Chapter 16: Problem 20
It is found that the stress tensor at any point \((x, y, z)\) in a certain continuous medium has the form (with an unspecified, convenient choice of units) \(\mathbf{\Sigma}=\left[\begin{array}{ccc}x z & z^{2} & 0 \\ z^{2} & 0 & -y \\ 0 & -y & 0\end{array}\right]\) Find the surface force on a small area \(d A\) of the surface \(x^{2}+y^{2}+2 z^{2}=4\) at the point (1,1,1)
Short Answer
Step by step solution
Calculate Surface Normal Vector
Calculate Surface Normal Unit Vector
Evaluate Stress Tensor at Point (1, 1, 1)
Calculate Surface Force Using Stress Tensor
Summarize the Result
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Surface Force
The surface force will usually depend on two main factors:
- The stress tensor, which gives the distribution of internal forces within a body.
- The normal vector, which indicates the orientation of the surface at a given point.
This results in a force vector that tells us not only the magnitude of the force but also its direction in relation to the surface. For example, in the exercise, surface force calculations reveal the stress and tension experienced by the surface at a point, guiding engineers and scientists in understanding material behavior under stress conditions.
Normal Vector
The process of finding a normal vector involves calculating the gradient of the surface's mathematical equation. The gradient is a vector operation that points in the direction of the steepest ascent and its magnitude gives the rate of increase of the function's value.
- In the original exercise, the surface equation, given by \(x^2 + y^2 + 2z^2 = 4\), needs a gradient calculation to find the normal vector.
- The gradient vector \([2x, 2y, 4z]\) evaluated at the point \((1,1,1)\) yields the normal vector \([2, 2, 4]\).
Stress Tensor Evaluation
In mathematics and physics, a stress tensor is a 3x3 matrix that describes the stress applied to a point within a medium across three dimensions. Each element of the matrix represents a component of force experienced by an area element in the material.
The process involves these steps:
- Identify the components of the stress tensor for the specific point of interest. In the given exercise, the stress tensor \(\Sigma\) represents these components at point \((1,1,1)\).
- To evaluate how these stresses affect the surface, the tensor is multiplied by the unit normal vector derived from the surface equation. This multiplication gives the actual surface force, which can be interpreted to understand the material's response at that point.