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Modeling of microdamage accumulation in anisotropic metals and alloys
M. N. Krivosheina, M. A. Kozlova, S. V. Kobenko, E. V. Tuch, and A. I. Lotkov
Citation: AIP Conference Proceedings 1683, 020105 (2015);
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Published by the American Institute of Physics
Modeling of Microdamage Accumulation in Anisotropic
Metals and Alloys
M. N. Krivosheina1, a), M. A. Kozlova1, b), S. V. Kobenko2, c),
E. V. Tuch1, d), and A. I. Lotkov3, e)
Institute of Strength Physics and Materials Science SB RAS, Tomsk, 634055 Russia
Nizhnevartovsk State University, Nizhnevartovsk, Russia
National Research Tomsk State University, Tomsk, 634050 Russia
Corresponding author:
Abstract. This paper studies the processes of fracture of barriers using the fracture criterion, allowing for consideration
of large anisotropy for ultimate values of total plastic strains in various symmetry axes of an anisotropic material in
conditions of sign-variable strain. All calculations were performed using the finite-element method under conditions of
dynamic loading in three-dimensional graphics with the use of original software. It was shown that when modeling
processes of fracture in barriers made of anisotropic and isotropic materials the differences in size and localization of
fracture zones increase with the increase in the initial speed of shock loading of barriers.
Strain anisotropy of mechanical characteristics occurs in metals and alloys as a result of formation, each of them
occurring to different extents. As a rule, strength characteristics have the maximum anisotropy. When breaking-off,
the values of the ultimate stress of samples which are cut out in three mutually perpendicular directions may differ
by 10–15%, and the values of ultimate strains corresponding to them by 10 times.
Calculating the cumulative plastic strain in sign-variable processes allows for consideration of accumulation of
microdamage in dynamic processes of material loading. For modeling the fracture of metals and alloys within the
limits of isotropic deformation models under dynamic conditions of loading, especially under conditions of
compression, the fracture criterion containing the value of cumulative plastic strain—the Odkvist parameter is often
used. The Odkvist parameter for anisotropic materials reflects only the anisotropy of elastic and plastic properties,
but does not reflect anisotropy of strength characteristics. Therefore, the fracture criterion using values of ultimate
cumulative plastic strains in symmetry axes of a material was used. This criterion allows for modeling of the
microdamage accumulation in these axes in anisotropic materials. The relative residual strain under tension and
shear in constructional materials is used as values of ultimate cumulative plastic strains. The strain in samples was
calculated in three symmetry axes of materials. It is especially important for materials with a high degree of
anisotropy of plastic properties because when fracturing in some directions they can be considered low-plastic, and
in others, under fracture, they have developed a plastic strain [1–4].
This research aims to study the fracture of barriers using a fracture criterion which allows for the consideration
of large anisotropy of ultimate values of total plastic strains in various symmetry axes of an anisotropic material in
conditions of sign-variable strain occurring under wave pattern of deformation.
Advanced Materials with Hierarchical Structure for New Technologies and Reliable Structures
AIP Conf. Proc. 1683, 020105-1–020105-4; doi: 10.1063/1.4932795
© 2015 AIP Publishing LLC 978-0-7354-1330-6/$30.00
FIGURE 1. Initial volumetric configuration of the projectile and the barrier
All calculations were performed with the finite-element method [5] in three-dimensional graphics with the use of
original software. Discretization of computational regions of the projectile (D1) and the barrier (D2) was carried out
by applying tetrahedrons (Fig. 1).
The system of equations describing non-stationary adiabatic motions of the compressible anisotropic medium
includes [6] equation of continuity:
div U v 0,
equation of continuous medium motion:
dv k
wV ki
energy equation:
dE 1
Vij eij .
dt U
Here U—medium density, v—velocity vector, V ij —contravariant components of symmetric stress tensor, E—
specific internal energy;
(’ i v j ’ j vi ),
eij —components of symmetric strain velocity tensor, vi —components of velocity vector, i, j = 1, 2, 3.
Let us suppose that full strain is representable in the form of the sum of elastic and plastic strains, the plastic
flow of an anisotropic material does not depend on hydrostatic pressure, and the elastic properties of a material do
not change under plastic deformation.
Elastic deformation of a material is described by the generalized Hooke’s law
Cijkl ekl ,
Cijkl —components of the elastic constant tensor.
In the field of plastic strains the full tension tensor is decomposed into spherical and deviatoric parts
Vki PGki S ki ,
where Ɋ—hydrostatic pressure, S —components of stress deviator and G ki —Kronecker delta.
The dependence of hydrostatic stress on current density and specific internal energy has the form (GrüneisenMie EOS):
· ª
· º
P ¦ K n ¨ 1¸ «1 K 0 ¨ 0 1¸ 2 » K 0UE ,
n 1
¹ ¼
© 0 ¹
where K 0 , K1 , K 2 , K3 —constants of a material, V0 , V—initial and current volumes.
Take the associated flow law in the form of
parameter dO = 0 under elastic strain, under plastic strain it is always positive, it is determined by the yield criterion,
H ijp —components of plastic strain, F—plasticity function.
The von Mises–Hill ’48 criterion is used as a plasticity function.
The fracture criterion is used
T kl
d E kl , k , l
1, ..., 3,
where Ekl —relative residual strain under fracture in conditions of tension and shear, and
increment strain of every component of plastic strain tensor separately
T pkl
—cumulative plastic
³ d Hkl , k , l 1, ..., 3.
Tensions determined in the element which is rigidly turned in space are recalculated by the Jaumann derivative.
Elasto-plastic deformation of a material of the projectile was determined within the limits of the presented model
for a special case of isotropic material.
By means of the suggested criterion [7] the fracture of barriers made of transtropic aluminum alloy D16T is
modeled as well as ones made of a material having isotropic characteristics received by averaging the characteristics
of transtropic material. The results of numerical modeling of the fracture of barriers are given at initial loading
velocities of 200 and 600 m/s with a steel compact projectile of cylindrical form and 20 g in mass. The elastic,
plastic and strength characteristics of a barrier material [8] are the following: U = 2700 kg/m3, Ex = 86.7GPa,
E y Ez = 92.1 GPa, Q xy = 0.32, Q zx = 0.34, Q yz = 0.33, Gxy Gxz = 33 GPa, Gyz = 31 GPa, V xs = 290 MPa,
V ys
V zs = 350 MPa, W xys
fracture were E x = 0.14, E y
W xzs = 150 MPa, W yzs = 180 MPa. The values of ultimate cumulative strains under
E z = 0.2, E xy
E xz = 0.07, E yz = 0.1. Here Ei —Young’s moduli, Gij —shear moduli,
Qij —Poisson’s ratios of a transtropic material, Vis —tensile yield limits, Wijs —shear yield points, Eij —values of
ultimate strains under fracture in conditions of tension and shear.
The characteristics of isotropic material of the barrier received by averaging the characteristics of a transtropic
material are the following: E = 87883MPa, G = 32934 MPa, Vs = 330 MPa, E = 0.18.
The direction of shock loading is oriented along the axis OZ (Fig. 1), and a special axis of a transtropic material
of the barrier is the axis 0X so the non axisymmetric stress-strain state occurs in the barrier. To illustrate the
differences arising in various sections of the barrier, Figure 2 presents two halves of barrier sections completed in
planes ZOY (left parts) and ZOX (the right parts). Figures 2a and 2b present distributions of differences in mass
fractions of the fractured material Rd in sections of the barrier which were received in the calculations with
transtropic mechanical characteristics of the barrier deducting distributions of mass fractions of the fractured
material in similar barrier sections with isotropic mechanical characteristics. In the ZOY sections (the left parts of
Fig. 2) the values of mechanical characteristics in isotropic material are less than in transtropic material. Therefore,
negative values for Rd arise, i.e. in this section there are more fracture zones in isotropic barrier material. In the ZOX
sections (the right parts) the values of mechanical characteristics in isotropic material are larger than in transtropic
material. Therefore, the distribution of positive values for Rd presents distribution of additional fracture zones in
transtropic barrier material as compared to isotropic material.
FIGURE 2. Differences in mass fractions of the fractured material in sections of the barrier made of transtropic and isotropic
materials at the time moment 24 μs: (a) 200 m/s, (b) 600 m/s
Sections of deformable projectiles in Fig. 2 are removed. With the initial velocity of the projectile at 600 m/s the
fracture of the barrier occurs like “knocking-out a stopper”. It can be seen at the back surface of the barrier (Fig. 2b).
Additional zones of the fractured material in the barrier (Rd) made of isotropic material are observed in the ZOY
section (Fig. 2b, left part). In the ZOX section additional zones of the fractured material arise in the barrier made of
transtropic material, especially in the back area of the barrier and near the projectile (Fig. 2b, right part). Figure 2
illustrates that with the initial velocity of the projectile at 600 m/s the differences in distribution Rd in fracture zones
in the barriers made of isotropic and transtropic materials increase.
The increase in speed of shock loading of the barrier leads to an increase in the cumulative plastic strain in the
direction of each of the axes of transtropic material of the barrier. This explains the increase in differences in the
volume of zones of the fractured material of the barrier for cases of isotropic and transtropic materials. The
distribution of differences of the fractured material in the volume of barrier material is determined by the wave
pattern of stress state of the projectile and the barrier.
The use of the fracture criterion allows for the consideration of the process of microdamage accumulation and
fracture, taking into account anisotropy of ultimate strains under fracture. This criterion also allows for those
fracture zones in the barrier detecting zones that are localized, which are not considered when modeling fracture
processes in barriers made of isotropic materials. It has been shown in the case when the minimum elastic, plastic
and strength properties of a transtropic material of the barrier are oriented perpendicular to the direction of rolling.
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Deformation of Highly Porous Metals and Alloys under Dynamic Loading, in Physical Mesomechanics of
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of Physics, Melville, NY, 2014), pp. 315–318, doi 10.1063/1.4901487.
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