Effective transient behaviour of heterogeneous media in diffusion problems with a large contrast in the phase diffusivities

This paper presents a homogenisation-based constitutive model to describe the effective tran- sient diffusion behaviour in heterogeneous media in which there is a large contrast between the phase diffusivities. In this case mobile species can diffuse over long distances through the fast phase in the time scale of diffusion in the slow phase. At macroscopic scale, contrasted phase diffusivities lead to a memory effect that cannot be properly described by classical Fick's second law. Here we obtain effective governing equations through a two-scale approach for composite materials consisting of a fast matrix and slow inclusions. The micro-macro transition is similar to first-order computational homogenisation, and involves the solution of a transient diffusion boundary-value problem in a Representative Volume Element of the microstructure. Different from computational homogenisation, we propose a semi-analytical mean-field estimate of the composite response based on the exact solution for a single inclusion developed in our previous work [Brassart, L., Stainier, L., 2018. Effective transient behaviour of inclusions in diffusion problems. Z. Angew Math. Mech. 98, 981-998]. A key outcome of the model is that the macroscopic concentration is not one-to-one related to the macroscopic chemical potential, but obeys a local kinetic equation associated with diffusion in the slow phase. The history-dependent macroscopic response admits a representation based on internal variables, enabling efficient time integration. We show that the local chemical kinetics can result in non-Fickian behaviour in macroscale boundary-value problems.Comment: 36 pages, 14 figure

Effective transient behaviour of inclusions in diffusion problems

This paper is concerned with the effective transport properties of heterogeneous media in which there is a high contrast between the phase diffusivities. In this case the transient response of the slow phase induces a memory effect at the macroscopic scale, which needs to be included in a macroscopic continuum description. This paper focuses on the slow phase, which we take as a dispersion of inclusions of arbitrary shape. We revisit the linear diffusion problem in such inclusions in order to identify the structure of the effective (average) inclusion response to a chemical load applied on the inclusion boundary. We identify a chemical creep function (similar to the creep function of viscoelasticity), from which we construct estimates with a reduced number of relaxation modes. The proposed estimates admit an equivalent representation based on a finite number of internal variables. These estimates allow us to predict the average inclusion response under arbitrary time-varying boundary conditions at very low computational cost. A heuristic generalisation to concentration-dependent diffusion coefficient is also presented. The proposed estimates for the effective transient response of an inclusion can serve as a building block for the formulation of multi-inclusion homogenisation schemes.Comment: 24 pages, 9 figures. Submitted to ZAMM (under review

Mechanics of Supercooled Liquids

Pure substances can often be cooled below their melting points and still remain in the liquid state. For some supercooled liquids, a further cooling slows down viscous flow greatly, but does not slow down self-diffusion as much. We formulate a continuum theory that regards viscous flow and self-diffusion as concurrent, but distinct, processes. We generalize Newton’s law of viscosity to relate stress, rate of deformation, and chemical potential. The self-diffusion flux is taken to be proportional to the gradient of chemical potential. The relative rate of viscous flow and self-diffusion defines a length, which, for some supercooled liquids, is much larger than the molecular dimension. A thermodynamic consideration leads to boundary conditions for a surface of liquid under the influence of applied traction and surface energy. We apply the theory to a cavity in a supercooled liquid and identify a transition. A large cavity shrinks by viscous flow, and a small cavity shrinks by self-diffusion.Engineering and Applied Science

Interfacial electro-mechanical behaviour at rough surfaces

International audienceIn a range of energy systems, interfacial characteristics at the finest length scales strongly impact overall system performance, including cycle life, electrical power loss, and storage capacity. In this letter, we experimentally investigate the influence of surface topology on interfacial electro-mechanical properties, including contact stiffness and electrical conductance at rough surfaces under varying compressive stresses. We consider different rough surfaces modified through polishing and/or sand blasting. The measured normal contact stiffness, obtained through nanoindentation employing a partial unloading method, is shown to exhibit power law scaling with normal pressure, with the exponent of this relationship closely correlated to the fractal dimension of the surfaces. The electrical contact resistance at interfaces, measured using a controlled current method, revealed that the measured resistance is affected by testing current, mechanical loading, and surface topology. At a constant applied current, the electrical resistance as a function of applied normal stress is found to follow a power law within a certain range, the exponent of which is closely linked to surface topology. The correlation between stress-dependent electrical contact and normal contact stiffness is discussed based on simple scaling arguments. This study provides a first-order investigation connecting interfacial mechanical and electrical behaviour, applicable to studies of multiple components in energy systems

Strain rate dependence of the contribution of surface diffusion to bulk sintering viscosity

Modeling of bulk sintering viscosity usually neglects the contribution of pore surface diffusion with respect to grain‐boundary diffusion. This approximation is questionable at the high densification rates used today in advanced fast sintering techniques. A two‐dimensional analysis of the problem shows that the influence of surface diffusion on bulk viscosity at high strain rate can be decomposed as the sum of two terms: a term linked to the change in pore surface curvature and a term linked to the change in grain‐boundary size. The computational procedure relies on the partition of pore profile evolution into a transient component accounting for non‐densifying phenomena and an asymptotic component accounting for strain‐rate‐controlled phenomena. The largest impact of surface diffusion is found to arise from the change in grain‐boundary size. It follows a transition from Newtonian viscosity at low strain rate to non‐Newtonian viscosity which, during densification, increases nearly linearly with strain rate. In some conditions, viscosity can then reach more than twice the value estimated when neglecting pore surface diffusion. Reversely, expansion is accompanied by a decrease in grain‐boundary size which causes a decrease in viscosity and can lead to grain separation at high strain rate

Variational analysis of the influence of grain shape anisotropy on shear viscosity in Nabarro-Herring-Coble creep

The effect of strain-induced grain shape anisotropy on diffusional creep viscosity is analysed in two dimensions via a model representing grains by cylinders with elliptical cross section. Both cases of dominance of grain boundary diffusion and lattice diffusion are considered. Anisotropic creep viscosity is described by two coefficients calculated by considering different loading configurations with respect to the ellipse axes. Upper and lower bounds on these coefficients are obtained using kinematic and statical variational principles and assuming affine velocity, or uniform stress trial boundary fields, respectively. The analysis emphasises the dependence of the viscosity coefficients on aspect ratio and grain boundary viscosity. The difference between the bounds increases with grain elongation. A method is proposed for deriving estimates for the effective viscosity coefficients by coupling the two bounds. The strain hardening effect is analysed. Lattice diffusion contributes less to viscosity anisotropy than diffusion and sliding at grain boundaries