Abstract
A successful prediction of the response of poroelastic material to external forces depends critically on the use of appropriate constitutive relation for the material. The most commonly used stress–strain relation that captures the behavior of poroelastic materials saturated with a liquid is that proposed by Biot (J Appl Phys 12:155–164, 1941). It is a linear theory akin to the generalized Hooke’s law containing material constants such as the effective bulk and shear modulus of the porous material. However, the effective elastic coefficients are not known a priori and need to be determined either via separate experiments or by fitting predictions with measurements. The main cause for this drawback is that Biot’s theory does not account for the microstructural details of the system. This limitation in Biot’s model can be overcome by utilizing the constitutive relation proposed by Russel et al. (Langmuir 5(24):1721–1730, 2008) for the case of colloidal packings. We show that in the linear limit, the constitutive relation proposed by Russel and coworkers is equivalent to that of Biot. The elastic coefficients obtained from such a linearization are related to the micro-structural details of the packing such as the particle modulus, the packing concentration and the nature of packing, thereby enabling a more effective utilization of Biot’s model for problems in the linear limit. The derivation ignores surface forces between the particles, which makes the results also applicable to particles whose sizes are beyond the colloidal range. We compare the predictions of Biot’s model to those of the linearized model of Russel and coworker’s for two different one-dimensional model problems: fluid outflow driven by an applied mechanical load, also termed as the consolidation problem, and wave propagation in a saturated colloidal packing.
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Acknowledgements
A.B. thanks IIT Bombay for financial support. The project was funded in part by Department of Science and Technology, India.
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Appendices
Appendix
A: List of Variables
This section lists all variables and their definitions used in the main text.
Variables | Definitions |
|---|---|
\(\alpha\) | Biot–Willis coefficient |
\(\varDelta \varvec{\sigma }\) | Incremental stress of bulk material |
\(\varDelta \varvec{\sigma }^f\) | Incremental stress in the fluid phase |
\(\varvec{\epsilon ^p}=\varvec{\epsilon }\) | Strain in the solid/particle network/macroscopic strain |
\(\epsilon _o\) | Initial isotropic compressive strain |
\(\varvec{\epsilon ^{'}}\) | Perturbed strain |
\(\eta\) | Viscosity of liquid |
\(\varvec{\dot{\gamma }}\) | Shear rate |
\(\gamma _{pl}\) | Surface energy of the particle–liquid interface |
\(\kappa _p\) | Permeability of the packing |
\(\varvec{\delta }\) | Second-order identity tensor |
\(\lambda\) | Lame’s first constant |
\(\mu\) | Shear modulus |
\(\mu _{HS}^{\pm }\) | Hashin–Shtrikman’s upper and lower bounds for effective shear modulus |
\(\mu _{V}\) | Voigt bound for effective shear modulus bounds |
\(\mu _{R}\) | Reuss bound for effective shear modulus bounds |
\(\mu _{i}\) | Shear modulus of the \(i{\rm th}\) phase |
\(\nu _p\) | Poisson’s ratio of particle |
\(\nu\) | Poisson’s ratio of packing |
\(\omega\) | Angular frequency |
\(\phi\) | Solid volume fraction/particle phase volume fraction |
\(\phi ^o\) | Volume fraction in the prestress state |
\(\phi ^{'}\) | Perturbed/incremental volume fraction |
\(\rho _p\) | Mass density of particle phase |
\(\rho _f\) | Mass density of fluid phase |
\(\rho\) | Bulk density |
\(\varvec{\sigma }\) | Total effective stress/macroscopic stress |
\(\varvec{\sigma ^p}\) | Particle network stress |
\(\varvec{\sigma ^f}\) | Fluid stress |
\(\varvec{\sigma ^o}\) | Stress in the prestress/base state |
\(\varvec{\sigma ^{'}}\) | Perturbed/incremental stress |
\(\varvec{\tau _d}\) | Shear stress in fluid phase |
\(\varvec{\tau _v}\) | Viscoelastic stress in the particle |
\(\tau\) | Characteristic time scale in soil consolidation and one-dimensional wave problems |
\(\tau _r\) | Relaxation time in Russel’s model |
\(\zeta\) | Change in fluid or liquid content |
A | \(=\lambda\), obtained by linearization in the vicinity of prestressed state by Biot |
a | Particle radius |
\(a_R\) | Inverse of longitudinal wave modulus from Russel’s model |
\(a_B\) | Inverse of longitudinal wave modulus from Biot’s model |
\(\varvec{F}\) | Force acting on the particle pair |
\(f_{i}\) | Volume fraction of \(i{\rm th}\) phase |
G | Particle shear modulus |
\(\bar{G}\) | Effective elastic modulus |
H | Initial thickness |
h | Height of the soil consolidation column |
\(\varvec{I}\) | Identity tensor |
\(K_R\) | Diffusion coefficient of consolidation from Russel’s model |
\(K_B\) | Diffusion coefficient of consolidation from Biot’s model |
\(K_1\) | Bulk modulus of material termed 1 |
L | Thickness of saturated colloidal system in 1D wave problem |
\(L_c\) | Characteristic length scale for macroscopic dimensions |
M | Biot modulus |
m | Effective mass in Biot’s model |
N | Number of contacting neighbor |
\(N_b\) | \(=\mu\), obtained by linearization in the vicinity of prestressed state by Biot |
\(\varvec{n}\) | Unit normal vector on the surface of the particle |
P | Fluid pressure in the pores/pressure in the solvent phase packing |
\(P^o\) | Fluid pressure in the prestress state |
\(P^{'}\) | Fluid pressure in the perturbed/incremental state |
\(p_o\) | Normal load in soil consolidation problem |
\(Q, Q^{'}, R\) | Elastic constants obtained by linearization in the vicinity of prestressed state by Biot |
\(\varvec{S}\) | Surface dipole |
\(U_o\) | Sinusoidal vertical displacement in 1D wave problem |
\(\varvec{u}^p\) | Displacement field in the solid/particle network |
\(\varvec{u}^f\) | Displacement field in the fluid |
V | Representative volume in Russel’s model |
\(\varvec{V}\) | Fluid velocity |
\(\varvec{V}^p\) | Velocity of particle phase |
\(\varvec{V}^f\) | Velocity of fluid phase |
B: Soil Consolidation
C: Linearization of Russel's Model
The radial vector on the surface of the particle is written in Cartesian coordinates,
and then substituted in the integral expressions to obtain,
D: Mass Balance
The mass balance equations for the particle and the fluid phase are given by (see Fig.10 for shell balance):
-
1.
Particle phase
$$\begin{aligned}&({\text{Particle mass in}})-({\text{Particle mass out}})\nonumber \\&\quad = ({\text{Accumulation of particle mass inside the volume V}} )\; ;\nonumber \\&\bigg (\rho _p\phi V_{1}^{p}\varDelta x_2 \varDelta x_3|_{x_1}-\rho _p\phi V_{1}^{p}\varDelta x_2 \varDelta x_3|_{x_1+\varDelta x_1}\bigg )\\&+\bigg (\rho _p\phi V_{2}^{p}\varDelta x_1 \varDelta x_3|_{x_2}-\rho _p\phi V_{2}^{p}\varDelta x_1 \varDelta x_3|_{x_2+\varDelta x_2}\bigg )\\&\quad + \bigg (\rho _p\phi V_{3}^{p}\varDelta x_2 \varDelta x_1|_{x_3}-\rho _p\phi V_{3}^{p}\varDelta x_2 \varDelta x_1|_{x_3+\varDelta x_3}\bigg ) = \frac{\partial (\rho _p \phi )}{\partial t}\varDelta x_1 \varDelta x_2 \varDelta x_3 \end{aligned}$$(62)On dividing by \(\varDelta x_1 \varDelta x_2 \varDelta x_3\) and taking the limit as \(\varDelta x_i\rightarrow 0\) gives,
$$\begin{aligned} \frac{\partial (\rho _p\phi )}{\partial t}= & {} -\bigg (\frac{\partial (\rho _p\phi V_{1}^{p}) }{\partial x_{1}}+\frac{\partial (\rho _p\phi V_{2}^{p}) }{\partial x_{2}}+\frac{\partial (\rho _p\phi V_{3}^{p}) }{\partial x_{3}}\bigg ) \end{aligned}$$(63) -
2.
Fluid phase
$$\begin{aligned}&\big ({\text{Fluid mass in}}\big )-({\text{Fluid mass out}})=\\&\quad \big ({\text{Accumulation of fluid mass inside the volume }}V\big )\\&\quad \bigg (\rho _f(1-\phi ) V_{1}^{f}\varDelta x_2 \varDelta x_3|_{x_1}-\rho _f(1-\phi )V_{1}^{f}\varDelta x_2 \varDelta x_3|_{x_1+\varDelta x_1}\bigg )\\&\quad +\bigg (\rho _f(1-\phi ) V_{2}^{f}\varDelta x_1 \varDelta x_3|_{x_2}-\rho _f(1-\phi ) V_{2}^{f}\varDelta x_1 \varDelta x_3|_{x_2+\varDelta x_2}\bigg )\\&\quad +\bigg (\rho _f(1-\phi )V_{3}^{f}\varDelta x_2 \varDelta x_1|_{x_3}-\rho _f(1-\phi )V_{3}^{f}\varDelta x_2 \varDelta x_1|_{x_3+\varDelta x_3}\bigg )\\&\quad =\frac{\partial (\rho _f (1-\phi ))}{\partial t}\varDelta x_1 \varDelta x_2 \varDelta x_3\; ; \end{aligned}$$On dividing by \(\varDelta x_1 \varDelta x_2 \varDelta x_3\) and taking the limit as \(\varDelta x_i\rightarrow 0\) gives,
$$\begin{aligned}&\frac{\partial (\rho _f(1-\phi ))}{\partial t}=\nonumber \\&\quad -\bigg (\frac{\partial (\rho _f(1-\phi ) V_{1}^{f}) }{\partial x_{1}}+\frac{\partial (\rho _f(1-\phi ) V_{2}^{f}) }{\partial x_{2}}+\frac{\partial (\rho _f(1-\phi ) V_{3}^{f}) }{\partial x_{3}}\bigg ) \end{aligned}$$(64)
Mass balance a colloidal packing saturated with liquid
E: Momentum Balance
Next, the shell balance for the momentum is considered where both body force and convective acceleration are ignored (Fig. 11),
Momentum balance in the saturated colloidal system on particle phase
-
1.
Particle phase (along \(x_1\)):
$$\begin{aligned}&\bigg (\mathrm{Particle} \; \mathrm{momentum}\; \mathrm{coming}\; \mathrm{in}\bigg ) - \bigg (\mathrm{Particle}\; \mathrm{momentum}\; \mathrm{going}\; \mathrm{out}\bigg ) \\&\quad +\bigg (\mathrm{Net}\; \mathrm{solvent}\; \mathrm{pressure}\; \mathrm{acting}\; \mathrm{on}\; \mathrm{the}\; \mathrm{particle}\; \mathrm{phase}\bigg ) \\&\quad =\bigg (\mathrm{Acceleration}\; \mathrm{in}\; \mathrm{the} \; \mathrm{particle}\; \mathrm{phase}\bigg )\; ;\\&\quad \bigg (-\phi \sigma _{11}^{p}\varDelta x_2\varDelta x_3|_{x_1}+\big (\phi \sigma _{11}^{p}+\frac{\partial \phi \sigma _{11}^{p}}{\partial x_1}\varDelta x_1\big )\varDelta x_2\varDelta x_3|_{x_1+\varDelta x_1}\bigg )\\&\quad +\bigg (-\phi \sigma _{21}^{p}\varDelta x_1\varDelta x_3|_{x_2}+\big (\phi \sigma _{21}^{p}+\frac{\partial \phi \sigma _{21}^{p}}{\partial x_2}\varDelta x_2\big )\varDelta x_1\varDelta x_3|_{x_2+\varDelta x_2}\bigg )\\&\quad +\bigg (-\phi \sigma _{31}^{p}\varDelta x_2\varDelta x_1|_{x_3}+\big (\phi \sigma _{31}^{p}+\frac{\partial \phi \sigma _{31}^{p}}{\partial x_3}\varDelta x_3\big )\varDelta x_2\varDelta x_1|_{x_3+\varDelta x_3}\bigg )\\&\quad +\bigg ((1-\phi )P\varDelta x_2\varDelta x_3|_{x_1}-\big ((1-\phi )P+\frac{\partial (1-\phi )P}{\partial x_1}\varDelta x_1\big )\varDelta x_2\varDelta x_3|_{x_1+\varDelta x_1}\bigg )\\&\quad =\frac{\partial }{\partial t}(\phi \rho _p V_{1}^{p})\varDelta x_1\varDelta x_2\varDelta x_3 \end{aligned}$$As before, the above equation is divided by \(\varDelta x_1 \varDelta x_2 \varDelta x_3\) and the limit \(\varDelta x_i\rightarrow 0\) is taken to obtain the differential form of the equation. The Darcy’s law,
$$\begin{aligned} (1-\phi )(V_{i}^f-V_{i}^p)=-\frac{\kappa _p}{\eta }\frac{\partial (1-\phi )P}{\partial x_i}\frac{1}{1-\phi }\; \end{aligned}$$is substituted in the momentum balance equation, to obtain,
$$\begin{aligned} \frac{\partial \phi \sigma _{11}^{p}}{\partial x_1}+\frac{\partial \phi \sigma _{21}^{p}}{\partial x_2}+\frac{\partial \phi \sigma _{31}^{p}}{\partial x_3}+\frac{\eta }{\kappa _p}(1-\phi )^2(V_{1}^f-V_{1}^p)=\frac{\partial }{\partial t}(\phi \rho _p V_{1}^{p}) \end{aligned}$$(65)Likewise, along \(x_2\) and \(x_3\) momentum balance is
$$\begin{aligned}&\frac{\partial \phi \sigma _{12}^{p}}{\partial x_1}+\frac{\partial \phi \sigma _{22}^{p}}{\partial x_2}+\frac{\partial \phi \sigma _{32}^{p}}{\partial x_3}+\frac{\eta }{\kappa _p}(1-\phi )^2(V_{2}^f-V_{2}^p)=\frac{\partial }{\partial t}(\phi \rho _p V_{2}^{p}) \end{aligned}$$(66)$$\begin{aligned}&\frac{\partial \phi \sigma _{13}^{p}}{\partial x_1}+\frac{\partial \phi \sigma _{23}^{p}}{\partial x_2}+\frac{\partial \phi \sigma _{33}^{p}}{\partial x_3}+\frac{\eta }{\kappa _p}(1-\phi )^2(V_{3}^f-V_{3}^p)=\frac{\partial }{\partial t}(\phi \rho _p V_{3}^{p}) \end{aligned}$$(67) -
2.
Fluid phase (along \(x_1\)):
$$\begin{aligned}&\bigg (\mathrm{Fluid}\; \mathrm{momentum} \;\mathrm{coming}\; \mathrm{in}\bigg ) -\bigg (\mathrm{fluid}\; \mathrm{momentum} \;\mathrm{going} \;\mathrm{out}\bigg ) \\&\quad - \bigg (\mathrm{Net} \;\mathrm{solvent} \mathrm{pressure}\; \mathrm{acting}\; \mathrm{on} \;\mathrm{the} \;\mathrm{particle}\; \mathrm{phase}\bigg )\\&\quad = \bigg (\mathrm{Acceleration}\; \mathrm{in}\; \mathrm{the}\; \mathrm{fluid}\; \mathrm{phase}\bigg )\; ;\\&\quad \bigg (-(1-\phi )\sigma _{11}^{f}\varDelta x_2\varDelta x_3|_{x_1}+\big ((1-\phi )\sigma _{11}^{f}+\frac{\partial (1-\phi ) \sigma _{11}^{f}}{\partial x_1}\varDelta x_1\big )\varDelta x_2\varDelta x_3|_{x_1+\varDelta x_1}\bigg )\\&\quad + \bigg (-(1-\phi )\sigma _{21}^{f}\varDelta x_1\varDelta x_3|_{x_2}+\big ((1-\phi )\sigma _{21}^{f}+\frac{\partial (1-\phi ) \sigma _{21}^{f}}{\partial x_2}\varDelta x_2\big )\varDelta x_1\varDelta x_3|_{x_2+\varDelta x_2}\bigg )\\&\quad + \bigg (-(1-\phi )\sigma _{31}^{f}\varDelta x_2\varDelta x_1|_{x_3}+\big ((1-\phi )\sigma _{31}^{f}+\frac{\partial (1-\phi )\sigma _{31}^{f}}{\partial x_3}\varDelta x_3\big )\varDelta x_2\varDelta x_1|_{x_3+\varDelta x_3}\bigg )\\&\quad - \bigg ((1-\phi )P\varDelta x_2\varDelta x_3|_{x_1}-\big ((1-\phi )P+\frac{\partial (1-\phi )P}{\partial x_1}\varDelta x_1\big )\varDelta x_2\varDelta x_3|_{x_1+\varDelta x_1}\bigg )\\&\quad =\frac{\partial }{\partial t}((1-\phi ) \rho _f V_{1}^{f})\varDelta x_1\varDelta x_2\varDelta x_3 \end{aligned}$$As before, the above equation is divided by \(\varDelta x_1 \varDelta x_2 \varDelta x_3\) and the limit \(\varDelta x_i\rightarrow 0\) is taken to obtain the differential form of the equation. The Darcy’s law is substituted in the above momentum balance equation, to obtain,
$$\begin{aligned}&\frac{\partial (1-\phi )\sigma _{11}^{f}}{\partial x_1}+\frac{\partial (1-\phi )\sigma _{21}^{f}}{\partial x_2}+\frac{\partial (1-\phi )\sigma _{31}^{f}}{\partial x_3}-\frac{\eta }{\kappa _p}(1-\phi )^2(V_{1}^f-V_{1}^p)\nonumber \\&\quad =\frac{\partial }{\partial t}((1-\phi ) \rho _f V_{1}^{f}) \end{aligned}$$(68)Doing same analysis along \(x_2\) and \(x_3\) direction
$$\begin{aligned}&\frac{\partial (1-\phi )\sigma _{12}^{f}}{\partial x_1}+\frac{\partial (1-\phi )\sigma _{22}^{f}}{\partial x_2}+\frac{\partial (1-\phi )\sigma _{32}^{f}}{\partial x_3}-\frac{\eta }{\kappa _p}(1-\phi )^2(V_{2}^f-V_{2}^p)\nonumber \\&\quad =\frac{\partial }{\partial t}((1-\phi ) \rho _f V_{2}^{f}) \end{aligned}$$(69)$$\begin{aligned}&\frac{\partial (1-\phi )\sigma _{13}^{f}}{\partial x_1}+\frac{\partial (1-\phi )\sigma _{23}^{f}}{\partial x_2}+\frac{\partial (1-\phi )\sigma _{33}^{f}}{\partial x_3}-\frac{\eta }{\kappa _p}(1-\phi )^2(V_{3}^f-V_{3}^p)\nonumber \\&\quad =\frac{\partial }{\partial t}((1-\phi ) \rho _f V_{3}^{f}) \end{aligned}$$(70)
F: One-Dimensional Wave Propagation Governing Field Equations in Saturated Colloidal Packing
To derive the governing equation for wave propagation, we substitute \(\varvec{w}^{'}=(1-\phi ^{o})(\varvec{u}^{f'}-\varvec{u}^{p'})\) in the overall momentum balance Eq. (43) and fluid momentum balance Eq. (44) to obtain,
and,
respectively. Using equations (24) and (25), the incremental volume fraction can be related to the dilatation term for the particle phase,
Substituting the above relation in the total mass balance (39) and particle balance (37), and combining the two gives,
On integrating the above equation with respect to time relates the variation of the fluid content to the dilatation term for the fluid phase and volume of injected fluid (Q),
In our analysis \(Q=0\) since no external source supplies fluid to the packing. Next, we substitute \(\zeta\) in terms of the dilatation term for the particle phase (from (25)) into the above equation,
Taking divergence of equations (71) and (72) and replacing \(\varvec{\nabla }\cdot {\varvec{w}}^{'}\) with \(\varvec{\nabla }\cdot {\varvec{u}}^{p'}\) from (76) and rearranging, reduces the number of independent variables from 7 to 4 in the two equations,
and,
The above exercise results in an arbitrary constant of integration in both equations, which is neglected without loss of generality. The above two equations can be further combined into one by substituting \(\varvec{\nabla }P^{'}\) from (78) into (77), resulting in the final form of governing equation for wave propagation,
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Badar, A., Tirumkudulu, M.S. Mechanics of Saturated Colloidal Packings: A Comparison of Two Models. Transp Porous Med 135, 457–486 (2020). https://doi.org/10.1007/s11242-020-01483-0
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DOI: https://doi.org/10.1007/s11242-020-01483-0