拘束在彈性圓管內的彈性圓桿受軸向負載作用下的變形

Abstract

先前對於彈性圓桿被拘束在圓管的研究都假設圓管管壁為剛性管壁，這樣的假設在醫學上不適用，像是血管或是人體組織多是彈性體而非剛體，本篇論文中將介紹一根兩端夾持(clamped-clamped)且被拘束在彈性直管或是彈性彎管中的圓桿在端點推力作用下的變形。我們假設彈性圓桿只受到在圓管半徑方向所給的反力，且不考慮摩擦力與重力的影響。在圓管管壁的部份我們假設彈性管壁所給的反力會與接觸點的半徑方向之位移成正比，一般而言在剛性的情況，從接觸轉變到未接觸的過程只會在一個非常小的區域發生，但若為彈性的情況下，這種小區域的轉變會擴散開成為一段線段。當彈性係數 較大時，彈性管壁所給的反作用力的形式會非常接近剛性管壁的形式。舉例來說，在剛性管壁線接觸時，線接觸段的兩端會有一對集中力，若在彈性管壁的情況下也可以從分佈力中看出此現象。另外，被拘束的彈性圓桿其凸出量與分佈力的大小成正比，彈性圓桿的最大凸出量發生的位置不一定要在彈性圓桿的中點，另外，有趣的是軸向負載在最大值時，不會有最大凸出量的發生。在彈性彎管的部分，若以控制推進量的方式施加負載， 的情況會發生跳躍(jump)的現象。若管壁的彈性係數低於 或是實際轉角較大時，則不會發生跳躍的現象。在凸出量的部分，當推力達到最大負載時並不會有最大凸出量，反而是在三線接觸時圓桿的中點位置有最大凸出量。

Most previous works on spatial elastica constrained inside a tube assumed that the tube wall is rigid. This assumption is not adequate when it involves human artery and tissue. In this paper we study the deformation of a clamped-clamped rod constrained inside a straight or curved flexible tube. The reactive force exerted on the rod is assumed to be only in the radial direction and proportional to the radial displacement of the tube wall at the contact point. The results are compared with those of a rigid tube. The complicated deformations, which change from contact to non-contact within a tight range, can exist only when the tube is rigid. As the tube becomes more flexible, the dramatic variation of contact and non-contact tends to be eased off. It is shown that the peculiar phenomenon of concentrated force pairs on the edges of line-contact segment in rigid tube model can be captured by the springy wall model as the spring constant approaches infinity. It is also found that the maximum lateral protrusion of the flexible tube does not necessarily occur at the midpoint of the rod. In the case of a curved tube, jump phenomenon can be observed from the load-deflection diagram if the initial twist is specified at certain level. However, if the initial twist increases to a certain level or the spring constant becomes lower, the jump phenomenon may disappear.