In a tension test on a ductile material, a diffuse necking - so called because its spatial extension is much larger than the sheet thickness - begins to develop in the sample when the strain hardening is no longer able to compensate for the weakening due to the reduction of the cross section. After some elongation under decreasing load, a localized neck usually appears in the region of the diffuse neck. In the localized neck, severe thinning occurs leading to ultimate failure. This work focuses on studying the diffuse and localized necking under plane stress conditions in visco-plastic materials under dynamic loading. By means of a DOE analysis the main material parameters that influence the occurrence of local and diffuse necking were determined. A material model is then validated by systematic comparison of simulation results with physical tests carried out at different strain rates. One methodology to achieve good correlation between test and experiments is the use of a damage model. The damage model chosen by the authors is GISSMO (Generalized Incremental Stress-State dependent damage Model), due to its widespread usage in the crash community. The GISSMO model is defined in terms of a critical plastic strain that indicates the start of damage coupling and a failure plastic strain indicating fracture, both are defined as a function of stress triaxiality. As the necking or more generally plastic instability will result in mesh dependency of the simulation results, regularization is introduced by defining both the failure plastic strain and the damage exponent as a function of the mesh size. As the spatial extension of the diffuse neck differs as a function of strain rate, one way to achieve correlation between test and simulation is to express these parameters as function of both the mesh size and the strain rate. As consequence of the findings of this study, a number of new options were developed in the GISSMO model.

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
1
Paul Du Bois1
, Marcus Feucht2
, Salvatore Scalera3
1) Consultant, Freiligrahstr. 6, 63071 Offenburg, Germany
2) Daimler AG, EP/SPB, 71059 Sindelfingen, Germany
markus.feucht@daimler.com
3) DYNAmore Italia S.r.l., Piazza Castello, 139, 10124 Turin, Italy
salvatore.scalera@dynamore.de
Diffuse and localized necking under
plane stress in visco-plastic material
models

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
ABSTRACTIn a tension test on a ductile material, a diffuse necking - so called because its spatial extension is
much larger than the sheet thickness - begins to develop in the sample when the strain hardening is
no longer able to compensate for the weakening due to the reduction of the cross section. After
some elongation under decreasing load, a localized neck usually appears in the region of the diffuse
neck. In the localized neck, severe thinning occurs leading to ultimate failure.
This work focuses on studying the diffuse and localized necking under plane stress conditions in
visco-plastic materials under dynamic loading. By means of a DOE analysis the main material
parameters that influence the occurrence of local and diffuse necking were determined. A material
model is then validated by systematic comparison of simulation results with physical tests carried
out at different strain rates. One methodology to achieve good correlation between test and
experiments is the use of a damage model. The damage model chosen by the authors is GISSMO
(Generalized Incremental Stress-State dependent damage Model), due to its widespread usage in the
crash community.
The GISSMO model is defined in terms of a critical plastic strain that indicates the start of damage
coupling and a failure plastic strain indicating fracture, both are defined as a function of stress
triaxiality. As the necking or more generally plastic instability will result in mesh dependency of the
simulation results, regularization is introduced by defining both the failure plastic strain and the
damage exponent as a function of the mesh size. As the spatial extension of the diffuse neck differs
as a function of strain rate, one way to achieve correlation between test and simulation is to express
these parameters as function of both the mesh size and the strain rate.
As consequence of the findings of this study, a number of new options were developed in the
GISSMO model.

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
OVERVIEW
• INTRODUCTION
– Definition of diffuse & local necking and failure
– Failure models available in LS-DYNA. Why GISSMO?
• Visco-plastic material model
– Diffuse necking: new mathematical condition and its validation
– Local necking: DOE analysis to understand the influence of numerical
and phisical parameters
• GISSMO: short description
– Damage evolution
– Regularization
– Parameters identification
• GISSMO extension to include viscosity
– Description
– Correlation
• Conclusion

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
Notions in failure and instability theories
• Diffuse necking: the point where we
observe a loss of the homogeneous
state of deformation, a pretty CLEAR
notion, at least at a simulation level
• Local necking: basically the formability
limit, a rather FUZZY notion that needs
to be defined for every application, can
depend on the size of imperfection and
the size of the grid (numerical). To
better understand this phenomena, a
DOE analysis was carried out. The
results will be discussed further in this
presentation.
• Failure: the point where a simply
connected part becomes multiply
connected: cracks appear, also a pretty
CLEAR notion
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
Some failure models in LS-DYNA. Why GISSMO?
• GISSMO is a simple yet powerful damage and failure concept within LS-DYNA developed at Daimler in
cooperation with DYNAmore. Currently it is used and evaluated in most automotive companies (and
even copied by other codes).
• It is based on the Johnson-Cook approach but generalized to suite various crash and metal forming
demands.
• It can be used a simple failure criteria or a rather complex damage and failure model. Hence it can be
coupled to stresses and act as damage model.
• It might be used in sheet metal forming and crashworthiness and in both applications to close the gap
between forming and crashworthiness simulations.
• Existing Material Models are kept unaltered (usually MAT24 for crash)
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
Visco-plastic material: diffuse necking
𝐸𝑡 𝜖 + η 𝜀, 𝜀
𝜀
𝜀
≥ 𝜎 𝐸𝑡 =
𝜕𝜎
𝜕𝜀
η =
𝜕𝜎
𝜕𝜀
• Material model 1 (MAT24 tabulated)
𝜎 𝜎
𝜀 𝜀Observation:
• the uni-axial tensile test, carried out using a specimen characterized by the
material described above, gives ambiguous output also in the case of a simply
elasto-plastic behaviour.
• Considering two different material – both without strain-rate dependency - with
Et1 Et2, the shape of the necking point is completely different.
Conclusion:
The hypotesis of linear bahavior of the stress has to be removed
introduction diffuse necking local necking failure
The equation was written starting from the algorithmic expression of the stress increment,
combined with the condition of instability for having the necking point

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
𝐸𝑡 𝜖 + η 𝜀, 𝜀
𝜀
𝜀
≥ 𝜎 → 𝑛 ∗ 𝑏 ∗ 𝜀 𝑛−1
+
η ∗ 𝜀
𝜀
𝜎 = 𝑎 + 𝑏 ∗ 𝜖 𝑛
→ 𝐸𝑡 =
𝑑𝜎
𝑑𝜀
= 𝑛 ∗ 𝑏 ∗ 𝜀 𝑛−1
𝜎
𝜎
• Material model 2 (MAT24 tabulated)
Observation:
• Also in this case, a preliminary investigation only considering the
elasto-plastic behaviour, was conducted. As expected, the
condition for the localization
𝑑𝜎
𝑑𝜀
= 𝜎 is accurately verified and the
extension of the necking point is confined to the centre of the
specimen like in the quasi-static physical tests.
• The simulated tests conducted with the the visco-plastic material
shows again a 𝑑 ≫ 𝑑0, the longitudinal extension of the neck is
much grater than the one commonly observed in real dynamical
tests conducted on different type of steels
Conclusion:
The hypothesis of linear behaviour of the stress as function of the
strain rate has to be removed. In other terms
𝑑 = 𝑑 𝐸𝑡 𝜖 , η 𝜀, 𝜀
𝜀
𝜀
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
𝐸𝑡 𝜖 + η 𝜀, 𝜀
𝜀
𝜀
≥ 𝜎 → 𝑛 ∗ 𝑏 ∗ 𝜀 𝑛−1 +
𝜀
𝜀2
∗
𝜎2 − 𝜎1
ln 𝜀2 − ln(𝜀1)
≥ 𝜎
𝜎(𝜀) = 𝑎 + 𝑏 ∗ 𝜖 𝑛
𝜎 𝜀 = 𝜎1 +
ln 𝜀 − ln(𝜀1)
ln 𝜀2 − ln(𝜀1)
∗ 𝜎2 − 𝜎1
𝜎 𝜎
• Material model 3 (MAT24 tabulated)
𝜀 𝜀
𝐸𝑡 =
𝑑𝜎
𝑑𝜀
= 𝑛 ∗ 𝑏 ∗ 𝜀 𝑛−1
η =
𝑑𝜎
𝑑𝜀
=
1
𝜀
∗
𝜎2 − 𝜎1
ln 𝜀2 − ln(𝜀1)
𝜎1
𝜎2
𝜀1
𝜀2
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
• Material model 3 (MAT24 tabulated)
Observation:
• Assigning to the constants, the following set of the values
𝜎1 = 0.5 𝜎2 = 0.7 𝑏 = 0.8 𝑛 = 0.5 𝜀1 = 1𝑒−4
𝜀2 = 0.1,
the condition of no necking assumes the simplified formula:
𝐸𝑡 𝜖 + η 𝜀, 𝜀
𝜀
𝜀
≥ 𝜎 → 𝑛 ∗ 𝑏 ∗ 𝜀 𝑛−1
+
𝜀
𝜀2 ∗
𝜎2−𝜎1
ln 𝜀2 −ln 𝜀1
≥ 𝜎
0.4
𝜖
+ 0.03
𝜀
𝜀2 ≥ 𝜎
Conclusion:
When 𝑑𝐹 = 0 →
0.4
𝜖
+ 0.03
𝜀
𝜀2 = 𝜎; the new local condition of no necking for visco-
plastic material is confirmed by this experience.
𝜎 𝐹
𝑡 𝑡
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
The local condition
𝐸𝑡 𝜖 = 𝜎 elasto-plastic material
𝐸𝑡 𝜖 + η 𝜀, 𝜀
𝜀
𝜀
≥ 𝜎 visco-plasti material
is able to predict when the instability occurs, but of
course it can’t describe the instability itself.
Visco-plastic material: brittle & ductile behaviour
From that moment on, a diffuse neck begins to develop
in the sample because the strain hardening is not able
to compensate anymore the weakening due to the
reduction in width and thickness of the cross section.
After some elongation under decreasing load, a
localized neck suddenly appears in the region of the
diffuse neck. In the localized neck, mainly thinning
occurs

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
• Material model 4 (MAT 24 C&S)
Visco-plastic material: local neckingTo identify which are the factors that influence the local necking, or in other terms, that lead to a brittle or ductile
behaviour of the material, a DOE analysis was carried out.
The material model is a piecewise liner plasticity and the strain rate is accounted for, using the Cowper and
Symonds model:
𝜎 = 𝑎 + 𝑏 ∗ 𝜀 𝑛
∗ 1 +
𝜀
𝐶
1
𝑝
𝜎
𝜀
nominal Max Min
a 0.4 0.5 0.7
b 0.8 0 1
n 0.5 0 1
C 150 40 6500
p 3 2 10
Target of the DOE analysis: identify the parameters
which have major influence on the diffuse necking
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
Main results of the DOE analysis
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
introduction diffuse necking local necking failure
a=0.5
C= 40
p=5
Main results of the DOE analysis
𝜎 = 𝑎 + 𝑏 ∗ 𝜀 𝑛
∗ 1 +
𝜀
𝐶
1
𝑝

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
Failure model: the philosophy of GISSMO
• The user defines a failure curve (the
onset of cracks) and a critical strain
curve (the loss of uniformity in the
strain field)
• Between the critical strain curve and
the failure curve, we assume a
continuous process of localization,
inducing mesh dependency.
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
Failure model: the philosophy of GISSMO
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
Regularization in GISSMO
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
Extension of GISSMO to visco-plasticity
engineeringstress
REGULARIZATION
engineering strain
CORRELATION
New extension of LS-DYNA: strain rate dependecies for
fade exponent and regularization factor
introduction diffuse necking local necking failure

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9th European LS-DYNA User‘s Conference – Manchester - 2-4. June 2013
Acknowledgements
• Diffuse necking, local necking, failure were discussed for
visco-plastic material
• Diffuse necking: new condition with a preliminary validation
of the mathematical model. Further investigation needed
• Local necking: DOE analysis to start a numerical interpretation
of such fuzzy phenomena
• Failure: GISSMO extension
• Correlation with experimental tests
Conclusions
Thanks to Dr. Tobias Erhart for implementing the extensions in LS-DYNA and to Dr.
Andre Haufe, Dr. Tobias Graf and Dr. Filipe Andrade for their active collaboration in the
understanding and interpretation of the physical phenomena emerging from the real
and the virtual tests.