# Finite Element Method & Fracture Mechanics

University of Queensland

## Course Description

• ### Course Name

Finite Element Method & Fracture Mechanics

• ### Host University

University of Queensland

• ### Location

Brisbane, Australia

• ### Area of Study

Mechanical Engineering

• ### Language Level

Taught In English

MECH2300

### Hours & Credits

• Host University Units

2
• Recommended U.S. Semester Credits
4
• Recommended U.S. Quarter Units
6
• ### Overview

Course Description
Finite element method: strain energy, Castigliano's theorem, interpolation functions, element types, 2-D analysis types, plate & shell models; introduction to nonlinear finite element models. Fracture mechanics: linear-elastic fracture mechanics, Griffith's energy balance, stress intensity estimation, fatigue, elastic-plastic fracture mechanics, J-integral.

Course Introduction
The course deals with the prediction of failure of machine components and structures, by computer modelling using the finite element method and by assessment of criteria to predict fracture.

Part A - Finite element method:  The lectures on the finite element method will focus on issues in modelling and on the interpretation of results, with underlying mathematical theory only as needed for interpretation and error avoidance. Some lectures are also given on energy methods of structural analysis, to improve your ability to perform analytical checks on computed results.
(1) Castigliano’s theorem - finding deflections and indeterminate reactions from the energy of a structure.
(2) The finite element method - overall solution process for a linear elastic problem.
(3) Modelling with beam elements.
(4) 2D planar elasticity elements - types of 2D problems, finding stiffness matrices for 2D elements, numerical integration used to evaluate element stiffness and problems with it.
(5) Force and displacement boundary conditions - representing distributed loading, constraint equations used to relate displacements.
(6) Shell elements, their use and their limitations.
(7) Modelling with solid elements.
(8) Introduction to other types of analysis in a typical package program - eigenvalue analyses, nonlinear analyses, dynamic analysis and numerical optimisation.

Part B - Fracture mechanics and fatigue:
(1) The Griffith energy balance and the Griffith equation;
(2) Stress intensity factor (K) formulation of the Griffith equation, geometric calibration factors;
(3) Validity and limitations of linear elastic fracture mechanics - the plastic zone, validity of LEFM;
(4) LEFM for the professional engineer - ASTM E399, real cracks & complex geometries, getting it right;
(5) The weakest link principle - what determines failure mode in practice, notional stress concentration factors, fail-safe design;
(6) Fatigue - fatigue cracking & fractography, application of fracture mechanics to fatigue, rate equations, limitations of K-controlled fatigue;
(7) The Irwin-Kies energy balance, Irwin-Kies equation;
(8) Brief introduction to elastic-plastic fracture mechanics - crack tip opening displacement, the J-integral.

Learning Objectives
After successfully completing this course you should be able to:

1. APPLY ENERGY METHODS TO SOLID MECHANICS PROBLEM
1.1  Understand the types of problems that are best solved using energy methods
1.2  Calculate structural deflection and reactions by hand using energy methods
1.3  Analyse the behaviour of statically indeterminate structures using energy methods
1.4  Understand the role energy methods play in finite element methods

2. UNDERSTAND FINITE ELEMENT METHOD THEORY
2.1  Derive ‘direct’ system equations for simple spring, truss and beam structures
2.2  Describe the fundamental ideas/concepts that are required to derive the weak variational formulation of a differential equation
2.3  Explain how shape functions govern the displacement field the element can accurately reproduce
2.4  Recall the key properties of the stiffness matrix
2.5  List different solving methods that can be employed to solve the system equations
2.6  List key characteristics of different element types
2.7  Understand how the finite element method deal with different sources of nonlinearity

3. PERFORM STRUCTURAL FINITE ELEMENT ANALYSIS
3.1  Understand the necessary steps of a structural finite element analysis
3.2  Understand how simple elements can be used to reduce the computational expense
3.3  Apply boundary conditions to appropriately constrain a structure
3.4  Select appropriate element types for a given simulation problem
3.5  Employ numeric optimisation algorithms to optimise a design
3.6  Identify the different sources of non-linearities
3.7  Select appropriate modelling strategies to account for non-linearity
3.8  Explain how different validation approaches can be used to increase the confidence in the simulation results

4. EVALUATE FINITE ELEMENT ANALYSIS RESULTS
4.1  Recall common sources of errors in a finite element analysis
4.2  Justify assumptions made during the analysis with fundamental knowledge of finite element method theory
4.3  Perform post-processing of simulation results to answer different engineering questions related to the performance of the simulated structure
4.4  Identify meshing issues and employ mesh refinement strategies to improve mesh quality
4.5  Identify simple tests that can be performed to validate the accuracy of the simulation results.

5. UNDERSTAND AND APPLY THE GRIFFITH EQUATION
5.1  Understand how the tendency for a body to fracture is controlled by a balance between energy terms
5.2  Appreciate that certain simplifying assumptions may facilitate derivation of fracture mechanics equations, but that the equations do not ultimately depend on these assumptions
5.3  Understand and remember all of the terms in the full version of the Griffith equation
5.4  Understand the concepts of stress intensity factor (K) and fracture toughness (Kc)
5.5  Analyse geometries and identify & apply published solutions for the Y calibration factor to quantify stress intensity factor
5.6  Perform practical laboratory measurements to determine the fracture toughness of materials

6. UNDERSTAND AND APPLY LEFM
6.1  Understand how the existence of a plastic zone at the crack tip affects the validity of linear elastic fracture mechanics (LEFM) methodology
6.2  Apply quantitative guidelines for assessing LEFM validity
6.3  Identify the effects of LEFM invalidity and borderline validity as seen in practical laboratory measurements of the fracture toughness of materials
6.4  Understand and remember the requirements of international standards for the determination of a material’s fracture toughness using LEFM methodologies
6.5  Analyse more complex geometries and identify & apply published solutions to the evaluation of fracture conditions in these geometries
6.6  Remember common sources of error and how to ensure accuracy
6.7  Debate and make judgements regarding the fuzziness of LEFM validity guidelines and the ambiguities in terminology governing standard testing methodologies

7. ASSESS DEVELOPMENT AND CONSEQUENCES OF FRACTURE
7.1  Understand the concept of fail-safe design
7.2  Predict both the mode of failure and the practical strength of a component made from a material of known properties
7.3  Understand the physical processes occurring during the development and growth of sub-critical cracks under cyclic loading (i.e. fatigue)
7.4  Predict the rate of growth of fatigue cracks
7.5  Design and specify safe inspection intervals for components subject to fatigue;
7.6  Appreciate and debate the limitations of LEFM-based fatigue predictions for short cracks

8. MOVE BEYOND RELIANCE ON ELASTIC STRESS FORMULATONS
8.1  Understand the Irwin-Kies energy balance
8.2  Apply the Irwin-Kies methodology for the evaluation of a material’s fracture toughness in terms of energy release rate (G) rather than stress intensity factor (K);
8.3  Understand the four concentric zones around a crack as described by elastic-plastic fracture mechanics (EPFM)
8.4  Apply the crack tip opening displacement (CTOD) methodology for assessing the safety of a component
8.5  Understand the concepts underlying the J-integral

### Course Disclaimer

Courses and course hours of instruction are subject to change.

Eligibility for courses may be subject to a placement exam and/or pre-requisites.

Some courses may require additional fees.

Credits earned vary according to the policies of the students' home institutions. According to ISA policy and possible visa requirements, students must maintain full-time enrollment status, as determined by their home institutions, for the duration of the program.