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Research Papers

Mixed Convolved Action Variational Methods for Poroelasticity

[+] Author and Article Information
Bradley T. Darrall

Mechanical and Aerospace Engineering,
University at Buffalo,
The State University of New York,
Buffalo, NY 14260

Gary F. Dargush

Mechanical and Aerospace Engineering,
University at Buffalo,
The State University of New York,
Buffalo, NY 14260
e-mail: gdargush@buffalo.edu

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received April 16, 2016; final manuscript received June 1, 2016; published online July 13, 2016. Assoc. Editor: Harold S. Park.

J. Appl. Mech 83(9), 091011 (Jul 13, 2016) (12 pages) Paper No: JAM-16-1190; doi: 10.1115/1.4033753 History: Received April 16, 2016; Revised June 01, 2016

Although Lagrangian and Hamiltonian analytical mechanics represent perhaps the most remarkable expressions of the dynamics of a mechanical system, these approaches also come with limitations. In particular, there is inherent difficulty to represent dissipative processes, and the restrictions placed on end point variations are not consistent with the definition of initial value problems. The present work on the time-domain response of poroelastic media extends the recent formulations of the mixed convolved action (MCA). The action in this proposed approach is formed by replacing the inner product in Hamilton's principle with a time convolution. As a result, dissipative processes can be represented in a natural way and the required constraints on the variations are consistent with the actual initial and boundary conditions of the problem. The variational formulation developed here employs temporal impulses of velocity, effective stress, pore pressure, and pore fluid mass flux as primary variables in this mixed approach, which also uses convolution operators and fractional calculus to achieve the desired characteristics. The resulting MCA is formulated directly in the time domain to develop a new stationary principle for poroelasticity, which applies to dynamic poroelastic and quasi-static consolidation problems alike. By discretizing the MCA using the finite element method over both space and time, new computational mechanics formulations are developed. Here, this formulation is implemented for the two-dimensional case, and several numerical examples of dynamic poroelasticity are presented to validate the approach.

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Figures

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Fig. 1

Isoparametric master triangle element

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Fig. 2

Poroelastic half-space with applied surface traction pulse

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Fig. 3

Poroelastic half-space solution domain and boundary conditions

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Fig. 4

(a) Pore pressure at x=1 versus time for half-sine pulse and (b) horizontal displacement at x=1 versus time for half-sine pulse

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Fig. 5

(a) Pore pressure at x=1 versus time for sine-squared pulse and (b) horizontal displacement at x=1 versus time for sine-squared pulse

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Fig. 6

Explosion in a cylindrical cavity problem schematic

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Fig. 7

Explosion in a cylindrical cavity mesh

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Fig. 8

Convergence study for pore pressure at point A versus time, impermeable cavity

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Fig. 9

Pore pressure at point A versus time, impermeable cavity

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Fig. 10

Vertical displacement at points O and A versus time, impermeable cavity

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Fig. 11

Vertical displacement at points O and A versus time, permeable cavity

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