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A grain boundary damage model for delamination

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Abstract

Intergranular failure in metallic materials represents a multiscale damage mechanism: some feature of the material microstructure triggers the separation of grain boundaries on the microscale, but the intergranular fractures develop into long cracks on the macroscale. This work develops a multiscale model of grain boundary damage for modeling intergranular delamination—a failure of one particular family of grain boundaries sharing a common normal direction. The key feature of the model is a physically-consistent and mesh independent, multiscale scheme that homogenizes damage at many grain boundaries on the microscale into a single damage parameter on the macroscale to characterize material failure across a plane. The specific application of the damage framework developed here considers delamination failure in modern Al–Li alloys. However, the framework may be readily applied to other metals or composites and to other non-delamination interface geometries—for example, multiple populations of material interfaces with different geometric characteristics.

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Acknowledgments

M. C. Messner’s contribution was in part supported by the Department of Defense (DoD) through the National Defense Science & Engineering Graduate Fellowship (NDSEG) Program and in part performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344 (LLNL-JRNL-666218). A. J. Beaudoin recieved support through the U.S. Department of Energy under Contract DEFG36-05GO15049. R. H. Dodds received support from the M. T. Geoffrey Yeh Chair Fund.

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  1. M. C. Messner

    Present address: Lawrence Livermore National Laboratory, P.O. Box 808, L-227, Livermore, CA, 94551, USA

Authors and Affiliations

  1. Department of Civil and Environmental Engineering, 2113 NCEL, University of Illinois at Urbana-Champaign, MC 250, 205 N. Mathews Avenue, Urbana, IL, 61801, USA

    M. C. Messner & R. H. Dodds Jr.

  2. Department of Mechanical Science and Engineering, 354 MEB, University of Illinois at Urbana-Champaign, MC 244, 1204 Green Street, Urbana, IL, 61801, USA

    A. J. Beaudoin

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Projection matrices

Projection matrices

This appendix lists the components of the three projection tensors describe above—\(\mathbf {P}_{3\times 6}^{e}\), \(\mathbf {P}_{3\times 6}^{c}\), and \(\mathbf {P}_{6\times 6}^{D}\). Each matrix is a combination of the components of the orthogonal coordinate system describing the interface plane/grain boundaries with normal vector \(\mathbf {n}\) and transverse vectors \(\mathbf {s}\) and \(\mathbf {t}\). Let the components of these vectors be:

$$\begin{aligned} \mathbf {n}= & {} \left[ \begin{array}{c@{\quad }c@{\quad }c} n_{1}&n_{2}&n_{3}\end{array}\right] ^{T}\\ \mathbf {s}= & {} \left[ \begin{array}{c@{\quad }c@{\quad }c} s_{1}&s_{2}&s_{3}\end{array}\right] ^{T}\\ \mathbf {t}= & {} \left[ \begin{array}{c@{\quad }c@{\quad }c} t_{1}&t_{2}&t_{3}\end{array}\right] ^{T} \end{aligned}$$

As described above, under large deformations and rotations these vectors remain constant in the corotational configuration. The form of the projection matrices depends on the Voigt notation used in the finite element framework. The particular Voigt notation in this work is:

$$\begin{aligned}&{\varvec{\varepsilon }}\rightarrow \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \varepsilon _{11}&\varepsilon _{22}&\varepsilon _{33}&2\varepsilon _{12}&2\varepsilon _{23}&2\varepsilon _{13}\end{array}\right] ^{T}\\&{\varvec{\sigma }}\rightarrow \left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} \sigma _{11}&\sigma _{22}&\sigma _{33}&\sigma _{12}&\sigma _{23}&\sigma _{13}\end{array}\right] ^{T}. \end{aligned}$$

The components of each projection are:

$$\begin{aligned}&\mathbf {P}_{3\times 6}^{e}=\left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} n_{1} &{}0 &{}0 &{}n_{2} &{}0 &{}n_{3}\\ 0 &{}n_{2} &{}0 &{}n_{1} &{}n_{3} &{}0\\ 0 &{}0 &{}n_{3} &{}0 &{}n_{2} &{}n_{1} \end{array}\right] \\&\mathbf {P}_{3\times 6}^{c}\\&\quad =\left[ \begin{array}{llllll} s_{1}^{2} &{}s_{2}^{2} &{}s_{3}^{2} &{}s_{1}s_{2} &{}s_{2}s_{3} &{}s_{1}s_{3}\\ t_{1}^{2} &{}t_{2}^{2} &{}t_{3}^{2} &{}t_{1}t_{2} &{}t_{2}t_{3} &{}t_{1}t_{3}\\ s_{1}t_{1} &{}s_{2}t_{2} &{}s_{3}t_{3} &{}\frac{1}{2}\left( s_{2}t_{1}+s_{1}t_{2}\right) &{}\frac{1}{2}\left( s_{2}t_{3}+s_{3}t_{2}\right) &{}\frac{1}{2}\left( s_{3}t_{1}+s_{1}t_{3}\right) \end{array}\right] \end{aligned}$$

and

$$\begin{aligned} \mathbf {P}_{6\times 6}^{D}=\left[ \begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} p_{11} &{}\quad p_{12} &{}\quad p_{13} &{}\quad p_{14} &{}\quad p_{15} &{}\quad p_{16}\\ p_{21} &{}\quad p_{22} &{}\quad p_{23} &{}\quad p_{24} &{}\quad p_{25} &{}\quad p_{26}\\ p_{31} &{}\quad p_{32} &{}\quad p_{33} &{}\quad p_{34} &{}\quad p_{35} &{}\quad p_{36}\\ p_{41} &{}\quad p_{42} &{}\quad p_{43} &{}\quad p_{44} &{}\quad p_{45} &{}\quad p_{46}\\ p_{51} &{}\quad p_{52} &{}\quad p_{53} &{}\quad p_{54} &{}\quad p_{55} &{}\quad p_{56}\\ p_{61} &{}\quad p_{62} &{}\quad p_{63} &{}\quad p_{64} &{}\quad p_{65} &{}\quad p_{66} \end{array}\right] \end{aligned}$$

with:

$$\begin{aligned} p_{11}&=n_{1}^{2}\left( n_{1}^{2}+2\left( s_{1}^{2}+t_{1}^{2}\right) \right) \\ p_{12}&=n_{1}n_{2}\left( n_{1}n_{2}+2s_{1}s_{2}+2t_{1}t_{2}\right) \\ p_{13}&=n_{1}n_{3}\left( n_{1}n_{3}+2s_{1}s_{3}+2t_{1}t_{3}\right) \\ p_{14}&=2n_{1}\left( n_{1}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{2}\left( s_{1}^{2}+t_{1}^{2}\right) +n_{2}n_{1}^{2}\right) \\ p_{15}&=2n_{1}\left( n_{3}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{2}\left( s_{1}s_{3}+t_{1}t_{3}\right) +n_{1}n_{2}n_{3}\right) \\ p_{16}&=2n_{1}\left( n_{1}\left( s_{1}s_{3}+t_{1}t_{3}\right) +n_{3}\left( s_{1}^{2}+t_{1}^{2}\right) +n_{3}n_{1}^{2}\right) \\ p_{21}&=n_{1}n_{2}\left( n_{1}n_{2}+2s_{1}s_{2}+2t_{1}t_{2}\right) \\ p_{22}&=n_{2}\left( 2n_{2}\left( s_{2}^{2}+t_{2}^{2}\right) +n_{2}^{3}\right) \\ p_{23}&=n_{2}n_{3}\left( n_{2}n_{3}+2s_{2}s_{3}+2t_{2}t_{3}\right) \\ p_{24}&=2n_{2}\left( n_{2}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{1}\left( n_{2}^{2}+s_{2}^{2}+t_{2}^{2}\right) \right) \\ p_{25}&=2n_{2}\left( n_{2}\left( s_{2}s_{3}+t_{2}t_{3}\right) +n_{3}\left( s_{2}^{2}+t_{2}^{2}\right) +n_{3}n_{2}^{2}\right) \\ p_{26}&=2n_{2}\left( n_{3}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{1}\left( n_{2}n_{3}+s_{2}s_{3}+t_{2}t_{3}\right) \right) \\ p_{31}&=n_{1}n_{3}\left( n_{1}n_{3}+2s_{1}s_{3}+2t_{1}t_{3}\right) \\ p_{32}&=n_{2}n_{3}\left( n_{2}n_{3}+2s_{2}s_{3}+2t_{2}t_{3}\right) \\ p_{33}&=n_{3}^{2}\left( n_{3}^{2}+2\left( s_{3}^{2}+t_{3}^{2}\right) \right) \\ p_{34}&=2n_{3}\left( n_{2}\left( s_{1}s_{3}+t_{1}t_{3}\right) +n_{1}\left( n_{2}n_{3}+s_{2}s_{3}+t_{2}t_{3}\right) \right) \\ \end{aligned}$$
$$\begin{aligned} p_{35}&=n_{3}\left( 2n_{3}\left( s_{2}s_{3}+t_{2}t_{3}\right) +2n_{2}\left( n_{3}^{2}+s_{3}^{2}+t_{3}^{2}\right) \right) \\ p_{36}&=n_{3}\left( 2n_{3}\left( s_{1}s_{3}+t_{1}t_{3}\right) +2n_{1}\left( n_{3}^{2}+s_{3}^{2}+t_{3}^{2}\right) \right) \\ p_{41}&=n_{1}\left( n_{1}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{2}\left( s_{1}^{2}+t_{1}^{2}\right) +n_{2}n_{1}^{2}\right) \\ p_{42}&=n_{2}\left( n_{2}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{1}\left( n_{2}^{2}+s_{2}^{2}+t_{2}^{2}\right) \right) \\ p_{43}&=n_{3}\left( n_{2}\left( s_{1}s_{3}+t_{1}t_{3}\right) +n_{1}\left( n_{2}n_{3}+s_{2}s_{3}+t_{2}t_{3}\right) \right) \\ p_{44}&=n_{1}^{2}\left( 2n_{2}^{2}+s_{2}^{2}+t_{2}^{2}\right) +2n_{2}n_{1}\left( s_{1}s_{2}+t_{1}t_{2}\right) \\&\quad +n_{2}^{2}\left( s_{1}^{2}+t_{1}^{2}\right) \\ p_{45}&=n_{2}\left( n_{3}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{2}\left( s_{1}s_{3}+t_{1}t_{3}\right) \right) \\&\quad + n_{1}\left( n_{2}\left( s_{2}s_{3}+t_{2}t_{3}\right) +n_{3}\left( s_{2}^{2}+t_{2}^{2}\right) +2n_{3}n_{2}^{2}\right) \\ p_{46}&=n_{1}^{2} \left( 2n_{2}n_{3}+s_{2}s_{3}+t_{2}t_{3}\right) \\&\quad + n_{1}\left( n_{3}\left( s_{1}s_{2}+t_{1}t_{2}\right) + n_{2}\left( s_{1}s_{3}+t_{1}t_{3}\right) \right) \\&\quad + n_{2}n_{3}\left( s_{1}^{2}+t_{1}^{2}\right) \\ p_{51}&=n_{1}\left( n_{3}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{2}\left( s_{1}s_{3}+t_{1}t_{3}\right) +n_{1}n_{2}n_{3}\right) \\ p_{52}&=n_{2}\left( n_{2}\left( s_{2}s_{3}+t_{2}t_{3}\right) +n_{3}\left( s_{2}^{2}+t_{2}^{2}\right) +n_{3}n_{2}^{2}\right) \\ p_{53}&=n_{3}\left( n_{3}\left( s_{2}s_{3}+t_{2}t_{3}\right) +n_{2}\left( n_{3}^{2}+s_{3}^{2}+t_{3}^{2}\right) \right) \\ \end{aligned}$$
$$\begin{aligned} p_{54}&=n_{2}\left( n_{3}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{2}\left( s_{1}s_{3}+t_{1}t_{3}\right) \right) \\&\quad + n_{1}\left( n_{2}\left( s_{2}s_{3}+t_{2}t_{3}\right) +n_{3}\left( s_{2}^{2}+t_{2}^{2}\right) +2n_{3}n_{2}^{2}\right) \\ p_{55}&=n_{2}^{2}\left( 2n_{3}^{2}+s_{3}^{2}+t_{3}^{2}\right) +2n_{3}n_{2}\left( s_{2}s_{3}+t_{2}t_{3}\right) \\&\quad +n_{3}^{2}\left( s_{2}^{2}+t_{2}^{2}\right) \\ p_{56}&=n_{3}\left( n_{3}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{2}\left( s_{1}s_{3}+t_{1}t_{3}\right) \right) \\&\quad + n_{1}\left( n_{3}\left( s_{2}s_{3}+t_{2}t_{3}\right) +n_{2}\left( 2n_{3}^{2}+s_{3}^{2}+t_{3}^{2}\right) \right) \\ p_{61}&=n_{1}\left( n_{1}\left( s_{1}s_{3}+t_{1}t_{3}\right) +n_{3}\left( s_{1}^{2}+t_{1}^{2}\right) +n_{3}n_{1}^{2}\right) \\ p_{62}&=n_{2}\left( n_{3}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{1}\left( n_{2}n_{3}+s_{2}s_{3}+t_{2}t_{3}\right) \right) \\ p_{63}&=n_{3}\left( n_{3}\left( s_{1}s_{3}+t_{1}t_{3}\right) +n_{1}\left( n_{3}^{2}+s_{3}^{2}+t_{3}^{2}\right) \right) \\ p_{64}&=n_{1}^{2}\left( 2n_{2}n_{3}+s_{2}s_{3}+t_{2}t_{3}\right) \\&\quad + n_{1}\left( n_{3}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{2}\left( s_{1}s_{3}+t_{1}t_{3}\right) \right) \\&\quad +n_{2}n_{3}\left( s_{1}^{2}+t_{1}^{2}\right) \\ p_{65}&=n_{3}\left( n_{3}\left( s_{1}s_{2}+t_{1}t_{2}\right) +n_{2}\left( s_{1}s_{3}+t_{1}t_{3}\right) \right) \\&\quad +n_{1}\left( n_{3}\left( s_{2}s_{3}+t_{2}t_{3}\right) +n_{2}\left( 2n_{3}^{2}+s_{3}^{2}+t_{3}^{2}\right) \right) \\ p_{66}&=n_{1}^{2}\left( 2n_{3}^{2}+s_{3}^{2}+t_{3}^{2}\right) +2n_{3}n_{1}\left( s_{1}s_{3}+t_{1}t_{3}\right) \\&\quad +n_{3}^{2}\left( s_{1}^{2}+t_{1}^{2}\right) . \end{aligned}$$

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Messner, M.C., Beaudoin, A.J. & Dodds, R.H. A grain boundary damage model for delamination. Comput Mech 56, 153–172 (2015). https://doi.org/10.1007/s00466-015-1164-z

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