### Abstract

An equivalent new expression of the triphasic mechano-electrochemical theory [9] is presented and a mixed finite element formulation is developed using the standard Galerkin weighted residual method. Solid displacement u^{s}, modified electrochemical/chemical potentials ε^{w}, ε^{+} and ε^{-} (with dimensions of concentration) for water, cation and anion are chosen as the four primary degrees of freedom (DOFs) and are independently interpolated. The modified Newton-Raphson iterative procedure is employed to handle the non-linear terms. The resulting first-order Ordinary Differential Equations (ODEs) with respect to time are solved using the implicit Euler backward scheme which is unconditionally stable. One-dimensional (1-D) linear isoparametric element is developed. The final algebraic equations form a non-symmetric but sparse matrix system. With the current choice of primary DOFs, the formulation has the advantage of small amount of storage, and the jump conditions between elements and across the interface boundary are satisfied automatically. The finite element formulation has been used to investigate a 1-D triphasic stress relaxation problem in the confined compression configuration and a 1-D triphasic free swelling problem. The formulation accuracy and convergence for 1-D cases are examined with independent finite difference methods. The FEM results are in excellent agreement with those obtained from the other methods.

Original language | English |
---|---|

Pages (from-to) | 1375-1402 |

Number of pages | 28 |

Journal | International Journal for Numerical Methods in Engineering |

Volume | 45 |

Issue number | 10 |

State | Published - Aug 10 1999 |

Externally published | Yes |

### Fingerprint

### Keywords

- Finite element method
- Soft tissue
- Triphasic mechano-electrochemical theory

### ASJC Scopus subject areas

- Engineering (miscellaneous)
- Computational Mechanics
- Applied Mathematics

### Cite this

*International Journal for Numerical Methods in Engineering*,

*45*(10), 1375-1402.

**A mixed finite element formulation of triphasic mechano-electrochemical theory for charged, hydrated biological soft tissues.** / Sun, D. N.; Gu, Weiyong; Guo, X. E.; Lai, W. M.; Mow, V. C.

Research output: Contribution to journal › Article

*International Journal for Numerical Methods in Engineering*, vol. 45, no. 10, pp. 1375-1402.

}

TY - JOUR

T1 - A mixed finite element formulation of triphasic mechano-electrochemical theory for charged, hydrated biological soft tissues

AU - Sun, D. N.

AU - Gu, Weiyong

AU - Guo, X. E.

AU - Lai, W. M.

AU - Mow, V. C.

PY - 1999/8/10

Y1 - 1999/8/10

N2 - An equivalent new expression of the triphasic mechano-electrochemical theory [9] is presented and a mixed finite element formulation is developed using the standard Galerkin weighted residual method. Solid displacement us, modified electrochemical/chemical potentials εw, ε+ and ε- (with dimensions of concentration) for water, cation and anion are chosen as the four primary degrees of freedom (DOFs) and are independently interpolated. The modified Newton-Raphson iterative procedure is employed to handle the non-linear terms. The resulting first-order Ordinary Differential Equations (ODEs) with respect to time are solved using the implicit Euler backward scheme which is unconditionally stable. One-dimensional (1-D) linear isoparametric element is developed. The final algebraic equations form a non-symmetric but sparse matrix system. With the current choice of primary DOFs, the formulation has the advantage of small amount of storage, and the jump conditions between elements and across the interface boundary are satisfied automatically. The finite element formulation has been used to investigate a 1-D triphasic stress relaxation problem in the confined compression configuration and a 1-D triphasic free swelling problem. The formulation accuracy and convergence for 1-D cases are examined with independent finite difference methods. The FEM results are in excellent agreement with those obtained from the other methods.

AB - An equivalent new expression of the triphasic mechano-electrochemical theory [9] is presented and a mixed finite element formulation is developed using the standard Galerkin weighted residual method. Solid displacement us, modified electrochemical/chemical potentials εw, ε+ and ε- (with dimensions of concentration) for water, cation and anion are chosen as the four primary degrees of freedom (DOFs) and are independently interpolated. The modified Newton-Raphson iterative procedure is employed to handle the non-linear terms. The resulting first-order Ordinary Differential Equations (ODEs) with respect to time are solved using the implicit Euler backward scheme which is unconditionally stable. One-dimensional (1-D) linear isoparametric element is developed. The final algebraic equations form a non-symmetric but sparse matrix system. With the current choice of primary DOFs, the formulation has the advantage of small amount of storage, and the jump conditions between elements and across the interface boundary are satisfied automatically. The finite element formulation has been used to investigate a 1-D triphasic stress relaxation problem in the confined compression configuration and a 1-D triphasic free swelling problem. The formulation accuracy and convergence for 1-D cases are examined with independent finite difference methods. The FEM results are in excellent agreement with those obtained from the other methods.

KW - Finite element method

KW - Soft tissue

KW - Triphasic mechano-electrochemical theory

UR - http://www.scopus.com/inward/record.url?scp=0000779366&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000779366&partnerID=8YFLogxK

M3 - Article

VL - 45

SP - 1375

EP - 1402

JO - International Journal for Numerical Methods in Engineering

JF - International Journal for Numerical Methods in Engineering

SN - 0029-5981

IS - 10

ER -