# FRACTURE MECHANICS & FINITE ELEMENT ANALYSIS - 2024/5

Module code: ENGM250

## Module Overview

The module covers the principles of linear elastic and elastic plastic fracture mechanics and their application to predicting the performance of different materials and associated structural components under short-term and long term loading.  Further the concepts and principles underlying finite element stress analysis and its application are presented.

Students digital capabilities are developed throughout this module including some basic data handling and programming skills, specifically using mathematical formulae and functions, including: variables, assignments, operators, built-in functions and plotting, Matlab scripts, and functions.

A key element of the module is problem solving, thus developing students' resourcefulness. Efficient mechanical design, covering a full range of engineering materials, facilitates lightweighting, and in this way supports a sustainability agenda.

### Module provider

Mechanical Engineering Sciences

VIQUERAT Andrew (Mech Eng Sci)

### Module cap (Maximum number of students): N/A

Independent Learning Hours: 70

Lecture Hours: 30

Tutorial Hours: 10

Guided Learning: 10

Captured Content: 30

Semester 2

NA

## Module content

Indicative content includes:

a) Fracture Mechanics

• Competition between yield and fracture under plane-stress/plane-strain conditions. Ideal strength of a material. Effect of stress concentrators and cracks.

• The use of the stress intensity factor to characterise near-crack-tip stress fields leading to a material property (fracture toughness) for prediction of crack propagation. Expressions for stress intensity factors for a range of geometries.

• Modes of crack tip opening and corresponding stress intensity factors/crack tip stress fields. Mixed mode fracture problems. Failure of brittle solids in compression.

• Griffith energy balance analysis and its relation to stress intensity approach.

• Limitations of LEFM - role of crack tip plasticity. The 'thickness' criterion for plane strain conditions to apply. Measurement of fracture toughness.

• Crack-opening displacement approach to fracture. Derivation of expression for COD and compatibility with LEFM. COD-based design curves.

• J Integral approach to fracture. Definition, measurement and calculation from computer-based stress analysis including ideas of path-independence. Applications.

• Failure by ductile rupture - micro-void initiation, growth and coalescence.  Ductile brittle transition in un-notched bars. Cleavage failure.

• Fatigue failure. Cyclic stress/strain behaviour leading to hardening or softening. Crack initiation and growth mechanisms. Empirical laws for fatigue failure - Coffin-Manson, Basquin, Goodman and Miner. Crack growth (Paris) relation - application and limitations.

b) Finite Element Analysis

• Finite element principles: Outline of FE method, shape functions, element and global stiffness matrix, a simple example

• 1D FEA: Bars and beams, shape functions and elements stiffness matrix, assembling and solving the global stiffness matrix

• 2D FEA: 3-noded triangular element, shape function and element stiffness matrix, 4-noded iso-parametric element, shape functions, Jacobian, element stiffness matrix, numerical integration and solution.

• Modal Analysis using FEA: Introduction to dynamics in FEA and simple modal analysis of linear elastic structures.

## Assessment pattern

Assessment type Unit of assessment Weighting
Coursework FINITE ELEMENT CODING COURSEWORK 20
Examination 2 HR INVIGILATED EXAM 80

## Alternative Assessment

Alternative coursework for FE coding: written assignment {20%}

## Assessment Strategy

The assessment strategy is designed to provide students with the opportunity: (i) to show an understanding of the properties of the main classes of engineering materials and the principles of fracture mechanics and then apply this knowledge to a range of engineering problems (ii) to demonstrate understanding of the underlying principles of the finite element method and an ability to apply these principles to solve simple finite element problems.

Thus, the summative assessment for this module consists of:

• Coursework comprising assignment in FEA, at end of relevant sessions [Learning Outcomes 5 - 7]

• Written examination [Learning Outcomes 1 - 6]

Formative assessment and feedback

Formative assessment and feedback is provided by the weekly supported tutorial work – students complete a set of worked solutions to a range of questions in the tutorial classes and are provided with tutor support in comment and feedback in the sessions.  Fully marked and commented coursework are returned to the students and further feedback is provided through an in-class summary.

## Module aims

• To review the theoretical background to the key linear elastic fracture mechanics parameters (stress intensity factory, strain energy release rate) and elastic/plastic fracture mechanics parameters (crack opening displacement, J-integral) and to explain the relevance of these parameters to different engineering materials.
• To demonstrate to the students the range of failure mechanisms that can be shown by the various classes of engineering materials as a function of stress and temperature
• To describe the phenomena associated with fatigue deformation and crack growth together with the empirical equations that have been used to quantify these processes.
• To provide students with an understanding of finite element stress analysis enabling the student to appreciate the inherent approximate nature of the technique and the principles that underpin all modern FEA software packages.

## Learning outcomes

 Ref Attributes Developed 001 Identify when the application of linear-elastic fracture mechanics methods is relevant for a simple component C M1 002 Identify when the application of elastic-plastic fracture mechanics methods is relevant for a simple component C M3 003 Describe the deformation and fracture characteristics of different materials, with reference to the associated material property data C M2,M13 004 Make appropriate calculations using these data C M1 005 Explain how the finite element method works C M1,M2 006 Carry out simple FE calculations by hand C M2,M4 007 Write simple FE code in Matlab KC M1,M4,M5

Attributes Developed

C - Cognitive/analytical

K - Subject knowledge

T - Transferable skills

P - Professional/Practical skills

## Methods of Teaching / Learning

The learning and teaching strategy is designed to:

• Introduce the principles of linear elastic and elastic-plastic fracture mechanics and to explain their relevance to solving engineering problems relating to crack propagation in structures under short-term and long term (fatigue, creep) conditions.  In doing this, the full range of engineering materials will be considered

• Introduce the student to the theorem of stationary energy, the role of shape functions and their application to a range of finite elements.

These strategic aims are achieved through lectures that present the underlying theory, worked examples that illustrate the application of this theory and tutorial sessions where students can develop and enhance their understanding of the subject.

The learning and teaching methods include:

• Lectures

• Tutorials

• Captured content

• Revision class

Indicated Lecture Hours (which may also include seminars, tutorials, workshops and other contact time) are approximate and may include in-class tests where one or more of these are an assessment on the module. In-class tests are scheduled/organised separately to taught content and will be published on to student personal timetables, where they apply to taken modules, as soon as they are finalised by central administration. This will usually be after the initial publication of the teaching timetable for the relevant semester.

Upon accessing the reading list, please search for the module using the module code: ENGM250

## Other information

The School of Mechanical Engineering Sciences is committed to developing graduates with strengths in  Employability,  Digital  Capabilities,  Global  and  Cultural  Capabilities,  Sustainability,  and Resourcefulness  and Resilience.  This  module  is  designed  to  allow  students  to  develop knowledge, skills, and capabilities in the following areas:

• Digital capabilities
• Understand how digital technology is changing practices in their discipline.
• Judge whether digital information is trustworthy and relevant; distinguish different kinds of information.
• Sustainability
• To reflect on how the discipline/topic is impacted by, and impacts, aspects of social and environmental wellbeing, both now and into the future.
• Resourcefulness and resilience
• Demonstrate an ability to share informed opinions constructively, using evidence, and be open to different opinions and able to adapt perspectives.

## Programmes this module appears in

Programme Semester Classification Qualifying conditions
Advanced Mechanical Engineering MSc 2 Compulsory A weighted aggregate mark of 50% is required to pass the module
Aerospace Engineering MEng 2 Compulsory A weighted aggregate mark of 50% is required to pass the module
Mechanical Engineering MEng 2 Compulsory A weighted aggregate mark of 50% is required to pass the module

Please note that the information detailed within this record is accurate at the time of publishing and may be subject to change. This record contains information for the most up to date version of the programme / module for the 2024/5 academic year.