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2.7 Biphasic Material

Biphasic materials may be used to model deformable porous media. A biphasic material represents a mixture of a porous permeable solid and an interstitial fluid. Each constituent is intrinsically incompressible, but the mixture may change volume as interstitial fluid is exchanged with the pore space of the solid. Biphasic materials require the explicit modeling of fluid that permeates the solid. The biphasic material model is useful to simulate materials that show flow-dependent viscoelastic behavior resulting from the frictional interactions of the fluid and solid. Several biological materials such as cartilage can be described more accurately this way.

Governing Equations

Consider a mixture consisting of a solid constituent and a fluid constituent. Both constituents are considered to be intrinsically incompressible, but the mixture can change volume when fluid enters or leaves the porous solid matrix 12. According to the kinematics of the continuum 3, each constituent \(\alpha\) of a mixture (\(\alpha=s\) for the solid and \(\alpha=w\) for the fluid) has a separate motion \(\boldsymbol{\chi}^{\alpha}\left(\mathbf{X}^{\alpha},t\right)\) which places particles of each mixture constituent, originally located at \(\mathbf{X}^{\alpha}\), in the current configuration \(\mathbf{x}\) according to

\[ \begin{equation} \mathbf{x}=\boldsymbol{\chi}^{\alpha}\left(\mathbf{X}^{\alpha},t\right)\,.\label{eq102} \end{equation} \]

For the purpose of finite element analyses, the motion of the solid matrix, \(\alpha=s\), is of particular interest.

The governing equations that enter into the statement of virtual work are the conservation of linear momentum and the conservation of mass, for the mixture as a whole. Under quasi-static conditions, the conservation of momentum reduces to

\[ \begin{equation} \divg\boldsymbol{\sigma}+\rho\mathbf{b}=\mathbf{0}\,,\label{eq103} \end{equation} \]

where \(\sigma\) is the Cauchy stress for the mixture, \(\rho\) is the mixture density and \(\mathbf{b}\) is the external mixture body force per mass. Since the mixture is porous, this stress may also be written as

\[ \begin{equation} \boldsymbol{\sigma}=-p\mathbf{I}+\boldsymbol{\sigma}^{e}\,,\label{eq104} \end{equation} \]

where \(p\) is the fluid pressure and \(\sigma^{e}\) is the effective or extra stress, resulting from the deformation of the solid matrix. Conservation of mass for the mixture requires that

\[ \begin{equation} \divg\left(\mathbf{v}^{s}+\mathbf{w}\right)=0\,,\label{eq105} \end{equation} \]

where \(\mathbf{v}^{s}=\partial\boldsymbol{\chi}^{s}/\partial t\) is the solid matrix velocity and \(\mathbf{w}\) is the flux of the fluid relative to the solid matrix. Let the solid matrix displacement be denoted by \(\mathbf{u}\), then \(\mathbf{v}^{s}=\mathbf{\dot{u}}\).

To relate the relative fluid flux \(\mathbf{w}\) to the fluid pressure and solid deformation, it is necessary to employ the equation of conservation of linear momentum for the fluid,

\[ \begin{equation} -\varphi^{w}\grad p+\rho^{w}\mathbf{b}^{w}+\mathbf{\hat{p}}_{d}^{w}=\mathbf{0}\,,\label{eq106} \end{equation} \]

where \(\varphi^{w}\) is the solid matrix porosity, \(\rho^{w}=\varphi^{w}\rho_{T}^{w}\) is the apparent fluid density and \(\rho_{T}^{w}\) is the true fluid density, \(\mathbf{b}^{w}\) is the external body force per mass acting on the fluid, and \(\mathbf{\hat{p}}_{d}^{w}\) is the momentum exchange between the solid and fluid constituents, typically representing the frictional interaction between these constituents. This equation neglects the viscous stress of the fluid in comparison to \(\mathbf{\hat{p}}_{d}^{w}\). The most common constitutive relation is \(\mathbf{\hat{p}}_{d}^{w}=-\varphi^{w}\mathbf{k}^{-1}\cdot\mathbf{w}\), where the second order, symmetric tensor \(\mathbf{k}\) is the hydraulic permeability of the mixture. When combined with eq.\eqref{eq106}, it produces

\[ \begin{equation} \mathbf{w}=-\mathbf{k}\cdot\left(\grad p-\rho_{T}^{w}\mathbf{b}^{w}\right)\,,\label{eq107} \end{equation} \]

which is equivalent to Darcy's law. In general, \(\mathbf{k}\) may be a function of the deformation.

  1. Bowen, Ray M.. "Incompressible porous media models by use of the theory of mixtures." Int J Eng Sci, vol. 18, pp. 1129-1148 (1980). ↩

  2. Mow, V.C.; Kuei, S.C.; Lai, W.M.; Armstrong, C.G.. "Biphasic creep and stress relaxation of articular cartilage in compression: Theory and experiments." J. Biomech. Eng., vol. 102, pp. 73-84 (1980). ↩

  3. Truesdell, C.; Toupin, R.. "The classical field theories." Springer, vol. III/1 (1960). ↩