dynamic and quasi-static bending of saturated poroelastic timoshenko cantilever beam

Appl. Math. Mech. -Engl. Ed. 31(8), 995–1008 (2010)DOI 10.1007/s10483-010-1335-6c©Shanghai University and Springer-Verlag

Berlin Heidelberg 2010

Applied Mathematicsand Mechanics(English Edition)

Dynamic and quasi-static bending of saturated poroelasticTimoshenko cantilever beam∗

Xiao YANG (� �), Qun WEN (� �)

(Department of Civil Engineering, Shanghai University, Shanghai 200072, P. R. China)

(Communicated by Xing-ming GUO)

Abstract Based on the three-dimensional Gurtin-type variational principle of theincompressible saturated porous media, a one-dimensional mathematical model for dy-namics of the saturated poroelastic Timoshenko cantilever beam is established with twoassumptions, i.e., the deformation satisfies the classical single phase Timoshenko beamand the movement of the pore fluid is only in the axial direction of the saturated poroe-lastic beam. Under some special cases, this mathematical model can be degenerated intothe Euler-Bernoulli model, the Rayleigh model, and the shear model of the saturatedporoelastic beam, respectively. The dynamic and quasi-static behaviors of a saturatedporoelastic Timoshenko cantilever beam with an impermeable fixed end and a permeablefree 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 skeletonas well as the slenderness ratio of the beam on the dynamic/quasi-static performancesof the beam are examined. It is shown that the quasi-static deflections of the saturatedporoelastic beam possess a creep behavior similar to that of viscoelastic beams. In dy-namic responses, with the increase of the slenderness ratio, the vibration periods andamplitudes of the deflections at the free end increase, and the time needed for deflectionsapproaching to their stationary values also increases. Moreover, with the increase of theinteraction coefficient, the vibrations of the beam deflections decay more strongly, and,eventually, the deflections of the saturated poroelastic beam converge to the static deflec-tions of the classic single phase Timoshenko beam.

Key words saturated porous media, saturated poroelastic Timoshenko beam, math-ematical model, Laplace transform, dynamic/quasi-static bending

Chinese Library Classification O321, O357.32000 Mathematics Subject Classification 35L35, 44A10

1 Introduction

With the widespread applications of saturated poroelastic structures in biological engineer-ing, building construction, and heat transfer industries, etc. in recent years, researchers havefocused attentions on the mechanical behaviors of various saturated poroelastic structures, inwhich the dynamic and quasi-static performance of the saturated poroelastic beams is one ofthe key research areas. Based on the Biot theory of the saturated porous media, Zhang and

∗ Received Mar. 1, 2010 / Revised Jun. 14, 2010Project supported by the National Natural Science Foundation of China (No. 10872124)Corresponding author Xiao YANG, Professor, Ph. D., E-mail: [email protected]

996 Xiao YANG and Qun WEN

Cowin[1] studied the steady-state vibrations of a saturated poroelastic isotropic beam undercyclic loadings. It was found that the gradient of the pore pressures in the transverse directionwas generally larger than that in the axial direction. Therefore, the axial fluid movement can benegligible in general. This study was extended by Kameo et al.[2], in which the transient dynam-ical behavior of a saturated poroelastic material was analyzed under the uniaxial cyclic loading.Based on the three-dimensional (3D) Biot model, Wang et al.[3] investigated the steady andtransient responses of pure bending of a compressible saturated poroelastic beam. The resultswere compared with those of the finite element method. However, in nature and engineering,there exist some anisotropic porous materials such as plant stems, skeletons of animals, andducts for heat transfers, in which their microstructures satisfy that the fluid movement in theaxial direction is prevailing and that in the transverse direction can be neglected. Based onthe Biot model, Li et al.[4–5] established linear and nonlinear mathematical models for com-pressible saturated poroelastic beams and rods, analyzed extensively quasi-static and dynamicbehaviors of the saturated poroelastic beams, and revealed several unique performances[6].Chakraborty[7] studied the governing equations for the axial and transverse couple deformationof the anisotropic saturated poroelastic beam, and analyzed the wave propagations in the fre-quency domain. Furthermore, Wang et al.[8] gave a 3D analytical solution for pure bending ofthe saturated poroelastic beam based on Saint-Venant’s condition of the traction free on thebeam surface.

In reality, there are some situations in which fluid or solid phases can be viewed as micro-scopic incompressible. For such problems, Yang and Cheng[9] and Yang and He[10] presentedthe generalized Gurtin-type variational principles and the finite element method of the in-compressible saturated porous media. Yang and Li[11] derived a mathematical model of theincompressible saturated poroelastic Euler-Bernoulli beam from general 3D governing equa-tions. Yang and Wang[12] established the corresponding Gurtin-type variational principles, andanalyzed the dynamical behavior of the saturated poroelastic cantilever beam subjected to aharmonic load at its free end with the finite element method. Furthermore, Yang and Wang[13]

obtained a mathematical model for nonlinear bending of the saturated poroelastic beam, andinvestigated the nonlinear responses of the saturated poroelastic cantilever beam subjected toa concentrated load at its free end. Nevertheless, to the best of the authors’ knowledge, there isno research on the Timoshenko model, the Rayleigh model, and the shear model for saturatedporoelastic beams.

Excluding the mass and heat exchange between solid and fluid phases and taking the effectof shearing strain of the beam skeleton into consideration, based on the 3D Gurtin-type vari-ational principle of incompressible saturated porous media, a system of one-dimensional (1D)governing equations of a saturated poroelastic Timoshenko cantilever beam is established withtwo assumptions, i.e., the movement of the pore fluid is only in the axial direction of the satu-rated poroelastic beam and the deformation satisfies the classical Timoshenko beam for beamskeleton. The corresponding initial and boundary conditions are presented. The model of thesaturated poroelastic Timoshenko beam can be degenerated into the Euler-Bernoulli model, theRayleigh model, and the shear model of the saturated poroelastic beam, respectively, by dif-ferent limits of physical parameters. Then, with a Laplace transform and its Crump numericalinverse transform, the dynamic and quasi-static bending of a saturated poroelastic Timoshenkocantilever beam is analyzed with an impermeable fixed end and a permeable free one subjectedto a step load at its free end. The variations of the deflections at the beam free end againsttime are shown in figures for dynamic and quasi-static bendings. The influences of the in-teraction coefficient between the pore fluid and the solid skeleton as well as the slendernessratio of the beam on the dynamic/quasi-static performances of the beam are analyzed. Thenumerical results show that the interaction between the solid skeleton and the pore fluid hasthe effect of viscosity, and the quasi-static deflections of the saturated poroelastic beam possessa creep behavior. Due to the property of permeability at the beam free end, the dynamical and

Dynamic and quasi-static bending of saturated poroelastic Timoshenko cantilever beam 997

quasi-static deflections at the beam free end will approach to those of the classic single phaseTimoshenko beam eventually.

2 Mathematical model for dynamic bending of saturated poroelastic Tim-oshenko cantilever beam

As shown in Fig. 1, a saturated poroelastic beam with length L and cross section A subjectedto a transverse load q(x, t) is composed of the immiscible microscopic incompressible fluid �F

and the porous solid skeleton �S. The beam is impermeable on its surface and at its fixed end,and is permeable at its free end. Based on the theory of saturated porous media, neglectingthe body forces of the pore fluid and solid skeleton as well as the mass and the heat exchangebetween the two phases, the 3D governing equations of the momentum balance of the fluid-solidmixture, the momentum balance of the pore fluid, and the conservation of the volume fractioncan be expressed as[9,14]

div T SE − gradp− (ρS + ρF)uS − ρFvFS = 0, (1)

− nFgrad p− ρF(uS + vFS) − SvvFS = 0, (2)

div(uS + nFvFS) = 0, (3)

where T SE is the effective stress of the solid skeleton �S, and p is the pressure of the porefluid �F. nF is the volume fraction of the fluid phase, and the volume fraction of the solidskeleton is nS = 1−nF. ρS and ρF are partial mass densities of the solid skeleton and the fluid,respectively, which can be expressed by their real densities ρSR and ρFR as

ρS = nSρSR, ρF = nFρFR.

The interaction coefficient Sv between the solid skeleton and the pore fluid can be expressedwith the Darcy permeability coefficient kF as Sv = (nF)2γFR/kF[14]. The displacement, velocity,and acceleration of the solid skeleton are uS, uS, and uS, respectively. The relative velocity andacceleration of the pore fluid with respect to the solid skeleton are vFS and vFS, respectively.

Fig. 1 Saturated poroelastic Timoshenko cantilever beam subjected to transverse load

Under the assumption of infinitesimal deformation and neglecting the displacement in theOy direction, the skeleton displacements in the Ox, Oy, and Oz directions of the saturatedporoelastic beam can be expressed as[15]

⎧⎪⎪⎨

⎪⎪⎩

uSx(x, y, z, t) = −zφ(x, t),

uSy(x, y, z, t) = 0,

uSz(x, y, z, t) = w(x, t),

(4)

where φ(x, t) and w(x, t) are the rotation angle of the cross section and the deflection of theaxes line of the beam skeleton, respectively.


Assume that the movement of the pore fluid is possible in the axial direction only, and therelative velocity vFS of the pore fluid satisfies

{vFS

x = vFSx (x, y, z, t),

vFSy = vFS

z = 0.(5)

Neglecting the inertial effect of the pore fluid, the momentum equation in the Ox direction of(2) and the volume fraction conservation (3) become

⎧⎪⎪⎨

⎪⎪⎩

− nF ∂p

∂x− Svv

FSx = 0,

nF ∂vFSx

∂x− z

∂2φ

∂x∂t= 0.

(6)

It can be obtained that the pressure p of the pore fluid and the relative velocity vFSx are in

direct proportion to the coordinate z. Therefore, it can be assumed that p = ζzΨ(x, t). Definethe equivalent couple Mp of the fluid pressure p as

Mp =∫∫

A

zpdydz. (7)

Then,

Mp =∫∫

A

zpdydz =∫∫

A

z2ζ Ψdydz = Iζ Ψ, (8)

where I is the inertia moment of the cross section of the beam skeleton with respect to theneutral axis, i.e., I =

∫∫

A z2dydz.

As a result, it has

{ p =z

IMp,

vFSx = −zn

F

SvI

∂Mp

∂x.

(9)

Consequently, the boundary conditions for the pore fluid at the impermeable fixed end (vFSx |x=0 =

0) and the permeable free end (p|x=L = 0) can be rewritten as

∂Mp

∂x= 0, x = 0, (10)

Mp = 0, x = L. (11)

The strains, the shear force, and the bending moment on the cross section of the saturatedporoelastic beam are[15]

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

εSx = −z ∂φ∂x,

γSxz =

∂w

∂x− φ,

MSEx = −ESI

∂φ

∂x,

F SEx = kGSA

(∂w

∂x− φ

),

(12)

where ES and GS are Young’s modulus and the shear modulus of the skeleton, respectively,and k is the modified factor of the shear strain. Let the shear force and the bending moment at


the free end of the cantilever beam be F (t) and M(t), respectively. The boundary conditionsfor the beam skeleton at the fixed end and the free end are

w = φ = 0, x = 0, (13)

MSEx = −ESI

∂φ

∂x= M(t), F SE

x = kGSA(∂w

∂x− φ

)= F (t), x = L. (14)

If the saturated poroelastic Timoshenko beam is undeformed initially, the initial conditions are

w = 0, φ = 0, vFSx = 0, t = 0, (15)

∂w

∂t= 0,

∂φ

∂t= 0, t = 0. (16)

From Ref. [9], it is known that in a function class satisfying the boundary condition (13) ofthe displacement, the boundary condition (11) of the equivalent couple, and the initial condition(15), the solution for the dynamics of the saturated porous medium makes the variation of thefollowing Gurtin-type functional be zero:

Π =∫

V

12((2GSεS + λS(εS · I)I) ∗ εS + (ρS + ρF)uS ∗ uS)dv

+∫

V

(ρFvFS ∗ uS + vFS ∗ g ∗

(nFgradp+

12Svv

FS)

+12ρFvFS ∗ vFS

)dv

−∫

V

p ∗ div uSdv −∫

ST

uS ∗ T ds−∫

SQ

p ∗ g ∗Qds, (17)

where λS and GS are the macroscopic Lame constants of the solid skeleton, and

λS =GS(ES − 2GS)

3GS − ES;

εS is the strain tensor of the solid skeleton; I is the unit tensor; T is the prescribed surface force;Q is the prescribed seepage quantity; g(t) = 1; and ∗ is the convolution integral of functions,e.g.,

φ ∗ ψ =∫ t

0

φ(t− τ)ψ(τ)dτ .

From the deformation hypothesis (4) of the beam skeleton, the relative velocity (5) of thepore fluid, and the relationship (9), and taking the effect of the shear correction factor k intoconsideration, the functional (17) can be further simplified as

Π =∫ L

0

(12

(ESI

∂φ

∂x∗ ∂φ∂x

+ kGSA(∂w

∂x− φ

)∗

(∂w

∂x− φ

))+Mp ∗ ∂φ

∂x

)dx

+∫ L

0

12

((ρS + ρF)I

∂φ

∂t∗ ∂φ∂t

+ (ρS + ρF)A∂w

∂t∗ ∂w∂t

)dx

+∫ L

0

( (nF)2ρF

2S2vI

∂Mp

∂x∗ ∂Mp

∂x− (nF)2

2SvI

∂Mp

∂x∗ g ∗ ∂Mp

∂x

)dx

+∫ L

0

nFρF

Sv

∂Mp

∂x∗ ∂φ∂tdx−

∫ L

0

q ∗ wdx − (F ∗ w)|x=L + (M ∗ φ)|x=L. (18)

From the variational principle[9], the properties of the convolution integral, and the identities{φ ∗ ψ = φ ∗ g ∗ ψ + ψ(0)g ∗ φ, φ ∗ g ∗ ψ = ψ ∗ g ∗ φ,φ ∗ ψ = φ ∗ ψ + φ(0)ψ(t) − φ(t)ψ(0),

(19)


the governing equation can be obtained from the variation of the functional (18), i.e., δΠ = 0,which gives

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

ESI∂2φ

∂x2+∂Mp

∂x+ kGSA

(∂w

∂x− φ

)− nFρF

Sv

∂2Mp

∂x∂t− (ρS + ρF)I

∂2φ

∂t2= 0,

kGSA(∂2w

∂x2− ∂φ

∂x

)− (ρS + ρF)A

∂2w

∂t2+ q = 0,

∂φ

∂x+

(nF)2

SvIg ∗ ∂

2Mp

∂x2− nFρF

Sv

∂2φ

∂x∂t+

(nF)2ρF

S2vI

∂2Mp

∂x∂t= 0

(20)

and⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

ESI∂φ

∂x+Mp +M(t) = 0, kGSA

(∂w

∂x− φ

)− F (t) = 0, x = L,

nFρF

Sv

∂φ

∂t− g ∗ (nF)2

SvI

∂Mp

∂x+

(nF)2ρF

S2vI

∂Mp

∂x= 0, x = 0,

(ρS + ρF)A∂w

∂t= 0, (ρS + ρF)I

∂φ

∂t+nFρF

Sv

∂Mp

∂x= 0, t = 0.

(21)

The effect of the rotation inertia of the pore fluid with respect to the beam neutral axis ispresented as ∫

A

ρFzvFSx dydz = −

∫

A

z2nFρF

SvI

∂2Mp

∂x∂tdydz = −n

FρF

Sv

∂2Mp

∂x∂t. (22)

Neglecting such an effect and with the properties of the convolution integral, the governingequation (20) can be further reduced to

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

ESI∂2φ

∂x2+∂Mp

∂x+ kGSA

(∂w

∂x− φ

)− (ρS + ρF)I

∂2φ

∂t2= 0,

kGSA(∂2w

∂x2− ∂φ

∂x

)− (ρS + ρF)A

∂2w

∂t2+ q = 0,

I∂2φ

∂x∂t+

(nF)2

Sv

∂2Mp

∂x2− nFρFI

Sv

∂3φ

∂x∂t2= 0.

(23)

From (9) and (15), ∂Mp

∂x |t=0 = 0 can be obtained. Then, with the boundary conditions(11) and (13), the boundary conditions (10) and (14) as well as the initial condition (16) areobtained. Therefore, the initial boundary value problem for dynamic bending of the saturatedporoelastic Timoshenko cantilever beam is to seek the deflection w(x, t), the rotation angleϕ(x, t), and the equivalent couple Mp(x, t) of the pore pressure, which satisfy the governingequation (23), the boundary conditions (10) and (11), (13), and (14), and the initial conditions(15) and (16).

3 Dynamic bending of the Euler-Bernoulli, shear, and Rayleigh models ofthe saturated poroelastic beam

From the first two equations of the governing equation (23), it can be obtained that

ESI∂3φ

∂x3+∂2Mp

∂x2− (ρS + ρF)I

∂3φ

∂x∂t2+ (ρF + ρS)A

∂2w

∂t2− q = 0. (24)

Neglect the effects of the rotation inertia of the beam and the shear strain, i.e.,

(ρS + ρF)I → 0, ρFI → 0, kGS → ∞.


Then, it can be obtained that φ = ∂w∂x from the first equation of the governing equation (23).

Substituting it into the equation (24) and the third equation of governing equation (23) yields⎧⎪⎪⎨

⎪⎪⎩

ESI∂4w

∂x4+∂2Mp

∂x2+ (ρF + ρS)A

∂2w

∂t2− q = 0,

I∂3w

∂x2∂t+

(nF)2

Sv

∂2Mp

∂x2= 0.

(25)

This set of the governing equations is for the dynamic bending of the Euler-Bernoulli modelof the saturated poroelastic beam. If the effect of the transverse deformation of the beam isneglected, the governing equations in Ref. [11] are the same as those in (25).

Neglecting the effect of the rotation inertia of the beam, i.e.,

(ρS + ρF)I → 0, ρFI → 0,

the governing equation (23) is reduced to⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

ESI∂2φ

∂x2+∂Mp

∂x+ kGSA

(∂w

∂x− φ

)= 0,

I∂2φ

∂x∂t+

(nF)2

Sv

∂2Mp

∂x2= 0,

kGSA(∂2w

∂x2− ∂φ

∂x

)−

(ρS + ρF

)A∂2w

∂t2+ q = 0.

(26)

This set of the governing equations is for the dynamic bending of the shear model of thesaturated poroelastic beam.

Neglect the effect of the shear strain, i.e., kGS → ∞. Then, the governing equation (23) isreduced to

⎧⎪⎪⎨

⎪⎪⎩

ESI∂4w

∂x4+∂2Mp

∂x2+ (ρF + ρS)A

∂2w

∂t2− (ρS + ρF)I

∂4w

∂x2∂t2− q = 0,

I∂3w

∂x2∂t+

(nF)2

Sv

∂2Mp

∂x2− nFρFI

Sv

∂4w

∂x2∂t2= 0.

(27)

This set of the governing equations is for the dynamic bending of the Rayleigh model of thesaturated poroelastic beam.

If the influence of the pore fluid is neglected, i.e., Mp →0�ρF → 0, and Sv → 0, thenthe governing equations (23), (25), (26), and (27) are degenerated into the Timoshenko model,the Euler-Bernoulli model, the shear model, and the Rayleigh model of the classic single phaseelastic beam[16], respectively.

4 Bending of the saturated poroelastic Timoshenko cantilever beam sub-jected to a concentrated loading at its free end

Now, the dynamic responses of a saturated poroelastic Timoshenko cantilever beam sub-jected to a concentrated loading at its free end are considered. In this case, q(x, t) = 0 andM(t) = 0. Introduce the dimensionless variables and parameters as follows:

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

x∗ =x

L, t∗ =

t

T, w∗ =

w

L, β = ηs2,

η = 1 +ρF

ρS, s2 =

AL2

I, γ =

ρF

nFρS, κ = k

GS

ESs2,

M∗p =

L

ESIMp, F =

L2F

ESI, T =

L√ES/ρS

, α =SvL

2

(nF)2EST,

(28)


where s is the slenderness ratio of the beam, and κ describes the shearing effect of the beam.For the sake of convenience, x, t, w, and Mp are still used to denote x∗, t∗, w∗, and M∗

p ,respectively. Then, the dimensionless initial boundary value problem of a saturated poroelasticTimoshenko cantilever beam can be expressed as follows:

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

η∂2φ

∂t2− ∂2φ

∂x2− ∂Mp

∂x− κ

(∂w

∂x− φ

)= 0,

γ∂3φ

∂x∂t2− ∂2Mp

∂x2− α

∂2φ

∂x∂t= 0,

β∂2w

∂t2− κ

(∂2w

∂x2− ∂φ

∂x

)= 0,

(29)

⎧⎪⎨

⎪⎩

w = 0, φ = 0,∂Mp

∂x= 0, x = 0,

∂φ

∂x= 0, κ

(∂w

∂x− φ

)= F, Mp = 0, x = 1,

(30)

w = 0, φ = 0,∂w

∂t= 0,

∂φ

∂t= 0, t = 0. (31)

The initial boundary value problem (29)–(31) can be solved with a Laplace transform. Denotefunctions w(x, ξ), φ(x, ξ), and Mp(x, ξ) to be the Laplace transforms of functions w(x, t), φ(x, t),and Mp(x, t) with respect to time t, respectively. Then, the Laplace transform of the initialboundary value problem (29)–(31) are

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

d2φ

dx2+dMp

dx+ κ

dw

dx− (κ+ ηξ2)φ = 0,

d2Mp

dx2+ (αξ − γξ2)

dφ

dx= 0,

κ(d2w

dx2− dφ

dx

)− βξ2w = 0,

(32)

⎧⎪⎪⎨

⎪⎪⎩

w = 0, φ = 0,dMp

dx= 0, x = 0,

dφ

dx= 0,

dw

dx− φ =

F

κ, Mp = 0, x = 1,

(33)

where

F (ξ) = L[F (t)] =∫ +∞

0

F (t)e−ξtdt. (34)

From the first and third equations of (32), it can be obtained that⎧⎪⎪⎪⎨

⎪⎪⎪⎩

dφ

dx=d2w

dx2− βξ2

κw,

dMp

dx= (κ+ ηξ2)φ− d2φ

dx2− κ

dw

dx.

(35)

Substituting them into the second equation of (32) yields

d4w

dx4−

(αξ +

(η − γ +

β

κ

)ξ2

)d2w

dx2+βξ2

κ(κ+ αξ + (η − γ)ξ2)w = 0. (36)


Denote ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

f1(ξ) = αξ +(η − γ +

β

κ

)ξ2,

f2(ξ) =βξ2

κ

(κ+ αξ + (η − γ)ξ2

),

l1(ξ) =(f1(ξ) +

√f21 (ξ) − 4f2(ξ)2

) 12,

l2(ξ) =(f1(ξ) −

√f21 (ξ) − 4f2(ξ)2

) 12.

(37)

Then, the general solution of (36) is

w(x, ξ) = C1(ξ)el1x + C2(ξ)e−l1x + C3(ξ)el2x + C4(ξ)e−l2x. (38)

Furthermore, from (35), the following results can be obtained:

φ(x, ξ) = f3(ξ)(C1(ξ)el1x − C2(ξ)e−l1x

)+ f4(ξ)

(C3(ξ)el2x − C4(ξ)e−l2x

)+ C5(ξ), (39)

Mp(x, ξ) = f5(ξ)(C1(ξ)el1x + C2(ξ)e−l1x

)+ f6(ξ)

(C3(ξ)el2x + C4(ξ)e−l2x

)

+ (κ+ ηξ2)xC5(ξ) + C6(ξ),(40)

where Ci(ξ) (i = 1, 2, · · · , 6) are undetermined coefficients, and⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

f3(ξ) = l1(ξ) − βξ2

κl1(ξ),

f4(ξ) = l2(ξ) − βξ2

κl2(ξ),

f5(ξ) = (κ+ ηξ2 − l21(ξ))(1 − βξ2

κl21(ξ)

)− κ,

f6(ξ) = (κ+ ηξ2 − l22(ξ))(1 − βξ2

κl22(ξ)

)− κ.

(41)

The linear algebraic equation for determining the Ci(ξ) (i = 1, 2, · · · , 6) can be obtained bysubstituting (38)–(40) into (33), and can be expressed as

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

1 1 1 1 0 0

f3 −f3 f4 −f4 1 0

l1f3el1 l1f3e−l1 l2f4el2 l2f4e−l2 0 0

βξ2

κl1el1 −βξ2

κl1e−l1 βξ2

κl2el2 −βξ2

κl2el2 −1 0

l1f5 −l1f5 l2f6 −l2f6 κ+ ηξ2 0

f5el1 f5e−l1 f6el2 f6e−l2 κ+ ηξ2 1

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎝

C1

C2

C3

C4

C5

C6

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

000

F /κ

00

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

. (42)

Consequently, the solution can be expressed as

(C1 C2 C3 C4 C5 C6)T = F (C1 C2 C3 C4 C5 C6)T. (43)

Functions w(x, ξ), φ(x, ξ), and Mp(x, ξ) can be obtained by substituting (43) into (38)–(40).Furthermore, the Laplace transform of the dimensionless bending moment

mx =LMSE

x

ESI= −∂φ

∂x


of the beam skeleton is mx = −∂ eφ∂x . Then, functions w(x, t), φ(x, t), Mp(x, t), and mx(x, t) can

be obtained by the inverse Laplace transforms.In general, it is very difficult to obtain the analytical expressions of the inverse Laplace trans-

forms of functions w(x, ξ), φ(x, ξ), Mp(x, ξ), and mx(x, ξ). Therefore, the numerical method ofthe inverse Laplace transform can be employed. In numerous numerical methods, the Crumpmethod[17] will be employed. Let the function F (ξ) be the Laplace transform of the functionf(t). Then, the numerical Crump method of the inverse Laplace transform of function F (ξ) is

f(t) ≈

eat

T ∗(1

2F (a) +

∞∑

k=1

(Re

(F

(a+

kπiT ∗

))cos

kπt

T ∗ − Im(F

(a+

kπiT ∗

))sin

kπt

T ∗)). (44)

If |f(t)| < Meαt, then the error is

|ε| � Meαte−2T∗(a−α), T ∗ >t

2.

If the inertia terms in the governing equation (29) are ignored, i.e., η = γ = β = 0, then thequasi-static bending of the saturated poroelastic Timoshenko cantilever beam can be analyzedsimilarly. Due to the limitation of space, the concrete expressions are omitted.

5 Numerical results and discussions

Now, the dynamic responses of the saturated poroelastic Timoshenko cantilever beam sub-jected to a step load at its free end will be examined, i.e.,

F (t) = q0H(t), (45)

where the dimensionless load q0 = 1, and H(t) is the Heaviside function.The dimensionless parameters are taken as follows[16,18]:

ρFR

ρSR=

12.66

= 0.376, nF = 0.23, k =6(1 + νS)7 + 6νS

, νS = 0.3, (46)

where νS is the Poisson ratio of the beam skeleton.Figures 2 and 3 show the variations of the dimensionless deflection w0 = w(1, t) at the free

end of the saturate poroelastic Timoshenko cantilever beam against the dimensionless time tfor different slenderness ratios s for α = 2.0 and α = 4.0, respectively. It can be seen that withthe increase of t, the vibrations of w0 at the beam free end attenuate gradually and approachto their steady state values. Comparing Fig. 2 with Fig. 3, it is shown that these steady statevalues are independent of α, γ, and η, and increase with the decrease of s. In fact, the steadystate values of the initial boundary value problem (29)–(31) can be obtained as follows:

wst(x) = q0

(12x2

(1 − 1

3x)

+x

κ

), φst(x) = q0x

(1 − 1

2x), Mpst = 0. (47)

Therefore, the steady deflections at the free end are

wst(1) = q0

(13

+1κ

). (48)

Since Mpst=0, the steady state solution (47) is actually the static solution of the classic singlephase elastic Timoshenko beam. When s = 10, 20, and 30, the exact values (RES) obtainedfrom (48) and the numerical results (RCM) obtained from the numerical Crump method (44)


Fig. 2 Deflections w0 at the free end ofthe saturated poroelastic Timoshenkocantilever beam vs. time t for differentslenderness ratios s when α=2.0

Fig. 3 Deflections w0 at the free end ofthe saturated poroelastic Timoshenkocantilever beam vs. time t for differentslenderness ratios s when α=4.0

are listed in Table 1, respectively. It can be seen that the Crump method of the inverse Laplacetransform can give the quite accurate results. Moreover, the influence of the slenderness s onthe vibrations of the free end is great, and the amplitudes and periods of the deflection and thetime needed to approach to their steady state values increase with the increase of slendernessratio s.

Table 1 Results of the numerical Crump method and exact solution

s 10 20 30

RCM 0.362 67 0.340 67 0.336 67

RES 0.362 67 0.340 67 0.336 59

Figures 4 and 5 show the variations of the dimensionless deflection w0 at the beam free endagainst the dimensionless time t for different parameters α when s = 20 and s = 10, respectively.It can be seen that α, which describes the interaction between the solid skeleton and the porefluid, has a damping effect. With the increase of α, the attenuation of the deflection vibrationis accelerated, and the time needed to approach its steady state value decreases.

Fig. 4 Deflections w0 at the free end ofthe saturated poroelastic Timoshenkocantilever beam vs. time t for differentparameters α when s=20

Fig. 5 Deflections w0 at the free end ofthe saturated poroelastic Timoshenkocantilever beam vs. time t for differentparameters α when s=10


From (28), it is known that when η → 0 and γ → 0, the governing equation (29) is degen-erated into the model of the saturated poroelastic shear beam; when κ → ∞, the governingequation (29) is degenerated into the model of the saturated poroelastic Rayleigh beam; andwhen η → 0, γ → 0, and κ → ∞, the governing equation (29) is degenerated into the modelof the saturated poroelastic Euler-Bernoulli beam. Therefore, with the same procedure, theresponses of the saturated poroelastic shear, Rayleigh, and Euler-Bernoulli cantilever beamssubjected to the step load (45) at their free ends can be obtained. The variations of the dimen-sionless deflection w0 at the free end of the saturated poroelastic Timoshenko, shear, Rayleigh,and Euler-Bernoulli beams against t are shown in Figs. 6 and 7 for s = 20, α = 2.0 and s = 10,α = 2.0, respectively. It can be seen that the effect of the shearing on w0 at the free endis remarkable, while the rotation inertia effect of the cross section is small. Moreover, theshearing effect and rotation inertia effect of the cross section increases with the decrease of theslenderness ratio.

Fig. 6 Deflections w0 at the free end ofthe saturated poroelastic Timoshenko,shear, Rayleigh, and Euler-Bernoullicantilever beams vs. time t whenα=2.0 and s=20

Fig. 7 Deflections w0 at the free end ofthe saturated poroelastic Timoshenko,shear, Rayleigh, and Euler-Bernoullicantilever beams vs. time t whenα=2.0 and s=10

The responses of the deflections w0 at the free end for the quasi-static bending of thesaturated poroelastic Timoshenko cantilever beam against time t are shown in Figs. 8 and 9 for

Fig. 8 Deflections w0 at the free end ofthe saturated poroelastic Timoshenkobeam for quasi-static bending vs. timet when α=2.0

Fig. 9 Deflections w0 at the free end ofthe saturated poroelastic Timoshenkobeam for quasi-static bending vs. timet when α=4.0


different slenderness ratios s when α = 2.0 and α = 4.0, respectively. It is shown that when thestep load is applied at the free end, w0 gradually increase from zero and converge to their steadystate values eventually with the increase of t, and there is no oscillation for the deflections. Thesteady state values of the quasi-static deflections are the same as the limitations of the dynamicdeflections, i.e., the steady state deflections (48). However, the time needed to approach thesteady state values increases with the increase of α, and it is much smaller than that for dynamicresponses.

6 Conclusions

With the assumptions that the movement of the pore fluid is only in the axial direction ofthe saturated poroelastic beam and the deformation satisfies the classical Timoshenko beam forbeam skeleton, based on the Gurtin-type variational principle of the incompressible saturatedporous media, a 1D mathematical model for the dynamic bending of the saturated poroelasticTimoshenko cantilever beam is established, in which the basic unknowns are the deflection,the rotation angle of the cross section, and the equivalent couple of the pore fluid pressure.This mathematical model can be degenerated into the Euler-Bernoulli model, the Rayleighmodel, and the shear model of the saturated poroelastic beam, respectively. On the basis ofthe 1D mathematical model, the dynamic and quasi-static bending of the saturated poroelasticTimoshenko cantilever bean is investigated with an impermeable fixed end and a permeable freeend subjected to a step load at its free end by the Laplace transform method. It is revealed thatthe interaction coefficient between the beam skeleton and the pore fluid has a damping effect.Moreover, when subjected to a step load, the dynamic deflection at the free end of the saturatedporoelastic Timoshenko cantilever beam has a behavior of damping vibration. With the increaseof the interaction coefficient, the attenuation of the deflection vibration is accelerated, and thetime needed to approach its steady state decreases. Furthermore, the steady state values ofthe deflections are the solution of the classic single phase Timoshenko cantilever beam. At thesame time, it is found that the shearing effect has a significant influence on the deflection at thefree end, while the effect of the rotation inertia of the cross section is small. The shearing effectand the rotation inertia effect of the cross section increase with the decrease of the slendernessratio. In the quasi-static bending, with the increase of the interaction coefficient, the time forthe deflection to approach its steady state values increases, but it is much smaller than that fordynamic responses.

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dynamic and quasi-static bending of saturated poroelastic timoshenko cantilever beam

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