On the quasi-static effective behaviour of poroelastic media containing elastic inclusions

Royer, Pascale; Recho, Pierre; Verdier, Claude

doi:10.1016/j.mechrescom.2019.02.004

Cited by 10 publications

(27 citation statements)

References 18 publications

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“…[43,44] in the context of elastic composites). In this case, as the coefficients are to be computed at the porescale, our results could be exploited as a starting point to model the interaction between a poroelastic matrix and another fluid or elastic compartment, as done in the recent works [14,41,45,52] for vascularized and fibre/inclusion reinforced poroelastic materials, respectively.…”

Section: Discussionmentioning

confidence: 99%

“…In this section, we introduce the two-scale asymptotic homogenization technique which is used to derive a macroscale model for Eqs. (38)(39)(40)(41)(42)(43)(44)(45)(46)(47). We first assume that the microscale (where the pores and individual inclusions/fibres are clearly resolved), denoted by d, is small compared to average size of the domain L, i.e.…”

Section: The Asymptotic Homogenization Techniquementioning

confidence: 99%

“…We are now interested in obtaining a closed system of PDEs in terms of the leading (zeroth)-order pressure, velocity, and displacement fields. This is done by applying relationships (57) and (58) to system (38)(39)(40)(41)(42)(43)(44)(45)(46)(47) and constitutive equations (48)(49)(50), and in turn to relationships (51)(52)(53). This way, by equating coefficients of the same power of , we can obtain various sets of differential conditions which can be combined to obtain a system of PDEs which holds on the macroscale x ∈ Ω H .…”

Section: Remark 1 (Microscale Periodicity)mentioning

confidence: 99%

“…We apply the asymptotic homogenization assumptions (57) and (58) to Eqs. (38)(39)(40)(41)(42)(43)(44)(45)(46)(47)(48)(49)(50)(51)(52)(53) to obtain, accounting for periodicity, the following multiscale system of PDEs We have one subphase shown in blue that is in contact with the porous, solid, elastic matrix shown in red, and the fluid flowing in the pores is shown in white. We also highlight the interfaces Γ I which is shown in green between the subphase and the fluid, Γ II shown in black between the matrix and the fluid and Γ III shown in grey between the subphase and the matrix (colour figure online)…”

Section: Derivation Of the Macroscale Modelmentioning

confidence: 99%

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Effective balance equations for poroelastic composites

Miller

Penta

2020

Continuum Mech. Thermodyn.

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We derive the quasi-static governing equations for the macroscale behaviour of a linear elastic porous composite comprising a matrix interacting with inclusions and/or fibres, and an incompressible Newtonian fluid flowing in the pores. We assume that the size of the pores (the microscale) is comparable with the distance between adjacent subphases and is much smaller than the size of the whole domain (the macroscale). We then decouple spatial scales embracing the asymptotic (periodic) homogenization technique to derive the new macroscale model by upscaling the fluid-structure interaction problem between the elastic constituents and the fluid phase. The resulting system of partial differential equations is of poroelastic type and encodes the properties of the microstructure in the coefficients of the model, which are to be computed by solving appropriate cell problems which reflect the complexity of the given microstructure. The model reduces to the limit case of simple composites when there are no pores, and standard Biot's poroelasticity whenever only the matrix-fluid interaction is considered. We further prove rigorous properties of the coefficients, namely (a) major and minor symmetries of the effective elasticity tensor, (b) positive definiteness of the resulting Biot's modulus, and (c) analytical identities which allow us to define an effective Biot's coefficient. This model is applicable when the interactions between multiple solid phases occur at the porescale, as in the case of various systems such as biological aggregates, constructs, bone, tendons, as well as rocks and soil.

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Section: Discussionmentioning

confidence: 99%

Section: The Asymptotic Homogenization Techniquementioning

confidence: 99%

Section: Remark 1 (Microscale Periodicity)mentioning

confidence: 99%

Section: Derivation Of the Macroscale Modelmentioning

confidence: 99%

See 2 more Smart Citations

Effective balance equations for poroelastic composites

Miller

Penta

2020

Continuum Mech. Thermodyn.

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“…Previous studies have coupled fluid and solid mechanical systems to infer not only the impact that a solid inclusion would have on the fluid flow, but also the mechanical deformations that fluid flow would induce on the solid inclusion (Royer et al. 2019; Chen et al. 2019).…”

Section: Discussionmentioning

confidence: 99%

Multiple Scale Homogenisation of Nutrient Movement and Crop Growth in Partially Saturated Soil

et al. 2019

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In this paper, we use multiple scale homogenisation to derive a set of averaged macroscale equations that describe the movement of nutrients in partially saturated soil that contains growing potato tubers. The soil is modelled as a poroelastic material, which is deformed by the growth of the tubers, where the growth of each tuber is dependent on the uptake of nutrients via a sink term within the soil representing root nutrient uptake. Special attention is paid to the reduction in void space, resulting change in local water content and the impact on nutrient diffusion within the soil as the tubers increase in size. To validate the multiple scale homogenisation procedure, we compare the system of homogenised equations to the original set of equations and find that the solutions between the two models differ by . However, we find that the computation time between the two sets of equations differs by several orders of magnitude. This is due to the combined effects of the complex three-dimensional geometry and the implementation of a moving boundary condition to capture tuber growth.

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Double poroelasticity derived from the microstructure

Miller

Penta

2021

Acta Mech

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We derive the balance equations for a double poroelastic material which comprises a matrix with embedded subphases. We assume that the distance between the subphases (the local scale) is much smaller than the size of the domain (the global scale). We assume that at the local scale both the matrix and subphases can be described by Biot’s anisotropic, heterogeneous, compressible poroelasticity (i.e. the porescale is already smoothed out). We then decompose the spatial variations by means of the two-scale homogenization method to upscale the interaction between the poroelastic phases at the local scale. This way, we derive the novel global scale model which is formally of poroelastic-type. The global scale coefficients account for the complexity of the given microstructure and heterogeneities. These effective poroelastic moduli are to be computed by solving appropriate differential periodic cell problems. The model coefficients possess properties that, once proved, allow us to determine that the model is both formally and substantially of poroelastic-type. The properties we prove are a) the existence of a tensor which plays the role of the classical Biot’s tensor of coefficients via a suitable analytical identity and b) the global scale scalar coefficient $$\bar{\mathcal {M}}$$ M ¯ is positive which then qualifies as the global Biot’s modulus for the double poroelastic material.

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On the quasi-static effective behaviour of poroelastic media containing elastic inclusions

Cited by 10 publications

References 18 publications

Effective balance equations for poroelastic composites

Effective balance equations for poroelastic composites

Multiple Scale Homogenisation of Nutrient Movement and Crop Growth in Partially Saturated Soil

Double poroelasticity derived from the microstructure

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