Nonlinear deformation of a cantilever incompressible poroelastic beam under a uniform load

Nonlinear bending of cantilever incompressible poroelastic beams subjected to a uniform load is investigated with the constraint that fluid flow is only in the axial direction. The governing equations for large deflection of the poroelastic beam are derived from theory of incompressible saturated porous media. Then, nonlinear responses of a cantilever beam with impermeable fixed end and permeable free end are examined with the Galerkin truncation method. The deflections and bending moments of the poroelastic beam and the equivalent couples of the pore fluid pressures are shown in figures. The differences of the results between the large deflection and the small deflection theories are analyzed. It is shown that the results of the large deflection theory are smaller than those of the small deflection theory, and the time needed to approach their stationary states for the large deflection theory is shorter than that for the small deflection theory.

A nonlinear mathematical model for large deflection of incompressible saturated poroelastic beams

Nonlinear governing equations are established for large deflection of incompressible fluid saturated poroelastic beams under constraint that diffusion of the pore fluid is only in the axial direction of the deformed beams. Then, the nonlinear bending of a saturated poroelastic cantilever beam with fixed end impermeable and flee end permeable, subjected to a suddenly applied constant concentrated transverse load at its free end, is examined with the Gaierkin truncation method. The curves of deflections and bending moments of the beam skeleton and the equivalent couples of the pore fluid pressure are shown in figures. The results of the large deflection and the small deflection theories of the cantilever poroelastic beam are compared, and the differences between them are revealed. It is shown that the results of the large deflection theory are less than those of the corresponding small deflection theory, and the times needed to approach its stationary states for the large deflection theory are much less than those of the small deflection theory.

Quasi-static and dynamical bending of a cantilever poroelastic beam

Based on the theory of porous media, the quasi-static and dynamical bending of a cantilever poroelastic beam subjected to a step load at its free end is investigated, and the influences of its permeability on bending deformation is examined. The initial boundary value problems for dynamical and quasi-static responses are solved with the Laplace transform technique, and the deflections, the bending moments of the solid skeleton and the equivalent couples of the pore fluid pressure are shown in figures. It is shown that the dynamical and quasi-static behavior of the saturated poroelastic beam depends closely on the permeability conditions at the beam ends. Under the different permeability conditions, the deflections of the beam may oscillate or not. The Mandel-Cryer effect also exists in liquid-saturated poroelastic beams.

Bending of simply supported incompressible saturated poroelastic beams with axial diffusion

Based on the mathematical model of the bending of the incompressible saturated poroelastic beam with axial diffusion, the qUasi-static bendings of the simply supported poroelastic beam subjected to a suddenly applied constant load were investigated, and the analytical solutions were obtained for different diffusion conditions of the pore fluid at the beam ends. The deflections, the bending moments of the solid skeleton and the equivalent couples of the pore pressures were presented in figures. It is also shown that the behavior of the saturated poroelastic beams depends closely on the diffusion conditions at the beam ends, especially for the equivalent couples of the pore pressures. It is found that the Mandel-Cryer effect also exists in the bending of the saturated poroelastic beams under specific diffusion conditions at the beam ends.

Dynamic and quasi-static bending of saturated poroelastic Timoshenko cantilever beam

Based on the three-dimensional Gurtin-type variational principle of the incompressible saturated porous media, a one-dimensional mathematical model for dynamics of the saturated poroelastic Timoshenko cantilever beam is established with two assumptions, i.e., the deformation satisfies the classical single phase Timoshenko beam and the movement of the pore fluid is only in the axial direction of the saturated poroelastic beam. Under some special cases, this mathematical model can be degenerated into the Euler-Bernoulli model, the Rayleigh model, and the shear model of the saturated poroelastic beam, respectively. The dynamic and quasi-static behaviors of a saturated poroelastic Timoshenko cantilever beam with an impermeable fixed end and a permeable free end subjected to a step load at its free end are analyzed by the Laplace transform. The variations of the deflections at the beam free end against time are shown in figures. The influences of the interaction coefficient between the pore fluid and the solid skeleton as well as the slenderness ratio of the beam on the dynamic/quasi-static performances of the beam are examined. It is shown that the quasi-static deflections of the saturated poroelastic beam possess a creep behavior similar to that of viscoelastic beams. In dynamic responses, with the increase of the slenderness ratio, the vibration periods and amplitudes of the deflections at the free end increase, and the time needed for deflections approaching to their stationary values also increases. Moreover, with the increase of the interaction coefficient, the vibrations of the beam deflections decay more strongly, and, eventually, the deflections of the saturated poroelastic beam converge to the static deflections of the classic single phase Timoshenko beam.