Magnetorheology in saturating fields

Understanding magnetorheology in saturating fields is crucial for success in high torque applications. In this paper we use numerical computations, analytical developments, and experimental data (using a double-gap magnetocell) to study the saturation behavior of model magnetorheological fluids for different particle loadings. Numerical calculations demonstrate a nonlinear dependence of both shear and normal stresses with particle concentration in contrast with analytical predictions. These predictions are in very good agreement with numerical calculations at low volume fractions when the interchain interactions can be safely neglected. Numerical calculations for the (yield) shear stress overestimate experimental data for small and medium concentrations. However, a reasonably good qualitative agreement is found for the larger particle loadings. Normal stresses are extraordinarily sensitive to the particular microstructure; experiments suggest sample dilatation in good agreement with simulations in lattices with a body-centered basis.

Figure 1

Schematics of the particle arrangement. Simulated lattice under a strain $\gamma =tan\theta$ and an external applied magnetic field ${\vec{H}}_{\mathrm{ext}}$. Only the particles sheet at $x=0$ is plotted. Dashed lines determine the origin of the coordinate system. The upper (source) region of the lattice is represented with dotted circles while the bottom (field) region is represented with empty circles. In this sketch, the vector $\vec{R}$ joins the (0,3,2) source particle with the (0,0,–3) field particle. Due to the lattice symmetry, only the force acting over the red field particles needs to be computed. The sign criteria are represented by the black trapezium: blue (red) arrows represent positive field-driven shear (normal) stresses acting on the microstructure.

Figure 2

(a) According to Lorentz's model, a given field particle (black circle) will feel the force due to all source particles inside the cutoff sphere (dotted particles) plus a contribution from the continuous region (gray region) based on those source particles outside the cutoff sphere. As the continuous region is uniformly magnetized, its contribution will depend on its surface density distributed along its boundary surface (dotted line). This surface is axisymmetric and can be split in a flat plane with a hole of radius ${\rho }_0$ [(b)-red dotted line] and the inner surface of a spherical cap of radius $r_{\mathrm{cut}}$ and maximum polar angle ${\theta }_f$ [(c)-red dotted line].

Figure 3

Shear stress $\tau$ as a function of the applied shear strain $\gamma$ for different volume fractions $\varphi$. Numerical calculations are plotted with lines while results using FEM simulations for saturated dipoles as proposed in Ref. [24] are plotted with symbols. (a) Simple basis (see Sec. 2a). (b) BC basis (see Appendix pp1).

Figure 4

Yield stress ${\tau }_0$ dependence on particle volume fraction $\varphi$ according to simulations for simple (black squares) and BC (red circles) bases, analytical calculation for low concentrations (black line) and experiments (triangles). Blue up triangles correspond to experiments from this paper. Green down triangles correspond to experiments by Ref. [7] for a magnetic field strength of 979 kA/m. (a) lin-lin representation. (b) log-log representation.

Figure 5

Yield strain ${\gamma }_0$ dependence on particle volume fraction $\varphi$ according to simulations in simple (black squares) and BC (red circles) bases, analytic value for low concentrations (black dashed line) and approximation at high concentrations (magenta dotted line) only valid for simple basis. (a) lin-lin representation. (b) log-log representation.

Figure 6

Normal stress ${\sigma }_{zz}$ as a function of the applied shear strain $\gamma$ for different volume fractions $\varphi$. Results from the numerical calculations are plotted with lines while the results from FEM simulations proposed in Ref. [32] are plotted with symbols. Vertical arrows correspond to the yield strain ${\gamma }_0$ as obtained from Fig. 3. (a) Simple basis. (b) BC basis.

Figure 7

(a) Normal stresses at zero strain in the field and shear direction, ${\sigma }_{zz}\left(\gamma =0\right)$ and ${\sigma }_{yy}\left(\gamma =0\right)$, versus particle volume fraction $\varphi$ according to numerical simulations in simple basis (black squares and green triangles) and BC basis (red circles and blue diamonds). Simulations without the continuous region correction are plotted with open symbols. Analytical calculations for normal stress in the field direction at low concentrations are plotted with a black line. (b) and (c) log-log representation for normal stresses at zero strain in the field and shear direction respectively (original negative values are plotted with lighter colors).

Figure 8

Normal stress difference at zero strain $\left(N_1-N_2\right)/2$ versus particle volume fraction $\varphi$ according to numerical simulations in simple (squares) and BC (circles) bases. Also plotted with a line is the analytical approximation for low concentrations [Eq. (24)]. The maximum normal stress difference measured while straining the sample in start-up tests is plotted with triangles and the related value, once corrected using the Maxwell stress jump $\left[{\sigma }_M\right]$, with stars. (a) lin-lin representation. (b) log-log representation (negative values are plotted with lighter colors).

Figure 9

(a) Source and field regions used to compute the normal stress in the shear direction. A field particle (black circle) will feel the force due to all source particles inside the cutoff sphere (dotted circles) plus a contribution from the continuous region (gray region). (b) The continuous region contribution is due to only the cap surface (dotted red line).