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Multi-scale analysis of the early damage mechanics of ferritized ductile iron

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Abstract

A multi-scale analysis of the linear elastic and the early damage stages of ferritic ductile iron is introduced in this work. The methodology combines numerical and experimental analyses in the macro and micro scales. Experiments in the micro-scale are used for the characterization of the material micro constituents and the assessment of the micro-scale damage mechanisms; experiments in the macro-scale provide the data to calibrate and validate the models. The 2D multi-scale problem is modeled using the pre-critical regime of the Failure-Oriented Multi-Scale Variational Formulation, which is implemented via a FE\(^{2}\) approach. Finite element analysis in the micro-scale is customized to account for plastic deformation and matrix-nodule debonding. The multi-scale model is found effective for capturing the sequence and extent of the damage mechanisms in the micro-scale and to estimate, via inverse analyses, parameters of the matrix-nodule debonding law. Results allow to develop new insights for the better understanding of the ductile iron damage mechanics and to draw conclusions related to the modeling aspects of the multi-scale simulation.

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Abbreviations

\(\sigma ^{EXP}\) :

Stress

\(\varepsilon ^{EXP}\) :

Strain

\(\varepsilon _p^{EXP}\) :

Plastic strain

\(\sigma _s^{EXP} ,\varepsilon _s^{EXP}\) :

Stress and strain at the MND process start

\(\sigma _e^{EXP} ,\varepsilon _e^{EXP} \) :

Stress and strain at the MND process end

\(E_0^{EXP} \) :

Young’s modulus of the undamaged material

\(E^{\prime EXP} \) :

Young’s modulus as function of plastic strain

\(\sigma _{0.2}^{EXP} ,\varepsilon _{0.2}^{EXP} \) :

Offset yield stress and strain

\(\beta ^{EXP}\) :

Slope of the stress–strain curve in the elasto-plastic regime

\(\sigma ^{HOM}\) :

Stress

\(\varepsilon ^{HOM}\) :

Strain

\(\varepsilon _p^{HOM} \) :

Plastic strain

\(\sigma _s^{HOM} ,\varepsilon _s^{HOM}\) :

Stress and strain at the MND process start

\(\sigma _e^{HOM} ,\varepsilon _e^{HOM}\) :

Stress and strain at the MND process end

\(E_0^{HOM} \) :

Young’s modulus of the undamaged material

\(E^{\prime HOM} \) :

Young’s modulus as functions of plastic strain

\(\sigma _{0.2}^{HOM} ,\varepsilon _{0.2}^{HOM} \) :

Offset yield stress and strain

\(\beta ^{HOM}\) :

Slope of the stress–strain curve in the elasto-plastic regime

\({\varvec{\sigma }} _{\mu ,i}\) :

Micro-scale stress

\({\varvec{\varepsilon }} _{\mu ,i}\) :

Micro-scale strain

\(E_{\mu ,i} ,\nu _{\mu ,i}\) :

Elastic modulus and Poisson’s ratio

\(\sigma _{\mu ,i}^y ,H_{\mu ,i}\) :

Yield stress and hardening modulus

DI:

Ductile iron

FDI:

Ferritic ductile iron

FTF:

First to freeze zone

LTF:

Last to freeze zone

RVE:

Representative volume element

MNI:

Matrix-nodule interface

MND:

Matrix-nodule decohesion

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Acknowledgements

This research has been supported by grants awarded by CONICET (PIP 2013-2105 631), ANPYCT (PICT 2011-0159), the National University of Mar del Plata (ING 399-14) and the European Research Council under the European Union Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement No. 320815 (ERC Advanced Grant Project “Advanced tools for computational design of engineering materials” COMP-DES-MAT). The authors wish to express their gratitude to Prof. Alfredo Huespe (CIMEC-UNL-CONICET) for his valuable comments on the manuscript and for his assistance with the numerical simulations. The authors also acknowledge MEGAFUND S.A. for providing the material for the experimental analyses.

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Correspondence to D. O. Fernandino.

Appendices

Appendices

Table 5 Elasto-plastic model with isotropic hardening

In the following two appendices, subindices \(\left( . \right) _\mu \), indicating that variables are defined at the micro-scale, have been removed to alleviate the notation.

Appendix 1: Elasto-plastic model for the LTF and FTF matrix phases

The elasto-plastic model by Simo and Hughes (2000) is used for the matrix phases (LTF and FTF zones) at the micro-scale. The main equations governing this elasto-plastic model are presented in Box 1 and described next. The model parameters are identified in Fig. 8.

Table 6 Isotropic continuum tensile-damage model

Equation (6.1) is the additive decomposition of the strain tensor in its elastic, \({\varvec{\varepsilon }} ^{e}\) and plastic component \({\varvec{\varepsilon }} ^{p}\); equation (6.2) is the linear elastic relation between the elastic component of the strain and the stress tensor \({\varvec{\sigma }} \). The evolution of the plastic strain is given by the flow rule (6.3), where the consistency parameter \(\gamma \) is the norm of the plastic strain rate. The function \(g\left( {{\varvec{\sigma }} ,\alpha } \right) \) in equation (6.4) define the yield condition for \(g=0\). Equation (6.5) is the hardening law, which includes linear and exponential terms. The inequalities (6.7) describe the plastic loading/elastic unloading conditions.

Appendix 2: Damage model of the cohesive bands modeling the MNI

The constitutive relation for the micro-scale bands, modeling the MNI, is described by the continuum damage law presented in Box 2. This constitutive relation is from Simo and Ju (1987) and Oliver et al. (2002).

According with equation (7.1), the internal variables q and r determine the isotropic damage variable d, which defines the elastic stiffness degradation ruled by the elastic constitutive tensor \(\mathbf{C}^{e}\). Initial values of r and q are \(r_0 \) and \(q_0 \), respectively (see equation (7.3)). The variable r can be seen as the maximum value of a strain norm reached during the loading history. The relation between r and q is given by the intrinsic softening modulus \(\bar{H}^{d}\), as shown in equations (7.4). This parameter is depends on the fracture energy, \(G^{f}\), and the critical stress, \(\sigma ^{c}\), see equation (7.5). The remarkable aspect of the damage model is the regularization of the intrinsic softening modulus, with the parameter k (the thickness of the cohesive band). The damage criterion (7.6) is defined in terms of the positive part of the effective stress, \(\bar{{\varvec{\sigma }}}^{+}\), such that damage evolves under tensile stress only.

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Fernandino, D.O., Cisilino, A.P., Toro, S. et al. Multi-scale analysis of the early damage mechanics of ferritized ductile iron. Int J Fract 207, 1–26 (2017). https://doi.org/10.1007/s10704-017-0215-1

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