張常光+趙均海+杜文超

文章編號(hào)：16732049(2014)02000614

收稿日期：20131127

基金項(xiàng)目：國(guó)家自然科學(xué)基金項(xiàng)目(41202191)；中國(guó)博士后科學(xué)基金項(xiàng)目(2012M520079，2013T60868)；教育部高等學(xué)校博士學(xué)科點(diǎn)專項(xiàng)科研基金項(xiàng)目(20120205120001)；陜西省自然科學(xué)基礎(chǔ)研究計(jì)劃項(xiàng)目(2014JQ7290)；中央高?；究蒲袠I(yè)務(wù)費(fèi)專項(xiàng)資金項(xiàng)目(2013G2283007,2014G1281072)

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摘要：綜述巖石中間主應(yīng)力效應(yīng)試驗(yàn)和強(qiáng)度理論的研究進(jìn)展，總結(jié)了巖石中間主應(yīng)力效應(yīng)的基本規(guī)律和現(xiàn)有巖石強(qiáng)度理論的特點(diǎn)與不足，最后介紹了統(tǒng)一強(qiáng)度理論及其對(duì)巖石真三軸試驗(yàn)的預(yù)測(cè)與在巖土工程中的應(yīng)用。研究結(jié)果表明：中間主應(yīng)力效應(yīng)及其區(qū)間性是巖石強(qiáng)度的重要特性，且已是目前巖石強(qiáng)度準(zhǔn)則建立所必須考慮的基本問題之一；應(yīng)用統(tǒng)一強(qiáng)度理論并考慮更多復(fù)雜因素的綜合影響，可得到更符合實(shí)際情況的較理想解答。

關(guān)鍵詞：巖石；真三軸試驗(yàn)；中間主應(yīng)力；強(qiáng)度理論

中圖分類號(hào)：TU452 文獻(xiàn)標(biāo)志碼：A

Advances in Rock for Intermediate Principal Stress Effect and Strength Theory

ZHANG Changguang, ZHAO Junhai, DU Wenchao

（School of Civil Engineering, Changan University, Xian 710061, Shaanxi, China）

Abstract: Authors firstly summarized the advances in rock of the intermediate principal stress effect test and strength theory, then found out some basic laws of the intermediate principal stress effect for rock, and characteristics and shortcomings of the existing rock strength theories. Finally, authors introduced the unified strength theory, and its prediction for rock true triaxial tests and applications in geotechnical engineering. The study results show that the intermediate principal stress effect and its range is an important characteristic for rock strength and has been considered as one basic problem for proposing a new failure criterion; a better solution more consistent with actual situations can be obtained by using unified strength theory with taking combined effects of more complex factors into account.

Key words: rock; true triaxial test; intermediate principal stress; strength theory

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0引 言

巖石強(qiáng)度理論是研究巖體強(qiáng)度理論的基礎(chǔ)，是巖石本構(gòu)關(guān)系的重要組成部分。研究巖石強(qiáng)度理論的目的，就在于根據(jù)一定應(yīng)力狀態(tài)下巖石強(qiáng)度試驗(yàn)的結(jié)果，建立其強(qiáng)度準(zhǔn)則，從而得到一般應(yīng)力狀態(tài)下的巖石破壞判據(jù)。巖石真三軸試驗(yàn)是一種最全面的強(qiáng)度試驗(yàn)，它不僅是建立巖石強(qiáng)度準(zhǔn)則的重要資料，更是檢驗(yàn)巖石強(qiáng)度準(zhǔn)則的有效依據(jù)。巖石真三軸試驗(yàn)要求對(duì)試件施加3對(duì)相互獨(dú)立均勻的主應(yīng)力，對(duì)試驗(yàn)機(jī)的加載能力與控制、變形量測(cè)、端部摩擦約束效應(yīng)等要求都非常高。真三軸試驗(yàn)研究的一個(gè)重要內(nèi)容，就是中間主應(yīng)力效應(yīng)研究。巖石的中間主應(yīng)力效應(yīng)研究，不僅具有理論上的意義，而且還具有巨大的工程實(shí)際意義和社會(huì)經(jīng)濟(jì)效益。一方面，如果巖石強(qiáng)度與中間主應(yīng)力σ2無(wú)關(guān)，只需要考慮大主應(yīng)力σ1和小主應(yīng)力σ3，則巖石強(qiáng)度理論可以得到很大簡(jiǎn)化；在復(fù)雜應(yīng)力試驗(yàn)設(shè)備研制中，也只需要有能施加2個(gè)方向應(yīng)力的試驗(yàn)機(jī)；在工程實(shí)踐中只需要分析大主應(yīng)力σ1和小主應(yīng)力σ3，就可以很方便地應(yīng)用現(xiàn)有強(qiáng)度理論來解決實(shí)際問題。另一方面，巖石中間主應(yīng)力效應(yīng)研究十分困難，除了真三軸試驗(yàn)設(shè)備復(fù)雜、試驗(yàn)技術(shù)要求高、經(jīng)費(fèi)投入大外，還有靜水應(yīng)力效應(yīng)掩蓋了中間主應(yīng)力效應(yīng)［1］，要把它獨(dú)立出來，需要明確力學(xué)和物理概念；MohrCoulomb強(qiáng)度準(zhǔn)則和HoekBrown經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則可以適用于拉壓特性不同的巖石材料，又可以解釋靜水應(yīng)力效應(yīng)，已被廣泛地了解和接受，而這2個(gè)理論都沒有考慮中間主應(yīng)力σ2；在理論上要提出一個(gè)有一定的物理概念、數(shù)學(xué)表達(dá)式簡(jiǎn)單又能反映中間主應(yīng)力效應(yīng)的新強(qiáng)度理論并非易事，因此，巖石中間主應(yīng)力效應(yīng)問題已成為各國(guó)學(xué)者熱心研究而經(jīng)久不衰的一個(gè)問題［2］。本文中筆者在綜述巖石中間主應(yīng)力效應(yīng)試驗(yàn)和強(qiáng)度理論研究進(jìn)展的基礎(chǔ)上，總結(jié)了巖石中間主應(yīng)力效應(yīng)的基本規(guī)律和現(xiàn)有巖石強(qiáng)度理論的特點(diǎn)與不足，最后介紹了統(tǒng)一強(qiáng)度理論及其對(duì)巖石真三軸試驗(yàn)的預(yù)測(cè)與在巖土工程中的應(yīng)用。

[HS2][HT4H][STHZ][WTHZ]1巖石中間主應(yīng)力效應(yīng)試驗(yàn)進(jìn)展

從20世紀(jì)初Karman和Boker開始研究巖石中間主應(yīng)力效應(yīng)，到現(xiàn)在已有約100年的歷史，他們發(fā)現(xiàn)巖石三軸拉伸時(shí)的強(qiáng)度高于三軸壓縮時(shí)的強(qiáng)度，這是MohrCoulomb強(qiáng)度準(zhǔn)則和HoekBrown經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則所不能解釋的。巖石中間主應(yīng)力效應(yīng)試驗(yàn)在20世紀(jì)60年代取得了重要進(jìn)展，20世紀(jì)70年代有了比較明確的結(jié)論，代表性的研究者有Hobbs［3］，Murrell［4］，Handin等［5］，Hoskins［6］。在20世紀(jì)七八十年代，Mogi為巖石中間主應(yīng)力效應(yīng)的闡明做出了杰出的貢獻(xiàn)［711］，他改造原有軸對(duì)稱三軸試驗(yàn)機(jī)，成功研制了世界上第1臺(tái)巖石真三軸試驗(yàn)機(jī)，得到一系列不同小主應(yīng)力σ3時(shí)的中間主應(yīng)力效應(yīng)曲線，利用試驗(yàn)充分證明了巖石中間主應(yīng)力效應(yīng)的存在。Mechelis［1213］對(duì)大理巖進(jìn)行了真三軸試驗(yàn)，并指出中間主應(yīng)力效應(yīng)是巖石材料的重要特性。Takahashi等［14］對(duì)沉積巖進(jìn)行了真三軸強(qiáng)度和變形試驗(yàn)。Haimson等［1522］改進(jìn)了Mogi的真三軸試驗(yàn)機(jī)，提高了試驗(yàn)機(jī)的加載能力，簡(jiǎn)化了操作程序，對(duì)多種脆性硬巖進(jìn)行了真三軸試驗(yàn)，探討中間主應(yīng)力對(duì)強(qiáng)度、變形和剪脹的影響規(guī)律，同時(shí)利用電子掃描顯微鏡觀測(cè)裂紋發(fā)展和應(yīng)變局部化情況，并將巖石真三軸試驗(yàn)結(jié)果應(yīng)用于確定地應(yīng)力分布等。中國(guó)臺(tái)灣車籠埔斷層粉砂巖的真三軸試驗(yàn)結(jié)果［21］如圖1所示。

圖1粉砂巖的真三軸試驗(yàn)結(jié)果

Fig.1True Triaxial Test Results of Siltstone

從20世紀(jì)80年代以來，中國(guó)真三軸試驗(yàn)機(jī)的研制取得了很大進(jìn)展，張金鑄等［23］、許東俊等［2425］對(duì)多種不同巖石進(jìn)行真三軸試驗(yàn)，發(fā)現(xiàn)了中間主應(yīng)力效應(yīng)的區(qū)間性，尹光志等［26］對(duì)嘉陵江石灰?guī)r的真三軸試驗(yàn)也得出同樣的結(jié)論。許東俊等［27］和耿乃光等［2829］指出，中間主應(yīng)力σ2的改變(在σ1和σ3都不變的情況下，增加或減小σ2)，可以引起巖石的破壞，甚至可能引發(fā)地震。李小春等［30］對(duì)拉西瓦水電站的花崗巖進(jìn)行了較完整的真三軸試驗(yàn)，并驗(yàn)證了雙剪應(yīng)力強(qiáng)度理論的正確性。陶振宇等［31］和高延法等［32］對(duì)紅砂巖進(jìn)行了中間主應(yīng)力效應(yīng)試驗(yàn)，并收集了各國(guó)多種巖石真三軸試驗(yàn)資料，分析后指出中間主應(yīng)力影響系數(shù)(在σ3一定時(shí)，從σ2=σ3開始變化σ2的過程中，所得最大極限荷載σ1max相對(duì)于對(duì)稱三軸壓縮時(shí)強(qiáng)度的提高系數(shù))最低為18%，最高為75%，一般在25%~40%范圍內(nèi)。明治清等［33］研制了拉壓真三軸儀，[HJ2mm]進(jìn)行了巖石相似材料的真三軸試驗(yàn)，并驗(yàn)證了廣義雙剪應(yīng)力準(zhǔn)則的正確性。陳景濤等［34］和向天兵等［3536］對(duì)多種硬巖進(jìn)行了卸載與支護(hù)不同應(yīng)力路徑下的真三軸與聲發(fā)射試驗(yàn)。楊繼華等［37］和劉漢東等［38］采用LYC拉壓真三軸儀對(duì)完整巖體和節(jié)理巖體模型進(jìn)行了試驗(yàn)。

以上巖石真三軸試驗(yàn)結(jié)果和得出的曲線都是十分有價(jià)值的，但是有些試驗(yàn)的數(shù)量不夠或只集中于某一些特殊的應(yīng)力狀態(tài)，比較系統(tǒng)的巖石極限面試驗(yàn)包括中間主應(yīng)力效應(yīng)試驗(yàn)、不同應(yīng)力角的子午極限線試驗(yàn)和π平面極限線試驗(yàn)，可為工程中驗(yàn)證和選用強(qiáng)度理論提供更加全面的依據(jù)。

另外，Mogi［11］，Tiwari等［3940］對(duì)各向異性巖石進(jìn)行了真三軸試驗(yàn)，指出中間主應(yīng)力效應(yīng)與弱面傾向、巖石種類等密切相關(guān)。張強(qiáng)勇等［41］、朱維申等［4243］、陳安敏等［44］、孫曉明等［45］以及姜耀東等［46］對(duì)真三向應(yīng)力作用下的隧道變形與破壞開展了大型地質(zhì)力學(xué)模型試驗(yàn)研究，更加全面地認(rèn)識(shí)真實(shí)地應(yīng)力下巖石的強(qiáng)度特性、破裂過程及機(jī)制和隧道的錨固效應(yīng)。

總之，巖石的中間主應(yīng)力效應(yīng)已經(jīng)被大量的試驗(yàn)所證實(shí)，并認(rèn)為是巖石的一個(gè)重要特性。巖石中間主應(yīng)力效應(yīng)的基本規(guī)律為［1,25］：

(1)中間主應(yīng)力σ2對(duì)巖石強(qiáng)度有明顯的影響。在小主應(yīng)力σ3一定的應(yīng)力狀態(tài)下，增加σ2的各種應(yīng)力狀態(tài)(σ1≥σ2>σ3)下的巖石強(qiáng)度均大于軸對(duì)稱壓縮狀態(tài)（σ1>σ2=σ3）下的巖石強(qiáng)度，因此常規(guī)三軸壓縮狀態(tài)下所得出的巖石強(qiáng)度均偏低，考慮中間主應(yīng)力σ2效應(yīng)，巖石強(qiáng)度可以提高20%~30%。

(2)中間主應(yīng)力σ2效應(yīng)存在區(qū)間性。中間主應(yīng)力σ2從σ2=σ3的下限值增加到σ2=σ1的上限值過程中，巖石的強(qiáng)度先逐漸增加，達(dá)到一定峰值后隨著σ2的繼續(xù)增加而逐漸降低。三軸拉伸（σ2=σ1>σ3）時(shí)巖石的強(qiáng)度略高于三軸壓縮（σ2=σ3<σ1）時(shí)的巖石強(qiáng)度。

(3)在一定應(yīng)力狀態(tài)下，單獨(dú)改變(增加或減小)中間主應(yīng)力σ2可以引起巖石的破壞。巖石越致密堅(jiān)硬，中間主應(yīng)力σ2效應(yīng)越大，但是仍小于小主應(yīng)力σ3的圍壓效應(yīng)。

在工程應(yīng)用中，考慮巖石的中間主應(yīng)力效應(yīng)，可以充分發(fā)揮巖石材料的強(qiáng)度潛能，減弱支護(hù)強(qiáng)度或降低襯砌厚度，減小工程投資，這是對(duì)巖石中間主應(yīng)力效應(yīng)及其應(yīng)用的一個(gè)重要推動(dòng)。但是如何在理論上使用比較簡(jiǎn)單的數(shù)學(xué)表達(dá)式描述巖石強(qiáng)度，對(duì)其中間主應(yīng)力效應(yīng)的各個(gè)規(guī)律進(jìn)行解釋，并且能靈活地適用于各種巖石不同程度的中間主應(yīng)力效應(yīng)，這是20世紀(jì)90年代以來的又一難題［1］。

[HS2][HT4H][STHZ][WTHZ]2巖石強(qiáng)度理論研究進(jìn)展

建立一個(gè)科學(xué)合理的巖石強(qiáng)度理論，對(duì)于工程設(shè)計(jì)、災(zāi)害預(yù)防、資源開發(fā)等領(lǐng)域都具有重要意義。俞茂宏［47］將眾多強(qiáng)度理論劃分為單剪強(qiáng)度理論、雙剪強(qiáng)度理論和八面體剪應(yīng)力強(qiáng)度理論三大系列。沈珠江［48］?jiǎng)t將強(qiáng)度理論分為理論公式、經(jīng)驗(yàn)公式和內(nèi)插公式三大類。

在巖石強(qiáng)度理論的發(fā)展歷程中，最初引用金屬?gòu)?qiáng)度(屈服)理論和土體強(qiáng)度理論，后來隨著巖石試驗(yàn)技術(shù)的發(fā)展，逐漸發(fā)現(xiàn)巖石材料的基本力學(xué)特性［49］：?jiǎn)屋S拉壓強(qiáng)度不等，拉伸子午線與壓縮子午線不重合(應(yīng)力Lode角效應(yīng))，靜水應(yīng)力效應(yīng)，中間主應(yīng)力效應(yīng)及其區(qū)間性，不同的巖石材料具有不同程度的中間主應(yīng)力效應(yīng)以及屈服面的外凸性等。結(jié)合已有巖石試驗(yàn)結(jié)果，修正金屬和土體的強(qiáng)度理論，至今已提出幾十個(gè)巖石的屈服或破壞準(zhǔn)則。

目前巖土工程中最常用的當(dāng)數(shù)MohrCoulomb強(qiáng)度準(zhǔn)則和外接圓DruckerPrager準(zhǔn)則，前者忽略中間主應(yīng)力影響而計(jì)算偏保守，后者夸大中間主應(yīng)力影響而計(jì)算偏危險(xiǎn)，實(shí)際上二者對(duì)中間主應(yīng)力的處理為2個(gè)極端情況。

MohrCoulomb強(qiáng)度準(zhǔn)則的表達(dá)式為

σ1-σ3=σ1+σ32sin（φ）+ccos（φ）(1)

式中：c，φ分別為材料的粘聚力和內(nèi)摩擦角。

MohrCoulomb強(qiáng)度準(zhǔn)則屬于單剪強(qiáng)度理論，只考慮了最大剪應(yīng)力τ13=(σ1-σ3)/2及其面上的正應(yīng)力σ13=(σ1+σ3)/2對(duì)材料屈服或破壞的影響，只適用于σ2=σ3<σ1的軸對(duì)稱特殊應(yīng)力狀態(tài)，沒有考慮材料的中間主應(yīng)力σ2效應(yīng)，與很多材料的真三軸試驗(yàn)結(jié)果相差較大。

外接圓DruckerPrager準(zhǔn)則的表達(dá)式為

J2=βI1+k(2)

式中：β，k均為材料強(qiáng)度參數(shù)；I(xiàn)1為應(yīng)力張量第一不變量，也稱為靜水應(yīng)力,I(xiàn)1=σ1+σ2+σ3；J2為應(yīng)力偏量第二不變量,J2=［(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2］/6。

外接圓DruckerPrager準(zhǔn)則屬于八面體剪應(yīng)力強(qiáng)度理論，也稱為廣義Mises準(zhǔn)則，考慮了中間主應(yīng)力效應(yīng)和靜水應(yīng)力效應(yīng)，并且具有光滑圓錐極限面，在大型計(jì)算分析軟件中得到了廣泛應(yīng)用和推廣。但是外接圓DruckerPrager準(zhǔn)則認(rèn)為中間主應(yīng)力σ2對(duì)材料強(qiáng)度的影響和小主應(yīng)力σ3一樣，高估了σ2對(duì)材料強(qiáng)度的提高作用，同時(shí)沒有反映巖土材料拉壓異性、應(yīng)力Lode角效應(yīng)等基本力學(xué)特性，難以和巖土材料的真三軸試驗(yàn)相吻合。

在眾多巖石經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則中，HoekBrown經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則被工程所接受，廣泛應(yīng)用于巖石邊坡和地下隧道工程，其主要原因在于它與許多巖石軸對(duì)稱三軸試驗(yàn)結(jié)果相吻合，同時(shí)參數(shù)取值能反映巖體的結(jié)構(gòu)特征。HoekBrown經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則是Hoek等［50］通過試錯(cuò)法擬合大量軸對(duì)稱三軸試驗(yàn)結(jié)果提出的，最初應(yīng)用于完整巖石，后來曾進(jìn)行多次調(diào)整［51］，最新版（2002版）的廣義HoekBrown經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則［52］的表達(dá)式為

[JB(]σ1=σ3+σc(mbσ3 σc+s)a

mb=miexp(I(xiàn)GS-100 28-14D)

s=exp(I(xiàn)GS-100 8-3D)

a=0.5+1 6［exp(-I(xiàn)GS 15)-exp(-20 3)］

(3)

式中：σc為完整巖石的單軸抗壓強(qiáng)度；mb，s，a均為巖體的材料參數(shù)；mi為完整巖石的mb值，可根據(jù)巖體的巖性、結(jié)構(gòu)和構(gòu)造確定；D為巖體擾動(dòng)參數(shù)，與巖體的開挖方式及擾動(dòng)程度有關(guān)，取值范圍為0~1，0表示未擾動(dòng)狀態(tài)；I(xiàn)GS為地質(zhì)強(qiáng)度指標(biāo)，與巖體的巖性、結(jié)構(gòu)和不連續(xù)面等有關(guān)，可通過對(duì)表面開挖或暴露的巖體進(jìn)行肉眼觀察或經(jīng)驗(yàn)判斷來評(píng)定。

HoekBrown經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則和MohrCoulomb強(qiáng)度準(zhǔn)則以及很多其他經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則一樣，均忽略了中間主應(yīng)力效應(yīng)及其區(qū)間性，只適用于三軸等圍壓試驗(yàn)的應(yīng)力狀態(tài)，并不能代表巖石在一般三向應(yīng)力狀態(tài)下的強(qiáng)度特性。不少學(xué)者對(duì)HoekBrown經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則進(jìn)行了修正，以考慮中間主應(yīng)力σ2的影響，朱合華等［53］對(duì)其進(jìn)行了很好的總結(jié)。

Pan等［54］提出的經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則通式為

9 2σcτ2oct+3 22mbτoct-mbI(xiàn)1=sσc

(4)

昝月穩(wěn)等［55］和Yu等［56］提出的經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則通式為

[JB(]F=σ1-1 1+b(bσ2+σ3)-σc［mb (1+b)σc·(bσ2+σ3)+s］a=0 F≥F′

F′=1 1+b(σ1+bσ2)-σ3-σc(mbσ3 σc+s)a=0F′＞F

(5)

Zhang等［57］提出的經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則通式為

9 2σcτ2oct+3 22mbτoct-mbσ13=sσc

(6)

Zhang［58］提出的經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則通式為

1 σ1/a-1c(3 2τoct)1/a+mb 23 2τoct-mbσ13=sσc

(7)

式中：F，F(xiàn)′均為主應(yīng)力強(qiáng)度理論函數(shù)；τoct為八面體剪應(yīng)力，τoct=［(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2］0.5/6，；b為強(qiáng)度理論選擇參數(shù)，其取值的范圍為0~1。

Mogi［7］根據(jù)所做的巖石真三軸試驗(yàn)結(jié)果，提出的經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則通式為

σ1-σ3=f(σ1+μσ2+σ3) μ∈ ［0.1,0.2］

(8)

Mogi［8］修改的廣義Mises準(zhǔn)則通式為

τoct=f(σ1+σ3)或τoct=f(σ1+μσ2+σ3)

(9)

式中：f為單調(diào)遞增函數(shù)，可以采用一次直線、二次多項(xiàng)式或冪函數(shù)來表示；μ為強(qiáng)度擬合參數(shù)。

Mogi通過巖石真三軸試驗(yàn)，發(fā)現(xiàn)中間主應(yīng)力σ2對(duì)巖石強(qiáng)度的影響要比小主應(yīng)力σ3的影響小，因此式(8)，(9)中用μσ2來反映中間主應(yīng)力σ2的作用。

Haimson等［15］假定巖石破壞面平行于σ2方向，破壞面上的有效正應(yīng)力與中間主應(yīng)力σ2無(wú)關(guān)，建議的經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則通式為

τoct=f(σ1+σ3 2)=f(σ13)

(10)

AlAjmi等［5960］建立了巖石抗剪強(qiáng)度參數(shù)與直線型Mogi經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則中擬合參數(shù)之間的關(guān)系，修改后的直線型Mogi經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則稱為MogiCoulomb強(qiáng)度準(zhǔn)則，其表達(dá)式為

τoct=22 3sin（φ）σ13+223ccos（φ）

(11)

另外，Wiebols等［61］推導(dǎo)出一個(gè)基于巖石內(nèi)應(yīng)變能的強(qiáng)度準(zhǔn)則，但是應(yīng)用時(shí)需要知道難以量測(cè)的破裂面摩擦因數(shù)。Costamagna等［62］提出一個(gè)復(fù)雜的四參數(shù)巖土強(qiáng)度準(zhǔn)則，可以退化為多種非線性強(qiáng)度準(zhǔn)則。Mortara［63］基于MatsuokaNakai準(zhǔn)則［空間滑動(dòng)面準(zhǔn)則，即Spatially Mobilized Plane (SMP)準(zhǔn)則］和LadeDuncan準(zhǔn)則的相似形式，通過類比提出了一個(gè)新的非線性強(qiáng)度準(zhǔn)則。姚仰平等［64］利用插值方法建立了介于SMP準(zhǔn)則[HJ1.95mm]和Mises準(zhǔn)則之間的廣義非線性強(qiáng)度理論。胡小榮等［65］通過考慮十二面體單元主剪面上的3個(gè)主剪面應(yīng)力對(duì)的共同作用，提出了一個(gè)適用于巖土材料的三剪新強(qiáng)度準(zhǔn)則。周鳳璽等［66］同樣利用插值方法，提出了廣義DruckerPrager準(zhǔn)則。尤明慶［6769］提出了四參數(shù)的巖石指數(shù)型強(qiáng)度準(zhǔn)則。肖楊等［70］利用插值方法，提出了介于MatsuokaNakai準(zhǔn)則和LadeDuncan準(zhǔn)則之間的破壞準(zhǔn)則。Liu等［71］采用插值方法建立了介于MatsuokaNakai準(zhǔn)則和DruckerPrager準(zhǔn)則之間的非線性統(tǒng)一DPMNU準(zhǔn)則。

從上述各準(zhǔn)則的表達(dá)式和建立方法可以看出：①中間主應(yīng)力效應(yīng)已是目前巖石強(qiáng)度準(zhǔn)則建立所必須考慮的基本問題之一，除MohrCoulomb強(qiáng)度準(zhǔn)則和HoekBrown經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則外，其他準(zhǔn)則都包含了中間主應(yīng)力σ2，能在一定程度上反映中間主應(yīng)力的影響；②各準(zhǔn)則多是對(duì)真三軸試驗(yàn)數(shù)據(jù)的直接擬合，或通過插值、類比方法來建立新的強(qiáng)度準(zhǔn)則，公式的參數(shù)眾多且缺乏明確的物理意義；③新提出的各準(zhǔn)則之間沒有內(nèi)在的聯(lián)系，有的準(zhǔn)則極限面覆蓋范圍有限，公式非線性程度高，難以應(yīng)用于結(jié)構(gòu)的解析計(jì)算，且式(6)~(11)不滿足屈服面的外凸性，給數(shù)值計(jì)算帶來很大困難；④Mogi系列經(jīng)驗(yàn)強(qiáng)度準(zhǔn)則以τoct和σ13表示的擬合公式，即式(9)~(11)的計(jì)算值非常接近試驗(yàn)數(shù)據(jù)，但是并不能證明Mogi經(jīng)驗(yàn)準(zhǔn)則的正確性［72］，因?yàn)樵趲r石真三軸試驗(yàn)數(shù)據(jù)中τoct和σ13具有極高的正相關(guān)性。

3統(tǒng)一強(qiáng)度理論

20世紀(jì)60年代以來，俞茂宏基于雙剪單元體力學(xué)模型［12,7376］，建立了具有簡(jiǎn)單而統(tǒng)一的數(shù)學(xué)表達(dá)式、能適用于眾多材料的統(tǒng)一強(qiáng)度理論，且其參數(shù)可以由簡(jiǎn)單試驗(yàn)獲得。

以壓應(yīng)力為正、拉應(yīng)力為負(fù)時(shí)的統(tǒng)一強(qiáng)度理論表達(dá)式為

F=ασ1-1 1+b(bσ2+σ3)=σt

σ2≤σ3+ασ1 1+α

(12)

F′=α 1+b(σ1+bσ2)-σ3=σt

σ2≥σ3+ασ1 1+α

(13)

式中：σt為巖石單軸拉伸強(qiáng)度；α為巖石的單軸拉壓強(qiáng)度比；α，b取不同的值時(shí)，可以表示或線性逼近現(xiàn)有的各種強(qiáng)度準(zhǔn)則。

圖2為單軸拉壓特性不同材料(α≠1)的統(tǒng)一強(qiáng)

圖2 α≠1時(shí)統(tǒng)一強(qiáng)度理論在π平面上的極限線

Fig.2 Limit Loci of Unified Strength Theory on π Plane when α≠1

度理論在π平面上的極限線［1］，其中，σ′1，σ′2，σ′3分別為大主應(yīng)力、中間主應(yīng)力和小主應(yīng)力在π平面上的投影，θσ為應(yīng)力Lode角，θb為臨界應(yīng)力Lode角。圖3為α≠1，b=3/4時(shí)統(tǒng)一強(qiáng)度理論在主應(yīng)力空間的三維極限面［2］，其中，σ1=σ2=σ3表示靜水應(yīng)力軸。

圖3 α≠1,b=3/4時(shí)統(tǒng)一強(qiáng)度理論的空間極限面

Fig.3 Limit Loci of Unified Strength Theory in Principal Stress Space when α≠1,b=3/4

統(tǒng)一強(qiáng)度理論包含了現(xiàn)有各種主要強(qiáng)度理論和一些尚未發(fā)表過的新強(qiáng)度理論，將眾多已有強(qiáng)度理論作為其特例或是線性逼近，已形成一個(gè)全新的強(qiáng)度理論體系。統(tǒng)一強(qiáng)度理論很好地考慮了中間主應(yīng)力對(duì)巖土材料強(qiáng)度的影響，與多種巖土材料的真三軸試驗(yàn)結(jié)果相吻合，可以很方便地用于結(jié)構(gòu)解析計(jì)算，并且在數(shù)值模擬中很好地解決了角點(diǎn)奇異性。總之，統(tǒng)一強(qiáng)度理論同時(shí)具有周培源關(guān)于評(píng)價(jià)新理論的3個(gè)特點(diǎn)［77］，并且還有其他一些特點(diǎn)，具有重要的理論意義和工程應(yīng)用價(jià)值。

3.1巖石真三軸試驗(yàn)驗(yàn)證

應(yīng)用強(qiáng)度理論或破壞準(zhǔn)則的最終合理性以及其有效范圍依賴于所形成的模型預(yù)測(cè)試驗(yàn)數(shù)據(jù)的能力，[HJ1.8mm]現(xiàn)有復(fù)雜應(yīng)力狀態(tài)下不同特性巖土類材料的試驗(yàn)結(jié)果在π平面上的極限線多為凸形，并且位于0＜b≤1范圍內(nèi)，此處僅給出3種有代表性巖石真三軸試驗(yàn)結(jié)果的驗(yàn)證和比較。

圖4為Mogi［9］真三軸試驗(yàn)得出粗面巖的π平面極限線。由圖4可以看出，對(duì)于不同參數(shù)b的統(tǒng)一強(qiáng)度理論極限線：b=1時(shí)的極限線稍大，b=0時(shí)的極限線稍小，b=1/2時(shí)的極限線與試驗(yàn)結(jié)果吻合很好，圖4中的DruckerPrager準(zhǔn)則內(nèi)切錐與試驗(yàn)結(jié)果相差較大。

圖4粗面巖的π平面極限線

Fig.4Limit Loci on π Plane for Volcanic Rock

圖5為Mechelis［13］真三軸試驗(yàn)得出的3組靜水應(yīng)力p作用下大理巖的π平面極限線。圖6為李小春等［30］得出的在靜水應(yīng)力80~200 MPa范圍內(nèi)，6組花崗巖的真三軸試驗(yàn)結(jié)果。這9組試驗(yàn)結(jié)果均與b=1的統(tǒng)一強(qiáng)度理論相吻合。

圖5 大理巖的π平面極限線

Fig.5Limit Loci on π Plane for Mable

圖6花崗巖的π平面極限線

Fig.6Limit Loci on π Plane for Granite

3.2統(tǒng)一強(qiáng)度理論在巖土工程中的應(yīng)用

統(tǒng)一強(qiáng)度理論作為工程應(yīng)用的基礎(chǔ)理論，[HJ1.95mm]其生命力不僅在于能被試驗(yàn)結(jié)果所證實(shí)，還在于它具有應(yīng)用的可行性和廣泛性。統(tǒng)一強(qiáng)度理論已在土木工程的很多領(lǐng)域得到初步應(yīng)用，但是與單剪強(qiáng)度理論相比，統(tǒng)一強(qiáng)度理論的推廣應(yīng)用還只是一個(gè)開始，缺乏系統(tǒng)化研究，并且在很多方面應(yīng)該說還遠(yuǎn)遠(yuǎn)不夠或是空白，推廣應(yīng)用時(shí)將會(huì)碰到一系列新的問題。下述將簡(jiǎn)單介紹統(tǒng)一強(qiáng)度理論在土力學(xué)和地基工程、地下以及礦山等巖土工程中的一些代表性推廣與應(yīng)用。

1994年俞茂宏在《巖土工程學(xué)報(bào)》發(fā)表了關(guān)于巖土類材料的統(tǒng)一強(qiáng)度理論的論文［78］，標(biāo)志著統(tǒng)一強(qiáng)度理論在巖土材料中應(yīng)用的開始，后于1997年在《土木工程學(xué)報(bào)》又發(fā)表了關(guān)于統(tǒng)一滑移線場(chǎng)理論的論文［79］，進(jìn)一步拓展了統(tǒng)一強(qiáng)度理論在巖土結(jié)構(gòu)極限荷載求解方面的應(yīng)用范圍。

趙均海等［8081］、張永強(qiáng)等［82］、程彩霞等［83］用統(tǒng)一強(qiáng)度理論和統(tǒng)一滑移線場(chǎng)理論分析了某些塑性平面應(yīng)變問題，包括邊坡穩(wěn)定極限荷載、厚壁圓筒和球殼的彈性極限荷載及塑性極限荷載。此外，周小平等［84］、范文等［85］、高江平等［86］以及王祥秋等［87］用統(tǒng)一強(qiáng)度理論分析了飽和土條形地基的極限承載力，得出了考慮中間主應(yīng)力影響的飽和土地基極限承載力新公式。謝群丹等［88］及陳秋南等［89］用統(tǒng)一強(qiáng)度理論推導(dǎo)了飽和土的側(cè)向土壓力計(jì)算公式，范文等［90］推導(dǎo)了飽和土地基臨界荷載統(tǒng)一解，楊小禮等［91］探討了條形基礎(chǔ)下纖維加筋土的地基承載力統(tǒng)一解。

柱孔擴(kuò)張是地下和礦山等工程中的典型問題。蔣明鏡等［92］、曹黎娟等［93］、王亮［94］、汪鵬程等［9596］、羅戰(zhàn)友等［97］對(duì)理想彈塑性和應(yīng)變軟化模型的柱孔擴(kuò)張問題，采用統(tǒng)一強(qiáng)度理論進(jìn)行了較系統(tǒng)的研究。此后，王延斌等［98］應(yīng)用空間軸對(duì)稱統(tǒng)一特征線場(chǎng)理論又進(jìn)行了新的研究。

胡小榮等［99］應(yīng)用統(tǒng)一強(qiáng)度理論分析了理想彈塑性圍巖巷道問題，并假定塑性區(qū)體積應(yīng)變?yōu)?，不考慮剪脹和塑性區(qū)彈性應(yīng)變的影響，與巷道真實(shí)變形情況相差較大。徐栓強(qiáng)等［100101］、宋俐等［102］和范文等［103］得出了壓力隧洞彈塑性分析的統(tǒng)一解，蔡曉鴻等［104］和馬青等［105］基于統(tǒng)一強(qiáng)度理論分析了圍巖的抗力系數(shù)。張常光等［106108］結(jié)合水工壓力隧洞的特點(diǎn)，考慮多種因素的綜合影響，采用統(tǒng)一強(qiáng)度理論分析了施工期和運(yùn)行期水工隧洞的應(yīng)力場(chǎng)和位移場(chǎng)。曾開華等［109110］和張常光等［111115］提出了基于統(tǒng)一強(qiáng)度理論的理想彈塑性模型和彈脆塑性模型的隧道收斂約束分析新方法。張常光等［116120］和趙均海等［121］將統(tǒng)一強(qiáng)度理論和雙應(yīng)力狀態(tài)變量相結(jié)合，建立了非飽和土抗剪強(qiáng)度統(tǒng)一解，并將其用于非飽和土的側(cè)向土壓力、地基極限承載力和臨界荷載，得到一系列新的成果和有意義的結(jié)論。

總之，將統(tǒng)一強(qiáng)度理論應(yīng)用于巖土類材料，可以考慮巖土類材料的基本力學(xué)特性，并且與現(xiàn)有的巖土類材料真三軸試驗(yàn)數(shù)據(jù)相吻合，還可以充分發(fā)揮材料和結(jié)構(gòu)的強(qiáng)度潛能，能取得明顯的經(jīng)濟(jì)效益，其計(jì)算結(jié)果具有重要的工程應(yīng)用價(jià)值，為工程技術(shù)人員在各種工程應(yīng)用的發(fā)揮和創(chuàng)造性創(chuàng)造了條件。統(tǒng)一強(qiáng)度理論是更加合理和更符合試驗(yàn)結(jié)果的系列強(qiáng)度準(zhǔn)則，可以很方便地應(yīng)用于結(jié)構(gòu)彈塑性解析解和其他問題，但是在應(yīng)用統(tǒng)一強(qiáng)度理論時(shí)，應(yīng)更加注重對(duì)所研究問題的深入分析，考慮更多復(fù)雜因素的綜合影響，得到更符合實(shí)際情況的較理想解答，為工程設(shè)計(jì)和生產(chǎn)實(shí)踐提供更好的理論指導(dǎo)。

4結(jié)語(yǔ)

(1)中間主應(yīng)力效應(yīng)及其區(qū)間性是巖石材料強(qiáng)度的重要特性，單獨(dú)改變(增加或減小)中間主應(yīng)力可以引起巖石的破壞?？紤]巖石的中間主應(yīng)力效應(yīng)，可以充分發(fā)揮巖石材料的強(qiáng)度潛能，減小工程投資。

(2)中間主應(yīng)力效應(yīng)已是目前巖石強(qiáng)度準(zhǔn)則建立所必須考慮的基本問題之一，但是現(xiàn)有各準(zhǔn)則多是對(duì)真三軸試驗(yàn)數(shù)據(jù)的直接擬合，或通過插值、類比方法來建立的，各準(zhǔn)則之間沒有內(nèi)在的聯(lián)系，公式的參數(shù)眾多且缺乏明確的物理意義。

(3)統(tǒng)一強(qiáng)度理論很好地考慮了中間主應(yīng)力對(duì)巖土材料強(qiáng)度的影響，將其應(yīng)用于巖土類材料可以考慮巖土類材料的基本力學(xué)特性，并且與現(xiàn)有的巖土類材料真三軸試驗(yàn)數(shù)據(jù)相吻合，取得了明顯的經(jīng)濟(jì)效益，但是應(yīng)更加注重對(duì)所研究問題的深入分析，考慮更多復(fù)雜因素的綜合影響，得到更符合實(shí)際情況的較理想解答。

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ZENG Kaihua,JU Haiyan,ZHANG Changguang.Elastoplastic Unified Solution for Displacements Around a Deep Circular Tunnel and Its Comparative Analysis［J］.Rock and Soil Mechanics,2011,32(5):13151319.

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［115］趙均海,李愛國(guó),張常光.圓形隧道彈脆塑性圍巖自承載能力分析［J］.中國(guó)公路學(xué)報(bào),2014,27(4):8590.

ZHAO Junhai,LI Aiguo,ZHANG Changguang.Selfcarrying Capacity of Surrounding Rock for a Circular Tunnel Based on Elasticbrittleplastic Model［J］.China Journal of Highway and Transport,2014,27(4):8590.

［116］張常光,張慶賀,趙均海.非飽和土抗剪強(qiáng)度及土壓力統(tǒng)一解［J］.巖土力學(xué),2010,31(6):18711876.

ZHANG Changguang，ZHANG Qinghe，ZHAO Junhai.Unified Solutions of Shear Strength and Earth Pressure for Unsaturated Soils［J］.Rock and Soil Mechanics,2010,31(6):18711876.

［117］張常光,曾開華,趙均海.非飽和土臨界荷載和太沙基極限承載力解析解［J］.同濟(jì)大學(xué)學(xué)報(bào):自然科學(xué)版,2010,38(12):17361740.

ZHANG Changguang,ZENG Kaihua,ZHAO Jun[JP2]hai.Analytical Solutions of Critical Load and [JP]Terzaghis Ultimate Bearing Capacity for Unsaturated Soil［J］.Journal of Tongji University:Natural Science,2010,38(12):17361740.

［118］張常光,胡云世,趙均海.平面應(yīng)變條件下非飽和土抗剪強(qiáng)度統(tǒng)一解及其應(yīng)用［J］.巖土工程學(xué)報(bào),2011,33(1):3237.

ZHANG Changguang,HU Yunshi,ZHAO Junhai.Unified Solution of Shear Strength for Unsaturated Soil Under Plane Strain Condition and Its Application［J］.Chinese Journal of Geotechnical Engineering,2011,33(1):3237.

［119］張常光,趙均海,張冬芳.非飽和土強(qiáng)度非線性及對(duì)被動(dòng)土壓力的影響［J］.廣西大學(xué)學(xué)報(bào):自然科學(xué)版,2012,37(4):797802.

ZHANG Changguang,ZHAO Junhai,ZHANG Dong[JP]fang.Nonlinearity of Unsaturated Soils Strength and Its Influence on Passive Earth Pressure［J］.Journal of Guangxi University:Natural Science Edition,2012,37(4):797802.

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ZHANG Changguang,ZHAO Junhai,ZHU Qian.Classification and Summary of Shear Strength Formulae for Unsaturated Soils［J］.Journal of Architecture and Civil Engineering,2012,29(2):7482.

［121］趙均海,梁文彪,張常光,等.非飽和土庫(kù)侖主動(dòng)土壓力統(tǒng)一解［J］.巖土力學(xué),2013,34(3):609614.

ZHAO Junhai,LIANG Wenbiao,ZHANG Changguang,et al.Unified Solution of Coulombs Active Earth Pressure for Unsaturated Soils［J］.Rock and Soil Mechanics,2013,34(3):609614.



［115］趙均海,李愛國(guó),張常光.圓形隧道彈脆塑性圍巖自承載能力分析［J］.中國(guó)公路學(xué)報(bào),2014,27(4):8590.

ZHAO Junhai,LI Aiguo,ZHANG Changguang.Selfcarrying Capacity of Surrounding Rock for a Circular Tunnel Based on Elasticbrittleplastic Model［J］.China Journal of Highway and Transport,2014,27(4):8590.

［116］張常光,張慶賀,趙均海.非飽和土抗剪強(qiáng)度及土壓力統(tǒng)一解［J］.巖土力學(xué),2010,31(6):18711876.

ZHANG Changguang，ZHANG Qinghe，ZHAO Junhai.Unified Solutions of Shear Strength and Earth Pressure for Unsaturated Soils［J］.Rock and Soil Mechanics,2010,31(6):18711876.

［117］張常光,曾開華,趙均海.非飽和土臨界荷載和太沙基極限承載力解析解［J］.同濟(jì)大學(xué)學(xué)報(bào):自然科學(xué)版,2010,38(12):17361740.

ZHANG Changguang,ZENG Kaihua,ZHAO Jun[JP2]hai.Analytical Solutions of Critical Load and [JP]Terzaghis Ultimate Bearing Capacity for Unsaturated Soil［J］.Journal of Tongji University:Natural Science,2010,38(12):17361740.

［118］張常光,胡云世,趙均海.平面應(yīng)變條件下非飽和土抗剪強(qiáng)度統(tǒng)一解及其應(yīng)用［J］.巖土工程學(xué)報(bào),2011,33(1):3237.

ZHANG Changguang,HU Yunshi,ZHAO Junhai.Unified Solution of Shear Strength for Unsaturated Soil Under Plane Strain Condition and Its Application［J］.Chinese Journal of Geotechnical Engineering,2011,33(1):3237.

［119］張常光,趙均海,張冬芳.非飽和土強(qiáng)度非線性及對(duì)被動(dòng)土壓力的影響［J］.廣西大學(xué)學(xué)報(bào):自然科學(xué)版,2012,37(4):797802.

ZHANG Changguang,ZHAO Junhai,ZHANG Dong[JP]fang.Nonlinearity of Unsaturated Soils Strength and Its Influence on Passive Earth Pressure［J］.Journal of Guangxi University:Natural Science Edition,2012,37(4):797802.

［120］張常光,趙均海,朱 倩.非飽和土抗剪強(qiáng)度公式分類及總結(jié)［J］.建筑科學(xué)與工程學(xué)報(bào),2012,29(2):7482.

ZHANG Changguang,ZHAO Junhai,ZHU Qian.Classification and Summary of Shear Strength Formulae for Unsaturated Soils［J］.Journal of Architecture and Civil Engineering,2012,29(2):7482.

［121］趙均海,梁文彪,張常光,等.非飽和土庫(kù)侖主動(dòng)土壓力統(tǒng)一解［J］.巖土力學(xué),2013,34(3):609614.

ZHAO Junhai,LIANG Wenbiao,ZHANG Changguang,et al.Unified Solution of Coulombs Active Earth Pressure for Unsaturated Soils［J］.Rock and Soil Mechanics,2013,34(3):609614.



［115］趙均海,李愛國(guó),張常光.圓形隧道彈脆塑性圍巖自承載能力分析［J］.中國(guó)公路學(xué)報(bào),2014,27(4):8590.

ZHAO Junhai,LI Aiguo,ZHANG Changguang.Selfcarrying Capacity of Surrounding Rock for a Circular Tunnel Based on Elasticbrittleplastic Model［J］.China Journal of Highway and Transport,2014,27(4):8590.

［116］張常光,張慶賀,趙均海.非飽和土抗剪強(qiáng)度及土壓力統(tǒng)一解［J］.巖土力學(xué),2010,31(6):18711876.

ZHANG Changguang，ZHANG Qinghe，ZHAO Junhai.Unified Solutions of Shear Strength and Earth Pressure for Unsaturated Soils［J］.Rock and Soil Mechanics,2010,31(6):18711876.

［117］張常光,曾開華,趙均海.非飽和土臨界荷載和太沙基極限承載力解析解［J］.同濟(jì)大學(xué)學(xué)報(bào):自然科學(xué)版,2010,38(12):17361740.

ZHANG Changguang,ZENG Kaihua,ZHAO Jun[JP2]hai.Analytical Solutions of Critical Load and [JP]Terzaghis Ultimate Bearing Capacity for Unsaturated Soil［J］.Journal of Tongji University:Natural Science,2010,38(12):17361740.

［118］張常光,胡云世,趙均海.平面應(yīng)變條件下非飽和土抗剪強(qiáng)度統(tǒng)一解及其應(yīng)用［J］.巖土工程學(xué)報(bào),2011,33(1):3237.

ZHANG Changguang,HU Yunshi,ZHAO Junhai.Unified Solution of Shear Strength for Unsaturated Soil Under Plane Strain Condition and Its Application［J］.Chinese Journal of Geotechnical Engineering,2011,33(1):3237.

［119］張常光,趙均海,張冬芳.非飽和土強(qiáng)度非線性及對(duì)被動(dòng)土壓力的影響［J］.廣西大學(xué)學(xué)報(bào):自然科學(xué)版,2012,37(4):797802.

ZHANG Changguang,ZHAO Junhai,ZHANG Dong[JP]fang.Nonlinearity of Unsaturated Soils Strength and Its Influence on Passive Earth Pressure［J］.Journal of Guangxi University:Natural Science Edition,2012,37(4):797802.

［120］張常光,趙均海,朱 倩.非飽和土抗剪強(qiáng)度公式分類及總結(jié)［J］.建筑科學(xué)與工程學(xué)報(bào),2012,29(2):7482.

ZHANG Changguang,ZHAO Junhai,ZHU Qian.Classification and Summary of Shear Strength Formulae for Unsaturated Soils［J］.Journal of Architecture and Civil Engineering,2012,29(2):7482.

［121］趙均海,梁文彪,張常光,等.非飽和土庫(kù)侖主動(dòng)土壓力統(tǒng)一解［J］.巖土力學(xué),2013,34(3):609614.

ZHAO Junhai,LIANG Wenbiao,ZHANG Changguang,et al.Unified Solution of Coulombs Active Earth Pressure for Unsaturated Soils［J］.Rock and Soil Mechanics,2013,34(3):609614.

