以往需要使用高強度材料來達到工作環境要求時，都需利用不同材料製成複合板來增加強度，但材料間的不連續介面容易出現應力集中的現象，大幅降低可使用的壽命及強度，爾後出現了功能性材料，有材料性質為連續變化的特性，大幅改善了不連續面造成的問題，但不可避免的是材料內部還是會有缺陷，空缺或是裂縫等問題，由於功能性材料屬於非均質材料，獲得解析解有一定的困難，因此本文提出一套簡單且精確的方法以探討並測試含有裂縫之功能性材料之力學特性。
有鑒於此，本文乃針對裂縫問題，提出無網格徑向點插值法(Meshless Radial Point Interpolation Method)來求得數值解，以有效處理複雜邊界及裂縫週邊力學變化之行為，並透過弱形式(Weak form)轉換以形成系統統御方程式，再利用無網格徑向點插值法來離散弱形式方程，最終再透過背景網格數值化，以形成矩陣而快速求解。
本文首先探討一般均質裂縫平板受外拉力的破壞，比較RPIM法與ANSYS有限元素法之解、解析解之應力強度因子(Stress Intensity Factor)，RPIM法與解析解相比為1.15%的誤差值，在本文所設定之情況下，有限元素法則有 -22.5%的誤差值，接下來分析裂縫平板暫態熱傳也有相當好的擬合結果。最後分析功能性材料暫態熱破壞力學的特性，探討在兩材料之不同的體積分率下以及裂縫位置、長度對應力強度因子變化特性之影響，可從模擬結果得知裂縫位於強度較高的一端時，體積分率對應力強度因子的影響隨著裂縫長度增加而減少，裂縫長度越長，在不同的體積分率下所得之應力強度因子越來越接近；接著可從探討體積分率對暫態熱傳之影響時發現，儘管熱傳導性較低的材料占較大的體積分率，但由於比熱也相對較低，而導致升溫的速度反而比較快。本文針對非均質之功能性材料暫態熱傳及破壞力學提出一套模擬流程，提供更精準的破壞行為預測，希望提供業界在工件使用壽命及維修上提供有效的資訊。

SUMMARY
Functionally graded materials (FGMs) can be made by using two or more individual constituents and changing its mechanical properties by grading the volume fraction of constituents. However, it’s inevitable that there are some defects and cracks inside structures. It’s very difficult to get analytical solutions of an inhomogeneous material with cracks or complex boundary. So this research aims at simulating mechanics behaviors of FGMs plate with crack.
In this article, Meshless RPIM is applied. Compares to Finite Element Method, meshless method is independent of grid, so meshless method is much easier in dealing with complicated boundary, such as crack and large deformation. Then adopt weak form to governing equation and apply shape function of RPIM to discrete it.
From the simulation results, it can be obtained that when we calculating stress intensity factor, RPIM is much precise than FEM. Also, RPIM get a better results than FEM in simulating the homogeneous plate with hole. Finally, discuss the influences of different volume fraction of FGMs which bear time-dependent external force and temperature difference. Also discuss the influences of crack location and length to mechanics behaviors. In that case, we can predict much more accurate fracture characterizations of inhomogeneous materials.
Keywords: Meshless, MFree RPIM , Stress Intensity Factor, Functionally Graded Material, Fracture mechanics.

INTRODUCTION
In the past, composite plate was widely used to strengthen the structure. Owing to the discontinuous interface of composite plate, it’s likely to cause stress concentration which reduces the material’s life time and strength. To deal with such problems, functionally graded materials (FGMs) had been developed. Nowadays, FGMs gradually replace the use of composite plate. FGMs consist of two or more materials, usually uses ceramics and metal with continuous interface. There are several advantages of FGMs, including we can design desirable properties by adjusting the volume fraction of each components to satisfy the working conditions.

Meshless method had been developed over 20 years and it also got lots of researchers attentions. Several calculation methods had been developed to solve different problems, such as solid mechanics, fluid mechanics and aero dynamics. The fundamental concept of meshless method is removing the restrictions between nodes and grids which completely differs from finite element method. It just needs to construct the relevance between the arbitrary position nodes. Thus, meshless method has advantage in handling complex boundary, such as crack and large deformation and it can get much more accurate solution than FEM. Owing to the properties, meshless have been widely used in numerical simulation.

MATERIALS AND METHODS
This thesis employs meshless Radial Point Interpolation Method (RPIM) to analyze the characterizations of functionally graded high heat resistance plate with crack. The shape function of RPIM has radial function, it’s suitable for crack which is a singularity problem. First, we choose SiC and Steel as our two components of FGMs. Next we apply the weak form to equilibrium equation and heat transfer equation. Then we employ background mesh and nodes to do numerical integration. Also applying the boundary conditions and initial conditions. Finally we adopt the shape function of RPIM to the results of numerical integration. So that we can build up the system matrix to solve specific problem and get solutions.

RESULTS AND DISCUSSION
To show the accuracy of RPIM, first we compare the result with FEM. We analyze a plate with certain location and length under external load, calculating the stress intensity factor. From simulating result and comparing to exact solution, it can be obtained that RPIM has 1.15% error. Under the condition we set, FEM has -22.5% error. Obviously, RPIM has better result. Finally, discuss the influences of different volume fraction of FGMs which bear time-dependent external force and temperature difference. Also discuss the influences of crack location and length to mechanics behaviors. In first case, the direction of load is perpendicular to crack’s, we can obtain that when crack lies in stronger side which means SiC, in this case, the impact of volume fraction on stress decrease as the length of crack increase. If the crack lies in weaker side, get opposite result. Then we discuss the impact of external force form on stress. We use two types of forces, one is constant magnitude force, the other one is sine-waved force. We find that even the maximum values of two forces are the same, but sine-waved force gets smaller displacement and stress during impact time. Second case we discuss the characterization of FGMs under heat transfer situation. We can obtain that even though the rising percentage of SiC will reduce the value of thermal conductivity, but it also lows down the thermal specific. As a result, it can achieve steady state faster than higher percentage one can do. Finally, we consider the displacement causing by temperature difference. From the results, we can get when crack lies in weaker side, it will get a larger displacement than the one when crack lies in stronger side. Also, with the percentage of SiC arising, thermal displacement will decrease. In the end, we can use RPIM to predict much more accurate fracture characterizations of inhomogeneous materials. RPIM is valuable and reliable simulation method.

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