CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
ANISOTROPIC FAILURE MODEL DEVELOPMENT
AND IMPLEMENTATION
James D. Walker, Kathryn A. Dannemann and Charles E. Anderson, Jr.
Southwest Research Institute, P.O. Drawer 28510, San Antonio, TX, 78228
Abstract. Many materials are anisotropic in their failure. Examples of such materials include
hexagonal close-packed metals such as zirconium, directional structures such as fiber-reinforced
composites and extrusions of metals such as aluminum and beryllium. In general failure is difficult to
model, but there are added complications for materials with anisotropic failure behavior. This paper
presents a damage model that can be applied regardless of the magnitudes of the failure strains in the
various material directions. The damage model also applies to situations where the failure strains are
different in tension and compression. The actual anisotropic failure model was implemented in the
hydrocode CTH. The theory for the model is discussed and some representative calculations are
presented. The failure model is a strain-based model, and requires the use of Lagrangian strains for the
material. Extensions of the technique to stress-based failure models are discussed.
In addition, this paper addresses the placement
of an anisotropic failure model into the hydrocode
CTH. Representative calculations are presented.
Since the failure model is a strain-based model, it
requires the use of Lagrangian strains for the
material that are calculated with respect to the
original material orientation.
INTRODUCTION
Many materials are anisotropic in their failure.
Examples of materials include hexagonal closepacked metals such as zirconium, directional
structures such as fiber-reinforced composites and
extrusions of metals such as aluminum and
beryllium.
This paper focuses on the case of a material that
is isotropic and homogeneous in its elastic response,
but is anisotropic in its failure response. This
allows the calculation of the stresses with standard
numerical calculations available in most codes, but
requires the addition of an anisotropic failure model.
At the previous conference it was shown that
certain oft-discussed, stress-based anisotropic
failure surfaces have serious theoretical problems
that prevent their use in many situations [1]. These
difficulties carry over to the strain-based failure
models discussed here, and it will be pointed out
how the method presented here to model strainbased failure can also be extended to stress-based
failure surfaces.
APPROACH
Damage modeling has proven to be very difficult
in large-scale numerical simulations. One of the
inherent difficulties is that damage is typically
anisotropic either in its development or in its effects.
This model addresses the issue of damage that
accumulates at different rates depending upon the
predominant direction of strain.
The damage model was ultimately to be
implemented in the hydrocode CTH [2], and for this
reason a desired characteristic was that the
anisoptroic damage model default into one of the
isotropic damage models already implemented in
CTH. The choice was to have the anisotropic
damage model default into a strain-to-failure
275
deviator tensor, and therefore only the deviators of
the Lagrangian strain will be used in the
directionality determination. Thus,
damage model, where the strain used was the
equivalent plastic strain. Because of this choice, it
was decided to use the equivalent plastic strain,
even though it is isotropic, as the measure of strain,
and to use a multiplier to reflect the direction of the
strain, thus allowing anisotropy.
Currently within the Eulerian wavecode CTH
there is a capability of modeling failure of a material
based upon equivalent plastic strain. The equivalent
plastic strain is calculated according to the rate
equation
dt
L'L'
(3)
In order to calculate a direction, the choice is made
to work in the projected plane along the Ln=L22-L^
axis (for the stress, this projected plane is referred to
as the Ti-plane). In this plane, the direction of L\\
will be denoted by 9=0°, the L22 direction by
0=120°, and the L33 direction by 0=240° (Fig. 1). In
the projected plane the direction of each of the strain
deviators is given by (if X is chosen in the 0=0°
direction)
(1)
AV
where D? is the plastic part of the rate of
i . Vs.
deformation tensor. When the equivalent plastic
strain reaches a user defined critical value, the
strength of the material is reduced to zero over a
predetermined number of computational cycles.
Failed material cannot support any shear stresses
(the flow stress is set equal to zero) and cannot go
into tension.
The material still can support
hydrostatic compression.
Ongoing work at SwRI involves metals that are
essentially isotropic with regard to their elastic
properties, but show substantial directionality in
their strain-to-failure behavior. Thus, only the
damage model would display anisotropy, and the
elastic portion of the constitutive model could
remain unchanged. Using the equivalent plastic
strain, an isotropic measure of strain, as the strain
measure required the use of the Lagrangian strains
to provide a multiplier for the equivalent plastic
strain to include the directionality of the straining of
the material. Thus, the damage is written as
X+
~2
(4)
y
^
K V3 .
— x — —yv
2
2
and the direction angle of the strain L is computed
by the relation
(5)
•2C
In this expression, D is the damage and the Ly are
the Lagrangian strains, /is essentially the strain to
failure. The reason the shear strains are not
included here is that little information was available
about the shear failure strains, and so only the
primary axis strains are being used in the
directionality determination.
However, a later
section of the paper will discuss the inclusion of
shear strains.
A first simplification to be made regarding the
function / is that, in theory, the plastic strain is a
-2T
FIGURE 1. Geometry of strain deviators in the projected plane.
Once this direction is computed, then it is
straightforward to do a fit for the coefficient in
276
it is often the case that two of the directions are
strong, and for some materials, the failure strains
can differ by a factor of 10. It is for this reason that
the brute force approach of calculating a direction
angle was pursued.
terms of failure strain. If the various failure strains
are
L\\ compression: 8ic
Intension:
e1T
etc., then the directionality function can be
determined by linear interpolation on the arc length.
It is
3G£3C + (7r-3e)e17
(3e-7i)e 2r +(27t-3e)e 3C
3(e-7i)e 3r +(47i-3G)e lc
(3e-47c)e 2C +(57i-3e)e 3r
(39-57i )e ir +3(27i-0)e 2C
SAMPLE CALCULATIONS
To verify that the model works as expected,
calculations were performed to demonstrate the
directionality of the damage parameter.
In
particular, a flyer plate was impacted against a rigid
surface. The flyer plate is an aluminum alloy with
directionally dependent strains to failure. A MieGrrjneisen equation of state was used for the
aluminum with p0=2.804 g/cm3, c-5.2 km/s, ,9=1.36
and F0=2.2. The Johnson-Cook constitutive model
constants were used with A=496.4 MPa, B=310
MPa, n=0.30, c=0, m=1.2, and Tm=835°K.
The impact velocity was 1500 m/s. The strain in
the axial direction is given by -v/c - 0.29. For
uniaxial strain, the elastic and plastic rate of
deformation tensor components are
o<e<7i/3
7i/3<0<27i/3
27i/3<9 <n
n <0 <471/3
4 7 C / 3 < 0 < 571/3
(6)
It will be observed that this multiplier returns the
correct direction coefficient for strains along the
coordinate directions.
The CTH implementation of the damage
function follows the presentation above. The code
internally calculates the equivalent plastic strain. A
flag is set for the code to calculate the Lagrangian
strains. At each time step, for cells containing the
anisotropic material, the deviators of the Lagrangian
strains are determined and the direction / is
calculated. The damage is then found by dividing
the equivalent plastic strain by / If the damage
exceeds 1, the failed volume fraction is set equal to
1 over a predetermined number of computational
cycles, the strength of the material is reduced to
zero and the material is no longer allowed to support
tensile loads, just as with standard CTH damage
models.
It might be thought that there should be some
simpler formulation of/based on ideas from yield
and failure surfaces. However, it turns out that
simple approaches based on quadratic terms in the
stress or strain are unable to deal with situations
where a material is strong in two directions but
weak in one. In particular, it has been demonstrated
that when failure stresses or strains differ by a factor
of two or more, that in the case of two strong
directions and one weak direction that quadratic
polynomial fits to the failure coefficients do not
produce closed surfaces [1].
Since many
computations are in an axisymmetric environment,
1.
22
3
11'
11
3
11'
22
y
(7)
V /
Therefore, the equivalent plastic strain is
e »(2/3)|e n |
(8)
or 0.19 in this case. The equivalent plastic strain
reaching this value assumes that no failure occurs.
Two calculations are now considered. In the
first, the failure strain in the axial direction is 50%
and the failure strains in the normal plane are 10%.
Figure 2 shows the damage at 1 microsecond after
impact. It will be noted that there is no failure of
the material (save at the edges) since the failure
strain of 50% exceeds the equivalent plastic strain
of 19%. The second calculation has the axial failure
strain at a value of 10% and the failure strains in the
normal plane at 50%. The impact conditions are
identical. Figure 3 shows the damage contours at 1
microsecond after impact. The damage front can be
seen to be moving at roughly the bulk wave speed
of the aluminum. In this case, the fracture strain of
10% was less than the expected equivalent plastic
strain, and so the material failed.
These computations demonstrate that the
intended directionality of the yield surface works,
and that the damage measure is truly anisotropic.
277
1 .6
Here fj is the total strain to failure including the
shear terms. Given shear strains, this formulation
can provide a failure strain for any strain state.
The form of the failure strain given by Eqs. (6)
and (9) can also be extended to modeling failure
surfaces in stress space (this will be a different
approach to the lemniscate approach suggested in
[1]). In place of the strain deviators the stress
deviators sy- can be used. In place of the failure
strains, the failure stress for each direction can be
used. Also, the coefficient 2/>/3 must be replaced
by V3 (this is because uniaxial stress loading has
principle stresses (Y, 0, 0) while the corresponding
principle strains are (e,-£/2, -8/2)). The failure
surface is then given by
Plastic Strain or Damage
1 .4
1 .2
1 .0
0.8
0.6
0.4
0.2
0.0
.
-6
-4.
-2
0
2DC Block 1
6
2
X fern)
FIGURE 2. Damage contours for computation with large failure
strain in the axial direction and small failure strains in the normal
plane, at 1 microsecond.
^) 3 |Vff-J/r 2 (*») = 0
Ptostlc Strofn or Domoge_
-6-9-6-*-
1 .4 -
1
1
i
1
It will be noted that this failure surface is
independent of pressure. Hence, if the post-failure
or post-yield deformation direction was defined by
taking the stress derivative of the failure surface, the
deformation would be volume preserving, as is
assumed for plastic flow of metals.
-A- 1
1 .2 1 .0 -
0.8
*————————^
)|
I
0.6
0.4
'
0.2
n n
- 6
-
CONCLUSIONS
An anisotropic damage model has been presented
for damage in materials that show different strains
to failure in different directions. The model was
implemented in CTH.
Example calculations
demonstrated the directionality and anisotropy of
the damage model. An approach to extending the
method to stress-based failure surfaces was also
presented.
-
i
1
A I,
,
,
,
- 4 - 2
20C Block 1
,
,
0
,
2
,
,J A
4
f
6
X fern)
FIGURE 3. Damage contours for computation with small
failure strain in the axial direction and large failure strains in the
normal plane, at 1 microsecond.
EXTENSIONS
Shear strains were not included above because it
is unusual to have shear strain failure information.
However, below is an approach to including shear
strains. It is necessary to weight the damage
function above against the shear strains. One
possible weighting is to follow the form of the
second invariant of the strain deviator tensor:
2
= ( ( A ' , ) +(4)
(10)
ACKNOWLEDGMENTS
The authors thank Dick Sharron of SwRI for assistance
with the computations.
REFERENCES
1.Walker, J.D., and Thacker, B.H., "Yield Surfaces for
Anisotropic Plates," in Shock Compression in
Condensed Matter-1999, AIP Conference Proc. 505,
New York, pp. 567-570 (2000).
2. McGlaun, J.M., Thompson, S.L., Elrick, M.G., Int. J.
Impact Engng 10, pp. 351-360 (1990).
2
278