Fatigue Crack Growth - Lecture 15

Damage Tolerance Analysis and Crack Growth Analysis

Course Overview

This course, titled Damage Tolerance Analysis: Crack Growth Analysis, is a critical aspect of avionic systems design and analysis, specifically suited for Aero 1001 Level 6, offering 15 credits. It is facilitated by Dr. Khosru Rahman, a chartered engineer and distinguished member of several engineering bodies. For further inquiries or assistance, students can reach Dr. Rahman via email at A.K.Rahman@greenwich.ac.uk.

Learning Outcomes

The presentation serves to equip students with a foundational understanding of fracture mechanics and the analysis of crack growth in engineering materials. Upon completion of this course, students will be able to:

  • Understand the mechanical behavior of materials when subjected to loading.

  • Apply linear elastic fracture mechanics for analyzing cracked materials.

  • Analyze fatigue crack growth in aircraft flight control systems.

Contents Outline

  1. Fatigue Cracks

  2. Crack Size vs. Cycles

  3. Definitions for Applied Loading

  4. Overview of Crack Growth

  5. Stress Ratio Effect on Crack Growth Rate

  6. Growth Rate Relationships

  7. Failure Criteria

  8. Life Prediction

  9. Factor of Safety

  10. Crack Growth Using Paris-Erdogan Equation

  11. Summary

  12. References & Reading List

  13. Tutorial

1.0 Fatigue Cracks

Overview of Fatigue Cracks

  • Fatigue cracks are a common failure mode in engineered parts, significantly diminishing their load-bearing capabilities.

  • These cracks often initiate from pre-existing flaws within the component and extend during operational stress cycles.

  • Crack growth occurs due to cyclic applied loading or under steady loads in corrosive environments (not covered herein).

  • The focus of this section is on fatigue crack growth, where the principles of fracture mechanics are essential for analysis.

2.0 Crack Size vs. Cycles

Crack Growth Dynamics

  • A notable plot illustrates crack growth as a function of the number of applied load cycles (N).

  • The critical variables include:

    • Crack size (a): the current dimension of the crack.

    • Crack growth rate (da/dN): the slope of the crack growth curve, indicating how quickly the crack grows.

  • Initially, cracks grow slowly; however, as they increase in size, the growth rate accelerates due to the relationship between crack size and the stress intensity factor, which drives the growth process until reaching a critical failure size.

3.0 Definitions for Applied Loading

Stress Definitions

  • The cyclic loading concept utilizes maximum and minimum stress definitions:

    • σmax: Maximum applied stress.

    • σmin: Minimum applied stress.

    • σm: Mean stress, calculated as $( rac{ ext{σmax} + ext{σmin}}{2})$.

    • σa: Stress amplitude, defined as $( rac{ ext{σmax} - ext{σmin}}{2})$.

    • Δσ: Stress range, calculated as $( ext{σmax} - ext{σmin})$.

    • Stress ratio (R): The ratio of minimum stress to maximum stress, calculated as $( ext{R} = rac{ ext{σmin}}{ ext{σmax}})$.

4.0 Overview of Crack Growth

Mechanism of Crack Propagation

  • When a cyclic load is applied, a fluctuating stress intensity drives the crack growth at a determined rate.

  • The stress intensity range (ΔK) is linked with cyclical applications in specific increments of cycles (ΔN), leading to the incremental length growth of the crack (Δa).

  • The overall rate of crack growth is represented mathematically:
    extGrowthrate=racΔaΔNext{Growth rate} = rac{Δa}{ΔN}

  • In continuous terms, this is expressed as racdadNrac{da}{dN}.

  • Growth rates vary among materials, depicted logarithmically, usually showing a straight-line region correlating da/dN and ΔK values through the Paris curve defining the relationship:
    racdadN=C(ΔK)nrac{da}{dN} = C(ΔK)^n

  • Here, C is an intercept constant, and n signifies the slope on the log-log scale; C’s units vary depending on the value of n.

5.0 Stress Ratio Effect on Crack Growth Rate

Influence of Stress Ratio on Growth Rate

  • There is a direct relationship between the stress ratio (R) and the crack growth rate: as R increases, the crack growth rate also increases, and vice versa.

  • Empirical data from materials such as the 7075-T6 Aluminum Alloy Sheet indicates that the curves shift upward with increases in R, though the slope of the crack growth rate curve remains unchanged, emphasizing that the stress ratio alters the intercept constant C but not the exponent n in the Paris equation:
    racdadN=C(ΔK)nrac{da}{dN} = C(ΔK)^n

6.0 Growth Rate Relationships

Common Models

Several models exist to characterize fatigue crack growth rates, including:

  • Paris Equation: Defines a linear growth region without considering stress ratio. Adjustments to C are necessary for non-zero ratios.

  • Walker Equation: An extension of the Paris model that incorporates the effects of stress ratio R on crack growth rate.

    • racdadN=C0(ΔK)nf(R)rac{da}{dN} = C_0(ΔK)^{n}f(R), where γ influences the relationship.

  • NASGRO Equation: The most comprehensive model that incorporates stress ratio R, crack closure, and accounts for tails in the growth rate curve: racdadN=C<em>0f(R)(ΔK−ΔK</em>th)prac{da}{dN} = C<em>0f(R)(ΔK - ΔK</em>{th})^p

    • C0 represents intercept constants, Kmax symbolizes peak stress intensity, while Kc delineates critical stress intensity.

7.0 Failure Criteria

Determining Failure

  • As cracks propagate, the associated stress intensity factor at the tip escalates, leading to failure if it surpasses material’s critical stress intensity (Kc).

  • For cyclic stress scenarios, Kmax should be compared with Kc to ascertain failure potential.

  • Other failure criteria may include a comparison between applied maximum stress and flow stress ($σf$), which is derived from both yield strength and ultimate strength equations.