Embed Size (px)
Citation preview
Linear Elastic Fracture Mechanics
Fracture Mechanics
Linear Elastic Fracture MechanicsPresented by
Calvin M. Stewart, PhD
MECH 5390-6390
Fall 2020
Outline
• Crack Tip Singularity
• Williams Solution
• Westergaard Stress Function
• Stress Intensity Factor
Harold Malcolm Westergaard (1888-1950)
Crack Tip Singularity
Crack Tip Singularity
• Elastic solutions for stress concentraions, such as Inglis solution for the elliptical hole, can provide useful information about the stresses at a flaw.
• An interesting phenomenon is observed when those features are sharpened into cracks.
• As the vertical thickness, b reduces to zero, the stresses at the crack tip become infinite!!!
EllipseLine discontinuity (crack)
A =
Crack Tip Singularity
• It turns out that the strength-of-materials assumption, where fracture is controlled by the stress at the tip of the crack, is not a valid failure model in general.
• Any hairline flaw on a structure would result in instant failure.
• When there is a significant stress gradient in the structure, failure is generally not governed by a single high-stress point.
• An alternative method is needed.
Crack Tip Singularity
• Linear Elastic Fracture Mechanics can be divided into two approaches.
• Fracture Energy approach
• Stress Intensity Factor approach
• Both approaches are a part of LEFM.
• In this Lecture, we will focus on the Stress Intensity Factor approach.
Williams SolutionDerivation from Anderson 2nd Edition, Appendix 2A
Williams Solution
• For certain cracked configurations, it is possible to derive closed-form expressions for the stresses in the body, assuming isotropic linear elastic material behavior.
• Westergaard, Irwin, Sneddon, and Williams were among the first to publish such solutions. In cylindrical coordinates, the Stress Field at the crack tip can be defined as
where σij is the stress tensor; r and θ are the radius and angle from the crack tip; k is a constant; and fij is a dimensionless function of in the leading term.
Williams Solution
• Williams (1950’s) developed an approach for analyzing the general solution to two singular problems: plate corner and sharp crack.
• Key Assumptions: In-Plane Loading, Elastic, Isotropic Conditions
Plate corner with included angle Special case of a Sharp Crack
Williams Solution: Sharp Crack
• Williams postulated a form of the Airy Stress Function
• Such that
( ) ( ) ( )
( )
( )
* *
1
*
, , ,
,
, ......
r g r f
g r r
f
+
=
=
=
is determined as a part of the solution
Williams Solution: Sharp Crack
• To Satisfy Equilibrium, in cylindrical coordinates
• where the primes ‘ denote derivatives with respect to
• Traction free fractures surfaces produce
*
( ) ( )
( ) ( )
0 2 0
0 2 0r r
= =
= =
Williams Solution: Sharp Crack
• For the displacements in all parts of the body to be finite,
• The boundary conditions can only be satisfied when
• Thus,
• As such, we can express the components of as
0
( )sin 2 0 =
Williams Solution: Sharp Crack
• is a function that depends on F and its derivatives. When r is small, the first term dominates the solution and higher order terms are neligible, using the additional B.C.’s gives
Williams Solution: Sharp Crack
Replacing gives
The stresses become,
* = +
Williams Solution: Sharp Crack
• The terms si and ti are defined.
• si are multiplied by cosine functions => Symmetric
• ti are multiplied by sine functions => Antisymmetric
• Symmetric => Bending and/or Tension => ti=0 => Mode I
• Antisymmetric => Pure Shear => si =0 => Mode II
• Thus,
Williams Solution: Sharp Crack
Crack Tip Stress Fields
Mode I: Mode II:
Note: This is the solution if we neglect the higher order terms.
Williams Solution: Sharp Crack
The Solution can be converted from Cylindrical to Cartesian coordinates
Mode II
sin22
cos22
IIxz
IIyz
K
r
K
r
= −
=
Mode I
Westergaard Stress Function
Westergaard Stress Function
• Westergaard showed that a limited class of problems can be solved by introducing a complex stress function Z(z) where
• The Westergaard stress function is related to Airy stress function as
• where the base represents integration such that
Westergaard Stress Function
• Remember from Airy Stress Functions that
• Thus,
Westergaard Example
• Derive the crack tip stress field for the infinite plate loaded in equi-biaxial tension
• Assume the Origin is at the center of the crack.
• The Westergaard Stress function is given as
remote stress
a half crack length
=
=
Westergaard Example
• The boundary conditions are
0 0
0
y
y
x
y
a x a y
x
y
x a y
= − =
→ →
→ →
= = =
(Crack Faces are Traction Free)
(Remote Conditions)
(Crack Tip Singularity)
Westergaard Example
• Check BC’s
• 1)
• When,
• Then,
0 ( ) ( )y z x Z z Z x= = =
2 2 0aa x xa− −
2 2 2 2 2 2 2 2( )
x x x xZ z i
x a i a x i a x a x
= = = −
− − − −
Re ( ) 0Z z =
2
1 ii
i i= −
( ) ( )Re Im 0y Z z y Z z = + =
Note:
Westergaard Example
• Check BC’s
• 2)
• Then,(All Real)
2 2 x y z x y→ → = + →
2
2
( )1 a
z
Z z
= →−
( ) ( ) ( )
( ) ( ) ( )
Re Im Re
Re Im Re
x
y
Z z y Z z Z z
Z z y Z z Z z
= − = =
= + = =
11 1− →
Note:
Westergaard Example
• Check BC’s
• 3)
• Then,
& 0x a y z a→ → =
( ) ( ) ( )Re Im Rey Z z y Z z Z z = + = =
( )( )
22
limz a
aZ z
a a
+
+
→ +
= = →
−
( )( )
22
limz a
aZ z
a a
−
−
→ −
= = →
−
Westergaard Example
• Mode I Stress Intensity Factor
Near the crack tip,
z a = −
( )( )
( )
( )2 22 2
a aZ
aa a
+ += =
++ −
a
( )1
2
22
a aZ
a
−
= =
In Polar Coordinates
( )1 1 1
2 2 2,2 2
i ii a a
re Z r e er
− − −
= = =
Expression for K
Westergaard Example
• The factor was introduced to make the formulation conform to current notation. It was also used here because it appears in the energy release rate defined with K.
• Using,
• The Stress components of the equi-biaxially loaded plate can be found as…
( ) ( )cos sinire r i = +
( )1 1 1
2 2 2,2 2
i ii a a
re Z r e er
− − −
= = =
Expression for K
Westergaard Example
( )
( )
( ) ( )
Re cos22
3Re cos
22 2
3 3Im sin sin sin cos sin
2 2 2 22 2 2
I
I
I I
KZ z
r
KZ z
r r
K Ky Z z r
r r r
=
= −
= =
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
= −
= +
=
Thus,
Note: For Non-equi-biaxial loading. Where the vertical and horizontal loads are different. We must superimpose an opposing load, in the axial load in the x-axis, as follows
3cos 1 sin sin
2 2 22
Ix ox
K
r
= − −
Stress Intensity Factor
Stress Intensity Factor
• In the stress field equations derived thus far, a term k, also shown as KI, KII, and/or KIII, keeps appearing.
This parameter, called the Stress Intensity Factor, K remains finite even when the crack is sharp.
This is a useful parameter in LEFM.
Stress Intensity Factor
• In the next lecture, we will explore the usefulness of the Stress Intensity Factor.
• It will be shown that K is a fracture resistance parameter that can be evaluated against a material property KIC to determine if fracture has occured.
Summary➢ Several researchers proposed solutions to the Stress Field in the vicinity of a
crack.
➢ Any Stress field solution must satisfy both equilibrium and compatibility. This can be achieved by proposing a function Airy Stress function, φ(x,y) that simplifies to the Biharmonic equation.
➢ Considerable mathematical effort must be expended to derive these equations.
➢ The Stress Intensity Factor, K remains finite.
➢ It will be shown that K is a fracture resistance parameter that can be evaluated against a material property KIC to determine if fracture has occured.
4 0 =
Homework 4
• Verify that ANSYS Student Edition has been successfully installed on your computer.
• Perform literature review and investigate the Fracture Mechanics tools that are available inside of ANSYS. Write a brief 500-word summary of the tools that are available. Collect and provide links to your references including but not limited to: technical reports, user manuals, guides, tutorials, and instructional videos.
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
me.utep.edu/cmstewart/
