# Fracture Mechanics

Fracture mechanics develops methods of predicting the load-carrying capabilities of structures and components containing cracks.  In terms of repeated loading, fracture mechanics provides a fatigue-crack propagation approach known as the damage-tolerance approach.

Fracture is a problem that creates catastrophic failures on structures.  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.  The fundamentals of linear elastic fracture mechanics were established in 1960’s, and researchers turned their attention to crack-tip plasticity.

Although the load on a crack can be arbitrary, in 1957 G. Irwin found any state could be reduced to a combination of three independent stress intensity factors:

• Mode I – Opening mode (a tensile stress normal to the plane of the crack),
• Mode II – Sliding mode (a shear stress acting parallel to the plane of the crack and perpendicular to the crack front), and
• Mode III – Tearing mode (a shear stress acting parallel to the plane of the crack and parallel to the crack front).

Theoretically, the stress ahead of a sharp crack tip becomes infinite and cannot be used to describe the state around a crack. Fracture mechanics is used to characterize the loads on a crack, typically using a single parameter to describe the complete loading state at the crack tip. A number of different parameters have been developed. When the plastic zone at the tip of the crack is small relative to the crack length the stress state at the crack tip is the result of elastic forces within the material and is termed linear elastic fracture mechanics (LEFM) and can be characterized using the stress intensity factor K.  This factor represents a single parameter that includes the effect of the applied stress and the effect of the crack of a given size.

When the size of the plastic zone at the crack tip is too large, elastic-plastic fracture mechanics can be used with parameters such as the J-integral or the crack tip opening displacement.

The characterizing parameter describes the state of the crack tip which can then be related to experimental conditions to ensure similitude. Crack growth occurs when the parameters typically exceed certain critical values. Corrosion may cause a crack to slowly grow when the stress corrosion stress intensity threshold is exceeded. Similarly, small flaws may result in crack growth when subjected to cyclic loading. Known as fatigue, it was found that for long cracks, the rate of growth is largely governed by the range of the stress intensity ΔK experienced by the crack due to the applied loading. Fast fracture will occur when the stress intensity exceeds the fracture toughness of the material. The prediction of crack growth is at the heart of the damage tolerance mechanical design discipline.

The driving force for Mode I fracture is stress intensity factor K.  The stress intensity factor is usually given a subscript to denote the mode of loading, i.e., KI , KII, or KIII for Mode I, Mode II or Mode III respectively.  However, KI for mode I cracking is most commonly seen in engineering practice.

The figure above shows a crack of length 2a in an infinite plate subject to a remote tensile stress σ. For a linear elastic material, the stresses near the crack tip is a function defined by the stress intensity factor KI as follows:

σxx= KI f(r, θ)

σyy= KI g(r, θ)

τxy= KI h(r, θ)

The stress intensity factor KI is given by

As it can be seen that the stress intensity factor solution for this special case is very simple. In reality, the configurations of the components and the cracks are much more complex and the solutions are much more complicated.  FFSpro provides Fracture Mechanics solutions for plate, plate with a hole, cylinder, sphere, elbow and pipe bend, nozzle or piping tee, ring-Stiffened cylinder, sleeve reinforced cylinder, round bar or bolt, cracks at fillet welds, cracks in clad or weld overlayed.

Failure occurs when KI = KIc where the critical stress intensity factor KIc represents the material resistance to fracture.  The parameter KIc is a true material property in the same sense as is the yield strength of a material.  The parameter KIc is known as the plane-strain fracture toughness; any combination of applied stress and crack size that exceeds this value would produce unstable crack growth, as shown in the figure below.  It can be seen that depending on crack size and geometry, the fracture stress can be significantly lower than yield strength.

In work with tougher, lower-strength materials, stable slow crack growth would occur accompanied by considerable plastic deformation. Such phenomena led to the nonlinear J-integral and R-curve (Resistance curve) concepts, which can be used to predict the onset of stable slow crack growth (ductile tearing) and final instability under elastic-plastic conditions, as shown in the figure below.

The fracture mechanics approach is applied to characterize subcritical crack growth phenomena where time-dependent slow crack growth, da/dt, or cyclic crack growth, da/dN, may be induced by special environments or fatigue loading.  For combinations of stress and crack length above some environmental threshold, KIscc, or fatigue threshold, ΔKth, subcritical growth occurs, as shown in the figure below.

The resistance to fracture is called fracture toughness. The fracture toughness of a material measures its ability to resist crack initiation and propagation. Several fracture toughness parameters are available, including critical stress intensity factor (KIC) in the elastic range; and in the elastic/plastic regime, the critical value of the J-integral (Jcrit), and the critical crack tip opening displacement CTOD or 𝛿crit.  In practice, the J-integral and CTOD parameters are frequently converted to equivalent KIC to avoid tedious inelastic analyses.

FFSpro is a Fitness-For-Service (FFS) assessment software that utilizes Failure Assessment Diagram (FAD) to demonstrate the structural integrity of in-service components that contain crack-like flaws, following API 579-1/ASME FFS-1 Level 2 and Level 3 assessment procedures.

FFSpro evaluates the present integrity of the component by Fracture Mechanics for a given flaw, and also predicts the remaining life by analyzing fatigue crack growth or hydrogen assisted crack growth.  In addition, FFSpro evaluates the fracture tearing resistance by the amount of stable ductile tearing.