Model reduction by mean-field homogenization in viscoelastic composites. III. Dual theory

2021 ◽  
Vol 477 (2249) ◽  
pp. 20200869
Author(s):  
Noel Lahellec ◽  
Martín I. Idiart ◽  
Pierre Suquet
Keyword(s):  
Order Approximation ◽  
Mean Field ◽  
Companion Paper ◽  
Present Scheme ◽  
Reduced Order ◽  
Link Type ◽  
Potential Structure ◽  
Finite Set ◽  
Viscoelastic Composites ◽  
Kinetics Of

The mean-field homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2013 Int. J. Plasticity 42, 1–13 ( doi:10.1016/j.ijplas.2012.09.005 )) is revisited from the standpoint recently adopted in a companion paper (Idiart MI et al. 2020 Proc. R. Soc. A 20200407 ( doi:10.1098/rspa.2020.0407 )). It is shown that the scheme generates a reduced-order approximation wherein the microscopic kinetics of the composite are described in terms of a finite set of macroscopic forces identified with the phase averages and intraphase covariances of the various microscopic force fields, which can be evaluated by mean-field homogenization techniques. The approximation exhibits a two-potential structure with a convex complementary energy density but a non-convex force potential. The consequential properties of the approximation are exposed and their implications are discussed. The exposition is supplemented by proofs of equivalence between the present scheme and other candidate schemes proposed in the literature for composites with elementary local rheologies of Maxwellian type.

scholarly journals Model reduction by mean-field homogenization in viscoelastic composites. II. Application to rigidly reinforced solids

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200408
Author(s):  
Martín I. Idiart ◽  
Noel Lahellec ◽  
Pierre Suquet
Keyword(s):  
Elastic Moduli ◽  
Solid Phase ◽  
Mean Field ◽  
Free Energy Density ◽  
Companion Paper ◽  
Viscoelastic Solid ◽  
Complex Deformation ◽  
Link Type ◽  
The Mean ◽  
Viscoelastic Composites

The mean-field homogenization scheme proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44 , 507–529 ( doi:10.1016/j.ijsolstr.2006.04.038 )) and revisited in a companion paper (Idiart et al . 2020 Proc. R. Soc. A 20200407 ( doi:10.1098/rspa.2020.0407 )) is applied to random mixtures of a viscoelastic solid phase and a rigid phase. Two classes of mixtures with different microstructural arrangements are considered. In the first class the rigid phase is dispersed within the continuous viscoelastic phase in such a way that the elastic moduli of the mixture are given exactly by the Hashin–Shtrikman formalism. In the second class, both phases are intertwined in such a way that the elastic moduli of the mixture are given exactly by the Self-Consistent formalism. Results are reported for specimens subject to various complex deformation programmes. The scheme is found to improve on earlier approximations of common use and even recover exact results under several circumstances. However, it can also generate highly inaccurate predictions as a result of the loss of convexity of the free-energy density. An auspicious procedure to partially circumvent this issue is advanced.

Impact of pH on growth of Staphylococcus epidermidis and Staphylococcus aureus in vitro

2021 ◽  
Vol 70 (9) ◽  
Author(s):  
Vidula Iyer ◽  
Janhavi Raut ◽  
Anindya Dasgupta
Keyword(s):  
Staphylococcus Aureus ◽  
Growth Kinetics ◽  
Staphylococcus Epidermidis ◽  
Type Species ◽  
Ph Dependence ◽  
Skin Microbiome ◽  
Content Type ◽  
Link Type ◽  
Kinetics Of

The pH of skin is critical for skin health and resilience and plays a key role in controlling the skin microbiome. It has been well reported that under dysbiotic conditions such as atopic dermatitis (AD), eczema, etc. there are significant aberrations of skin pH, along with a higher level of Staphylococcus aureus compared to the commensal Staphylococcus epidermidis on skin. To understand the effect of pH on the relative growth of S. epidermidis and S. aureus , we carried out simple in vitro growth kinetic studies of the individual microbes under varying pH conditions. We demonstrated that the growth kinetics of S. epidermidis is relatively insensitive to pH within the range of 5–7, while S. aureus shows a stronger pH dependence in that range. Gompertz’s model was used to fit the pH dependence of the growth kinetics of the two bacteria and showed that the equilibrium bacterial count of S. aureus was the more sensitive parameter. The switch in growth rate happens at a pH of 6.5–7. Our studies are in line with the general hypothesis that keeping the skin pH within an acidic range is advantageous in terms of keeping the skin microbiome in balance and maintaining healthy skin.

scholarly journals Model reduction by mean-field homogenization in viscoelastic composites. I. Primal theory

2020 ◽  
Vol 476 (2242) ◽  
pp. 20200407 ◽  
Author(s):  
Martín I. Idiart ◽  
Noel Lahellec ◽  
Pierre Suquet
Keyword(s):  
Strain Field ◽  
Mean Field ◽  
Inelastic Strain ◽  
Mathematical Structure ◽  
Schwarz Inequality ◽  
Internal Variables ◽  
Legendre Transform ◽  
Time Step ◽  
Legendre Transforms ◽  
Viscoelastic Composites

A homogenization scheme for viscoelastic composites proposed by Lahellec & Suquet (2007 Int. J. Solids Struct. 44 , 507–529 ( doi:10.1016/j.ijsolstr.2006.04.038 )) is revisited. The scheme relies upon an incremental variational formulation providing the inelastic strain field at a given time step in terms of the inelastic strain field from the previous time step, along with a judicious use of Legendre transforms to approximate the relevant functional by an alternative functional depending on the inelastic strain fields only through their first and second moments over each constituent phase. As a result, the approximation generates a reduced description of the microscopic state of the composite in terms of a finite set of internal variables that incorporates information on the intraphase fluctuations of the inelastic strain and that can be evaluated by mean-field homogenization techniques. In this work we provide an alternative derivation of the scheme, relying on the Cauchy–Schwarz inequality rather than the Legendre transform, and in so doing we expose the mathematical structure of the resulting approximation and generalize the exposition to fully anisotropic material systems.

scholarly journals A simplified model of the Martian atmosphere - Part 1: a diagnostic analysis

2005 ◽  
Vol 12 (5) ◽  
pp. 603-623 ◽  
Author(s):  
S. G. Whitehouse ◽  
S. R. Lewis ◽  
I. M. Moroz ◽  
P. L. Read
Keyword(s):  
Total Energy ◽  
Order Approximation ◽  
Travelling Waves ◽  
Martian Atmosphere ◽  
Simplified Model ◽  
Horizontal Structure ◽  
Reduced Order ◽  
Temperature State ◽  
Model Equations ◽  
Baroclinic Modes

Abstract. In this paper we derive a reduced-order approximation to the vertical and horizontal structure of a simplified model of the baroclinically unstable Martian atmosphere. The original model uses the full hydrostatic primitive equations on a sphere, but has only highly simplified schemes to represent the detailed physics of the Martian atmosphere, e.g. forcing towards a plausible zonal mean temperature state using Newtonian cooling. Three different norms are used to monitor energy conversion processes in the model and are then compared. When four vertical modes (the barotropic and first three baroclinic modes) are retained in the reduced-order approximation, the correlation norm captures approximately 90% of the variance, while the kinetic energy and total energy norms capture approximately 83% and 78% of the kinetic and total energy respectively. We show that the leading order Proper Orthogonal Decomposition (POD) modes represent the dominant travelling waves in the baroclinically-unstable, winter hemisphere. In part 2 of our study we will develop a hierarchy of truncated POD-Galerkin expansions of the model equations using up to four vertical modes.

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