J.Cugnoni, LMAF-EPFL, 2016
Stress analysis
Fracture mech.
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Stress based criteria
(like Von Mises) usually
define the onset of
“damage” initiation in
the material
Once critical stress is
reached, what happens?
In this case, a defect is
now present (ie crack)
The key question is
now: will it propagate?
If yes, will it stop by
itself or grow in an
unstable manner.
Stress concentrator:
Critical stress is reached…
A crack is formed…
Will it extend further?
If yes, will it propagate abruptly
until catastrophic failure?
•Perform a stress analysis
Stress analysis
•Locate stress critical regions
•Assume the presence of a defect in those regions (one at a time)
•Consider different crack lengths and orientation
Crack
analysis
•For each condition, check if the crack would propagate and if yes if it is
stable or not
•Define operation safety conditions: maximum stress / crack length,…
before failure occurs
Design
evaluation
•Define damage inspection intervals / maintainance plan
•Perform a stress analysis
Stress analysis
•Locate stress critical regions
•Assume the presence of a defect in those regions (one at a time)
•Consider different crack lengths and orientation
Crack
analysis
•For each condition, check if the crack would propagate and if yes if it is
stable or not
•Define operation safety conditions: maximum stress / crack length,…
before failure occurs
Design
evaluation
•Define damage inspection intervals / maintainance plan
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Potential energy: P0=U0-V0
Assumed
Crack extension dA
Crack propagation : an “energetic” process
◦ Extend crack length: energy is used to create a
new surface (break chemical bonds).
◦ Driving “force”: internal strain energy stored in
the system
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“Energy release rate”:
◦ Change in potential energy P (strain energy and
work of forces) for an infinitesimal crack
extension dA. Units: J/m2, Symbol: G
◦ measure the crack “driving force”
Potential energy: P1= P0 –Er
And Er = G*dA
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“Critical Energy release rate”:
◦ Energy required to create an additionnal crack
surface. Is a material characteristic (but depends
on the type of loading). Units: J/m2, symbol: Gc
New crack surface dA:
Dissipates Ed=Gc*dA
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Crack propagation occurs if G > Gc
(see the rest of the course for more explanation on these
concepts)
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Using the “J-Integral” approach
(see course), it is possible to
calculate the ERR G as G = J in
linear elasticity.
If we know the displacement and
stress field around the crack tip,
we can compute J as a contour
integral:
How to get these fields: a Finite
Element simulation can be used
to evaluate the displacement and
stress field in any condition!
J-integral can be calculated in
Abaqus.
t σ n
W=strain energy density
u = displacement field
s = stress field
G = contour: ending and starting
at crack surface
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Can be calculated in elasticity
/ plasticity in 2D plane
stress, plane strain, shell and
3D continuum elements.
Requires a purely
quadrangular mesh in 2D
and hexahedral mesh in 3D.
J-integral is evaluated on
several “rings” of elements:
need to check convergence
with the # of ring)
Requires the definition of a
“crack”: location of crack tip
and crack extension direction
Crack tip and
extension direction
Quadrangle
mesh
Crack plane
Rings 1 & 2
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Create a linear elastic part, define an “independent” instance in Assembly
module
Create a sharp crack: use partition tool to create a single edge cut, then in
“interaction” module, use “Special->Crack->Assign seam” to define the crack
plane (crack will be allowed to open)
In “Interaction”, use “Special->Crack->Create” to define crack tip and
extension direction (can define singular elements here, see later for more
info)
In “Step”: Define a “static” load step and a new history output for J-Integral.
Choose domain = Contour integral, choose number of contours (~5 or more)
and type of integral (J-integral).
Define loads and displacements as usual
Mesh the part using Quadrangle or Hexahedral elements, if possible
quadratic. If possible use a refined mesh at crack tip (see demo). If singular
elements are used, a radial mesh with sweep mesh generation is required.
Extract J-integral for each contour in Visualization, Create XY data -> History
output.
!! UNITS: J = G = Energy / area. If using mm, N, MPa units => mJ / mm2 !!!
By default a 2D plane stress / plane strain model as a thickness of 1.
See demo1.cae example file
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To create a 1/sqrt(r) singular
mesh:
◦ In Interaction, edit crack definition and
set “midside node” position to 0.25
(=1/4 of edge) & “collapsed element
side, single node”
◦ In Mesh: partition the domain to create
a radial mesh pattern as show beside.
Use any kind of mesh for the outer
regions but use the “quad dominated,
sweep” method for the inner most
circle. Use quadratic elements to
benefit from the singularity.
◦ Refine the mesh around crack tip
significantly.
39
Regular mesh
Singular Mesh
38.8
38.6
38.4
38.2
38
37.8
37.6
Contour 1
Contour 2
Contour 3
Contour 4
Contour 5
•Perform a stress analysis
Stress analysis
•Locate stress critical regions
•Assume the presence of a defect in those regions (one at a time)
•Consider different crack lengths and orientation
Crack
analysis
•For each condition, check if the crack would propagate and if yes if it is
stable or not
•Define operation safety conditions: maximum stress / crack length,…
before failure occurs
Design
evaluation
•Define damage inspection intervals / maintainance plan
J(t)=G(t)
Crack propagation onset, G=Gc
Gc
•Perform a stress analysis
Stress analysis
•Locate stress critical regions
•Assume the presence of a defect in those regions (one at a time)
•Consider different crack lengths and orientation
Crack
analysis
•For each condition, check if the crack would propagate and if yes if it is
stable or not
•Define operation safety conditions: maximum stress / crack length,…
before failure occurs
Design
evaluation
•Define damage inspection intervals / maintainance plan
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Abaqus tutorials:
◦ http://lmafsrv1.epfl.ch/CoursEF2012
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Abaqus Help:
◦ http://lmafsrv1.epfl.ch:2080/v6.8
◦ See Analysis users manual, section 11.4 for fracture
mechanics
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Presentation and demo files:
◦ http://lmafsrv1.epfl.ch/jcugnoni/Fracture
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Computers with Abaqus 6.8:
◦ 40 PC in CM1.103 and ~15 in CM1.110