梯度飽和多孔材料中彈性波的截止頻率

周鳳璽, 張家齊, 曹小林

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

基于Biot多孔介質理論,應用WKB(Wentzel-Kramers-Brillouin)法,推導得到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式.求解過程中應用波數趨于零的極限條件,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性質以及非均勻性密切相關.考慮材料參數沿板厚按指數形式變化的飽和多孔板,通過數值算例分析彈性平面波在該類非均質飽和多孔材料中的截止頻率變化規律.數值結果表明,截止頻率隨著板的厚度、孔隙率和非均勻參數及滲透系數的不同均有顯著的變化,驗證了計算結果滿足精確性要求.

功能梯度材料；含液飽和材料；波的傳播；WKB法；截止頻率

多孔介質材料在巖土工程、地球物理以及生物工程等領域有著廣泛的應用,自Biot[1-2]提出描述飽和多孔介質動力特征方程以來,國內外許多專家學者從不同角度對飽和多孔介質中波的傳播問題進行研究[3-7],包括多孔介質動力響應問題的解析研究、數值模擬方法以及波的傳播特性等的研究.目前關于飽和多孔介質波動問題的研究主要集中在幾何特征為無限半空間區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落后于對單相連續彈性介質的研究.

20世紀80年代中期,由日本科學家提出了功能梯度材料的概念,國內外許多專家學者從不同角度,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容,基于Biot多孔介質波動模型,考慮到波數趨于零,在該極限條件下,應用WKB方法求解帶有變系數的微分方程[16],通過理論推導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式,且表達式滿足精確性要求.

1 基本方程

基于Biot多孔介質理論可知,均質飽和多孔介質的基本方程[1-7]如下.

物理方程為

σij=λεkkδij+2μεij-αpδij,

(1)

(2)

幾何關系為

(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;c為波速，c=ω/k,其中ω為圓頻率.

將式(5)代入式(1)～(4),經過整理可以得到液相位移與固相位移的關系為

(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,同時應用波數趨于零的極限條件,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法求解變系數微分方程(9),設

(10)

且令

(11)

將式(10)及式(11)的前3項代入式(9),經過整理可得

(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,飽和介質中SH和SV波的截止頻率相同.

3 數值分析與討論

3.1 精確性驗算

WKB的計算結果必須嚴格滿足精確性要求[16],即

(35)

(36)

(37)

(38)

將式(15)、(25)、(28)分別代入式(35)～(37),可得

3.2 數值計算

為了對平面彈性波的截止頻率進行參數分析,選取多孔飽和介質的材料參數[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.考慮材料力學參數按式(38)沿厚度方向連續變化,對于不同非均勻參數、不同板厚以及不同孔隙率、不同滲透系數下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kf=1.9×10-7m4/(kN·s),孔隙率n0為0.19,圖2、3分別給出當非均勻參數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的變化曲線.可以看出,當飽和多孔板的厚度、密度、孔隙率、滲透系數一定時,截止頻率隨著非均勻參數的增加總體呈現近似于線性的增大,截止頻率明顯受材料非均勻性的影響.有非均勻參數在相同的區間內變化,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 滲透系數和S波截止頻率之間的關系Fig.8 Relations between permeability and cut-off frequencies of S waves

圖9 滲透系數和P波截止頻率之間的關系Fig.9 Relations between permeability and cut-off frequencies of P waves

如圖6、7所示為當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,當滲透系數較小時,截止頻率隨滲透系數的增大而明顯增大；當滲透系數大到一定程度后,S波和P波的截止頻率隨滲透系數的變化不是很大.

4 結 語

基于Biot多孔介質理論,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,模量沿板厚方向按指數形式連續變化的情形,通過數值算例參數分析各影響因素對橫波以及縱波截止頻率的影響規律.數值結果表明：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,孔隙率和滲透系數對截止頻率的影響相對??；彈性波的截止頻率隨板厚度的增大而減小,隨多孔介質中孔隙率增大呈增大趨勢,并且截止頻率隨非均勻參數及滲透系數的增大而增大.

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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. 浙江大學學報(工學版)網址： www.journals.zju.edu.cn/eng

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

周鳳璽(1979—),男,教授,從事巖土力學和非均勻材料結構力學的研究.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