本文關(guān)鍵詞： 三維 彈塑性 角點(diǎn)問題 接觸問題 互補(bǔ)問題 有限元 牛頓法　出處：《武漢大學(xué)》2014年博士論文　論文類型：學(xué)位論文

【摘要】：彈塑性問題和接觸問題是巖土工程中兩種最常見的非線性問題,其中存在的一些問題,如Mohr-Coulomb中的角點(diǎn)問題、三維接觸中的方向問題等,至今還沒有得到很好的解決。但它們有一個(gè)共同點(diǎn),都可以歸結(jié)于互補(bǔ)問題。 本文從它們的互補(bǔ)問題入手進(jìn)行了研究,主要內(nèi)容及取得的研究成果如下： 1.針對(duì)Mohr-Coulomb準(zhǔn)則中存在的角點(diǎn)問題,基于Koiter法則在主應(yīng)力空間將Mohr-Coulomb準(zhǔn)則的多屈服面表達(dá)為其等價(jià)的互補(bǔ)模型,并進(jìn)一步用Fischer-Burmeister互補(bǔ)函數(shù)進(jìn)行描述,從而可以使用當(dāng)前常用的Newton算法進(jìn)行求解。所提出的算法克服了角點(diǎn)問題帶來的收斂困難,并且消除了常規(guī)對(duì)勢(shì)函數(shù)光滑處理所帶來的誤差,提高了三維Mohr-Coulomb問題的精度。 2.基于光滑接觸問題的等價(jià)互補(bǔ)模型,提出了一個(gè)新的光滑逼近模型,當(dāng)該模型中的參數(shù)趨于零時(shí),它等價(jià)于原來的互補(bǔ)模型。由于該模型中函數(shù)的導(dǎo)數(shù)存在且連續(xù),且相應(yīng)的Jacobian矩陣非奇異,因此,這使得常規(guī)的Newton法及Newton族算法可以順利進(jìn)行下去。 3.將Bathe在二維摩擦接觸問題中所提出的約束函數(shù)推廣到三維,通過方向向量的引入,解決了三維接觸問題中由于方向角的周期性帶來的求解穩(wěn)定性問題。該方法將接觸中的法向和切向歸納到統(tǒng)一的函數(shù)來逼近。 4.利用上述的方法,給出了巖土工程典型的三維接觸問題分析：對(duì)巖石的單軸試驗(yàn)、剛性基礎(chǔ)與彈性地基的接觸問題等進(jìn)行了分析,驗(yàn)證了該算法的正確性。 5.對(duì)巴西圓盤試驗(yàn)進(jìn)行了數(shù)值模擬,指出了其中的問題,并建議采用半徑為1.02倍圓盤直徑的弧形加載板進(jìn)行加載,可以大大消除應(yīng)力集中。 6.采用本文方法,對(duì)有限元強(qiáng)度折減問題中的采用線性單元和二次單元進(jìn)行了的對(duì)比研究,結(jié)果表明二次單元的誤差是線性單元的誤差最大值的1/7左右,且隨單元數(shù)量的增多,二次單元比一次單元收斂得更快；線性單元夸大了系統(tǒng)的安全系數(shù),偏于冒險(xiǎn),建議在強(qiáng)度折減計(jì)算中采用二次單元。
[Abstract]:Elasto-plastic problem and contact problem are the two most common nonlinear problems in geotechnical engineering, some of them exist, such as the corner problem in Mohr-Coulomb, the direction problem in three-dimensional contact and so on. But they have one thing in common, which can be attributed to complementarity. This paper starts with their complementary problems, the main contents and the research results are as follows: 1. Aiming at the corner problem in Mohr-Coulomb criterion. The multi-yield surface of Mohr-Coulomb criterion is expressed as its equivalent complementary model in principal stress space based on Koiter rule. Furthermore, it is described by Fischer-Burmeister complementary function. The proposed algorithm overcomes the convergence difficulty caused by the corner problem and eliminates the error caused by the conventional smoothing of potential function. The accuracy of 3D Mohr-Coulomb problem is improved. 2. Based on the equivalent complementary model of smooth contact problem, a new smooth approximation model is proposed when the parameters in the model tend to 00:00. It is equivalent to the original complementary model, because the derivative of the function in the model exists and is continuous, and the corresponding Jacobian matrix is nonsingular. This makes the conventional Newton method and Newton family algorithm go on smoothly. 3. The constraint function proposed by Bathe in the two-dimensional friction contact problem is extended to 3D, and the direction vector is introduced. The stability problem caused by the periodicity of directional angle in the three-dimensional contact problem is solved, and the normal and tangential directions in the contact are summed up to the unified function to approximate it. 4. Using the above method, the typical three-dimensional contact problem analysis of geotechnical engineering is given. The uniaxial test of rock and the contact problem between rigid foundation and elastic foundation are analyzed. The correctness of the algorithm is verified. 5. The numerical simulation of the Brazilian disc test is carried out, and the problems are pointed out, and it is suggested that the stress concentration can be greatly eliminated by using an arc loading plate with a radius of 1.02 times the diameter of the disc. 6. By using this method, the linear element and the quadratic element are compared in the finite element strength reduction problem. The results show that the error of the quadratic unit is about 1/7 of the maximum error of the linear unit, and with the increase of the number of units, the quadratic unit converges faster than the primary unit. The linear element exaggerates the safety factor of the system and tends to take risks. It is suggested that the quadratic unit should be used in the strength reduction calculation.
【學(xué)位授予單位】：武漢大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2014
【分類號(hào)】：TU45

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