冪定律用於循環彈塑性與黏彈性之建模(2/2)

Abstract

本計畫以兩年為期，探索固態系統之黏彈性與彈塑性行為反應規律，嘗試找出其共通的共 性，歸結基本原理，以適當的數學語言表達之。以此共性、原理、數學模式為平台，具體引入黏 彈性時間冪定律及塑性當量冪定律的觀念，藉由冪定律與分數導數及積分組成律核函數的關係來 系統化地模擬黏彈性與彈塑性應力應變關係。 本計畫研究各種不老化線性黏彈性組成模式，其中無參數的有積分組成律、分布Kelvin 與 Maxwell 模型、分布分數Kelvin 與Maxwell 模式等三種，有參數的有微分組成律、廣義Kelvin 與Maxwell 模型、各種特定分布Kelvin 與Maxwell 模型、狀態空間表示法、分數微積分組成模 式、各種特定分布分數Kelvin 與Maxwell 模式等六種。因為積分組成律可作為各種研究的平台， 本計畫進而研究它的16 種寫法，並且由線性不老化，推廣至線性老化，也研究其推廣為非線性老化的通式。 在黏彈性及彈塑性的各種應力應變關係中，循環棘齒行為是極難妥適建模描述的。一般認 為唯有在應力均值不為零的循環負載下才會發生棘齒，但由實際的文獻與實驗卻發現即使在應力 均值為零的對稱等振幅應力循環負載下，也能觀察到拉伸向的棘齒現象。本計劃在實驗部分嘗試 把實驗分為真實應力控制與標稱應力控制來分析在應力均值為零的循環負載下所觀察到的棘齒 現象。經比較分析後證實：不論是真實應力控制或標稱應力控制的實驗，在應力均值為零的循環負載下都會往拉伸向產生棘齒。 由進一步分析顯示，上述在應力均值為零時所觀察到的拉伸向棘齒現象是由拉、壓硬化不 對稱所造成，壓硬化大於拉硬化，因此隨著實驗循環圈數的累積，會逐漸往拉伸向產生棘齒。而 且先拉或先壓不同的控制歷時會影響拉、壓硬化不對稱的程度，先壓的控制歷時會讓壓硬化加 劇，因此往拉伸向的棘齒會較先拉控制歷時還快出現，棘齒現象也較明顯。對於循環硬化材料， 如鋁7075，拉伸向的棘齒現象在一開始的幾圈循環裡會被循環硬化所隱蔽。但隨著循環硬化的 影響逐漸變小，便可觀察到拉伸向棘齒。 另外，為了讓實驗達到真實應力控制，本研究也導出徑向位移的理論公式，並以自製的環向伸長計驗證其精確度。

The present project was proposed to study in a two-year period common characteristics and basic principles underpinning viscoelasticity and elastoplasticity. The power law is specifically introduced into the stress-strain relationship by converting the power law to the fractional derivative and to the kernel of the stress functional of the plastic strain increment. We studied various constitutive models of non-aging linear viscoelasticity. These include 3 kinds of non-parametric models: integral constitutive laws, distributed Kelvin and Maxwell models, and distributed fractional Kelvin and Maxwell models, and also 6 kinds of parametric models: differential constitutive laws, generalized Kelvin and Maxwell models, distributed Kelvin and Maxwell models of a variety of special distributions, state space methods, fractional derivative constitutive models, and distributed fractional Kelvin and Maxwell models of a variety of special distributions. In view of the role the integral constitutive laws play as a framework for constructing and uniting various forms of constitutive modeling, we went further to investigate their 16 expressions, generalizing them from non-aging linearity to aging linearity, and furthermore to a general expression for aging nonlinarity. Ratchetting behavior is one of the most difficult phenomena to model among the viscoelastic and elastoplastic stress-strain relations. Generally speaking, ratchetting can be found under cyclic loading with non-zero mean stress. However, some experiments showed that even under cyclic loading with zero mean stress, ratchetting in the direction of tension can still be found. The experimental part of this project analyzed the phenomena of ratchetting under nominal-stress -controlled cyclic loading and true-stress-controlled cyclic loading with zero mean stress. The results show that ratchetting in the direction of tension can be found in true-stress-controlled experiments as well as in nominal-stress-controlled experiments. The results of analysis also show that the ratchetting in the direction of tension as mentioned above was caused by the asymmetry of hardening between tension and compression, the hardening of compression being larger than the hardening of tension and hence the tensile strain being larger than the compressive strain in each cycle. As the cyclic contributions accumulated, the ratchetting in the direction of tension gradually developed. Furthermore, the difference of controlled path would also affect the asymmetry of hardening between tension and compression. If the controlled path started in the compression direction, the asymmetry of hardening between tension and compression would be more apparent than that of the controlled path starting in the direction of tension; therefore, ratchetting is more apparent. For cyclic hardening materials, e.g. Al 7075, the phenomenon of ratchetting in the direction of tension was shadowed in the first few cycles. Once the cyclic hardening effect phased out, ratchetting manifested itself. In order to conduct true-stress-controlled experiments, we developed formulae to calculate the theoretical value of radial strain, and checked its accuracy by a self-developed radian strain extensometer.