# 统计代写|主成分分析代写Principal Component Analysis代考|”Axisymmetric Conditions in Elastic Stress Analysis: Displacement, Strain, and Stress-Strain Relations”

In the context of finite element analysis, the passage discusses the application of axisymmetric conditions to problems involving elastic stress analysis. Axisymmetric problems assume that a solid object is symmetric about a central axis (the z-axis in cylindrical coordinates), and that all loads, boundary conditions, and material properties are also symmetric about this axis. Under these circumstances, the displacement field does not depend on the tangential coordinate (θ), which simplifies the analysis to a two-dimensional problem in the rz-plane.

The following are the key aspects from the excerpt:

Radial Displacement (u): The radial strain is directly proportional to the radial derivative of the radial displacement: ϵr=∂u∂r\epsilon_r = \frac{tial u}{tial r}ϵ r ​ = ∂r ∂u ​

Axial Displacement (w): The axial strain is the axial derivative of the axial displacement: ϵz=∂w∂z\epsilon_z = \frac{tial w}{tial z}ϵ z ​ = ∂z ∂w ​

Circumferential (Tangential) Displacement (v): Although the problem is independent of θ, the tangential strain exists due to the change in circumference. It is given by: ϵθ=ur\epsilon_{\theta} = \frac{u}{r}ϵ θ ​ = r u ​

Shear Strains: Only one non-zero shear strain component exists in the rz-plane: γrz=∂u∂z+∂w∂r\gamma_{rz} = \frac{tial u}{tial z} + \frac{tial w}{tial r}γ rz ​ = ∂z ∂u ​ + ∂r ∂w ​ While the shear strains in the radial-tangential and tangential-axial directions are zero.

Stress-Strain Relations: By substituting the strain components into Hooke’s law for isotropic materials (adjusted for axisymmetric conditions), we obtain the stress components:

\sigma_r &= \frac{E}{(1+\nu)(1-2\nu)} [(1-\nu)\epsilon_r + \nu(\epsilon_{\theta} + \epsilon_z)] \sigma_{\theta} &= \frac{E}{(1+\nu)(1-2\nu)} [(1-\nu)\epsilon_{\theta} + \nu(\epsilon_r + \epsilon_z)] \sigma_z &= \frac{E}{(1+\nu)(1-2\nu)} [(1-\nu)\epsilon_z + \nu(\epsilon_r + \epsilon_{\theta})] \tau_{rz} &= \frac{E}{2(1+\nu)} \gamma_{rz} \end{align*} \] Material Property Matrix ([D]): These stress-strain relationships can be written in matrix form with the material property matrix:

\sigma_r & \sigma_{\theta} & \sigma_z & \tau_{rz} \end{array}\right] = \frac{E}{(1+\nu)(1-2\nu)} \left[\begin{array}{ccc|c} 1-\nu^2 & \nu & \nu & 0 \nu & 1-\nu & \nu & 0 \nu & \nu & 1-\nu & 0 \hline 0 & 0 & 0 & \frac{1-2\nu}{2} \end{array}\right] \left[\begin{array}{c} \epsilon_r \epsilon_{\theta} \epsilon_z \gamma_{rz} \end{array}\right] \] This allows for a simplified treatment of three-dimensional stress analysis problems by reducing them to two dimensions in the rz-plane, making it computationally efficient for solving axisymmetric structures. However, special attention is required near the axis of symmetry (r=0), particularly for problems involving rotation and centrifugal forces, which introduce additional complexities in the analysis.

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