本文選題：矩形疊層厚板 + 固支/自由邊　； 參考：《合肥工業(yè)大學(xué)》2017年博士論文

【摘要】：經(jīng)典薄板理論和各種中厚板理論都是建立在某些人為假設(shè)基礎(chǔ)上的,若將這些理論用于求解疊層厚板問題,會(huì)產(chǎn)生不可忽略的誤差。為了獲得疊層厚板的三維精確解,一些基于三維彈性力學(xué)基本方程的厚板理論逐漸被提出來。狀態(tài)空間法作為厚板理論中最有效、最流行的解法,能很好地處理層間的連續(xù)性問題,且其解的形式簡(jiǎn)單而統(tǒng)一,便于理解和應(yīng)用。采用狀態(tài)空間法求解矩形疊層厚板問題時(shí),通過雙傅里葉級(jí)數(shù)展開進(jìn)行變量分離,恰好能嚴(yán)格滿足四邊簡(jiǎn)支的邊界條件。但是對(duì)于含非簡(jiǎn)支邊的矩形板,求解依然存在一些難度,常見的方法是通過在非簡(jiǎn)支邊上假定待定的邊界位移函數(shù),并采用分層的辦法進(jìn)行求解,所得到的解在非簡(jiǎn)支邊上并不能沿厚度方向嚴(yán)格滿足邊界條件。本文以含固支邊或自由邊的矩形疊層厚板作為研究對(duì)象,采用狀態(tài)空間法求解該類矩形疊層厚板靜力問題的三維精確解。在求解過程中,為了嚴(yán)格滿足固支邊或自由邊的邊界條件,在該邊上假定邊界位移函數(shù),并將其作為狀態(tài)變量引入狀態(tài)方程,建立不同邊界條件下的矩形單層與疊層厚板的齊次狀態(tài)方程,得到相應(yīng)靜力問題的三維精確解。整個(gè)求解過程簡(jiǎn)單清晰,無需處理大量未知量,便于應(yīng)用。在第三章到第七章中,針對(duì)不同邊界條件下的矩形單層與疊層厚板,分別建立了齊次狀態(tài)方程,得到了相應(yīng)靜力問題的三維精確解。算例表明,本文解與有限元解吻合得很好,具有很高的精度和很好的收斂性,而且具有很廣的適用性。與經(jīng)典薄板理論和各種中厚板理論相比,本文解嚴(yán)格滿足三維彈性力學(xué)基本方程,考慮了所有的彈性參數(shù),是正真意義上的三維精確解,能夠給出位移和應(yīng)力分量沿厚度方向的精確分布規(guī)律;而且該解不受板的厚度和材料屬性的限制,能很好地處理疊層板的層間連續(xù)性問題,充分體現(xiàn)了狀態(tài)空間法求解疊層厚板問題的優(yōu)越性。與現(xiàn)有三維精確解相比,本文完全采用解析方法建立了不同邊界條件下的矩形單層與疊層厚板的齊次狀態(tài)方程,使固支邊或自由邊也能嚴(yán)格滿足邊界條件,并在這些邊界上得到了非常精確的位移和應(yīng)力結(jié)果。這表明,本文解突破了現(xiàn)有三維精確解對(duì)于求解含非簡(jiǎn)支邊矩形疊層厚板的限制。此外,對(duì)與固支邊或自由邊的邊界位移函數(shù)相關(guān)聯(lián)的多項(xiàng)式函數(shù)的次數(shù)作了比較和研究,結(jié)果表明,多項(xiàng)式的次數(shù)對(duì)本文解的精度和收斂性影響不大。
[Abstract]:The classical thin plate theory and all kinds of plate theories are based on some artificial assumptions. If these theories are used to solve the laminated thick plate problems, the errors can not be ignored. In order to obtain the exact three-dimensional solution of thick laminated plates, some theories of thick plates based on the basic equations of three-dimensional elasticity have been proposed gradually. As the most effective and popular method in thick plate theory, the state space method can deal with the continuity problem between layers well, and the form of the solution is simple and uniform, which is easy to understand and apply. When the state space method is used to solve the problem of rectangular laminated thick plates, the variables are separated by double Fourier series expansion, which can exactly satisfy the boundary condition of simply supported on four sides. However, for rectangular plates with non-simply supported edges, there are still some difficulties in solving the problem. The common method is to assume the undetermined boundary displacement function on the non-simply supported edges, and to solve the problem by stratification. The obtained solution does not satisfy the boundary condition strictly in the direction of thickness on the edge of non-simple support. In this paper, a rectangular laminated thick plate with clamped or free edges is used as the research object. The state space method is used to solve the three-dimensional exact solution of the static problem of the rectangular laminated thick plate. In order to satisfy the boundary conditions of clamped or free edges strictly, the boundary displacement function is assumed to be a state variable and is introduced into the equation of state. The homogeneous state equations of rectangular monolayer and laminated thick plates under different boundary conditions are established and the exact three-dimensional solutions of the corresponding static problems are obtained. The whole solution process is simple and clear, it does not need to deal with a large number of unknown quantities, so it is easy to be applied. In the third to seventh chapters, the homogeneous equation of state is established for the rectangular monolayer and laminated thick plates under different boundary conditions, and the three dimensional exact solution of the corresponding static problem is obtained. The numerical examples show that the proposed solution is in good agreement with the finite element solution, has high accuracy and good convergence, and has a wide range of applicability. Compared with the classical thin plate theory and various plate theories, the solution in this paper satisfies the basic equations of three-dimensional elasticity strictly, and considers all the elastic parameters. It is a exact three-dimensional solution in the sense of positive truth. The displacement and stress components can be accurately distributed along the thickness direction, and the solution is not limited by the thickness of the plate and the properties of the material, so it can deal with the continuity problem between the laminated plates well. The advantages of state space method for solving thick laminated plates are fully demonstrated. Compared with the existing three dimensional exact solutions, the homogeneous state equations of rectangular monolayer and laminated thick plates under different boundary conditions are completely established by using analytical method, so that the clamped or free edges can satisfy the boundary conditions strictly. The results of displacement and stress on these boundaries are very accurate. It is shown that the solution in this paper breaks through the limitations of the existing three-dimensional exact solutions for solving thick rectangular laminated plates with non-simply supported edges. In addition, the degree of polynomial function associated with the boundary displacement function of fixed or free edges is compared and studied. The results show that the degree of polynomial has little effect on the accuracy and convergence of the solution in this paper.
【學(xué)位授予單位】：合肥工業(yè)大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：TB33

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