耦合原子與有限元素的創新理論與計算方法之研究

Abstract

正確計算材料在奈米尺度的受力與變形行為，能協助我們深入瞭解奈米尺度的力學性質，提供先進奈米機電元件的設計基礎。但對於當今引領許多創新研發的奈米科技而言，其關心的尺度所包含的原子數目，往往不是分子動力（molecular dynamics）分析所能負擔，而其於缺陷或原子結構生成或互動等的微觀物理現象又不是以連體力學為基礎的有限元素法（finite element）材料組成律所能描述。因此，妥善連接此兩大模擬理論的相關多尺度理論與計算模擬，是目前學術界或工業界急切需要的模擬理論。 緣此，本研究探討連結原子尺度分析與連體力學計算之多尺度擬連體法(quasicontinuum, QC)，此法透過引進有限元素的kinematic constraints (QC with kinematic constraint, QCKC) 耦合分子動力與有限元素，一方面利用原子尺度的微觀計算處理由材料瑕疵或局部外力造成之非均勻變形，另一方面則以有限元素保留連體力學架構處理較大區域的均勻變形，大幅減少原子的自由度數目，也提供了發展簡化力與能量的計算式以達到減少計算量的可能。目前簡化力與能量的計算方法可分為兩類，其一為利用Cauchy-Born法則來減少能量計算的能量近似擬連體理論(QCE)，另一則為利用cluster summation rule以減少力的計算的力近似擬連體理論(QCF)，本研究的主要目的是一方面探討此兩方法是否能有效率並準確近似QCKC，另一方面對其進行改進，進而提出完善的擬連體理論。 在原子區域(atomistic region)與有限元素區域(finite element region)的交界處，本研究發現不論Cauchy-Bron法則或是cluster summation法則都無法成功連結此兩區域，為了解決傳統擬連體法中，原子區域與有限元素區域的不協調，本研究提出過渡區（transition region）的觀念，將局部代表原子(local representative atom)進一步分為過渡區的代表原子(transition representative atom)以及純局部代表原子(pure local representative atom)，再將被代表原子(slaved atom)區分為過渡區的被代表原子(slaved transition atoms)及純被代表原子(pure slaved atom)，針對能量與力的計算方式加以改良，並驗證新的能量與力的計算方式能準確的傳遞原子區的不均勻至有限元素區，克服了此兩區域不協調的課題。 在有限元素區域，本研究探討利用QCE與QCF近似QCKC的誤差，結果顯示，QCE的近似誤差主要取決於有限元素的大小，有限元素越大則QCE的誤差越小，而其誤差主要來自於Cauchy-Born法則「局部」的假設，因此其誤差是無法避免的；QCF的近似誤差則主要決定於取樣原子(sampling atom)的多寡，結果顯示如果要將誤差控制在10%以下則必須取樣超過90%的原子，因此在準確性的考量下QCF法將變得很沒有效率，不過相較於QCE法，QCF法沒有「局部」假設所造成的誤差，因此其誤差可藉由較多的取樣原子屏除之。 有鑒於QCE與QCF法在有限元素區的近似都有其未盡之虞，本研究提出新的取樣方法來近似有限元素區的力，經由引進原子勢能的性質，以及分析同一元素中被代表原子的受力分布，純被代表原子(pure slaved atom)進一步被分為內部純被代表原子(interior pure slaved atom)與表面純被代表原子(exterior pure slaved atom)，結果顯示不論元素的大小為何，只需要20Å的取樣半徑即可將誤差控制在10%以下，相當於當元素的邊長為960Å時只需要0.07%的取樣原子，遠遠少於QCF的超過90%的取樣原子，因此新的取樣方法有效率並準確的近似QCKC法。 最後，經由以上成果的整合，本研究成功發展出在原子、過渡與有限元素區域都能有效率且準確近似QCKC的擬連體法，並將其應用於奈米壓印中硬度尺寸效應的探討，本研究使用圓形壓印子針對金金屬材料在奈米壓印下的硬度尺寸效應，一方面利用分子動力進行半徑15Å, 20Å與25Å的圓形壓印子壓印，另一方面利用擬連體法進行半徑50 Å的圓形壓印子壓印，結果顯示對於圓形的壓印子，硬度對於壓印深度沒有尺寸效應，而是對於壓印子的大小有尺寸效應，硬度隨著壓印子的變大而減少，相符於實驗觀察到的現象。

The theoretical and computational limitations of continuous description and atomistic modeling on capturing phenomena at the micro- or nano-scale have called for the need to fuse atoms with finite elements. Atomistic modeling has been used to address a wide variety of deformation processes in solids in the past decades. Nevertheless, the length scale restriction of atomistic modeling has long been a substantial obstacle in making useful prediction. In the past few years, a very promising method called the quasicontinuum (QC) method has been developed to circumvent this length scale problem. By using kinematic constraints through finite element interpolation, the QC method with kinematic constraints (QCKC) allows for developing fully atomistic scale resolution near defects while exploiting coarser description further away to reduce redundant degrees of freedom. Furthermore, utilization of kinematic constraints makes it possible to develop approximated energy and force formulations in an efficient way. Two approximated methods have been developed. One is based on energy approximation (QCE) and the other on force approximation (QCF). The QCE was developed by Tadmor et al. (1996) and Shenoy et al. (1999a) by using the Cauchy-Born rule to approximate the energy functional. The QCF was developed by Knap and Ortiz (2001) by introducing the cluster summation rule to approximate force calculations. The objective of this thesis is to revisit current practice in QC, analyze variants of QC to reproduce QCKC and derive novel formulations for improvement. For the interface of atomistic region and finite elements, improperly applying Cauchy-Born rule and cluster summation rule at the interface have been identified and the deficiency is removed by introducing transition entities to the QC method. New description of the force formulation that allows transition elements to correctly transmit inhomogeneity from the atomistic to the continuum regions is proposed. The formulation is verified through reproducing the QCKC. The errors of force formulation for finite element region of both the QCE and QCF methods are analyzed. For the QCE method, errors are mainly governed by the size of elements but not by the variation of deformation gradients. The error mainly results from the inherent “local” assumption of the Cauchy-Born rule and thus it is unavoidable. For the QCF method, errors are mainly affected by the ratio of the number of sampling atoms to the number of total slaved atoms. To control the error within 10% the sampling ratio should be greater than 90%, an indication of a very inefficient scheme. Generally speaking, the QCE method is only suitable for large elements while the QCF method is only suitable for small elements. By considering the property of central symmetry potential and by analyzing the distribution of atomic forces of represented atoms within an element, a novel summation rule based on decomposition of force contributions from slaved atoms in exterior cluster is developed. The new summation rule approximates the forces for finite element region accurately and efficiently. It is shown that for a constant sampling cutoff (20 angstroms), the error of computing force is controlled under 10% for all sizes of elements. Therefore, the force formulation using the summation rule proposed herein reproduces the QCKC method efficiently. Finally, QC with accurate and efficient formulations to reproduce the QCKC method at the atomistic, transition and finite element regions is compiled together and used to study indentation size effect of Au thin film with spherical indenters. The results show that for a spherical indenter, the hardness is not affected by indentation depth but by the sphere radius. The hardness decreases with the increase of the sphere radius.