Poroelasticity is a continuum theory for the analysis of a porous media consisting of an elastic matrix containing interconnected fluid-saturated pores. In physical terms the theory postulates that when a porous material is subjected to stress, the resulting matrix deformation leads to volumetric changes in the pores. Since the pores are fluid-filled, the presence of the fluid not only acts as a stiffener of the material, but also results in the flow of the pore fluid (diffusion) between regions of higher and lower pore pressure. If the fluid is viscous the behavior of the material system becomes time dependent.

Poroelastic theories were originally motivated by problems in soil and geomechanics. This is the point of departure of most of the literature cited above. These problems generally concern massive structures and are by nature three dimensional. Consolidation problems, seismic wave propagation, crustal dynamics, seabed mechanics, etc. are some examples. This application of poroelastic theory is relatively mature. In the past two decades the poroelastic model has also been extensively and successfully applied in biomechanics.

Relatively few papers have thus far investigated the poroelastic beams or plates, the light structures, for which the boundary conditions and the type of loading, and thus the behavior of the structure, are quite different from those for large formations. When such elements are subjected to bending, the stress gradients would generally be expected to be much greater in the perpendicular direction than in the axial or in-plane directions. Thus if the bulk material is considered to be isotropic, the diffusion in the transverse direction is dominant. Hence, in the studies reported in the literature the diffusion in the axial or in-plane directions is, justifiably, considered negligible and the fluid movement in the perpendicular direction has been taken as the prevailing diffusion effect.

This book is devoted to the analysis of fluid-saturated poroelastic beams, columns and plates made of materials for which diffusion in the longitudinal direction(s) is viable, while in the perpendicular direction(s) the flow can be considered negligible because of the micro-geometry of the solid skeletal material. Our initial motivation to investigate such structures is mainly related to plant stems and petioles. These elements of plants serve the dual functions of providing structural strength and stiffness, and also contain the vascular tissue, which conducts water from the root system to transpiring leaves. Living herbaceous stems and woody stem tissue are water saturated, the former often containing as much as 85% free water by weight, the latter as much as 60%. Such structural plant material is highly anisotropic. The axial stiffness is some 20 times greater than transverse stiffness for woody tissue, and for other plant tissue the anisotropy is probably much greater.

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The governing equations for a transversely isotropic poroelastic beam (transversely isotropic in the cross-section) subjected to transverse and/or axial loads, as obtained within the small deflection theory, are presented in Chapter 2, including the inertia of the bulk material. Biot's theory, with relative motion between the solid and fluid governed by Darcy's law, is adapted for the case considered. The governing differential equations can be separated into two groups, one for bending and another for extension; since they are not coupled they can be solved independently. Each group includes three equations for three unknown time-dependent functions: the total stress resultant, the pore pressure resultant and the displacement. The conditions for determination of solutions include the geometrical boundary conditions, the load boundary conditions and the diffusion boundary conditions, as well as the initial conditions.

Chapter 3 presents analytical solutions for the quasi-static bending problem of beams. The formulation is derived by deleting the inertia term from the partial differential equation, which governs the equilibrium of the beam. The elastic solutions, i.e. the solutions for the corresponding drained beams, are introduced in order to simplify the solution procedure so that various closed form solutions for the poroelastic beams can be found. Series solutions are found for normal loading with various mechanical and diffusion boundary conditions.

Due to the complexity of the boundaries and the governing differential equations, it is often difficult to get analytical solutions for general cases, especially when the boundary conditions are not homogenous or when they can not be decoupled. Therefore, finding suitable numerical methods for respective problems is an important part of the present work. The finite element method is employed for the quasi-static beams and columns under small deflection in Chapter 4. Variational principles are first developed. The variational functional is expressed in terms of integrals of the unknown time-dependent functions with respect to position on the beam and convolution integrals with respect to time. Two types of variables, the displacements and pore pressure resultants, are involved in the time-dependent functionals. The method of Lagrange multipliers is employed in order to include the flow equations (generalized Darcy's law) in the Euler-Lagrange equations of the functionals. Two functionals are given, of which one includes the initial values of the unknown functions and is more convenient for the interpolation of the displacement velocities; another functional is more convenient for the interpolation of the displacements. Both functionals are found to be equivalent to each other in terms of their stationary conditions, which give the governing differential equations and boundary conditions. A mixed finite element scheme is then presented based on one of the variational functionals obtained. Numerical solution examples for both types of variables are presented in order to test the finite element model, and good coincidence with the previously found analytical solutions is shown. The results also demonstrate some unique features of poroelastic beams and columns, which can not be shown by example for which analytical solutions can be found.

In Chapter 5 solutions are found for free and forced vibration situations of the poroelastic beams. Closed form solutions of the initial value problems are obtained for simply supported beams with general loading by use of Laplace transformation. It turns out that the fluid works as a damper. Similar to the classic vibration theory of damped elastic beams, the responses to initial deviations can be classified into three kinds: light damping, critical damping and over damping. The vibration patterns are also dependent on the nature of the initial conditions; observed behavior of this sort can not be explained by the classical vibration theory. Computations for the harmonic vibrations are carried out for different boundaries. The amplitude response versus the frequency of the loading, and the resonance areas, are investigated.

Chapter 6 deals with large deflections of beams. While in the previous chapters the deflection was considered to be small, and thus linear theories are sufficient when the constitutive law is linear, for some situations it may be necessary to employ a large deflection theory in order to correctly describe the behavior of the structure. On the other hand, the deformation can be still small and the skeletal material yet behaves elastically; the large deflection is made possible by the slenderness of the beam. Therefore, it is modeled as geometrically nonlinear and constitutively linear. Biot's constitutive law and Darcy's law are adopted as in the linear theory, while new geometrical relations and equilibrium equations are necessarily introduced. In the large deflection case the stretching and bending problems are coupled. The nonlinear boundary value problem is solved numerically by using the finite difference method with respect to the spatial coordinate and using a simple successive implicit formula (the trapezoid formula) to deal with the time variable. Several types of geometrical and diffusion boundary conditions are investigated by means of numerical solutions. Results are presented, for which observations are made, and some interesting features are found which do not occur when the problem is modeled as linear (i.e. small deflections).

The stability of poroelastic columns is investigated in Chapter 7. Three problems are considered: buckling, post-buckling and dynamic stability. For the buckling problem, the time dependent behaviors of the critical loads and deflections are considered for various diffusion and geometrical boundaries. Upper (short time) and lower (long time) limits for the time dependent critical load are found for the case of time-dependent load. It is also shown that buckling can be avoided during a loading procedure by properly choosing the loading path, even when the load at finite time is greater than the lower limit of the critical load. For the post-buckling problem, the time-dependent behavior of the columns, governed by three coupled equations, is obtained by using the large deflection theory. These equations are transformed into a single one, enabling the analytical derivation of the initial and the final responses. It is shown that unlike the quasi-static response obtained by using the small deflection theory, the long time response derived here is bounded. The imperfection sensitivity of these columns is also investigated. For the dynamic stability problem, stability conditions and boundaries are derived. It is shown that the stability regions are expanded with respect to the elastic (drained) case. The critical (minimum) loading amplitude for which instability occurs is also given.

Formulations are found in Chapter 8 for fluid-saturated poroelastic plates consisting of a material for which the diffusion is possible in the in-plane directions only, both for bending and for in-plane loading. The plates considered are isotropic in the plate plane, and the Kirchhoff hypotheses are assumed. Again Biot's constitutive law is adopted and Darcy's law is used to describe the fluid flow in pores. The basic equations are so derived that they could be easily extended for the situation of an orthotropic poroelastic plate. Closed form solutions are extracted for quasi-static problems and for vibrations. Observations are made on the types of deflection/vibration patterns, which are obtained.

Finally in Chapter 9 we present a beam model composed of a discrete elastic structure containing a fluid which together will behave in an identical manner to the beam composed of a poroelastic continuum used in the formulation of Chapter 2. This discrete model has heuristic value in appreciating the phenomena. We show how adjusting the physical parameters of the materials and the geometric parameters influence the extent of the poroelastic effect.

Throughout the book some very unique features of the proposed model are shown (in some cases a comparison is made with another time-dependent model, viscoelasticity). First, the mechanical behaviors of the structural elements are shown to be greatly dependent on the diffusion patterns which are in turn dependent on the loading, on the geometrical and diffusion boundary conditions, as well as on the material parameters. This is in contrast to the case of problems of the same kind of structures (as far as geometry and loading) with diffusion in the transverse direction, where the transverse diffusion patterns for different positions along the beam are all similar. Second, three time scales are required to describe the vibration system, as compared to two in viscoelasticity. Moreover, the vibration patterns of the present system are determined not only by the parameters of the material and the geometry of the structure, but also by the initial pore pressure conditions. This implies that even in the case of "light damping" oscillatory motion may not occur for some initial pore pressure conditions, which would not be explainable if the structures were modeled as viscoelastic. Third, the pore pressure at a given position does not necessarily decay monotonically after a suddenly applied and then constant loading; it may increase for some time and then decay toward the final value in some cases. This phenomenon is similar to the so-called Mandel-Cryer effect. Moreover, the pore pressure at other positions can possibly increase monotonically for all time; in some cases the sign of pore pressure can even change twice during the diffusion process. Fourth, the pore fluid works also as a damper. This damping mechanism is useful in reducing vibrations. By changing the properties of the fluid, which for instance can be changed by a temperature increment, or by altering the diffusion boundaries, a resonance can be avoided or an oscillatory motion could be made to disappear when so desired. It is also possible to avoid oscillatory motions by choosing initial pore pressure conditions, without changing the properties of the material and the geometry of the structural element. Again, this is quite different from a viscoelastic damping system.

As a result of the features mentioned above, it is recognized that the response of the poroelastic structural element to loading is sensitive to the properties of the fluid and to the diffusion boundaries, which can be easily altered in practice. Therefore, such structural elements and thus their features are potentially controllable. In other words, it could be possible to convert such elements into intelligent or smart structures. If this is so, it would be interesting that such structural elements could work as both sensors and actuators, e.g. the fluid can "feel" the change of the temperature by changing its viscosity and this results in a change of the behavior of the structure.

This book attempts to constitute a reasonably self-contained presentation of a wide spectrum of problems related to the analysis of the type of poroelastic structure considered. It is hoped that the book will serve as an inspiration, guide, and reference for applying such elements in mechanical, biomechanical, civil and aerospace engineering, as well as a textbook for graduate studies.