地質材料的非線性動力學探討

Investigation On The Nonlinear Dynamics Of Geological Material

黃存慧(Tsun-Hui Huang)

指導教授 ： 謝秋雰

國立中正大學/理學院/地球與環境科學系地震研究所/博士(2012年)

摘要

中文摘要

地震發生的循環大致可分成四個階段，即動力加壓、準靜態結核、動力破裂傳播和停止，還有伴隨發生的應力重新分配和斷層強度的恢復。一般研究較少關注斷層發生破裂前的應力累積階段，而此階段應力的累積方式與之後斷層破裂動力學之間的關係有待釐清。本研究由材料的非線性應力-應變著手，建立非線性波動方程式，以數值方法探求具有外力源模型破裂發生前應力累積的方式。
首先，由實驗室材料壓應力試驗所得之應力-應變關係曲線，經由曲線擬合的方法取得非線性多項式方程式，再進一步建立非線性波動方程式。在一維的模型中，大致可分為曲線凹向下(曲線趨向應力軸)與曲線凹向上(曲線趨向應變軸)兩種曲線模式；曲線凹向下模型所得之應力主要累積在接近受壓端的極小區段內，並有一個明顯的應變斜率轉折處，此結果不會因為曲線擷取擬合區段不同及外力施加值大小而有太大差異，推論岩層破裂極可能發生於此斜率轉折變化處。曲線凹向上的模型，應力與應變的傳遞方式以數個應力、應變孤立波的形式傳遞。此應力、應變孤立波可往前傳遞，並與擠壓處相隔一段距離；各個孤立波的能量傳遞方式複雜，主要的趨勢為強度隨擠壓時間增加，屬增長型應力、應變孤立波，移動速率約2m/s；變動的應力孤立波可能造成岩層的不穩定，使其易於破裂，而破裂位置也可發生在離施壓端較遠處。此現象可用以解釋非板塊邊界處地震發生的原因。
二維模型中，施予單位步階函數型外力與單純脈衝函數型外力源，此二者皆產生明、暗孤立波組成的應力孤立波群傳遞能量，兩種外力源產生的應力孤立波主要的差別在應力強度上，單純脈衝函數型外力施壓時間短，產生的應力孤立波強度較小；應力孤立波群移動速率約370m/s，隨傳遞距離增加強度略減。透過兩個脈衝函數型外力，檢視相對及垂直方向碰撞交會的過程發現，個別能量仍保持原波形、速度、方向前進，因此可確定這些應力波為孤立波。應力孤立波群在岩層中的傳遞使岩層短時間內承受應力的擠壓、拉張，降低岩層強度，甚至造成破裂；當岩層受到似脈衝(pulse-like)的應力衝擊後隨即恢復原應力強度，此特性與地震破裂動力學中Heaton(1990)所提之自體癒合滑移脈衝模型(self-healing slip pulse model)類似，因此本研究結果支持Heaton的理論，但也由研究結果可知，實際破裂瞬間的應力狀況遠較Heaton模型中單一應力脈衝複雜。

關鍵字

非線性動力學 ； 地質材料

並列摘要

ABSTRACT

The entire earthquake generation cycle can be divided into four stages, namely tectonic loading, quasi-static nucleation, dynamic rupture propagation and stop, and subsequent stress redistribution and fault strength restoration. The investigation of stress accumulation prior to fault rupture was rarely studied. At this stage, the relationship between the cumulative stress and the fault rupture dynamic can be clarified. A nonlinear stress-strain material was used to establish a nonlinear wave equation. A numerical method is used to explore the stress accumulation of the established model with external pressure sources before the stress fracture.
First, a curve fitting method was used to establish a non-linear constitutive equation of stress-strain relation with rock material, which is an approximation method. Following the obtained nonlinear polynomial equation, a nonlinear wave equation was established. In one dimensional, the stress-strain curve can be divided into concave downward curve (curve towards the strain axis) and concave upward curve (curve towards ordinate axis). The stress accumulation of the curve concave down model was near a very small section of the pressure side, and there is a clear turning point of the slope of strain. This result will not be much difference between the curve fitting at different segments and the values of external force. The inference of a rock rupture was occurred in slope change at a turning point.
In the model of concave upward curve, the transference of stress and strain was passed by a number of stress and strain in the form of solitary waves. The stress and strain solitary waves can transfer forward from a distance of the external force. The energy transfer of a solitary wave was in various complex ways. The main trend is the strength increasing with extrusion time. It is a growth-oriented stress, strain solitary wave, moving speed of about 2m/s. An alternating stress soliton can result in an unstable rock and be easily ruptured. The rupture position was far from the external force. This phenomenon can be used to explain the because of non-plate boundary earthquakes. In two-dimensional mathematical model, two kinds of external force were used. One is a unit step function; the other is impulse function. Both of them can generate blight and darken stress solitons for energy transferring. The difference of the two external forces is mainly in the strength of stress. The kind of impulse force was acted in very short time. The strength of the stress solitary wave is smaller by impulse force. The speed of solitary wave is 370m/s. Increases along with the transmission distance, the intensity slightly reduce. Through two kinds of impulse force, two solitary waves were generated in each orthogonal direction and were collided. After they were collided, their form, speed and direction were kept. Therefore, the stress wave was solitary wave in this work. The layer of a rock was bearing pressure and external stress based on the generated stress solitary and cut down the strength of the rock so as to make a rupture of the rock. When the stress solitary wave was generated by impulse force and passed a rock layer, its stress value was quickly restored. This characteristic was similar to the self-healing slip pulse model of Heaton’s theory in 1990. Therefore, the results of this paper support Heaton’s theory. However, the results of this paper were complex than single impulse of Heaton’s model.

並列關鍵字

Geological Material ； Nonlinear Dynamics

參考文獻

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