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Multiscale Dislocation-Based Plasticity

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Mesoscale Models

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 587))

Abstract

This chapter, outlines a multiscale dislocation-based plasticity framework coupling discrete dislocation dynamics (DDD) with continuum dislocation-based plasticity. In this framework, and guided by DDD, a continuum dislocation dynamics (CDD) plasticity model involving a set of spatio-temporal evolution equations for dislocation densities representing mobile and immobile species is developed. The evolution laws consist of a set of components each corresponding to a physical mechanism that can be explicitly evaluated and quantified from DDD analyses. In this framework, stochastic events such as cross-slip of screw dislocations and uncertainties associated with initial microstructural conditions are explicitly incorporated in the continuum theory based on probability distribution functions defined by activation energy and activation volumes. The result is a multiscale dislocation-based plasticity model which can predict not only the macroscopic material mechanical behavior but also the corresponding microscale deformation and the evolution of dislocation patterns, size and gradient-dependent deformation phenomena, and related material instabilities at various length and time scales.

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Notes

  1. 1.

    (i) Dislocation multiplication, term \( {\beta}_1={\alpha}_1\ {\overline{v}}_g^{\alpha }/\overline{\ell}. \)Let the rate of multiplication be \( {\left.{\dot{\rho}}_m^{+\alpha}\right|}_{(1)}=\kern0.5em {\alpha}_1{\rho}_m^{+\alpha }/\overline{t} \), and \( \overline{t} \) be some characteristic time for dislocation segment of characteristic length \( \overline{\ell} \) and having an average velocity of \( {\overline{v}}_g^{\alpha } \). Then \( \overline{t}=\overline{\ell}/{\overline{v}}_g^{\alpha } \), which can be substituted into the expression for \( {\left.{\dot{\rho}}_m^{+\alpha}\right|}_{(1)} \), yielding β 1.

    (ii) Mutual annihilation of mobile dislocations, term \( {\beta}_2=2{\alpha}_2{R}_c{\overline{v}}_g^{\alpha } \). Let \( f=1/\overline{t} \) be the frequency of which a mobile dislocation located in a circular region of radius R c (capture radius for annihilation) gets annihilated by a mobile dislocation of opposite sign sweeping through the circular region with an average velocity \( {\overline{v}}_g^{\alpha } \). Then \( \overline{t}=\overline{x} \) / \( {\overline{v}}_g^{\alpha } \) , where \( \overline{x} \) is the average distance traveled by dislocations sweeping through the circular region. Then let the rate of annihilation be \( {\left.{\dot{\rho}}_m^{+\alpha}\right|}_{(2)}={\alpha}_1{N}_m^{+\alpha }\ {N}_m^{-\alpha }/{A}_c\ \overline{t} \), where \( {A}_c=\pi {R}_c^2 \) and \( {N}_m^{+\alpha }={\rho}_m^{+\alpha }{A}_c \), is the number of mobile dislocations with opposite sign of burgers vector residing in A c , and \( {N}_m^{-\alpha }={\rho}_m^{-\alpha }{A}_c \) is the number of disloctions of opposite sign entering the area A c with velocity \( {\overline{v}}_g^{\alpha } \). Suppose a mobile dislocation is gliding on a slip plane intersecting the circular area A c and located at distance y < R c from the center of the circle, then the distance the dislocation travels through the area is equal to \( 2\sqrt{R_c^2-{y}^2} \). Then for all possible slip planes interesting the area A, the average distance swept through the circle by dislocations entering and exiting the circle is \( \overline{x}=\left(1/{R}_c\right){\int}_0^{R_c}2\sqrt{R_c^2-{y}^2} dy=\pi {R}_c/2 \), substituting this result in the expression for the rate of annihilation \( {\left.{\dot{\rho}}_m^{+\alpha}\right|}_{(2)} \) leads to β 2. β 3 is derived using the same arguments.

    (iii) Pinning of mobile dislocations by immobile, term \( {\beta}_3={\alpha}_3\pi {R}_c^3{\overline{v}}_g^{\alpha } \). In this case it is argued that an immobile dislocation resided within the capture area of radius R c may trap equally mobile dislocations of opposite signs entering the capture area. For each mobile dislocation entering the area, say form right to left, there is a dislocation of opposite sign entering the area from left to right. Furthermore, it is assumed that since immobile dislocation (e.g. dipole or junction) is formed from two dislocations (e.g. dipole or junction), and thus the term includes \( {\left({\rho}_i^{\alpha}\right)}^2 \). The derivation follows the same steps described above for term 2.

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Acknowledgement

The support provided by the National Science Foundation’s CMMI program to WSU under Grant No. 1434879 is gratefully acknowledged.

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Authors and Affiliations

  1. School of Mechanical and Materials Engineering, Washington State University, Pullman, WA, USA

    Hussein M. Zbib, Mehdi Hamid & Hao Lyu

  2. Department of Mechanical & Aeronautical Engineering, Clarkson University, Potsdam, NY, USA

    Ioannis Mastorakos

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  1. Hussein M. Zbib
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  2. Mehdi Hamid
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  3. Hao Lyu
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  4. Ioannis Mastorakos
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Correspondence to Hussein M. Zbib .

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Editors and Affiliations

  1. Department of Mechanical and Materials Engineering, Washington State University, Pullman, WA, USA

    Sinisa Mesarovic

  2. Centre des Matériaux, Mines ParisTech, Evry, France

    Samuel Forest

  3. Department of Mechanical and Materials Engineering, Washington State University, Pullman, WA, USA

    Hussein Zbib

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Zbib, H.M., Hamid, M., Lyu, H., Mastorakos, I. (2019). Multiscale Dislocation-Based Plasticity. In: Mesarovic, S., Forest, S., Zbib, H. (eds) Mesoscale Models. CISM International Centre for Mechanical Sciences, vol 587. Springer, Cham. https://doi.org/10.1007/978-3-319-94186-8_2

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  • DOI: https://doi.org/10.1007/978-3-319-94186-8_2

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