissipation (and dissipation from other p sources) per unit area of crack growth. The modified version of Griffith's energy criterion can then be written as For brittle materials such as glass, the su rface energy term dominates and . For ductile materials such as steel, the plastic dissipation term dominates and . For polymers close to the glass transition temperature, we have intermediate values of . Stress intensity factor Another significant achievement of Irwin and his colleagues was to find a method of calculating the amount of energy available for fracture in terms of the asymptotic stress and displacement fields around a crack front in a linear elastic solid. This asymptotic expression for the stress field around a crack tip is where σ are the Cauchy stresses, r is the distance from the crack tip, θ is the angle with ij respect to the plane of the crack, and f are functions that are independent of the crack ij geometry and loading conditions. Irwin called the quantity K the stress intensity factor. Since the quantity f is dimensionless, the stress intensity factor can be expressed in units ij of . When a rigid line inclusion is considered, a similar asymptotic expression for the stress fields is obtained. Strain energy release Irwin was the first to observe that if the size of the plastic zone around a crack is small compared to the size of the crack, the energy required to grow the crack will not be critically dependent on the state of stress at the crack tip. In other words, a purely elastic solution may be used to calculate the amount of energy available for fracture. The energy release rate for crack growth or strain energy release rate may then be calculated as the change in elastic strain energy per unit area of crack growth, i.e., where U is the elastic energy of the system and a is the crack length. Either the load P or the displacement u can be kept fixed while evaluating the above expressions. Irwin showed that for a mode I crack (opening mode) the strain energy release rate and the stress intensity factor are related by: where E is the Young's modulus, ν is Poisson's ratio, and K is the stress intensity factor I in mode I. Irwin also showed that the strain energy release rate of a planar crack in a linear elastic body can be expressed in terms of the mode I, mode II (sliding mode), and mode III (tearing mode) stress intensity factors for the most general loading conditions. Next, Irwin adopted the additional assumption that the size and shape of the energy dissipation zone remains approximately constant during brittle fracture. This assumption suggests that the energy needed to create a unit fracture surface is a constant that depends only on the material. This new material property was given the name fracture toughness and designated G . Today, it is the critical stress intensity factor K which is accepted as Ic Ic the defining property in linear elastic fracture mechanics. Limitations The S.S. Schenectady split apart by brittle fracture while in harbor (1944) But a problem arose for the NRL researchers because naval materials, e.g., ship-plate steel, are not perfectly elastic but undergo significant plastic deformation at the tip of a crack. One basic assumption in Irwin's linear elastic fracture mechanics is that the size of the plastic zone is small compared to the crack length. However, this assumption is quite restrictive for certain types of failure in structural steels though such steels can be prone to brittle fracture, which has led to a number of catastrophic failures. Linear-elastic fracture mechanics is of limited practical use for structural steels for another more practical reason. Fracture toughness testing is very expensive and engineers believe that sufficient information for selection of steels can be obtained from the simpler and cheaper Charpy impact test. Nonlinear elasticity and plasticity Vertical stabilizer, which separated from American Airlines Flight 587, leading to a fatal crash Most engineering materials show some nonlinear elastic and inelastic behavior under operating conditions that involve large loads. In such materials the assumptions of linear elastic fracture mechanics may not hold, that is, • the plastic zone at a crack tip may have a size of the same order of magnitude as the crack size • the size and shape of the plastic zone may change as the applied load is increased and also as the crack length increases. Therefore a more general theory of crack growth is needed for elastic-plastic materials that can account for: • the local conditions for initial crack growth which include the nucleation, growth, and coalescence of voids or decohesion at a crack tip. • a global energy balance criterion for further crack growth and unstable fracture. R-curve An early attempt in the direction of elastic-plastic fracture mechanics was Irwin's crack extension resistance curve or R-curve. This curve acknowledges the fact that the resistance to fracture increases with growing crack size in elastic-plastic materials. The