莫尔圆是二维图形表示 柯西应力张量它可视化了法线的变换。 压力 在任意方向的平面上,某一点的正应力(σn)和剪应力(τn)。圆上每一点的横坐标为正应力,纵坐标为剪应力,从而可以方便地确定主应力。

(图片仅供参考)
Mohr’s circle provides a powerful graphical tool to understand the state of stress at a point within a continuous body. For any given 2D stress state defined by normal stresses [latex]\sigma_x[/latex], [latex]\sigma_y[/latex] and 剪应力 [latex]\tau_{xy}[/latex], the circle allows one to find the stresses on any plane passing through that point. The center of the circle is located on the [latex]\sigma_n[/latex] axis at [latex]C = (\sigma_{avg}, 0)[/latex], where [latex]\sigma_{avg} = (\sigma_x + \sigma_y)/2[/latex]. The radius of the circle is calculated as [latex]R = \sqrt{\left(\frac{\sigma_x – \sigma_y}{2}\right)^2 + \tau_{xy}^2}[/latex]. Each point on the circumference of the circle represents the stress state ([latex]\sigma_n, \tau_n[/latex]) on a specific plane. A rotation of an angle [latex]\theta[/latex] of the physical plane corresponds to a rotation of [latex]2\theta[/latex] on Mohr’s circle in the same direction. This graphical method elegantly bypasses the need to solve the stress transformation equations directly for each angle, making it an intuitive and efficient method for engineers and physicists.
历史上,克里斯蒂安·奥托·莫尔于1882年发展了这种方法。它相比纯粹的解析方法是一项重大进步,提供了一种可视化的辅助手段,极大地简化了应力转换的复杂数学计算。在莫尔之前,工程师们依赖于奥古斯丁-路易·柯西的应力张量公式,该公式虽然功能强大,但在实际设计应用中却不够直观。莫尔的图形化方法使得主应力和最大剪应力的概念变得易于理解,而这些概念对于根据特雷斯卡准则或冯·米塞斯准则等理论预测材料失效至关重要。
莫尔压力圈
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