Peer-review under responsibility of the organizing committee of Implast 2016 doi: 10.1016/j.proeng.2016.12.178 ScienceDirect 11thInternational Symposium on Plasticity and Impact Mechani
Trang 1Procedia Engineering 173 ( 2017 ) 1021 – 1028
1877-7058 © 2017 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license
( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).
Peer-review under responsibility of the organizing committee of Implast 2016
doi: 10.1016/j.proeng.2016.12.178
ScienceDirect
11thInternational Symposium on Plasticity and Impact Mechanics, Implast 2016
On Rayleigh waves in self-reinforced layer embedded over an incompressible half-space with varying rigidity and density
a Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad, Jharkhand, India
Abstract
In this paper a mathematical model to study the Rayleigh wave propagation in a self-reinforced layer over an incompressible inhomogeneous elastic half-space with quadratically varying rigidity and linearly varying density has been represented The dis-placement components for both the media have been deduced separately Under suitable boundary conditions the secular equation has been established in closed form Numerical results analyzing the equation are discussed and presented graphically It is quite ocular that the reinforcement in the upper medium and variations in density and rigidity related to the lower semi-infinite medium has remarkable effect on the Rayleigh wave propagation Due to the importance of self-reinforced material, the above problem has certain realistic applications in geophysics and civil engineering.
c
2016 The Authors Published by Elsevier B.V.
Peer-review under responsibility of the organizing committee of Implast 2016.
Keywords: Rayleigh wave; Self-reinforced material; Incompressible, Whittaker function
1 Introduction
The study of elastic waves and the nature of earthquakes remained as paramount interest to the geophysicists The primary work on elastic waves received it impetus from the earlier studies by the scientists around the middle of the nineteenth century Owing to slower attenuation of energy compare to the body waves, Rayleigh type surface waves are the most destructive seismic surface waves that can unobstructedly transmit along the stress free surface with the phase velocities lying within the subsonic range and its amplitude decaying exponentially with depth from the surface
In case of layered earth media, measuring Rayleigh waves is one of the versatile and convenient tool to predict disas-ters caused by earthquakes and to find out various material properties of the layers present in earth’s crust The study
of these surface waves in heterogeneous medium with different boundaries is of great importance to the seismologists Rayleigh waves are generally non-dispersive in nature but exhibit dispersion in case of stratified semi-infinite medium Lord Rayleigh [1] came out with an pioneering work where he showed the how these kind of waves prop-agate in isotropic elastic solid Bromwich [2] demonstrated the Rayleigh type surface wave propagation in isotropic layered media which was found as dispersive
∗Corresponding author Tel.:+91-7870591051.
E-mail address: mostaidahmed@yahoo.in
© 2017 The Authors Published by Elsevier Ltd This is an open access article under the CC BY-NC-ND license
( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).
Peer-review under responsibility of the organizing committee of Implast 2016
Trang 2Two brittle materials of different physical and chemical properties are brought together to form a composite ma-terial Fiber composites produces a composition of much preferred mechanical properties that can not be achieved
by either of the ingredients alone and having fatigue resistance with high strength and stiffness of fibers Due to excessive temperature and pressure these fiber materials are transformed into self-reinforced fibers The ocular prop-erty retained by fiber-reinforced concretes is that its components act together as a single body such that there can be
no relative displacement between them under any external forces Nylon, aramid, vinylester etc lie on the class of fiber-reinforced composites Massive applications of fiber-reinforced composites can be found in the field of structural and civil engineering, aerospace engineering and automobile industries Belfield et al [3] first introduced the idea of continuous self-reinforcement at each point of elastic solid Later on, Spencer [4] established the constitutive equation for fiber-reinforced linearly anisotropic elastic solid in preferred direction Kaur et al [5] formulated the effect of reinforcement, gravity and liquid loading on Rayleigh wave propagation
The layered earth medium comprises of various hard/soft rocks with dissimilar elastic properties When these type
of rocks persist in the way of seismic waves they influence the transmission of waves Any disturbance in the earth’s interior may serve as the basic cause of seismic wave propagation The inhomogeneity plays a pivotal role in study
of dynamic problems on seismic waves In a pioneering work, Birch [6] noted down the nature of seismic waves with increasing depth below the earth’s crust Singh [7] discussed about Rayleigh waves in an inhomogeneous medium Vishwakarma and Gupta [8] demonstrated a case study on Rayleigh wave propagation in an incompressible crustal layer under rigid boundary over a dry sandy and elastic half-space with void pores Recently Vishwakarma and Xu [9] have shown the effect of quadratically varying rigidity and linearly varying density on the Rayleigh wave propagation Sharma and Sharma [10] modeled the phenomenon of thermoelastic Rayleigh waves in a solid over a fluid layer with varying temperature
With these geophysical notions and motivated by certain practical situations we have studied the propagation of Rayleigh waves in self-reinforced layer embedded over an incompressible inhomogeneous half-space by deducing the secular equation The effect of inhomogeneity parameter and thickness of layer on the phase velocity of Rayleigh wave has been identified Within the theoretical framework the outcome of the study bears a great significance in diverse fields like geophysics, geotechnical engineering and civil engineering etc
2 Formulation of the problem
In this present investigation, we consider an incompressible inhomogeneous half-space with quadratically varying
rigidity and density covered with a self-reinforced layer of thickness H Referring to the rectangular Cartesian co-ordinate system, O lies at the common interface of the layer and the semi-infinite medium, x-axis is taken along the propagation of Rayleigh wave and z-axis directed vertically downwards Owing to give it an algebraic formation, the
lower semi-infinite medium is considered to occupy the regionz ≥ 0, whereas the upper layer occupies the region
−H < z < 0 The uppermost boundary surface of the fiber-reinforced layer is defined as z = −H.
Fig 1 Geometry of the Problem
Trang 33 Dynamics of self-reinforced medium
The elasto-dynamical eqns of motion for a three dimensional elastic solid medium are given by
∂τxx
∂x +
∂τxy
∂y +∂τ
xz
∂z = ρ1∂2u1
∂τyx
∂x +
∂τyy
∂y +
∂τyz
∂z = ρ1∂2v1
∂τzx
∂x +
∂τzy
∂y +
∂τzz
∂z = ρ1∂2w1
whereρ1is the density of the material, (u1, v1, w1) are the displacement components and (x , y, z) be the coordinates at
any timet The governing eqn for a self-reinforced linearly elastic medium with preferred direction ‘a’is given by
Belfield [3] as
τi j = λe kkδi j+ 2μT e i j + α(a k a m e kmδi j + a i a j e kk)+ 2(μL− μT )(a i a k e k j + a j a k e ki)+ β(a k a m e km a i a j) (4)
where i, j, k, m = 1, 2, 3 τ i j are the components of stress, e i j= 1
2
u i , j + u j ,i
represent the components of infinitesimal strain,δi j denotes the Kronecker delta a iare components of −→a , all referred to rectangular Cartesian coordinates x
i,
−
→a = (a1, a2, a3) is the preferred directions of reinforcement under the restrictive assumption of a1 + a2 + a3 = 1. The vector −→a may be a function of position The coefficients λ, μ
T, α, β & μLare the elastic constants with dimension
of stress.μTrefers to the shear modulus in transverse shear across the preferred direction,μLas the shear modulus in longitudinal shear in preferred direction.α, β are the specific stress components that are taken into account in regard
of the composite material
If −→a is chosen so that its components are (1, 0, 0) i.e., the preferred direction of reinforcement is everywhere along
x− axis, then (4) leads to
τxx= (λ + 2α + 4μL− 2μT + β) e xx + (λ + α) e yy + (λ + α) e zz
τyy = (λ + α) e xx+ (λ + 2μT)e yy + λe zz
τzz = (λ + α) e xx + λe yy+ (λ + 2μT)e zz
τyz= 2μT e yz, τxz= 2μL e xz τxy= 2μL e xy
The Rayleigh waves are propagating in the x− direction, so we may write
u1= u1(x , z, t) , v1= 0, w1= w1(x , z, t) and ∂y∂ ≡ 0 With the aid of the above relation and eqn (4), the eqns (1)-(3) reduce to
P1∂2u1
∂x2 + P2∂2w1
∂x∂z + μL∂2u1
∂z2 = ρ1∂2u1
μL∂2w1
∂x2 + P2∂2u1
∂x∂z + P3∂2w1
∂z2 = ρ1∂2w1
where P1= (λ + 2α + 4μL− 2μT + β), P2= (α + λ + μL ) and P3= (λ + 2μT)
We assume that the solutions of (5) & (6) as
u1= Ae −skz+ik(x−ct)
w1= Be −skz+ik(x−ct)⎫⎪⎪⎬
Thus using (7) in (5) and (6) we get
s2μL+ ρ1c2− P1
Trang 4
For non-trivial solution of eqns (8) & (9) we get
P3
μL
s4+ M1s2+ N1= 0 The roots of the above eqn is
s j2= −M1± M1 − 4N1
P
3
μL
2P
3
μL
where
M1=
c2
β1
− 1
+ P3
μL
c2
β1
− P1
μL
− P2
μL2, N1=
c2
β1
− P1
μL
c2
β1
− 1
, β1=
μ
L
P1
Thus (8) & (9) yield
w1
u1 = s j
2+ c2
β 1 −P1
μL
i
s j.P2
μL
In light of (10) & (11), the displacement components of the fiber-reinforced layer is given by
u1=A1e −ks1z + A2e −ks2z + A3e −ks3z + A4e −ks4z
w1= η1
A1e −ks1z − A3e −ks1z
+ η2
A2e −ks2z − A4e −ks2z e ik(x −ct) (13)
4 Dynamics of incompressible inhomogeneous half-space
The two dimensional eqn of motion of an incompressible inhomogeneous semi-infinite medium are given by
∂
∂x
λΔ + 2μ2 ∂u2
∂x
+∂z∂
μ2 ∂u
2
∂z +
∂w2
∂x
= ρ2 ∂2u2
∂
∂x
μ2 ∂u
2
∂z +
∂w2
∂x
+ ∂z∂
λΔ + 2μ2 ∂w2
∂z
= ρ2 ∂2u2
whereλ is the Lame’s constant, Δ refers to dilatation, μ
2is shear modulus.ρ2is the density of the medium, u2, w2are
the displacement components in the x & z directions respectively So, for an incompressible medium, we have
Δ = ∂u2
∂x +∂w
2
We may write the displacement components u2andw2in terms of scalar and vector potentials as
u2=∂φ∂x +∂ψ∂z and w2= ∂φ∂z −∂ψ∂x (17) whereφ and ψ are the scalar and vector potentials respectively
We considerμ2 = μ2(1+ γz)2, ρ2 = ρ2(1+ γz) with μ2&ρ2are the rigidity and the density of the medium γ is chosen as an inhomogeneity parameter which has the dimension i.e., inverse of length Now we have the compress-ibility conditionΔ = 0 together with lim
λ→∞, Δ→0→ −P1, P1be the hydrostatic stress Now eqns (14) and (15) can be written as
∂
∂x
−P1+ 4μ2γ (1 + γz) w2− ρ2∂2φ
∂t2
+∂z∂
μ2∇2ψ − ρ2∂2ψ
∂t2
∂
∂z
−P1+ 4μ2γ (1 + γz) w2− ρ2∂2φ
∂t2
−∂x∂
μ2∇2ψ − ρ2∂2ψ
∂t2
Trang 5
which are identically satisfied by
P1= 4μ2γ (1 + γz) w2− ρ2∂2φ
μ2∇2ψ − ρ2∂2ψ
together with∇2φ = 0 where
Here Rayleigh waves are confined to propagate along the x− direction with phase velocity ‘c’ and wave length 2 π
k
u1, w2are the functions of z only apart from a factor e ik(x −ct), thus we may consider
φ = φ1(z) e ik(x −ct)
ψ = ψ1(z) e ik(x −ct)
⎫⎪⎪
⎬
Using the relations (23), eqn (20) and (21) now become equivalent to
d2ψ1
dz2 + k2
2c2
μ2(1+ γz)− 1
d2φ1
In order to solve eqn (24), we assume r= 2k
γ (1+ γz) , z > 0 and with the aid of this transformation eqn (24) reduces
to
d2ψ1
dr2 +
Q
r −1 4
where Q= 1
2
c2
c1
k
γ
and c1 = μ2
ρ 2
Equation (26) is commonly known as Whittaker’s eqn and the solution can be written as
ψ1= B1W Q,1(r) + B2W Q,1(−r) (27)
where W Q,1(r) is the Whittaker’s function and B1, B2are the arbitrary constants
We have considered the following asymptotic expansion of Whittaker’s function [11] upto three terms as
W k , m(z) = e−z
2.z k
⎡
⎢⎢⎢⎢⎢
⎢⎢⎢⎢⎢
⎣1+
m2−k−1
2
2
z.1! +
m2−k−1
2
2
m2−k−3
2
2
⎤
⎥⎥⎥⎥⎥
⎥⎥⎥⎥⎥
⎦
where the value z dominates the values of k, leading to each term very small compared to the previous term The
solution of (25) can be written as
With the help of (28) & (27), the eqn (23) becomes
φ =C1e kz + C2e −kz
e ik(x −ct)
ψ =B1W Q,1(r) + B2W Q,1(−r)e ik(x −ct)
⎫⎪⎪⎪
⎬
Since the displacement at z → ∞ should be zero, the arbitrary constant B2& C1must vanishes Therefore (19) takes
of the form
φ = C2e −kz e ik(x −ct)
ψ = B1W ,1(r) e ik(x −ct)
⎫⎪⎪⎬
Trang 6Thus we will have the displacement components of the form
u2=
C2(ik) e −kz + 2B1k d
dr W Q,1(r)
e ik(x −ct)
w2= C2(−k) e −kz − B1(ik) W Q,1(r) e ik(x −ct)
(31)
5 Boundary conditions and Secular equation
i) At z = −H the surface remains stress free, i.e.,
(τxz)I = 0 and (τzz)II = 0
ii) At the interface i.e., z= 0 of the two different elastic solid the displacement components remains continuous,
u1= u2 and w1= w2
iii) At the interface of the two media i.e., z= 0, the stress components also remains continuous, therefore,
(τxz)I= (τxz)II and (τzz)I = (τzz)II From the first boundary condition and using (12) & (13) we have
(−ks1+ ikη1)e ks1H A1+ (−ks2+ ikη2)e ks2H A2+ (ks1− ikη1).e −ks1H A3+ (ks2− ikη2)e −ks2H A4= 0 (32) Similarly from the second condition and using (12) & (13) we get
(λ + α) ik − ks1η1(λ + 2μT)
e ks1H A1+(λ + α) ik − ks2η2(λ + 2μT)
e ks2H A2+
(λ + α) ik − ks
1η1(λ + 2μT)
e −ks1H A3+(λ + α) ik − ks2η2(λ + 2μT)
e −ks2H A4= 0 (33)
Using (12) & (31) and substituting r= 2k
γ we have
A1+ A2+ A3+ A4− 2k d
dr W Q,1(r)
r=2k
γ
Similarly using (13) & (31) and substituting r=2k
γ we have
η1A1+ η2A2− η1A3− η2A4+ ik W Q, 1(r)
r=2k
γ
B1+ kC2= 0 (35)
With the help of (12), (13) & (31) and putting r= 2k
γ the fifth boundary condition reduces into
μL(−ks1+ ikη1).A1+ μL(−ks2+ ikη2).A2+ μL(ks1− ikη1).A3
+ μL(ks2− ikη2).A4− μ2k2
⎡
⎢⎢⎢⎢⎢
⎢⎣4 d
2
dr2
W Q,1(r) r=2k
γ
+ W Q, 1(r)
r=2k
γ
⎤
⎥⎥⎥⎥⎥
⎥⎦ B1+ i.2μ2k2C2= 0 (36)
Similarly with an aid of (12), (13) & (31) and putting r= 2k
γ, the fifth boundary condition reduces into
(λ + α) ik − ks
1η1(λ + 2μT)
A1+(λ + α) ik − ks2η2(λ + 2μT)
A2+(λ + α) ik − ks1η1(λ + 2μT)
A3
+(λ + α) ik − ks2η2(λ + 2μT)
A4+ i4μ2k2
d
dr W Q,1(r)
r=2k
γ
B1− 2μ2k2C2= 0 (37)
Now eliminating the arbitrary constants A1, A2, A3, A4, B1&C2we have the equation in determinant form
Real
θ11 θ16
θ61 θ66
whereθi j ’s (i , j = 1, 2, , 6) are given in Appendix-I We consider the real one and ignoring the imaginary part for
its attenuating nature Hence the eqn (38) refers the desired secular equation of Rayleigh waves in a self-reinforced layer over an inhomogeneous incompressible half-space
Trang 76 Numerical Computation and Discussion
In order to illustrate to the effect of reinforcement, inhomogeneity in rigidity and density on phase velocity and wave number, we perform some numerical computations taking the following data:
i) For self-reinforced medium [Markham [12]]:
μL= 5.66 × 109N /m2, μT= 2.46 × 109N /m2, α = −1.28 × 109N /m2, β = 220.9 × 109N /m2
λ = 5.65 × 109N /m2, ρ1= 2660Kg/m3
ii) For incompressible, inhomogeneous semi-infinite solid [Gubbins [13]]:
μ2= 78.4 × 109N /m2, ρ2= 3.5350 × 106Kg /m3
1 2 3
1 Γ = 0.18
2 Γ = 0.20
3 Γ = 0.22
0.02
0.04
0.06
0.08
0.10
Wave Number kH
Fig 2 Variation in dimensionless phase velocity c
β1
2
against dimensionless wave number kH for increasing
val-ues of inhomogeneity parameter γ
1 2 3
1 H 14.0
2 H 16.0
3 H 18.0
0.02 0.04 0.06 0.08
Wave Number kH
Fig 3 Variation in dimensionless phase velocity c
β1
2
against dimensionless wave number kH for increasing val-ues of the thickness of layer H
Due to the variation in elastic properties of various materials present in the earth’s media, the phase velocity of Rayleigh type waves is varying with wave number and the whole incident is known as dispersion In Fig 2 and 3, the dispersion curves expound the variation of non-dimensional quantityc
β 1
2
against non-dimensional wave number From figure 2, the dispersion curves delineate that the gradual increase in the inhomogeneity parameterγ present
in the lower semi-infinite medium increases the phase velocity of the Rayleigh type surface waves
In figure 3, it is clearly observed that the phase velocity shows a dual character with the increase of the thickness of
the upper layer With an increase of thickness H, first the phase velocity decreases and after a certain while it shows
an increasing nature
With an overview of this graphical representation, a conclusion can drawn that the inhomogeneity and thickness of layer substantially affects the phase velocities concerned with the proposed earth model
7 Conclusions
An analytical and numerical approach has been adopted to investigate the two layered earth model consists of self-reinforced layer and incompressible inhomogeneous half-space Accumulating the real terms, a closed form expression of the dispersion relation in terms of sixth order determinantal form has been established We have ignored the imaginary terms due to its attenuating nature The observations from the helps to arrive at the conclusions
(i) The inhomogeneity present in the lower half-space influences the phase velocity of the Rayleigh waves by gradually increasing it
(ii) The increasing value of average thickness of layer first affects the phase velocity by decreasing it and then shows
an increasing nature after a certain wave number
(iii) The phase velocity of Rayleigh waves is greatly affected by the wave number
Trang 8The outcome of the model has certain practical applications in order to predict the damage caused by the surface waves As the reinforced concrete material is considered to be the basic ingredient in heavy constructions, it holds
an important aspect for setting up highly efficient and disaster resistant buildings and bridges The study also bears a great significance in diverse fields like geotechnical and civil engineering also
Acknowledgements
The authors are thankful to University Grants Commission, New Delhi for providing fellowship to Mr Mostaid Ahmed through grant number F 7-79/2007(BSR)
References
[1] L Raleigh, J.W Strutt, On waves propagating along the plane surfaces of an elastic solid, Proc London Math Soc., 17 (1885) 4–11 [2] T.J Bromwich, On the influence of gravity on elastic waves, and, in particular on the vibrations of an elastic globe, Proceedings of the London Mathematical Society, 1 (1898) 98–120.
[3] A.J Belfield, T.G Rogers, A.J.M Spencer, Stress in elastic plates reinforced by fibres lying in concentric circles, J Mech Phys Solids, 31 (1983) 25–54.
[4] A.J.M Spencer, Deformations of Fiber-Reinforced Materials, Oxford University Press, London 1972.
[5] T Kaur, S.K Sharma, A.K.Singh, Effect of reinforcement, gravity and liquid loading on Rayleigh-type wave propagation, Meccanica, 51 (2016) 2449–2458.
[6] F Birch, Elasticity and constitution of the Earth’s interior, Journal of Geophysical Research, 57 (1952) 227–286.
[7] P.C Singh, Rayleigh waves in an inhomogeneous medium, Pure and Applied Geophysics, 98 (1972) 87–101.
[8] S.K Vishwakarma, S Gupta, Rayleigh wave propagation:A case wise study in a layer over a half space under the e ffect of rigid boundary, Archives of Civil and Mechanical Engineering, 14 (2014) 181–189.
[9] S.K Vishwakarma, R Xu, Impact of quadratically varying rigidity and linearly varying density on the Rayleigh wave propagation: An analytic solution, International Journal of Solids and Structures, 97 (2016) 182–188.
[10] J.N Sharma, R Sharma, Modelling of thermoelastic Rayleigh waves in a solid underlying a fluid layer with varying temperature, Applied Mathematical Modelling, 33 (2009) 1683–1695.
[11] E.T Whittaker, G.N Watson, A Course in Modern Analysis, Cambridge University Press, Cambridge, 1990.
[12] M.F Markham, Measurement of the elastic constants of fibre composites by ultrasonics, Composites, 1 (1969) 145–149.
[13] D Gubbins, Seismology and Plate Tectonics, Cambridge University Press, Cambridge, 1990.
Appendix-I
θ11= (−s1+ iη1)e ks1H, θ12= (−s2+ iη2)e ks2H, θ13= (s1− iη1).e −ks1H, θ14= (s2− iη2)e −ks2H, θ15= θ16= 0
θ21=(λ + α) i − s1η1(λ + 2μT)
e ks1H, θ22=(λ + α) i − s2η2(λ + 2μT)
e ks2H
θ23=(λ + α) i − s1η1(λ + 2μT)
e −ks1H, θ24=(λ + α) i − s2η2(λ + 2μT)
e −ks2H, θ25= θ26= 0
θ31= 1, θ32= 1, θ33= 1, θ34= 1, θ35= −2k d
dr W Q,1(r)
r=2k
γ, θ36= −ik
θ41= η1, θ42= η2, θ43= η3, θ44= η4, θ45= −ik W Q, 1(r)
r=2k
γ, θ46= k
θ51= μL(−s1+ iη1), θ52= μL(−s2+ iη2), θ53= μL(s1− iη1), θ54= μL(s2− iη2)
θ55= −μ2k
4dr d22
W Q,1(r) r=2k
γ + W Q, 1(r)
r=2k
γ
, θ56= i.2μ2k
θ61=(λ + α) i − s1η1(λ + 2μT), θ
62=(λ + α) i − s2η2(λ + 2μT)
θ63=(λ + α) i − s1η1(λ + 2μT), θ
64=(λ + α) i − s2η2(λ + 2μT)
θ65= i4μ2k
d
dr W Q,1(r)
r=2k
γ
, θ66= −2μ2k
... consists of self- reinforced layer and incompressible inhomogeneous half- space Accumulating the real terms, a closed form expression of the dispersion relation in terms of sixth order determinantal... Xu, Impact of quadratically varying rigidity and linearly varying density on the Rayleigh wave propagation: An analytic solution, International Journal of Solids and Structures, 97 (2016) 182–188.... self- reinforced layer over an inhomogeneous incompressible half- space Trang 76 Numerical Computation and