Principle of Superposition

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Principle of Superposition
For a linear elastic system, the individual
components of stress, strain, displacement are
additive. For example, two normal stress in x
direction caused by two different external load
can be added to get total stress
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Principle of Superposition
For example consider a plate having semicircular
crack subjected to an internal pressure
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Crack Tip Plasticity
LEFM assumes a sharp crack tip, inducing infinite
stress at the crack tip. But in real materials,
the crack tip radius is finite and hence the
crack tip stresses are also finite. In addition,
inelastic deformations due to plasticity in
metals, crazing in polymers leads to further
relaxation of stresses.
For metals with yielding, LEFM solutions are not
accurate. A small region around the crack tip
yields leading to a small plastic zone around it.
For moderately yielding metals, LEFM solutions
can be used with simple correction. For
extensively yielding metals, alternative fracture
parameters like CTOD, J-integral are to be used
taking into account material non-linearity.
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Small Scale Plasticity
Irwins Approach Normal stress syy based on
elastic analysis is given by On the crack
plane q 0 As a first approximation yielding
occurs when syy sys ry first order
estimate of plastic zone size. This is
approximate estimate, because of the fact that
syy is based on elastic analysis
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Irwins Approach When yielding occurs, stresses
redistribute in order to satisfy equilibrium
conditions. The cross hatched regions represents
forces active in the elastic analysis that cannot
be carried in elasto-plastic analysis, because of
the reason that the stresses cannot exceed sys.To
redistribute this excessive force, the plastic
zone size must increase. This is possible if the
material immediately ahead of plastic zone
carries more stress. Irwin proposed that
plasticity makes the crack behave as if it were
larger than actual physical size. Let the
effective crack size be aeff , such that
aeff ad, where d is the correction.
or
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To permit redistribution of stresses , the areas
A and B must be the same
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A long rectangular plate has a width of 100 mm,
thickness of 5 mm and an axial load of 50 kN. If
the plate is made of titanium Ti-6AL-4V,(KIC115
MPa-m1/2, sysp10 MPa) what is the factor of
safety against crack growth for a crack of length
a 20 mm.
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Strip Yield Method (Proposed by Dugdale and
Barrenblatt) They assumed a narrow long slender
plastic zone ahead of crack tip in a
non-hardening (ideal plastic) material in plane
stress for a through crack in a infinite plate.
The strip yield plastic zone assumes a crack
length of 2a 2r Where r is length of plastic
zone. Approximate elasto-plastic behavior is
obtained by superimposing two elastic solutions
(a) a through crack under remote tension (b) a
through crack under crack closure
stress. Concept stress at the crack tip is no
more a singular and it is a finite value (sys),
hence stress singularity term is zero. I.e. the
length of the plastic zone is such that the
stress intensity factor due to remote tension
cancels with crack closure.
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Strip Yield Method
SIF due to crack closure stress can be estimated
by considering a normal force P applied to the
crack at a distance x from the center line of
crack
SIF for two crack tips are given by
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Strip Yield Method
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Comparison of plastic corrections with LEFM
LEFM solutions are linear Irwin and strip yield
predictions deviate from LEFM theory at stresses
greater than 0.5 sys Two plasticty predictions
deviate at 0.85 sys
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Plastic Zone Shape In the earlier calculation
plastic zone size was calculated for q 0 (along
the crack plane), the plastic zone shape will be
quite different when all angles are considered.
Plasticity are based on von Mises and Trescas
yield criteria. For mode I problem stress field
are obtained using Westergaard stress function.,
then principal stresses are given by
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Using the principal stresses in von Mises yield
criteria
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Plane stress plastic zone sizes are larger than
plane strain plastic zone size. Tresca plastic
zones are larger than von Mises plastic zones.
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Plastic zone shapes for sliding mode and tearing
modes
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Plane strain or plane stress
In general, the conditions ahead of a crack tip
are neither plane stress nor plane strain. There
are limiting cases where a two dimensional
assumptions are valid, or at least provides a
good approximation. The nature of the plastic
zone that is formed ahead of a crack tip plays a
very important role in the determination of the
type of failure that occurs. Since the plastic
region is larger in PSS than in PSN, plane stress
failure will, in general, be ductile, while, on
the other hand, plane strain fracture will be
brittle, even in a material that is generally
ductile. This phenomenon explains the peculiar
thickness effect, observed in all fracture
experiments, that thin samples exhibit a higher
value of fracture toughness than thicker samples
made of the same material and operating at the
same temperature. From this it can be surmised
that the plane stress fracture toughness is
related to both metallurgical parameters and
specimen geometry while the plane strain fracture
toughness depends more on metallurgical factors
than on the others.
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Due to presence of crack tip, stress in a
direction to normal to crack plane syy will be
large near the crack tip. This stress would in
turn tries to contract in x and z direction. But
the material surrounding it will constraint it,
inducing stresses in x and z direction, there by
a triaxial state of stress exists near the crack
tip. This leads to plane strain condition at
interior. At the plate surface szz is zero and
ezz is maximum. This leads to plane stress
condition at exterior.
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The state of stress is also dependent on size of
plate thickness. If the plastic zone size is
small compared to the plate thickness, plane
strain condition exists. If the plastic zone
size is larger than the plate thickness, plane
stress condition prevails. As the loading is
increased, plastic zone size also increases
leading to plane stress conditions.
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Effect of plate thickness on fracture toughness
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Limits of LEFM
As per ASTM standard LEFM is applicable for
components of size
As per ASTM standard fracture toughness testing
can be done on Specimens of size
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Concept of Isoparametric Elements

• Elements are defined in local coordinate system
  (x,h) with straight well defined edges.
• Elements defined in local coordinates have
  advantage that numerical integration limits vary
  1 to 1
• Elements from local coordinated when mapped on to
  global cartesian coordinates, it can be distorted
  to a new curvilinear set as shown in the figure.
  
• In Isop. Formulation, with distorted elements,
  numerical integration are still carried with
  limits of local coorinates i.e. 1 to 1

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In the conventional FE formulations final
stiffness matrix is given by K ? BTDB
dv
and dv dxdy
If an appropriate transformation is obtained for
B and dv from (x,y) to (x,h) we have equation
for K set in (x,h) system where the integration
can be done with in the limits 1 to 1
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In order to achieve such a transformation ( B
and dv) two sets of nodes ate defined for each
element.
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Another set of nodes (marked as )are used to
interpolate displacements of a point with in the
element using nodal displacements. For example if
N1, N2, N3 are the interpolation function used
to interpolate shape variation, such that u N1
u1 N2 u2 N3 u3 v N1 v1 N2 v2 N3 v3
For a four node bilinear element N1 - ¼
.(1-x)(1-h) N2 - ½ .(1x)(1-h) The
interpolation functions N1, N2, N3 are
generally called here as Displacement function
as they define the displacement variation with in
the element.
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If number of nodes used to define shape are more
than the number of nodes used to define the
displacement variation say for example 8 nodes
for shape and 4 nodes for displacement, then such
elements are called SUPERPARAMETRIC ELEMENTS. If
number of nodes used to define shape are less
than the number of nodes used to define the
displacement variation say for example 4 nodes
for shape and 8 nodes for displacement, then such
elements are called SUBPARAMETRIC ELEMENTS.
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For any curved element formulations (iso, super
or sub) J is obtained using shape functions
(defined for shapes)
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Virtual Crack Extension Method to Evaluate SIF
For a two-dimensional cracked body in mode I, the
total potential in terms of FE solutions is given
by U ½ uTKu- uTF The strain energy
release rate Is defined as
FE mesh
crack tip
The first term in above expression is zero
(equilibrium condition). In the absence of
traction on crack face the third term is also
zero.
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The strain energy release rate is proportional to
the derivative of the stiff ness matrix with
respect to crack length.
For a given FE mesh for a body with crack length
a, and to extend the crack length by Da it is not
necessary to alter entire mesh. This can be
achieved by moving a few elements near the crack
tip and keeping rest intact. If N number of
elements that are effected then
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Stress Approach to Evaluate SIF
On the crack plane q 0
r1 r2 r3 r4
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Displacement Approach to Evaluate SIF
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Strain Energy Release Rate to Evaluate SIF