23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds) KTH, Stockholm, 2010c
Erik Lindfeldt, Magnus Ekh and H˚akan Johansson
for increasing RVE-size, i.e the number grains can be kept small. However, in order to limit the computational effort it is common to employ some sort of Taylor assumption for the displacement field. A possible approach, which is investigated in this paper, is to compute the diplacements along the grain boundaries from the macroscopic deformation gradient in the ”Taylor spirit”.
Whatever the choice, it is always necessary to solve the boundary value problems for the plastic slip fields including the gradient effect. The adopted algorithm employs the so-called dual-mixed FE formulation4.
Clearly, this type of problem comprises several length scales within which certain mechanisms occur. These length scales are schematically described, for a pearlitic steal, in the figure below.
Figure 1: Modelling approach: from macro to micro level
2 Preliminary results
The following results were obtained using a mesomodel consisting of 1 nodule with 7 colonies, all with different lamella orientations. Using the Taylor assumtion (constant strain field) at the meso level means that the constitutive response only needs to be solved once per colony. For the results presented below it is assumed that the meso domain is loaded by simple shear. The figure below shows the shear component, σ12, of the homogenised meso stress for three different sizes of the micro model.
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0
200 400 600 800 1000 1200
γ
[MPa]
a) Shear stress
L=4 L=2 L=3
Figure 2: Homogenised meso response, σ12, for different values of L
Erik Lindfeldt, Magnus Ekh and H˚akan Johansson
REFERENCES
[1] Modi, O. et al. Effect of interlamellar spacing on the mechanical properties of 0.65% c steel.
Materials Characterization 46, 347–352 (2001).
[2] Allain, S. & Bouaziz, O. Microstructure based modeling for the mechanical behavior of ferrite-pearlite steels suitable to capture isotropic and kinematic hardening. Materials Sci-ence and Engineering A 496, 329–336 (2008).
[3] Evers, L., Brekelmans, W. & Geers, M. Scale dependent crystal plasticity framework with dislocation density and grain boundary effects. International Journal of Solids and Structures 41, 5209–5230 (2004).
[4] Ekh, M., Grymer, M., Runesson, K. & Svedberg, T. Gradient crystal plasticity as part of the computational modelling of polycrystals. International Journal for Numerical Methods in Engineering 72, 197–220 (2007).
[5] Ekh, M., Lillbacka, R. & Runesson, K. A model framework for anisotropic damage coupled to crystal (visco)plasticity. International Journal of Plasticity 20, 2143–2159 (2004).
23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds) KTH, Stockholm, 2010c
ON THE MODELING OF DEFORMATION INDUCED ANISOTROPY OF PEARLITIC STEEL
N. LARIJANI, M. EKH, G. JOHANSSON AND E. LINDFELDT Department of Applied Mechanics
Chalmers University of Technology, Gothenburg, Sweden e-mail: nasim.larijani@chalmers.se
Key words: Plasticity, anisotropy, pearlitic steel, homogenization
Summary. A micromechanically based plasticity model for modeling of anisotropy in pearlitic steel is investigated. The model was proposed in Johansson and Ekh [1] and takes into account large strains as well as deformation induced anisotropy. The initially randomly oriented ce-mentite lamellae in the pearlitic steel will tend to align with the deformation which causes a development of anisotropy.
1 INTRODUCTION
Pearlite is a two-phase material where each grain has a preferred direction that is deter-mined by the cementite lamellae. The hard and brittle cementite lamellae are embedded in a softer ferrite matrix. Each grain can be considered to be transversally isotropic. The initial random orientation of the cementite lamellae gives an isotropic pearlitic material. After shear deformation, the orientations of individual grains tend to align with each other which causes a development of anisotropy. In this contribution, the modelled anisotropy on the macroscopic length scale is obtained from homogenization procedures of a micromechanical model of ”crystal plasticity”-type, proposed in Johansson and Ekh [1], of the pearlitic microstructure. In this model the plasticity is assumed to be driven by shear stress of the ferrite between the cemen-tite lamellae, and the re-orientation of the cemencemen-tite lamellae is assumed to be of areal-affine type, cf. Dafalias [2] Through the homogenization procedure all grains in the microstructure are assumed to be subjected to the same deformation gradient and the yield function of the grains have been replaced by a macroscopic yield function motivated from the micromechanical yield function. The macroscopic yield function is calculated by spherical integrations using an integration formula proposed by Baˇzant and Oh [3]. Finally, results showing the development of the yield surface, the reorientation of cementite lamellae and the macroscopic stress-strain response are given.
2 MICRO-MACROMECHANICAL MODEL
The point of departure is the micromechanical yield function Φµ which is formulated as follows:
Φµ= τµ2− Yµ2 (1)
N. Larijani, M. Ekh, G. Johansson and E. Lindfeldt
where Yµis the yield stress (taking into account hardening), and τµis the projected shear stress on the cementite lamella plane (or rather the ferrite in between the cementite lamellae) defined as:
τµ= τµ: [mµ⊗ nµ] . (2)
In this expression we introduced the Kirchhoff stress τµand the normal to the cementite lamella nµ. Further, the direction mµ is defined as the closest projection of the traction stress tµ = τµ,· nµ onto the cementite lamella plane.
The evolution of the cementite lamellae is assumed to be of an areal affine type determined by the deformation gradient, i.e.
nµ= F−tµ · nµ,0
|F−tµ · nµ,0| (3)
with nµ,0 being the initial normal to the cementite lamellae. We also propose to adopt an isotropic elastic law of Neo-Hooke type, an associative type of evolution law for the plastic deformation gradient, and a nonlinear hardening of the yield stress Yµ.
To compute the response of a microstructure of pearlitic steel for a given macroscopic defor-mation gradient, a finite element analysis using the micromechanical model summarized above with proper boundary conditions can be performed.
We homogenize the micromechanical yield function Φµto motivate a macroscopic yield func-tion Φ as
Φ =htra · τ2− τ : B : τi− Y2, (4) with
a = h aµi = h nµ⊗ nµi , B = h aµ⊗ aµi . (5) The current macroscopic yield stress Y takes hardening and lamella distance into account as discussed in Allain and Bouaziz [4]. The computational homogenization procedure to obtain the quantaties a and B is to carry out an integration over a unit sphere. In order to save the computational time an integration formula proposed in Baˇzant and Oh [3] was employed. A good example of integration over a unit sphere using the corresponding formula can be found in Miehe et.al. [5].
3 RESULTS
A technical application of pearlitic steel is in heavily drawn cords used for suspension of bridges. The required cold deformation of pearlitic wires is called wire-drawing in the liter-ature. Assume that an initially isotropic material is subjected to a severe uniaxial tension (wire-drawing). In Figure 1 we illustrate the stress response of the model when the pearlitic wire is drawn to a diameter 0.7 of its initial diameter. For this case, where we have assumed no isotropic hardening.
The reorientation of the normals of cementite lamellea for the corresponding loading type is shown in Figure 2.
N. Larijani, M. Ekh, G. Johansson and E. Lindfeldt
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0 500 1000 1500 2000 2500
εVM [ ] τVM [MPa]
Von-Mises stress v.s. strain during wire drawing
Anisotropy Isotropy
Figure 1: The stress-strain curve of pearlitic steel during wire drawing obtained from the model (com-paring the result with evolution of anisotropy and without)
-1 -0.5 0 0.5 1
0 0.2 0.4 0.6 0.8 1
nx [ ]
Reoientation of Normal Vectors of Cementite Lamellae
ny [ ]
F22 = 2 F22 =0
Figure 2: Reorientation of normals of cementite lamellae
N. Larijani, M. Ekh, G. Johansson and E. Lindfeldt
REFERENCES
[1] G. Johansson and M. Ekh, On the modeling of evolving anisotropy and large strains in pearlitic steel, European Journal of Mechanics/A Solids, 15, 1041-1060, 2006.
[2] Y.F. Dafalias, Orientation distribution function in non-affine rotations, Journal of Me-chanics and Physics of Solids, 49, 2493-2516, 2001.
[3] Z.P. Baˇzant and B.H. Oh, Efficient numerical integration on the surface of a sphere, Zeitschrift f¨ur angewandte mathematik und mechanik, 66, 37-49, 1986.
[4] S. Allain and O. Bouaziz, Microstructure based modeling for the mechanical behavior of ferrite-pearlite steels suitable to capture isotropic and kinematic hardening, Materials Sci-ence and Engineering A, 496, pp. 329-336, 2008.
[5] C. Miehe, S. Goktepe and F. Lulei, A micro-macro approach to rubber-like materialsPart I:
the non-affine micro-sphere model of rubber elasticity, Journal of the mechanics and physics of solids, 52, 2617-2660, 2004.
23rd Nordic Seminar on Computational Mechanics NSCM-23 A. Eriksson and G. Tibert (Eds)
c
KTH, Stockholm, 2010