Regen D M
Department of Molecular Physiology and Biophysics, Vanderbilt University School of Medicine, Nashville, TN 37232, USA.
Ann Biomed Eng. 1996 May-Jun;24(3):400-17. doi: 10.1007/BF02660889.
Equations for calculating wall tensions in an ellipsoidal chamber might be useful in analyses of elongated chambers whose transverse sections are not round, and they should be useful for examining the tension distribution associated with such shapes. Considering the forces changing a prolate spheroid (semiaxes a > b = c) into a general ellipsoid (semiaxes a > b > c) led to an equation for tensions at the poles of an ellipsoid. Considering the thickness distribution of a chamber of uniform average stress led to an equation for the average of orthogonal tensions at any point on an ellipsoidal chamber. Applying these equations with Laplace's law to points along an axis plane showed that tension normal to that plane is a weighted average of tensions normal to that plane at the intersections of the ellipsoid with the other two axis planes. It was postulated that this rule also would apply to the tension component normal to a plane coincident with any hoop (line of constant distance from one axis plane), and this postulate led to an equation for tension orthogonal to a hoop at any point. These three equations (pole tensions, local average tension, local hoop-orthogonal tension) allowed calculation of the tension tensor at any point. The equations and their algorithm were validated by four tests: the surface integral of the average of orthogonal tensions is as necessary for tensile work to equal hydraulic work in a symmetrical displacement (satisfying chamber equilibrium), the line integral of the component of tension normal to any hoop-coincident plane is equal to the product of pressure and area in the hoop (satisfying force balance), at any point the tensions predicted from the tensor for the directions of greatest and least curvature are compatible with Laplace's law (satisfying local equilibrium), and the calculated principal-tension lines relate properly to the nodes where tension is the same in all surface directions. These tests could be used to validate finite-element analyses of complex chambers.
计算椭圆形腔室壁张力的方程可能有助于分析横截面不是圆形的细长腔室，并且对于研究与这种形状相关的张力分布应该是有用的。考虑将长椭球体（半轴a > b = c）转变为一般椭球体（半轴a > b > c）时的力，得出了椭球体极点处张力的方程。考虑具有均匀平均应力的腔室的厚度分布，得出了椭圆形腔室任意点处正交张力平均值的方程。将这些方程与拉普拉斯定律应用于沿轴平面的点，结果表明垂直于该平面的张力是椭球体与其他两个轴平面相交处垂直于该平面的张力的加权平均值。据推测，该规则也适用于垂直于与任何环（与一个轴平面距离恒定的线）重合的平面的张力分量，并且这一推测得出了椭圆形腔室任意点处垂直于环的张力的方程。这三个方程（极点张力方程、局部平均张力方程、局部环向正交张力方程）允许计算任意点处的张力张量。这些方程及其算法通过四项测试得到了验证：正交张力平均值的表面积分对于拉伸功等于对称位移中的液压功（满足腔室平衡）是必要的，垂直于任何与环重合平面的张力分量的线积分等于环内压力与面积的乘积（满足力平衡），在任意点处从张量预测的最大和最小曲率方向的张力与拉普拉斯定律兼容（满足局部平衡），并且计算出的主张力线与所有表面方向上张力相同的节点正确相关。这些测试可用于验证复杂腔室的有限元分析。
