本文選題：哈密頓體系 + 粘彈性�。� 參考：《大連理工大學(xué)》2016年博士論文

【摘要】：隨著我國(guó)先進(jìn)材料科學(xué)和制造工藝的發(fā)展,先進(jìn)復(fù)合材料的制備與應(yīng)用對(duì)國(guó)家發(fā)展有重大意義。同時(shí)具備力、電、磁、熱、聲、光等兩種或多種特性的功能復(fù)合材料,諸如聚合物基材料、壓電材料、電磁材料、光伏材料等被廣泛應(yīng)用于航空航天、建筑工業(yè)、電子工業(yè)以及醫(yī)療器械等領(lǐng)域。層合板結(jié)構(gòu)是工程中的基本結(jié)構(gòu),該類結(jié)構(gòu)是由多層單層板通過(guò)聚合物膠黏劑粘合在一起組成整體的結(jié)構(gòu)板。由于其界面的連接特點(diǎn),脫粘是該類結(jié)構(gòu)的主要破壞模式。因此,研究粘彈性斷裂問題以及力電磁彈耦合界面斷裂問題具有重要的實(shí)際意義�，F(xiàn)有斷裂問題的研究方法主要有解析法(復(fù)勢(shì)函數(shù)法、積分變換法、權(quán)函數(shù)法等)和數(shù)值方法(有限單元法、邊界單元法、無(wú)網(wǎng)格法等)。解析研究主要?dú)w結(jié)為高階偏微分方程或積分方程的求解,在數(shù)學(xué)上受到求解體系(拉格朗日體系)的限制,它以提高控制方程的階數(shù)為代價(jià)來(lái)減少變量個(gè)數(shù),從而造成方程難以求解。在這種情況下,本博士論文將問題導(dǎo)入全新的哈密頓體系下進(jìn)行求解,利用辛方法以增加基本變量為代價(jià)來(lái)降低微分方程階數(shù),利用高性能計(jì)算機(jī)來(lái)求解低階微分方程。本博士論文以復(fù)合材料的層間界面斷裂問題為研究對(duì)象,以粘彈性斷裂問題為突破口,采用辛方法對(duì)電磁彈性材料弱界面斷裂問題展開研究,主要研究工作如下：(1)提出一種求解粘彈性材料斷裂問題的解析方法。首先,利用Laplace變換將粘彈性斷裂問題轉(zhuǎn)換為頻域相關(guān)問題。然后,在頻域內(nèi),通過(guò)拉格朗日函數(shù)和哈密頓函數(shù),建立粘彈性材料斷裂問題的哈密頓正則方程組和求解體系,將求解問題歸結(jié)為辛本征值和辛本征解問題。最后,利用本征解之間的辛共軛正交關(guān)系以及展開定理,解析地表征出裂紋尖端的奇異性和域內(nèi)的物理場(chǎng),直接獲得應(yīng)力強(qiáng)度因子和J積分的解析表達(dá)式。數(shù)值結(jié)果驗(yàn)證了方法的收斂性和有效性,并且反映出粘彈性材料特有的應(yīng)力松弛現(xiàn)象和蠕變現(xiàn)象,特別指出溫度荷載作用對(duì)粘彈性斷裂參數(shù)的影響。(2)將弱界面模型引入到平面彈性斷裂問題中,并采用辛方法進(jìn)行求解。將復(fù)合材料層間粘結(jié)問題簡(jiǎn)化為弱連接問題,彈簧模型作為弱連接條件的數(shù)學(xué)模型。首先推導(dǎo)出兩種彈性材料的辛本征解形式,然后由弱連接條件以及裂紋面條件確定兩種本征解共用的辛本征值,從而全域內(nèi)的辛本征解由以上兩種材料所在區(qū)域的本征解組成,并且在全域內(nèi)滿足辛共軛正交關(guān)系。最后利用辛本征解的展開定理、外邊界邊界條件和辛共軛正交關(guān)系,可直接得到裂紋尖端處的奇異性,域內(nèi)應(yīng)力等分布和廣義應(yīng)力強(qiáng)度因子等解析表達(dá)式。數(shù)值結(jié)果表明,該方法與已有經(jīng)典問題結(jié)果相吻合,說(shuō)明本文方法是有效的,并且該方法具有較高的精度。(3)將弱界面模型引入到電磁彈性材料反平面斷裂問題中,并采用辛方法對(duì)多場(chǎng)耦合作用情況進(jìn)行求解。首先,采用能量方法和勒讓德變換,確定原變量(位移、電勢(shì)和磁勢(shì))的對(duì)偶變量(應(yīng)力、電位移、磁感應(yīng)強(qiáng)度),從而將問題導(dǎo)入到哈密頓體系中。然后,以彈簧模型作為弱連接條件的數(shù)學(xué)模型,結(jié)合不同的力學(xué)、電學(xué)、磁學(xué)裂紋面條件,推導(dǎo)出彈簧模型連接的兩區(qū)域內(nèi)的辛本征解向量和共用的辛本征值,并驗(yàn)證全域內(nèi)辛本征解之間存在辛共軛正交關(guān)系。最后,利用外邊界條件和辛共軛正交關(guān)系,確定問題解的解析表達(dá)式以及斷裂參數(shù)。數(shù)值結(jié)果表明,由于弱連接界面的存在,使得廣義強(qiáng)度因子分為兩類,且弱界面參數(shù)影響強(qiáng)度因子的大小,而裂紋面條件影響廣義強(qiáng)度因子的存在與否。
[Abstract]:With the development of advanced materials science and manufacturing technology in China, the preparation and application of advanced composites have great significance for the development of the country. At the same time, two or more functional composite materials, such as polymer based materials, piezoelectric materials, electromagnetic materials and photovoltaic materials, are widely used in aerospace and aerospace. In the fields of construction industry, electronic industry and medical equipment. Laminate structure is the basic structure in engineering. This kind of structure is composed of multi layer monolayer bonded together by polymer adhesive. Because of its interface characteristics, debonding is the main failure mode of this kind of structure. Therefore, the study of viscoelastic fracture is studied. The problem and the fracture problem of the force electromagnetic elastic coupling interface are of great practical significance. The main research methods of the existing fracture problems are analytic method (complex potential function method, integral transformation method, weight function method etc.) and numerical methods (finite element method, boundary element method, meshless method, etc.). The analytical study is mainly attributed to the high order partial differential equation or integral square. The solution of the process is limited by the mathematical solution system (Lagrange system). It reduces the number of variables by increasing the order of the control equation. Thus, the equation is difficult to solve. In this case, the doctoral thesis introduces the problem into a new Hamilton system and uses the symplectic method to increase the basic variables. In order to reduce the order of differential equations and use high performance computer to solve the low order differential equation, this thesis takes the interlayer interface fracture problem of composite material as the research object, taking the problem of viscoelastic fracture as the breakthrough point, using symplectic method to study the problem of weak interface fracture of electromagnetic elastic material. The main research work is as follows: (1) proposed An analytical method for solving the fracture problem of viscoelastic materials is solved. First, the problem of viscoelastic fracture is converted into a frequency domain problem by Laplace transformation. Then, in the frequency domain, the Hamilton regular equation set and solution system for the fracture of viscoelastic materials are established by Lagrange's function and Hamiltonian function. The solution is reduced to symplectic problem. The eigenvalues and the symplectic eigensolutions are solved. Finally, the analytical expressions of the stress intensity factors and the J integral are obtained by using the symplectic conjugate orthogonal relations between the eigensolutions and the expansion theorem, and the analytical expressions of the stress intensity factors and the J integral are obtained. The numerical results verify the convergence and effectiveness of the square method and reflect the viscoelasticity. The special stress relaxation and creep phenomenon of the sexual material, especially the effect of the temperature load on the viscoelastic fracture parameters. (2) the weak interface model is introduced into the plane elastic fracture problem, and the symplectic method is used to solve the problem. The problem of interlayer bonding is simplified to the weak connection problem, and the spring model is used as the weak connection condition. The mathematical model. First, the symplectic eigensolution of two kinds of elastic materials is derived. Then the symplectic eigenvalues shared by the two eigensolutions are determined by the weak connection condition and the condition of the crack surface, thus the symplectic eigensolution in the whole domain is composed of the eigensolutions of the regions where the above two materials are located, and the symplectic conjugate orthogonal relationship is satisfied in the whole domain. The expansion theorem of eigensolution, the boundary boundary condition of the outer boundary and the symplectic conjugate orthogonality relation can directly obtain the analytic expressions of the singularity at the crack tip, the distribution of the stress in the domain and the generalized stress intensity factor. The numerical results show that the method is in agreement with the results of the existing classical problems, which shows that the method is effective and the method has a good effect. High precision. (3) the weak interface model is introduced to the anti plane fracture of electromagnetic elastic material and the symplectic method is used to solve the multi field coupling. First, the dual variable (stress, potential shift, magnetic induction intensity) of the original variable (displacement, potential and magnetic potential) is determined by the method of energy and Legendre transformation. In the Hamilton system, the symplectic eigenvalues and symplectic eigenvalues in the two regions connected by the spring model are derived by using the spring model as the mathematical model of the weak connection condition, combining the different mechanics, electrical and magnetic crack surface conditions, and the symplectic conjugate orthogonal relationship exists between the symplectic eigenvalues in the whole domain. Finally, the benefit of the symplectic conjugate is verified. By using the external boundary condition and the symplectic conjugate orthogonal relation, the analytic expression of the solution and the fracture parameters are determined. The numerical results show that the weak interface is divided into two types, and the weak interface parameters affect the intensity factor, while the cracked noodles affect the existence or not of the generalized intensity factor.
【學(xué)位授予單位】：大連理工大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類號(hào)】：O346.1

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