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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) 452 [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.