Poroelasticity in the presence of active fluids

TL;DR

The paper addresses designing porous, deformable media infused with magnetorheological fluids whose viscosity and microstructure can be tuned by a magnetic field, producing a threshold behavior for flow and stiffness. It develops a linear poroelastic Biot-based formulation that incorporates a field-dependent fluid phase via a threshold function $f(η)$ and a permeability that scales as $k = k_{geo}/μ(η) (1 - f(η))$. Constitutive relations for the drained solid, the fluid, and coupling terms (Biot coefficient $A$ and Biot modulus $M$) are derived, and a stress correction $\mathbf{T} = (1-φ)^β \mathbb{C}_{mat}:\mathbf{E} + φ^{β} f(η) \mathbb{C}_{F}^{O}:\mathbf{E} - α(1 - f(η))(p - p_0) \mathbf{I}$ accounts for the phase transition. The approach is validated against oedometric and three-point bending benchmarks, showing magnetically tunable stiffness and flow, with good agreement to experiments, and suggests pathways for design of adaptive exosuits and tissue rehabilitation devices.

Abstract

This work presents a model for characterizing porous, deformable media embedded with magnetorheological fluids (MRFs). These active fluids exhibit tunable mechanical and rheological properties that can be controlled through the application of a magnetic field, which induces a phase transition from a liquid to a solid-like state. This transition profoundly affects both stress transmission and fluid flow within the composite, leading to a behaviour governed by a well-defined threshold that depends on the ratio between the pore size and the characteristic size of clusters of magnetic particles, and can be triggered by adjusting the magnetic field intensity. These effects were confirmed through an experimental campaign conducted on a prototype composite obtained by imbibing a selected MRF into commercial sponges. To design and optimize this new class of materials, a linear poroelastic formulation is proposed and validated through comparison with experimental results. The constitutive relationships, i.e. overall elastic constitutive tensor and permeability, of the model are updated from phenomenological observations, exploiting the experimental data obtained for both the pure fluid and the composite material. The findings demonstrate that the proposed simplified formulation is sufficiently robust to predict and optimize the behaviour of porous media containing MRFs. Such materials hold significant promise for a wide range of engineering applications, including adaptive exosuits for human tissue and joint rehabilitation, as well as innovative structural systems.

Figures (8)

• Figure 1: On the left: viscosity measurements of a specific magnetorheological fluid (MRF). The fluid is characterized by: a base silicon oil with a viscosity of 100 cSt; $3\%$wt hydrophilic Fumed FS(HL) as hydrophilic additive; $20\%$wt of carbonyl iron (CI) powder in the form of microparticles with an average diameter of 8 µm. Measurements were performed with a controlled-stress Physica MCR 702 Rheometer (Anton Paar, Graz, Austria), equipped with the magnetorheological device of Physica (MRD 70/1T). A plate-plate geometry with plates of 20 mm diameter and a gap of 0,5 mm was used. A homogeneous magnetic field with different strengths (0, 10, 25, 40, 60, 80, 170, 350, 510, 670, 800 mT), perpendicular to the shear flow direction, was applied, and have then been normalized with respect to a reference value $\mu(0)$ which depends on the shear rate of the test: $\mu(0)=0.86\textup{Pa}\ \textup{s}$, $\mu(0)=1.01\textup{Pa}\ \textup{s}$ and $\mu(0)=3.36\textup{Pa}\ \textup{s}$ for a shear rate of $\dot{\gamma}=1000\ s^{-1}$, $\dot{\gamma}=250\ s^{-1}$ and $\dot{\gamma}=10\ s^{-1}$, respectively. The magnetic field has been normalized with respect to a value, called $\eta_{cr}$ in the following that, as will be explained in the text, emerges from the interaction of the fluid with the microstructure of the solid in which it is embedded. The ad hoc fitting curve in the graph is obtained from the expression of the viscosity in equation (\ref{['eq:permeability']}). On the right: (a) the analyzed MRF upon activation of an external magnetic field builds internal structures made by the metallic particles that (b) assemble along the direction of the magnetic field with shape and length that depend on (c) the micro particle's size and magnetization properties.
• Figure 2: Tensile test carried out on the chosen sponge samples. On the left, average stress vs average strain measured during the test: the dots report the actual experimental measurements, while the continuous red line is obtained through a Neo-Hookean fitting (see Appendix for more details). On the right, an optical microscopy (above) and a scanning electron microscopy (below) displaying the partially open porosity with characteristic void size.
• Figure 3: Mechanical response of the deformable sponge embedded with MRF (10cP, 20$\%$ wt of CI) tested in oedometric conditions using the ad hoc prepared set-up. The magnetic field is turned off at first (red lines), and then turned on (green lines). The stiffness exhibited by the specimen in such conditions corresponds to the so-called oedoetric modulus $E_{oe}=\lambda+2G$ ($\lambda$ and $G$ being the two Lamè constants) of the deformable porous sponge, which are typically referred to in poroelasticity and the dry elastic properties. The magnetic field $H$, when present, is applied by means of a thin cylindrical neodymium magnet placed beneath the oedometer in between the lower walls and the anchoring to the testing machine. As such, the magnetic field is not everywhere parallel to the loading direction but exhibits classical curved paths in space. On the right, optical microscopy shows the variation in particles' direction and alignment when a magnetic field is applied upon the composite sponge-MRF.
• Figure 4: Three different pore channel structures (on the left) to which the same total porosity corresponds. On the right, a qualitative curve representing the pore channel occlusion that occurs at a critical value of magnetization. Increasing magnetization increases the size of the cluster $\delta_{\textup{cluster}}$ formed by the magnetic particles in the pore, until at a certain threshold $\delta_{\textup{cluster}}=d$, flow gets impeded.
• Figure 5: Graphs of the two functions $f$ and $\mu$ for varying viscosity parameter $\eta$. The plots have been obtained by choosing a representative value for $s=20$, $c=0.2/\mu(0)$.