Materials Science Forum Vols. 697-698 (2012) pp 204-207 © (2012) Trans Tech Publications, Switzerland doi:10.4028/www.scientific.net/MSF.697-698.204
Online: 2011-09-21
An Explicit Integration Algorithm for Introducing User-Defined Thermo-Viscoplastic Constitutive Models in FE Simulations C. Y. Gao 1,2,a, P. H. Liu 1 and L. C. Zhang 2,b 1
Department of Mechanics, Zhejiang University, Hangzhou, 310027, China
2
School of Mechanical and Manufacturing Engineering, The University of New South Wales, NSW 2052, Australia a
[email protected], b
[email protected]
Keywords: FE simulation, Integration algorithm, Metallic constitutive model, Abaqus, Vumat
Abstract. Material models in the libraries of commercially available finite element codes often cannot satisfy the needs of some special material property descriptions, such as those of materials under ultra-high-speed cutting. To overcome the difficulty, this paper introduces an efficient explicit algorithm to implement a generalized user-defined model of metal plasticity into the Abaqus/Explicit program. This algorithm makes the computation quite efficient by avoiding updating the tensor of elastic-plastic modulus in each increment. Introduction An accurate description of the dynamic response of materials and structures under impact loading coupled with high strain rate, high temperature as well as large strain is important to many engineering applications. Among the rate-dependent plastic constitutive models in the past decades, the phenomenological Johnson-Cook (J-C) equation has been the most widely used [1]: m
σ y = ( A + Bε n）[1 + C ln(ε / ε0 )](1 − T * )
(1)
where A, B, C, n and m are material constants; ε is the equivalent plastic strain, ε is the plastic strain rate and ε0 is the reference strain rate. T * =（T − Troom）/(Tmelt − Troom ) , and T is the absolute temperature. The material constants can be determined by tension or torsion tests. The model has successfully described the dynamic thermo-mechanical behaviors of a variety of materials by using the well-known material dynamic code EPIC-2 or general FE software such as Abaqus. In dealing with the deformation under very high strain rates such as high speed machining (HSM), penetration, or dynamic impact, however, the J-C model cannot provide satisfactory results of flow stress [2]. The primary reason is that this type of constitutive relationship is established purely on the empirical observation of experimental phenomena and is lack of microscopic physics. Thus, some physically-based models are of increasing interest. Zerilli and Armstrong [3] developed a constitutive model based on the thermal activation analysis of dislocation motion in the plastic deformation of crystal materials. They proposed different constitutive models for describing the behavior of body-centered-cubic (b.c.c.) and face-centered-cubic (f.c.c.) metals. Gao and Zhang [4] established a physical plastic constitutive model for f.c.c. metals based on the concept of mechanical threshold stress (MTS):
σ y = (σ G + k S d −1 / 2 ) + Yˆε n exp[c3 T ln(ε / ε s 0 )] ⋅ {1 − [−c 4 T ln(ε / ε0 )]1 / q }1 / p 1
(2)
where σˆ a (= σ G + kS d −1/ 2 ) is the athermal component of flow stress which is regarded as constant for f.c.c. metals, Yˆ , n , c , c , p, q are parameters to be determined for the thermal component of flow 1
3
4
stress. ε0 and εs 0 are reference strain rates which can be evaluated in advance. These material constants were all related directly with micromechanical characteristics. In reality, any physical or complex thermo-viscoplastic constitutive model, which considers the coupled strain hardening effect, rate hardening effect and thermal softening effect together, can be generally expressed as: All rights reserved. No part of contents of this paper may be reproduced or transmitted in any form or by any means without the written permission of Trans Tech Publications, www.ttp.net. (ID: 129.94.237.47, University of New South Wales, Sydney, Australia-27/11/15,07:38:56)
Materials Science Forum Vols. 697-698
σ y = f (ε , ε, T )
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(3)
Although there are various constitutive models in the material libraries of general FE software, none can well describe the behavior of a material under HSM as mentioned above. Abaqus has provided the robust self-extension ability for users, which allows users to introduce newly-defined material models into the main program by a user-material subroutine (umat or vumat). This paper will present how to realize a new generalized plastic constitutive model of metals in Abaqus/Explicit program by using the vumat subroutine. Formulation of the algorithm The algorithm is based on the explicit time-domain finite-difference method. The stress in each increment is updated based on a simplified stress-compensation method [5]. The basic governing equations and the principal procedure are described below. During elastic deformation, the stress is expressed by the Green-Naghdi rate form in a corotational framework. The generalized Hooke’s law can therefore be written as
∆σ ije = 2µ∆ε ije + λδ ij ∆ε kke
(4)
Mises yield criterion is adopted for the plastic flow in the case of isotropic strain hardening (the formulation of the hybrid hardening model can be found in [6]). The yield function is
3 S ij S ij − σ y (ε , ε, T ) = 0 2
(5)
where σ y is the dynamic yield stress which can be determined by the user-defined constitutive equation of any complex form similar to Eq. 3. S is the deviatoric stress, i.e., S ij = σ ij − δ ij σ kk / 3 . The trial stress, σ tr , can be calculated based on the old stress at the beginning of the increment by assuming an elastic behavior on the total strain increment ∆ε , i.e,
σ ijtr = σ ijold + 2 µ∆ε ij + λδ ij ∆ε kk
(6)
If the Mises equivalent deviatoric trial stress, S tr (= (3 / 2) S ijtr S ijtr ) , is less than the current yield stress σ y , the deformation is elastic, and the trial stress is the new stress to be updated at the end of the increment. Otherwise, plastic flow occurs, and the new stress should be updated by
σ ijnew = σ ijtr − (2µ∆ε ijp + λδ ij ∆ε kkp )
(7)
where the key variable to be determined is the plastic strain increment ∆ε p . Based on the orthogonal principle of the plastic flow rule (i.e., the direction of the plastic strain increment is along the external normal of the yielding surface), ∆ε p can be determined by [6]:
∆ε p =
1−ω S tr 2 2ωµ + (2 / 3)ω h
(8)
where ω = σ y / S tr ; h = ∂σ y / ∂ε
p
is the strain hardening rate at the beginning of the increment. By
the definition, the equivalent plastic strain increment is
∆ε p = (2 / 3)∆ε p : ∆ε p =
S tr − σ y 3ωµ + ω 2 h
(9)
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Thus, the new equivalent plastic strain, ε p (new) , and its rate, ε p (new) , can be easily obtained by using ∆ε p , and they should be saved as state variables to be used in the next time increment. Furthermore, since the time increment in a dynamic analysis is always very small ( ~ 10 -6 s ), ∆ε p can be calculated approximately by assuming ω ≈ 1 in the denominator (but not in the numerator) in Eq. (9) for simplification [7]. Finally, the procedure of the integration algorithm in the vumat subroutine of Abaqus is illustrated in Fig. 1. The user subroutine is called by the main program at each increment for every integration point of materials. START of Subroutine vumat
Subroutine Interface (state data transfer)
YES
If steptime .eq. zero ? NO
σ tr by σ tr = σ t + D e : ∆ε
Calculate new elastic stress
Calculate the trial stress
Calculate the yield surface size i.e., dynamic yield stress by J-C constitutive equation:
σ y = f (ε p , ε p , T ) S tr ≥ σ y ? (Mises yield criterion) NO Update the stress by set new stress = trial stress
END of Subroutine vumat (Return to main program)
YES
Calculate the increment of plastic strain
∆ε p
Update the stress by
σ t + ∆t = σ tr − D e : ∆ε p Update specific internal energy, inelastic energy, state variables e.g. equivalent plastic strain (rate).
Fig. 1 The flowchart of the vumat subroutine
Validation To validate the above algorithm, the J-C constitutive model was adopted here because this model has already been included in the material library of Abaqus. The numerical results of our user subroutine (vumat) can be compared with those computed by Abaqus itself. The implicit coding of J-C model in umat was given in [8] but the implicit algorithm in umat and the explicit algorithm in vumat are very different. The testing example is a plain strain cantilever beam (2.0 × 0.2 m) fixed at its left end and loaded at its right end by a perpendicular high speed impact (a down displacement of 0.1 m within 0.01 s). The material is AISI4142 steel. It can be seen from the comparison in Fig. 2 that the program with vumat subroutine gives a satisfactory result equivalent to that produced by the program without the subroutine (max stress error < 3.5 %).
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vumat
(b) with
vumat
Fig. 2 Comparison of the Mises stress calculated in Abaqus with and without the vumat subroutine.
Summary This paper has introduced an explicit stress-updating algorithm for integrating the user-defined thermo-viscoplastic constitutive models with a commercially available finite element code. The new algorithm makes the computation more efficient compared with the conventional ones, because it does not need to update the tensor of elastic-plastic modulus in each increment. Acknowledgements This research work is supported by the Australian Research Council and the Sci. & Tech. Research and Development Program of Ministry of Railway of China under Grant No. 2010J003-D. References [1] G. Johnson, W. Cook: Eng. Fract. Mech. Vol. 21 (1985), p. 31 [2] J.A. Arsecularatne, L.C. Zhang: Key Eng. Mater. Vol.274-276 (2004), p. 277 [3] F.J. Zerilli, R.W. Armstrong: J. Appl. Phys. Vol.5 (1987), p. 1816 [4] C.Y. Gao, L.C. Zhang: Materials Science and Engineering A Vol.527 (2010), p. 3138 [5] H.W. Li, H. Yang, Z.C. Sun: Trans. Nonferrous Metals Society China Vol. 16 (2006), p. 624 [6] H.W. Li, Doctoral dissertation, Xi’an: Northwestern Polytechnical University (2007) [7] D. Hibbit, B. Karlsson, P. Sorenson: Abaqus Theory Manual (ver6.5), HKS Inc., USA (2004) [8] J.F. Lu, Z. Zhuang, F. Zhang: Proceeding of Abaqus User Conf., Tsinghua Univ., Beijing (2003)
Advances in Materials Manufacturing Science and Technology XIV 10.4028/www.scientific.net/MSF.697-698
An Explicit Integration Algorithm for Introducing User-Defined Thermo-Viscoplastic Constitutive Models in FE Simulations 10.4028/www.scientific.net/MSF.697-698.204 DOI References [1] G. Johnson, W. Cook: Eng. Fract. Mech. Vol. 21 (1985), p.31. http://dx.doi.org/10.1016/0013-7944(85)90052-9 [2] J.A. Arsecularatne, L.C. Zhang: Key Eng. Mater. Vol. 274-276 (2004), p.277. doi:10.4028/www.scientific.net/KEM.274-276.277 [3] F.J. Zerilli, R.W. Armstrong: J. Appl. Phys. Vol. 5 (1987), p.1816. http://dx.doi.org/10.1063/1.338024 [4] C.Y. Gao, L.C. Zhang: Materials Science and Engineering A Vol. 527 (2010), p.3138. doi:10.1016/j.msea.2010.01.083 [5] H.W. Li, H. Yang, Z.C. Sun: Trans. Nonferrous Metals Society China Vol. 16 (2006), p.624. doi:10.1016/S1003-6326(06)60301-4
