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Linear Elastic Fracture Mechanics II
Fracture Mechanics
Linear Elastic Fracture Mechanics IIPresented by
Calvin M. Stewart, PhD
MECH 5390-6390
Fall 2020
Outline
• Definition
• Fracture Modes
• Useful Expressions
• Finite Width
• Superposition
• Determination of SIF
Definition
Stress Intensity Factor and Fracture Toughness
• Definition: The Stress Intensity Factor (SIF) is a fracture mechanics parameter used to characterize the strength of the singularity at the crack tip, denoted by K.
• The subscripts I,II, or III correspond to the opening, sliding, and tearing modes of fracture, respectively.
• Definition: The Critical Fracture Toughness is a cracked material’s ability to resist fracture, denoted by Kc. It is considered a material property and commonly used in design.
• Units are or1.099ksi in MPA m=
ksi in MPA m
LEFM Inequality
• We can apply these two parameters to make decision concerning the fracture resistance of a structure.
• If this inequality is true, Fracture has occurred
I cK K
Equation. Function of• Loading Conditions• Body Geometry• Crack Geometry
Material Property Function ofEnvironmentSurface RoughnessMicrostructural Mechanisms
Stress Intensity Factor (SIF) Equation Critical Fracture Toughness Property
Fracture Modes
Stress Intensity Factor
• All stresses in the vicinity of the crack tip can be divided into three basic types, each associated with a local mode of crack surface displacement.
• KI (Opening Mode)
• KII (In-plane shear)
• KIII (Out-of-plane shear
• / tearing)
Fracture MOdes
• Mode 1 – Opening
• Mode 2 – In-Plane Shear
• Mode 3 – Out-of-Plane Shear / Tearing
Useful ExpressionsCartesian and Principal Stresses; Elastic Displacement; Crack Flank Displacement; Mode II; Mode III
Useful Expressions
• Let us remembers Westergaards solution for an infinite width plate under equi-biaxial loading.
• Cartesian Notation (x,y,z)
3cos 1 sin sin
2 2 22
3cos 1 sin sin
2 2 22
3sin cos cos
2 2 22
Ix
Iy
Ixy
K
r
K
r
K
r
= −
= +
=
• The stress field equations can be expressed in principal stresses by apply the transformation equation from Mohr’s circle.
• Principal Stresses
Principal Stresses
For Plane Strain
For Plane Stress
3 0 =
Principal Directions
• The elastic displacement field around a crack can be derived from the elastic stress field using the two-dimensional form of Hooke’s law, valid for either a plane stress or a plane strain condition
• Where u and v are the displacement in the x and y directionsrespectively.
Elastic Displacement
For Plane Strain For Plane Stress
Crack Flank Displacement
• Another useful expression is the Crack Flank Displacement, v which is the displacement at any position along the crack flank.
For Plane Strain
For Plane Stress
Stress Field Equations for Modes II and III
• The stress field equations for mode II and mode lII loading may be obtained in a similar manner to Westergaard’s solution for mode I.
Finite Width
Finite Width Crack Bodies
• Closed form solutions (like those found for infinite boundary problems) are either• Difficulty or impossible to
achieve for finite boundary problems
• As the crack length, a increase or width, W decreases the outer boundaries have an increased influence on the crack tip
Finite Width Crack Bodies
• Free boundary affects the crack tip stress distribution
• Most Solutions to finite boundary problems are expressed in the general form
• C=fitting constant
• f=dimensionless geometric fractor
( )aK C a fW
=
Center Cracked Plate in Tension (CCT)
• The earliest attempt to obtain a solution to a finite-body problem is attributed to Irwin (1957) for the infinite plate in tension with co-linear cracks of equal length and spacing
Center Cracked Plate in Tension (CCT)
• The Westergaard stress function is proposed as
• noting,
( )( )( )
2
2
sin1
sin
a
W
z
W
Z z
=
−
2 2
2 2Re Imrr
d Z dZ y Z
dx dx = = +
Center Cracked Plate in Tension (CCT)
• At the crack tip A, the stress intensity factor can be expressed as
• After manipulation Irwin solution,
• As a/W approaches 1
• K approaches infinite
( ) ( )( )
2 2
2 2
sin2
2sin sinx a
W W
x
WK x a
= −−
2tan
2
W aK a
a W
=
A
Finite Width Centre Crack Specimen
• The solution for a Finite Width Centre Cracked specimen can be distilled to
• where note, the width is now described as W instead of 2W.
tanW a
K aa W
=
Center Cracked Plate in Tension (CCT)
• The formulation was later improved with a polynomial expression Brown Solution
• Or by Fedderson Solution,
• Or by Isidia Solution,
2 3
1 0.256 1.152 12.200a a a
K aW W W
= + − +
seca
K aW
=
2 4
sec 1 0.025 0.06a a a
K aW W W
= − +
Edge Notched Specimen
The single and double edge notched specimens.
Edge Notched Problems
• The slight increase in the S.I.F. of both cases can be compared to the CCT case.
• The Closing effect is absent in the edge crack case,
• The Edge crack opens more because it is less restrained.
• A through crack must be longer (a increased) to achieve the same opening displacement
• Solutions to S.I.F.’s for more complex geometries are determined experimentally, numerically, and analytically or with mixed methods.
More Restraint
Less Restraint
Edge Notched Specimen
• Approximate Solution
• For longer cracks, the finite geometry results in stress enhancement. Correction factors have taken both the free edge effect and finite geometry effect into account.
Edge Notched Specimen
• Double edge notched specimen
Superposition
Superposition
• The total stress field due to two or more different mode I loading systems can be obtained by an algebraic summation of the respective stress intensity factors. This is called the superposition principle.
• Only valid for combinations of the same mode of loading, i.e. all mode I, all mode II or all mode III.
• By using the superposition principle the stress intensity factor for a number of seemingly complicated problems can be readily obtained.
Through Crack under Internal Pressure
a −Subsisting, P=-σ in case D gives the required result, P(πa)^1/2
Semi-Elliptical Surface Crack in Cylindrical Pressure Vessel
• Hoop Stress
• Stress due to internal pressure on crack surfaces.
• Superposition
Determination of SIF
Determination of SIF
• Often, when applying fracture mechanics you will find that there is no standard solution for your problem of interest.
• For example, a cracked turbine blade.
• Unique geometry, Load conditions, and location of the crack results in the stress fields being difficult to derive.
https://amses-journal.springeropen.com/articles/10.1186/s40323-016-0083-7
Determination of SIF
• Recommended Approach1. Literature Review - Check the Literature for a direct solution (i.e. analytical
solution).2. Analytical Solution - If no solution is directly available, assess the effort needed to
solve the problem analytically. This effort depends on the seriousness of the problem, the desired accuracy, and how many times the solution will be useful. If worthwhile, derive the analytical solution. A strong mathematical background is required.
3. Computational Solution – If deriving the analytical solution is not worthwhile and/or the problem is too complex to obtain a closed-form solution, applying computational tools to approximate the for your problem.
4. Experimental Solution – Alternatively, a more straightforward approach is to perform a series of fracture/fatigue experiments with different initial crack lengths.
Analytical Methods
• Books and Various Design Codes and Standards
• Sih, G.C. (Editor), Methods of Analysis and Solutions of Crack Problems, Noordhoff International Publishing (1973): Leiden.
• Paris, P.C., McMeeking, R.M. and Tada, H., The Weight Function Method for Determining Stress
• Intensity Factors, Cracks and Fracture, ASTM STP 601, American Society for Testing and Materials, pp. 471 489 (1976): Philadelphia.
• Rooke, D.P. and Cartwright, D.J., Compendium of Stress Intensity Factors, Her Majesty’s Stationery Office (1976): London.
• Tada, H., Paris, P.C. and Irwin, G.R., The Stress Analysis of Cracks Handbook, Paris Productions Incorporated (1985): St. Louis, Missouri.
• Murakami, Y., Stress Intensity Factors Handbook, Vols. 1, 2 and 3, Pergamon Press (1987 vols. 1 and 2, 1992 vol. 3): Oxford.
Computational Methods➢ Finite Element Method (FEM)➢ Boundary Element Method (BEM)➢ Extended Finite Element Method (XFEM)
Experimental Methods
• Def: Standard – an accepted guide that governs concepts, procedure, definitions, etc.
• Ex: ASTM E399 - Standard Test Method for Plane- Strain Fracture Toughness of Metallic Materials
• Ex: ISO 12135:2002- Metallic materials -- Unified method of test for the determination of quasistatic fracture toughness
Compendium of SIF SolutionsEach reference has a library of direction solution.
In our textbook by Janssen see pages 53 -59
And many More
Summary➢ The Stress field equations can be expressed in different notations, Cylindrical,
Cartesian, Principal, etc.
➢ The analytical solution to Finite Width Specimen are more difficult to obtain but solution do exist.
➢ Superposition can be applied to combine the effect of multiple loading conditions. Note, these loading conditions must be all of the same type. All mode 1 ; all mode 2; or all mode 3.
➢ The determination of SIF should generally proceed as follows: literature review, analytical solution, computational solution, experimental solution.
➢ Cost/Value should be considered in the determination of SIF.
Homework 5
• Recreate figure 2.8 from the textbook which compares the correction factors for a single edge notch specimen. Read the book and associated literature for each solution. Discuss the origins of each solution and compare their accuracy.
• Define the following three parameters (stress concentration factor, stress intensity factor, and critical fracture toughness). Compare how they are applied in Mechanics.
• Used published sources (e.g. your textbook, journal articles, books, Internet) to determine the fracture toughness versus strength properties for the following materials. Cite the references that were used in each case: • Aluminum 7075-T651 • Silicate glasses • Stainless Steel 347L• Inconel 718
References
• Janssen, M., Zuidema, J., and Wanhill, R., 2005, Fracture Mechanics, 2nd Edition, Spon Press
• Anderson, T. L., 2005, Fracture Mechanics: Fundamentals and Applications, CRC Press.
• Sanford, R.J., Principles of Fracture Mechanics, Prentice Hall
• Hertzberg, R. W., Vinci, R. P., and Hertzberg, J. L., Deformation and Fracture Mechanics of Engineering Materials, 5th Edition, Wiley.
• https://www.fracturemechanics.org/
CONTACT INFORMATION
Calvin M. Stewart
Associate Professor
Department of Mechanical Engineering
The University of Texas at El Paso
500 W. University Ave, Suite A126, El Paso, TX 79968-0521
Ph: 915-747-6179
cmstewart@utep.edu
me.utep.edu/cmstewart/
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