Fracture Mechanics
SIB-96-154, Rev. 1

Introduction
When cracks are identified in structures or components during service, they must be evaluated to determine suitability for continued operation. Fracture mechanics provides a methodology evaluating the structural integrity of components containing such defects, and demonstrating whether they are capable of continued, safe operation.

Basic Criterion in Fracture Mechanics
The basic criterion in any fracture mechanics analysis is to prevent failure. To do so, the crack driving force must be less than the material resistance to cracking, as illustrated in the next figure.

The crack driving force and the material resistance depend on the fracture mechanics regime. Under a given set of stresses, the flaw size at which the crack driving force equals the material resistance is called the critical flaw size. In general, a safety factor is applied to the material resistance in comparison with the crack driving force to determine the allowable flaw size for a component.

Regimes of Fracture Mechanics
As illustrated in the figure to the right, there are three broad regimes of fracture mechanics analysis.

Linear Elastic Fracture Mechanics
In this regime, the crack driving force is measured by a parameter called the stress intensity factor (K_I). is generally a function of the applied stress, the crack size and the geometry of the component. In its simplest form, K_I is calculated from the relationship:

A wide variety of handbook solutions are available for the determination of K in various cracked bodies. Structural Integrity Associates (SI) has developed a convenient computer program (pc-CRACK™) containing a large number of such handbook solutions in a convenient electronic format.

The material resistance is measured by a parameter called the fracture toughness (K_Ic). K_Ic for a particular material is a function of temperature, the loading rate and the environment. Published values of K_Ic for most commonly used materials are available in several handbooks. K_Ic can also be estimated from conventional materials test report properties such as yield and tensile strengths and impact properties.

Elastic-Plastic Fracture Mechanics
The crack driving force in this regime is measured by a parameter called the J Integral (J_I). J_I defines the work done under the applied stresses in the vicinity of the crack in an elastic-plastic stress and strain field. J_I depends upon the geometry of the component, the applied stress, the crack size and the elastic-plastic stress-strain relationship of the material. In lieu of using detailed finite element stress analyses to calculate J_I, a number of handbook solutions have been developed and published. For electric generating equipment, the Electric Power Research Institute (EPRI) has published an EPFM handbook of most crack configurations encountered in practice. These solutions are in the form:

pc-CRACK™ contains a complete set of the EPRI EPFM handbook solutions.

In its simplest form, the material resistance in this regime is measured by the elastic-plastic fracture toughness (J_Ic). Because of the ductile nature of materials in the EPFM regime, there may also be considerable stable crack extension of the material even when the applied J_I reaches the J_Ic value.

Hence, another material resistance property becomes important in this regime. This property is represented by the J-Resistance (J-R) material curve which measures the resistance of the material to stable tearing. Comparison of applied J_I versus the J-R curve allows the determination of the crack size or stress at which unstable tearing occurs.

Limit Load Analysis
For materials that are highly ductile, limit load analysis may be used to determine the failure stress in the presence of a flaw. This analysis method assumes that the entire cross section of the component becomes fully plastic before the onset of failure, as illustrated in the following figure.

pc-CRACK contains a number of limit load solutions applicable to power plant piping.

Subcritical Crack Growth Analysis
Before comparing the crack driving force to the material resistance in any of the three regimes, subcritical crack growth analysis should be performed to determine the crack size at the end of the evaluation interval. Crack growth methodology has been developed only for the LEFM regime, but may be conservatively applied in all three regimes. In most cases, crack growth can be attributed to either fatigue or stress corrosion cracking (SCC). For fatigue, the crack growth is determined from the relationship:

For SCC evaluation, the crack growth is determined from the expression:

Values of C and n, as well as A and m, can be found in literature for most commonly used materials. pc-CRACK incorporates convenient algorithms for both fatigue and SCC crack growth computations.

Software
To facilitate the evaluation of flawed components, SI has developed several special-purpose computer software programs. A summary of the most widely used fracture mechanics software is provided below.

• pc-CRACK is the most widely used fracture mechanics software in the nuclear and fossil industry. pc-CRACK performs LEFM, EPFM, Limit Load and Crack Fracture Mechanics; also performs crack growth analysis and ASME Code, Section XI flaw evaluation.
• APPENDA/MAPPA performs vessel flaw evaluation per the requirements of ASME Code, Section XI to provide acceptable flaw sizes during in-service inspections.
• RPVIMS is an Automated Pressure-Temperature moni-toring system which uses state-of-the-art fracture-mechanics technology to evaluate reactor vessel temperature transients, and determines allowable pressure-temperature limits per the requirements of ASME Code, Section XI.
• K-Solver is a numerical tool for the analytical prediction of crack-tip stress intensity factors (Mode I, II, and III) in elliptical or partial elliptical cracks of any orientation in a flat plate or pipe subjected to arbitrary crack surface loading.
• RRing-Life estimates the proba-bility of crack initiation and failure in generator retaining rings, allowing utilities to make decisions relative to continued operation or replacement. This software was developed under the sponsorship of EPRI.
• ANSC performs net section collapse analysis for arbitrarily-flawed sections which is required for the evaluation of austenitic stainless steels per Appendix C of ASME Code, Section XI.
• Viper (Vessel Inspection Program Evaluation for Reliability) was developed under the sponsorship of the BWR Vessel Internals Project (BWRVIP). VIPER performs a probabilistic fracture mechanics evaluation of a BWR pressure vessel to determine the probability of failure associated with a specific vessel in-service inspection program.

SI's Qualifications
SI is a highly recognized company in the application of fracture mechanics to the evaluation of flawed components. SI has applied the above fracture mechanics principles in the evaluation of numerous components in both nuclear and fossil power plants. Some of the components SI has evaluated include:

• Reactor pressure vessels
• Reactor internals
• Piping components
• Pumps and valves
• Generator retaining rings
• Turbine disks and blades
• Boiler tubes and headers

Several of SI's associates are members of various working groups of ASME Section XI, and have contributed to the development of evaluation methodologies in ASME Section XI. SI's collection of experts and software provides them with the opportunity to respond in a timely manner to its clients' needs.

If you have a flawed component and are looking for an innovative and timely solution, please contact SI.