A numerical method for the simulation of ductile fracture

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A numerical method for the simulation
of ductile fracture
A.E. Huespe(1), J. Oliver (2), A. Needleman(3), P.J. Sánchez(1)
(1) CIMEC,
CIMEC Intec (UNL-CONICET),
(UNL CONICET) Santa Fe
Fe, Argentina,
Argentina
(2) ETSECCP, Technical University of Catalonia, Barcelona, Spain
(3) Dept. of Materials Sc. and Eng. , College of Engineering and Center for
Advanced Scientific Computing and Modeling (CASCaM), Univ. of North Texas, U.S.A.
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A typical mechanism of ductile fracture
Strain localization
band
Fracture surface displaying
void coalescence (Garrison
1987)
Stress triaxiality ratio
: mean stress
:equivalent stress
Coalescence of voids (Cox et al. 1974)
Fracture mode (I, II, etc.),
depends on T
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Motivation
Gurson porous plastic constitutive relation
(as a continuum model) to simulate material ductile failure
The Gurson model lacks of a characteristic length
Standard numerical approaches fail (FE presents mesh dependence)
Some previous approaches:
i) Cohesive models: (Tvergaard & Hutchinson, 1992)
The standard approach, tt=g(β)
g(β) does not
depend on the stress triaxility T;
Siegmund et al.
al (2002) propose a cohesive law,
which parameters depends on T:
t=g(β,T)
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The present model proposes:
To embed into the continuum, a weak discontinuity kinematics mode
characterized by a band width D;
To regularize the constitutive model by including a characteristic length
in the model (the band width D of the weak discontinuous mode);
To include those numerical aspects allowing for embedding arbitrary
weak discontinuities into the FE meshes:
By using specific FE (E-FEM technique);
By defining different methodologies for evaluating the crack
propagation through the FE mesh;
By using robust integration scheme (Impl-Ex)
Huespe, Needleman, Oliver and Sánchez, 2009, Int.J.Plast.
g deformation
Large
setting is included
in this work
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Large deformation kinematics with an embedded weak
discontinuity mode
Di l
Displacement
field:
fi ld
Weak discontinuity mode
Smooth part
Displacement jump
Do
across the band
D
Finite band width
(j
(jump
iin strains)
t i )
The deformation gradient and its multiplicative decomposition
(Armero et al. 1996, Oliver et al. 2003)
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Gurson constitutive relation formulated in principal stretches
Multiplicative decomposition of the elastic and plastic gradient of
deformation (Simo 1992 Comp. Met. Appl. Mech. Eng.)
Hyperelastic stress-strain
relation
Equivalent stress (Cauchy stress !)
Void fraction
Yield criterion
(Gurson model)
Matrix yield stress(Cauchy stress !)
Plastic flow law and
internal variable evolution
Loading-unloading
condition
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Integration IMPL-EX of the Gurson constitutive model in principal stretches
(Oliver et al. 2008, CMAME)
Implicit
stresses
Return mapping in the
principal stress space
Impl-Ex stresses
Set of internal variables
ω >0
0
Important consequences in the algorithmic tangent tensor
increasing the numerical robustness !!!
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Gurson constitutive relation formulated in principal stretches
Momentum balance equation and material bifurcation
condition:
Equilibrium condition on a discontinuity surface
is the failure criteria
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Algorithm for solving the material bifurcation condition:
• In the present large deformation context, this problem can be solved
using the iterative algorithm of Oliver et al. 2010 Int.J.for Num. Meth Eng.:
Eigenvalue of iteration i
END IF
Eigenvector of iteration i
Convergence takes 10-15
iterations!!
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A Finite Element Formulation with embedded weak discontinuities (E-FEM)
Underlying finite element: BBAR
Elemental parameter
(condensed)
Interpolation of the displacement field
Ni(x),i=1,…,4,
standard shape
function of the
bilineal
quadrilateral
element
Specified thickness of
the discontinuity band
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Additional Issues addressed to evaluate the crack path
The EFEM technique requires to know which FE of
the mesh are intersected by the crack path and
which nodes belong
g to Ω+
¾
A global technique: all possible crack paths are
simultaneously evaluated in the FE mesh (Oliver et al 2004,
Int. J. Num. Meth Geom.)
¾
An alternative procedure based on the injection of strain localized
d
di a ttracking
ki algorithm
l ith (Oliver
(Oli
l 2010,
2010 EUROCEUROC
modes,
nott needing
ett al.
2010- quasi-brittle material)
simplifies the weak (or strong) discontinuity based
method implementation
enhanced strain methods
implemented
in two steps:
p
p
a) injection of an elemental constant strain mode
b) injection of specific strain localization modes
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A procedure based on the injection of strain localization modes
a) STEP 1: injection of a constant strain mode into the FE
Constant in the FE
Domain of injection (
)
+ use of a consistent variational formulation
Under plastic
loading conditions
This provides a more flexible FE:
enhanced capacity for capturing and
propagating localization modes in the initial
postcritical stages (with respect to the standard
underlying
y g element).
)
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a) STEP 1: injection of a constant strain mode inside the element
b) STEP 22: iinjection
j ti off a specific
ifi strain
t i localization
l li ti mode:
d
Domain of injection :
+ plastic loading conditions
Initial bifurcation
time
To define this mode
mode, we need to know:
The normal vector of the injected mode :
Detection of which nodes are in Ω+ :
A criterion based on the (rate of) stretching of every FE side is adopted
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Numerical examples
CASE 1:
1 regularization
l i ti off th
the constitutive
tit ti model
d l
Specimen subjected to biaxial stresses
“n”: number of
elements
Characteristic lengths D is independent of the FE size !!!
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Ductile rupture of a plane strain notched bar
Specimen with notch
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Ductile rupture of a plane strain notched bar
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Selection of the crack path direction
Injection of strain localized modes
Two possible
tangent directions
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Fracture resistance curves: small scale yielding of a mode I crack growth
Crack tip
advance
Mode I linear elastic singular crack tip field
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J-integral versus the crack growth (Δa ): Different values of the
g D and FE meshes.
characteristic length
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Conclusions
A new numerical
i l approach
h based
b d on th
the FE method
th d for
f large
l
deformation ductile fracture simulation has been presented.
Its main features are:
i)
the methodology is based on a finite thickness embedded weak
di
discontinuity
i i band,
b d which
hi h induces
i d
a characteristic
h
i i length
l
h (D).
(D)
i)
the same constitutive relation (Gurson plastic model) describes the
material response in the pre and the postcritical stage;
iii) Implicitly, the evolution of the fracture process zone takes into
account the stress triaxiality ratio T;
Th necessary ingredients
The
i
di
to generalize
li the
h code
d to 3D are there!
h !
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Gurson elastoplastic constitutive relation
Asociative plastic flow law:
Yield function:
Matrix hardening law:
Void fraction evolution
Modified void-fraction function:
(Void-coalescence phenomenon
Tvergaard & Needleman, 1982)
Loading-unloading
Conditions:
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Stress:
Total strain rate
Plastic strain rate
Equivalent stress
Matrix eq. stress
V id ffraction
Void
i
Matrix eq. plast. strain
Plastic multiplier
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Large deformation kinematics with embedded weak discontinuity mode
The deformation gradient and its multiplicative decomposition
(Armero et al. 1996, Oliver et al. 2003)
D
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Large deformation kinematics with embedded weak discontinuity mode
The deformation gradient and its multiplicative decomposition
(Armero et al. 1996, Oliver et al. 2003)
D
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A Finite Element Formulation with embedded weak discontinuities (E-FEM)
B-Bar FE approach
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A Finite Element Formulation with embedded weak discontinuities (E-FEM)
1
Petrov-Galerkin approach:
2
Q d t
Quadrature
points
i t
1.
Standard Gauss points 1;..;4 (position and weights)
g
the terms of eq.
q
integrates
1
2.
Gauss points 5 and 6 are in the center of the FE:
PG 5:
PG 6:
3.
2
Once the strain localization mode is active,
PG1,…,5 are constrained to behave elastically.
PG6 follows a plastic loading path
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Motivation
ii) Computational Cell Model (Xia et al. 1995)
is based on the observation that planar fracture process zone displays
a wide of 1 or 2 void spacing:
(D is a length compared with the
mean void spacing, and induces
a characteristic length in the
model)
Embedded computational
cell in the FE mesh (1 row
of FE of width D)
•Gurson constitutive model into
th computational
the
t ti l cell
ll
•The stress triaxiality influences
the response of material failure