Download Theoretical and computational aspects of implementation of

CMM-2009 – Computer Methods in Mechanics
18–21 May 2009, Zielona Góra, Poland
Theoretical and computational aspects of implementation of anisotropic constitute model for
metals with microstructural defects∗
Wojciech Sumelka and Adam Glema
Poznan University of Technology, Faculty of Civil and Environmental Engineering
Institute of Structural Engineering, Division of Computer Aided Design
ul. Piotrowo 5, 60-965 Poznań, Poland
e-mail: [email protected]
Abstract
The material model is stated in terms of the continuum mechanics, in the framework of thermodynamics. The aim of the paper is the
implementation of the anisotropic material model as a user subroutine in Abaqus/Explicit environment. The rate form of the anisotropic
material model and the applied Eulerian description have enforced the use of the, so called, objective rate, to fulfil the frame indifference
requirements. The Lie derivative is accepted to obtain the covariant description of the constitutive structure while in Abaqus/Explicit
the Green-Naghdi rate is implemented. The fundamental results of the mathematical anisotropic material model description is given,
detailed discussion of the implementation of the Lie derivative in Abaqus/Explicit program is presented and the numerical results for
an adiabatic process for anisotropic body with strain rate reaching 104 s−1 are shown.
Keywords: anisotropy, objective rate, frame indifference, constitutive relation
1.
Introduction
One of the most important requirement, concerning each
physical theory, is that it must be objective (frame indifferent)
[4, 5, 7]. It means that each theory can not depend on the observer
or in terms of mathematics, on the selected coordinate system in
which governing equations are written.
Objectivity, can be obtained in physical theory, if the mathematical model, used to describe the analysed phenomenon, is
based on the tensorial calculus [5]. Unfortunately, even if we use
tensorial calculus to reach objectivity, the obtained theory will
not be unique. The objective rate of defined tensorial fields is
not uniquely defined. There is infinite set of objective rates to be
considered.
Definition of the objective rates is of the crucial importance
in rate type constitutive structures. The objective tensor field,
regarding constitutive theories, e.g. stress field, do not have objective material time rate (or direct flux). To reach objectivity one
have to choose, so called “objective rate”.
It will be shown that the consequent use of the objective Lie
derivative gives us the covariant anisotropic material model description. The anisotropic rate type constitutive structure is applied to high strain rate thermo-mechanical (adiabatic) process
by taking into account the user subroutine capability of Abaqus
finite element software. During numerical analysis the explicit
time integration scheme is accepted as a most efficient solution
for analysis where wave effect plays crucial role [6, 3].
2.
2.1.
Anisotropic constitutive structure
The concept of microdamage tensor
The discussed rate type anisotropic material model structure
bases on the phenomenological approach and the fundamental
variables are introduced as a state variable. Among others well
established in literature e.g. equivalent plastic strain, back stress,
the new state variable that governs the local anisotropic properties of microdamage is proposed [2, 3, 6]. This variable is called
microdamage tensor and is symmetric second order tensor field
- denoted in the paper by ξ. The physical interpretation of this
variable is that its norm (defined as Euclidean norm)
p
ξ = ξ : ξ,
(1)
defines the scalar ξ called volume fraction porosity which is the
ratio of the void volume to the volume of a material element.
The evolution equation (rate type based on Lie derivative) for
microdamage is proposed in an additive form where the nucleation and growth are considered (the growth term is inactive before the nucleation)
Lυ ξ =
∂h∗ 1
In
− 1]i
hΦ[
∂τ Tm
τn (ξ, ϑ, ∈p )
{z
}
|
nucleation
Ig
∂g ∗ 1
− 1]i,
hΦ[
+
∂τ Tm
τeq (ξ, ϑ, ∈p )
|
{z
}
(2)
growth
where the scalar functions of tensorial arguments h∗ and g ∗ describe the microcrack interaction for nucleation and the microcrack interaction for growth process, respectively while Tm denotes the relaxation time for mechanical disturbances, In , Ig
are the stress intensity invariants, τn , τeq denote two threshold
stresses and τ is the Kirchhoff stress.
2.2.
Fundamental relations
The fundamental results of the presented theory for adiabatic
process are governed by the following relations describing the
evolution of stress tensor τ and temperature ϑ
Lυ τ = Le : d − Lth ϑ̇ − (Le + gτ + τ g) : dp ,
(3)
∗
The support of Polish Ministry of Higher Education and Science under grant N N519 419435 “The evolution of properties and failure criteria of materials and structures
under fast dynamic loadings” is kindly acknowledged.
451
CMM-2009 – Computer Methods in Mechanics
18–21 May 2009, Zielona Góra, Poland
and
∂τ
:d
ρcp ϑ̇ = −divq + ϑ
ρRef ∂ϑ
+ ρχ∗ τ : dp + ρχ∗∗ K : Lυ ξ
(4)
where Le is linear elastic operator, Lth is thermal expansion operator, d denotes the spatial rate of deformation, dp is the rate
of viscoplastic deformation, ρ is density, cp is specific heat and
χ∗ , χ∗∗ are irreversibility coefficients.
ρ
3.
where Υ |i = −∆t |i (Ω |i ·τ |i −τ |i ·Ω |i ) and τ |i = R |i
τ̃ |i RT |i .
4.
Numerical results for adiabatic process
Material model implementation
The material model structure has been implemented as a user
subroutine VUMAT in Abaqus/Explicit program. As stated previously, in the presented material model, for stress rate (and for
all other rates), the fundamental variable in VUMAT subroutine,
Lie derivative has been taken into account, thus
Lυ τ = τ̇ − lT · τ − τ · l,
(5)
while in contrast, in Abaqus/Explicit VUMAT user subroutine,
the Green-Naghdi rate is calculated, through the following formula [1]
◦
τ (G−N ) = τ̇ + τ · Ω − Ω · τ ,
(G−N )
(6)
T
where Ω = Ω
= Ṙ · R represents the angular velocity
of the material and R denotes the rotation tensor.
It should be pointed out that material model in
Abaqus/Explicit VUMAT user subroutine must be defined in
so called corotational coordinate system, which is defined by
the spin tensor Ω (see Fig. 1). To give the physical meaning
of the corotational coordinate system, one can say that in this
coordinate system the stress tensor τ becomes τ̃ = RT τ R and
what is very important that the material time derivative of the
◦
corotational stress tensor τ̃ ◦ = RT τ (G−N ) R [8].
Figure 2: The evolution of microdamage tensor in subsequent
time steps
The numerical results of IBVP for an adiabatic process, with
strain rates reaching 104 s−1 will be presented. The evolution of
the microdamage tensor will be shown to predict the potential
degradation path before fracture appearance (see Fig. (2)).
References
[1] Abaqus Version 6.8 Documentation Collection, 2008.
[2] Glema, A., Łodygowski, T., Perzyna, P., Sumelka, W. Constitutive Anisotropy Induced by Plastic Strain Localization, 35th SOLID MECHANICS CONFERENCE, Kraków,
Poland, September 4-8, pp. 139-140, 2006.
Figure 1: Initial (X Y Z) and corotational (X̃ Ỹ Z̃) coordinate
systems
If we assume, that in the iterative procedure forward difference scheme is taken to calculate the material derivative of the
second order tensor, from Eqns (5) and (6) we obtain
τ̃ |i+1 = RT |i+1 [τ |i +∆tLυ τ |i
i
+ ∆t(lT |i ·τ |i +τ |i ·l |i ) R |i+1
and
T
h
τ̃ |i+1 = R |i+1 τ |i +∆tτ
(G−N )◦
(7)
|i
+ ∆t(Ω |i ·τ |i −τ |i ·Ω |i )] R |i+1 ,
(8)
in corotational coordinate system respectively. Thus, it is clear
that the Green-Naghdi rate, produces an additional term
∆t (Ω |i ·τ |i −τ |i ·Ω |i ) .
(9)
That is why, one have to subtract term (9) to compute Lie derivative. In the procedure we have formulated the stress update as
follows
τ̃ |i+1 = RT |i+1 [τ |i +∆t (2τ |i ·d |i
+Lυ τ |i ) + Υ |i ] R |i+1 ,
(10)
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[3] Glema, A., Łodygowski, T., Sumelka, W., Perzyna, P., The
Numerical Analysis of the Intrinsic Anisotropic Microdamage Evolution in Elasto-Viscoplastic Solids, International
Journal of Damage Mechanics, first published on November 21, 2008 as doi:10.1177/1056789508097543.
[4] G.A. Holzapfel. Nonlinear Solid Mechanics - A Continuum
Approach for Engineering. Wiley, 2000.
[5] J.E. Marsden and T.J.H Hughes. Mathematical Foundations
of Elasticity. Prentice-Hall, New Jersey, 1983.
[6] Perzyna, P., The Thermodynamical Theory of ElastoViscoplasticity Accounting for Microshear Banding and Induced Anisotropy Effects, 35th SOLID MECHANICS CONFERENCE, Kraków, Poland, September 4-8, pp. 35-36,
2006.
[7] C. Truesdell and W. Noll. The non-linear field theories
of mechanics, volume in: Handbuch der Physik III/3.
Springer-Verlag, Berlin, S: Flüugge, edition, 1965.
[8] H. Xiao, O.T. Bruhns, and A. Meyers. Logarithmic strain,
logarithmic spin and logarithmic rare. Acta Mechanica,
124:89–105, 1997.