Stress-Strain-Diffusion Interactions in Solids J. Svoboda 1 and F.D. Fischer 2 1 Institute of Physics of Materials, Brno, Czech Republic 2 Institute of Mechanics, Montanuniversität Leoben, Austria DSL 2014

CONTENT 1.Introduction 2.Generalized Manning theory for diffusion 3.Solution of 1-D example – two assembled sheets 4.Simulation of system evolution 5.Summary/conclusions

1. Introduction - A solid state system often involves defects, which can significantly influence its kinetics at elevated temperatures. - Defects can influence both effective diffusion coefficients as well as activity of sources and sinks for vacancies causing the Kirkendall effect and internal stress development with the feed back on diffusion kinetics. - Manning’s theory considering the vacancy wind effect has been generalized to account also for influence of sources and sinks for vacancies and the stress field, providing the evolution of the chemical composition coupled with deformation state of the system. - Motivation: to demonstrate the generalized Manning’s theory on a simple example and to show the influence of stress-strain-diffusion interactions on the system evolution kinetics.

2. Theory System with substitutional components Site fractions Volume corresponding to 1 mole of lattice positions Chemical potential of vacancies Chemical potential of components Coupling terms

Diffusion of substitutional components – Manning’s concept for fcc alloys, for bcc alloys

Conservation laws Generation/annihilation of vacancies at non-ideal sources/sinks Generalized creep (Fischer, Svoboda: Int. J. Plasticity 27, , 2011)

3. Solution of 1-D example – two assembled sheets Interface at Axial (z-axis) symmetry Elastic strain components Creep strain rate components Total strain components

Total thickness of the specimen No force and no moment acting on the outer boundary of the specimen Volume elements of actual configuration ________

Curvature of the deformed system obtaining the shape of a spherical shell Shift of the Kirkendall plane measured from the lower end of the system Solution of equations provides time evolution of - profiles of diffusive fluxes inclusive those of vacancies - profile of rate of vacancy generation/annihilation - profiles of site fractions of components inclusive vacancies - profiles of creep rates - profiles of strains and stresses - Kirkendall shift and curvature

4. Simulation of system evolution Material parameters (I) Starting values

Material parameters (II) Starting values

5. Summary/conclusions -Generalized Manning theory for diffusion of substitutional components accounting for sources and sinks for vacancies and interaction with developing internal stress field are presented. -The theory is demonstrated on simulation of evolution of a diffusion couple consisting of two assembled sheets; activity of sources and sinks for vacancies and diffusion coefficients are taken as system parameters. -For nearly the same diffusion coefficients of all components the influence of activity of sources and sinks for vacancies on system evolution kinetics is not significant. -For significantly different diffusion coefficients the influence of activity of sources and sinks for vacancies is evident. In such a case the measurement of diffusion coefficients MUST be completed by the characterization of the microstructure! If this is NOT done, the measured diffusion coefficients loose their credibility.