周鳳璽, 張家齊, 曹小林

(1.蘭州理工大學(xué) 土木工程學(xué)院, 甘肅 蘭州 730050;2.西部土木工程防災(zāi)減災(zāi)教育部工程研究中心, 甘肅 蘭州 730050)

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梯度飽和多孔材料中彈性波的截止頻率

周鳳璽1,2, 張家齊1, 曹小林1

(1.蘭州理工大學(xué) 土木工程學(xué)院, 甘肅 蘭州 730050;2.西部土木工程防災(zāi)減災(zāi)教育部工程研究中心, 甘肅 蘭州 730050)

基于Biot多孔介質(zhì)理論,應(yīng)用WKB(Wentzel-Kramers-Brillouin)法,推導(dǎo)得到SH波和Lamb波在梯度非均勻含液飽和材料中截止頻率的解析表達(dá)式.求解過程中應(yīng)用波數(shù)趨于零的極限條件,通過簡化控制方程,獲得平面SH波和P-SV波的截止頻率.問題的解答揭示了截止頻率與材料的物理力學(xué)性質(zhì)以及非均勻性密切相關(guān).考慮材料參數(shù)沿板厚按指數(shù)形式變化的飽和多孔板,通過數(shù)值算例分析彈性平面波在該類非均質(zhì)飽和多孔材料中的截止頻率變化規(guī)律.數(shù)值結(jié)果表明,截止頻率隨著板的厚度、孔隙率和非均勻參數(shù)及滲透系數(shù)的不同均有顯著的變化,驗證了計算結(jié)果滿足精確性要求.

功能梯度材料；含液飽和材料；波的傳播；WKB法；截止頻率

多孔介質(zhì)材料在巖土工程、地球物理以及生物工程等領(lǐng)域有著廣泛的應(yīng)用,自Biot[1-2]提出描述飽和多孔介質(zhì)動力特征方程以來,國內(nèi)外許多專家學(xué)者從不同角度對飽和多孔介質(zhì)中波的傳播問題進行研究[3-7],包括多孔介質(zhì)動力響應(yīng)問題的解析研究、數(shù)值模擬方法以及波的傳播特性等的研究.目前關(guān)于飽和多孔介質(zhì)波動問題的研究主要集中在幾何特征為無限半空間區(qū)域或有限厚度的巖土類材料,而對于多孔飽和材料,特別是非均勻飽和材料中彈性波的截止頻率分析研究遠(yuǎn)落后于對單相連續(xù)彈性介質(zhì)的研究.

20世紀(jì)80年代中期,由日本科學(xué)家提出了功能梯度材料的概念,國內(nèi)外許多專家學(xué)者從不同角度,包括功能梯度材料的制備[8-9],彈性波在功能梯度材料中的傳播特性[10-15]等問題展開了一系列的研究工作,并取得了豐碩的成果.本文以梯度非均勻含液飽和板中彈性波的截止頻率為研究內(nèi)容,基于Biot多孔介質(zhì)波動模型,考慮到波數(shù)趨于零,在該極限條件下,應(yīng)用WKB方法求解帶有變系數(shù)的微分方程[16],通過理論推導(dǎo)可以獲得非均勻含液飽和板中彈性波截止頻率的解析表達(dá)式,且表達(dá)式滿足精確性要求.

1 基本方程

基于Biot多孔介質(zhì)理論可知,均質(zhì)飽和多孔介質(zhì)的基本方程[1-7]如下.

物理方程為

σij=λεkkδij+2μεij-αpδij,

(1)

(2)

幾何關(guān)系為

(3)

運動方程為

(4)

(5)

考慮飽和材料中的簡諧波,則固相位移分量可以表示為

(6)

式中：k為x1方向的波數(shù);c為波速，c=ω/k,其中ω為圓頻率.

將式(5)代入式(1)～(4),經(jīng)過整理可以得到液相位移與固相位移的關(guān)系為

(7)

2 控制方程及通解

2.1 SH波傳播的截止頻率

圖1 非均勻含液飽和材料Fig.1 Functionally graded fluid-saturated materials

將式(3)和(1)代入式(4),考慮SH波的位移矢量,同時應(yīng)用波數(shù)趨于零的極限條件,可得

(8)

將式(7)代入式(8),可以得到SH波傳播的控制方程為

(9)

式中：

采用WKB法求解變系數(shù)微分方程(9),設(shè)

(10)

且令

(11)

將式(10)及式(11)的前3項代入式(9),經(jīng)過整理可得

(12)

省略ω-1和ω-2項,并考慮ω2、ω、ω0項前的系數(shù)為零,可得

(13)

(14)

(15)

將式(13)、(14)代入式(10),可得

(16)

(17)

通過求解方程(17),可得SH波的截止頻率為

(18)

2.2 Lamb波傳播的截止頻率

(19)

(20)

(22)

將式(21)、式(22)的前3項分別代入式(19)、(20),計算同SH波的截止頻率求解中的式(12),并考慮系數(shù)項為零可以得到

(23)

(24)

(25)

(26)

(27)

(28)

式中：

將式(23)、(24)和(26)、 (27)分別代入式(21),可得

(29)

(30)

(31)

(32)

飽和介質(zhì)中的P波和SV波的截止頻率分別為

(33)

(34)

比較式(33)、(18)可知,飽和介質(zhì)中SH和SV波的截止頻率相同.

3 數(shù)值分析與討論

3.1 精確性驗算

WKB的計算結(jié)果必須嚴(yán)格滿足精確性要求[16],即

(35)

(36)

(37)

(38)

將式(15)、(25)、(28)分別代入式(35)～(37),可得

3.2 數(shù)值計算

為了對平面彈性波的截止頻率進行參數(shù)分析,選取多孔飽和介質(zhì)的材料參數(shù)[17]為：λ0=2.0×107kN/m2,μ0=0.6×107kN/m2,α=0.908,M=1.29×107kN/m2,ρf=1 000 kg/m3,ρ=2 458 kg/m3,η=10-3Pa·s.考慮材料力學(xué)參數(shù)按式(38)沿厚度方向連續(xù)變化,對于不同非均勻參數(shù)、不同板厚以及不同孔隙率、不同滲透系數(shù)下P波以及S波的截止頻率ωP、ωS進行分析.

取板厚h=15 cm,滲透系數(shù)kf=1.9×10-7m4/(kN·s),孔隙率n0為0.19,圖2、3分別給出當(dāng)非均勻參數(shù)β為-0.9～0.9時,S波和P波的前4階截止頻率隨非均勻參數(shù)的變化曲線.可以看出,當(dāng)飽和多孔板的厚度、密度、孔隙率、滲透系數(shù)一定時,截止頻率隨著非均勻參數(shù)的增加總體呈現(xiàn)近似于線性的增大,截止頻率明顯受材料非均勻性的影響.有非均勻參數(shù)在相同的區(qū)間內(nèi)變化,P波的截止頻率明顯大于S波的截止頻率.

圖2 S波截止頻率隨非均勻參數(shù)的變化Fig.2 Relations between gradient coefficient and cut-off frequencies of S wave

圖3 P波截止頻率隨非均勻參數(shù)的變化Fig.3 Relations between gradient coefficient and cut-off frequencies of P wave

圖4 S波截止頻率隨板厚的變化Fig.4 Relations between plate thickness and cut-off frequencies of S wave

圖5 P波截止頻率隨板厚的變化Fig.5 Relations between thickness and cut-off frequencies of P wave

為了分析板厚對截止頻率的影響,在取kf=1.9×10-7m4/(kN·s),n0=0.19,β=0.5的情形下,當(dāng)板厚h為10～20 cm時,圖4、5分別繪出S波和P波的前4階截止頻率隨板厚的變化曲線.可以看出,截止頻率隨著板的厚度增加呈現(xiàn)非線性減小且減小速率越來越小,截止頻率明顯受材料厚度的影響.

圖6 S波截止頻率隨孔隙率的變化Fig.6 Relations between porosity and cut-off frequencies of S wave

圖7 P波截止頻率隨孔隙率的變化Fig.7 Relations between porosity and cut off frequencies of P wave

圖8 滲透系數(shù)和S波截止頻率之間的關(guān)系Fig.8 Relations between permeability and cut-off frequencies of S waves

圖9 滲透系數(shù)和P波截止頻率之間的關(guān)系Fig.9 Relations between permeability and cut-off frequencies of P waves

如圖6、7所示為當(dāng)h=15 cm,β=0.5,n0為0.15～0.25時,S波和P波截止頻率的變化.可以看出,當(dāng)孔隙率較小時,截止頻率隨孔隙率的增加明顯增大,而隨著孔隙率的進一步增加,兩類波的截止頻率均變化不大.取h=15 cm,β=0.5,圖8、9分別給出當(dāng)kf為10-6～10-3m4/(kN·s)時S波和P波的截止頻率.由圖8、9可以看出,當(dāng)滲透系數(shù)較小時,截止頻率隨滲透系數(shù)的增大而明顯增大；當(dāng)滲透系數(shù)大到一定程度后,S波和P波的截止頻率隨滲透系數(shù)的變化不是很大.

4 結(jié) 語

基于Biot多孔介質(zhì)理論,通過理論分析獲得SH和Lamb波在梯度非均勻含液飽和板中傳播的截止頻率.考慮材料密度為常數(shù),模量沿板厚方向按指數(shù)形式連續(xù)變化的情形,通過數(shù)值算例參數(shù)分析各影響因素對橫波以及縱波截止頻率的影響規(guī)律.數(shù)值結(jié)果表明：在相同條件下,P波的截止頻率總大于S波的截止頻率；材料的非均勻性和板的厚度以及材料的密度對截止頻率的影響較大,孔隙率和滲透系數(shù)對截止頻率的影響相對?。粡椥圆ǖ慕刂诡l率隨板厚度的增大而減小,隨多孔介質(zhì)中孔隙率增大呈增大趨勢,并且截止頻率隨非均勻參數(shù)及滲透系數(shù)的增大而增大.

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Analysis of cut-off frequencies for functionally graded fluid-saturated materials

ZHOU Feng-xi1,2, ZHANG Jia-qi1, CAO Xiao-lin1

(1.DepartmentofGeotechnicalEngineering,LanzhouUniversityofTechnology,Lanzhou730050,China;2.TheWesternCivilEngineeringDisasterPreventionandMitigationEngineeringResearchCenteroftheMinistryofEducation,Lanzhou730050,China)

The analytical expression for the cut-off frequencies of the horizontal shear waves (SH wave) and Lamb waves (P-SV wave) in a functionally graded inhomogeneous fluid-saturated media was deduced by applying the WKB (Wentzel-Kramers-Brillouin) method based on Biot’s theory of poroelastic medium. In the process of solving, the cut-off frequencies of SH wave and P-SV wave were obtained with the limiting condition of the wave number approaching zero by simplifying the governing equation. The solution to the problem revealed that the cut-off frequencies were closely associated with the physico-mechanical properties and the heterogeneity of the material. The material parameter of the fluid-saturated poroelastic plate changing along the thickness direction as an exponential form was considered. The changing regularity of the elastic plane wave’s cut-off frequencies in the inhomogeneous fluid-saturated porous plate was analyzed by numerical examples. The numerical results showed that the cut-off frequencies were related to the material properties of the fluid-saturated material, including thickness of the plate, porosity, gradient index and permeability. The accuracy of the numerical solution was validated.

functionally graded materials; fluid-saturated poroelastic plate; propagation of wave; Wentzel-Kramers-Brillouin method; cut-off frequency

2015-01-10. 浙江大學(xué)學(xué)報(工學(xué)版)網(wǎng)址： www.journals.zju.edu.cn/eng

國家自然科學(xué)基金資助項目(51368038,11162008)；甘肅省環(huán)保廳科研資助項目(GSEP-2014-23)；甘肅省教育廳研究生導(dǎo)師基金資助項目(1103-07).

周鳳璽(1979—),男,教授,從事巖土力學(xué)和非均勻材料結(jié)構(gòu)力學(xué)的研究.ORCID; 0000-0003-4709-2419. E-mail: geolut@163.com

10.3785/j.issn.1008-973X.2016.04.020

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1008-973X(2016)04-0744-06