Presentation Engineering template (1).pdf

Presentation Engineering template (1).pdf

1. An extended strain energy density failure criterion by
differentiating volumetric and distortional deformation
Group J
Indian Institute of Space Science and Technology
November 24, 2023
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2. Overview
1
Introduction
2
The extended strain energy density factor criterion
3
Application of extended SED model
4
Predictability of the SED model for kinks in inclined cracks
5
Pressure Dependent Failure criterion for extended SED
6
Conclusion
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3. Introduction
1
This paper is regarding crack propagation and the various ways to
study it.
2
Our main focus is to determine the crack kinks and their direction
when a crack is under far-field mixed-mode loading.
3
Theoretically, there are two broad ways to find crack kinks namely,
the Maximum Circumferential Stress Criterion (MCSC) suggested by
Erdogan and Sih (1963) and the Strain Energy Density (SED) Factor
Criterion developed by Sih (1974).
4
The mixed-mode criterion of fracture is given by,
f(k1,k2) =fcr (1)
which states that the combination of Mode I and Mode II stress-
intensity factors will cause crack initiation upon reaching some critical
value.
5
MCSC works well for brittle materials but not for the crack under
mixed-mode loading or under large plastic deformation as it also
depends on the geometry of the specimen along with the six
stress-strain components.
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4. The extended strain energy density factor criterion
The extended strain energy density factor criterion
Consider a material is under the general stress state in (x,y,z) coordinate,
the strain energy density per unit volume is:
dW
dV
=
1 +ν
2E
[σ
2
x+σ
2
y+σ
2
z−
ν
1 +ν
(σx+σy+σz)
2
+ 2(τ
2
xy+τ
2
yz+τ
2
xz]
The volumetric and distortional part of the
dW
dV
is:
dWd
dV
=
1 +ν
6E
[(σx−σy)
2
+ (σy−σz)
2
+ (σz−σx)
2
+ 6(τ
2
xy+τ
2
yz+τ
2
xz]
and
dWv
dV
=
1−2ν
6E
(σ
2
x+σ
2
y+σ
2
z)
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5. The extended strain energy density factor criterion
In principal plane, both the volumetric and distortional part can be
rewritten as:
dWd
dV
=
3(1 +ν)
2E
τ
2
oct=
3
4µ
τ
2
oct
and
dWv
dV
=
3(1−2ν)
2E
P
2
=
P
2
2κ
whereτoctis the octahedral shear stress and P is the hydrostatic tension.
Unified fracture criterion which couples the influences by distortion and
hydrostatic tension can be written as:
dWd
dV
ro
Sd
+
dWv
dV
ro
Sv
⩽1
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6. The extended strain energy density factor criterion
For a crack filed under mixed-mode loading taking the stress state ahead
of the crack tip in (r, θ) coordinates, the generalized strain energy density
factor¯Sis defined as:
¯S=r0
dWd
dV
+βr0
dWv
dV
= (βb11+c11)k
2
1+ (βb12+c12)k1k2+ (βb22+c22)k
2
2
To extend the Sih’s failure criteria we tookβ= 1 and made two
hypothesis:
1
The initial crack growth occurs in the direction (θ0) along which the
strain energy density factor¯Sis minimized, i.e.
∂¯S/∂θ= 0|
θ=θ0
= 0
whereθ0satisfies−π < θ0< π.
2
Crack extends if the strain energy density factor¯Sreaches the critical
value¯Scatθ=θ0.
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7. The extended strain energy density factor criterion
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8. Application of extended SED model
Central crack in tension
For this case, the stress intensity factorsk1andk2areσ
√
aand 0
respectively.
¯S=
(1+cosθ)k
2
1
48µ
[4β(1−2ν)(1 +ν) + 3(1−cosθ) + 2(1−2ν)
2
]
θ0= 0orθ0=acos
ı
2(1−2ν)
2
+2β(1−2ν)(1+ν)
3
ȷ
0< β <
3−(1−2ν)
2
2(1−2ν)(1+ν)
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9. Application of extended SED model
Central crack in shearing
For this case, the stress intensity factorsk1andk2are 0 andτ
√
a
respectively.
¯
S=
k
2
2
48µ
[2(1−2ν)(1−cosθ)(1−2ν) + 2β(1 +ν) + 3 + 9cos
2
θ]
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10. Application of extended SED model
θ0= 0
or
θ0=acos
ˇ
2β(1−2ν)(1 +ν) + (1−2ν)
2
9
˘
0< β≤
9−(1−2ν)
2
2(1−2ν)(1 +ν)
≈ −1
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11. Application of extended SED model
Central crack in mixed-mode loading
For this case, the stress intensity factorsk1andk2areσ
√
aandτ
√
a
respectively.
¯Sc=
(1−2ν)[2β(1+ν)+(1−2ν)]K
2
Ic
12µ
ı
KIIc
KIc
ȷ
2
=
4(1−2ν)[2β+1+2(β−1)ν]
2(1−2ν)(1−cosθ0)[(1−2ν)+2β(1+ν)]+3+9cos
2
θ0
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12. Application of extended SED model
Strain energy release rate is defined as,G=
(1−ν
2
)(K
2
Ic
+K
2
IIc
)
E
For kinked cracks, the stress intensity factorsKIcandKIIcare
according to the formulation given by Amestoy and Leblond.
KI=a11k1+a12k2
KII=a21k1+a22k2
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13. Predictability of the SED model for kinks in inclined cracks
Predictability of the SED model for kinks in inclined cracks
For an inclined crack under mixed-mode loading we have,k1=σ
√
asin
2
ϕ
andk2=σ
√
asinϕcosϕ, the direction of crack kinks is given by,
∂¯S/∂θ= 0|
θ=θ0
= 0 (2)
Forϕ̸= 0 and stationary value of¯Swe have
2(1−2v)[1 + 2β+ 2(β−1)v] sin (θ0−2ϕ)−3 sin (2θ0)
−6 sin (2θ0−2ϕ) = 0
forθ0. Corresponding solutions for severalβatv= 0.33 are shown in
figure below.
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14. Predictability of the SED model for kinks in inclined cracks
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15. Predictability of the SED model for kinks in inclined cracks
The predicted kink angles by different theories and experimental data is
shown here.
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16. Pressure Dependent Failure criterion for extended SED
Pressure Dependent Failure criterion for extended SED
In this section we are trying to explore the possibility of of SED
failure model to relate pressure distribution and yielding behaviour.
Notably we try to include the von mises criteria in our failure model
since it is derived from SED focusing solely on Distortional SED.
Earlier we derived the expression for distortional SED.
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17. Pressure Dependent Failure criterion for extended SED
dWd
dV
=
3τ
2
oct
4µ
≤
3
4µ
√
2τy
√
3
!
2
(3)
We can substitute theτoctin terms ofτyso that we have a failure
criteria term which is based on distortional strain energy in our
cohesive model.
Further we introduce a term Cavitation strength (pc) to accomodate
pressure dependent yielding.
dWv
dV
=
p
2
2κ
≤
`
p
2
c
2κ
´
(4)
after considering both terms in our yielding envelope our equation
takes the form
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18. Pressure Dependent Failure criterion for extended SED
3
2
`
τoct
τy
´
2
+
`
p
pc
´
2
sgn(p)≤1 (5)
Here theτoctrefers to the shear component applied upon the
interface with normal [111] in the principal stress coordinate, while P
refers the to the normal traction on the same plane which is
analogous to normal stress
this gives us a way to divide the failure of a material in two different
direction,τygives gives us the failure limit for macroscopic shear and
pcgives an idea of maximum resistance to interfacial sepration.
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19. Pressure Dependent Failure criterion for extended SED
The presence of dislocations in an atomic plane drastically alter the
resistance to gliding between the top and bottom blocks, resulting in
the drop in theτybut minor impact onpc.
τoct≤
√
2
√
3
τy
r
1−sgn(p)(
p
pc
)
2
(6)
this equation bears resemblance to the Mohr Coulomb law for
pressure-dependent shearing.
τf=C
′
+σtan(ϕ) (7)
hereσis analogous to p in our equation so we get a relation for shear
stress limit dependent on pressure.
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20. Pressure Dependent Failure criterion for extended SED
pcmeasures the separation strength of atomic layers and
approximates the ideal strength, approximately E/3 to E/10 for
crystalline metals.Commonly,σorτyis at least one to two orders of
magnitude smaller thanpc. Since in eq(36), pc being in the
denominator and being much larger than the other terms, the
dependence of yielding on the pressure in crystalline solids is neglible.
While in amorphous or granular solids, which have higher and
excessive amounts of cavities and free volume present,pcis
comparable toτyand the material yields more easily.
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21. Conclusion
Conclusion
We introduce an extended Strain Energy Density (SED) factor
criterion for material failure, addressing the weaknesses of the original
model by differentiating volumetric and distortional contributions.
This extension considers the physical differences between microscopic
shearing and separation during material fracture, providing a more
accurate prediction compared to the original SED factor criterion.
The model demonstrates good predictability for kink problems in
inclined cracks.
Material failure can be determined by applying the model to regular
geometries and using computational techniques for arbitrary
geometries.
While the extended SED model considers microscopic distortion and
separation during fracture, it currently lacks the capability to bridge
micro- to atomic-scale deformation mechanisms in crack tips for a
macroscopic description.
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22. Conclusion
We propose a material yielding criterion based on the extended SED
idea, accounting for microscopic shearing and separation mechanisms
differently.
The generalized yielding criterion aligns with the von Mises criterion
for high cavitation strength and resembles the Mohr–Coulomb law for
pressure-dependent shearing
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23. Conclusion
The End
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