What are the components of the stress tensor based on the given formulation?

[gsh84]

The following information is given:

The Cartesian components of stress tensor are: $$\sigma_{ij}=m(\hat \lambda _{i}\hat \mu _{j}+\hat \lambda _{j}\hat \mu _{i}) , (i=1,2,3; \ j=1,2,3)$$.

$$\hat \lambda _{i}$$ and $$\hat \mu_{j}$$ are the Cartesian components of the unit vectors $$\hat \lambda$$ and $$\hat \mu$$ , who enclose an angle of $$2 \alpha$$.

m is a scalar with a stress dimension.

Now my question. What are the components($$\sigma_{11}, \sigma_{12}.. \sigma_{33}$$) of the the stress tensor based based on this formulation?

According to the information you can say: $$\hat \lambda \cdot \hat \mu = \cos 2\alpha$$

Is for $$\sigma_{11}$$ example: $$(\cos 2\alpha+\cos 2\alpha)p$$?
And what would $$\sigma_{12}$$ be? 0?

 

[jbsteele]

Yes, \sigma_{11} would be (\cos 2\alpha+\cos 2\alpha)m and \sigma_{12} would be 0. The other components of the stress tensor would be as follows:\sigma_{11} = (\cos 2\alpha+\cos 2\alpha)m\sigma_{12} = 0\sigma_{13} = (\cos 2\alpha-\cos 2\alpha)m\sigma_{21} = 0\sigma_{22} = (\cos 2\alpha+\cos 2\alpha)m\sigma_{23} = (\cos 2\alpha-\cos 2\alpha)m\sigma_{31} = (\cos 2\alpha-\cos 2\alpha)m\sigma_{32} = (\cos 2\alpha-\cos 2\alpha)m\sigma_{33} = (\cos 2\alpha+\cos 2\alpha)m

 

[nlightner]

I would like to clarify that the given formulation represents the general form of the stress tensor in Cartesian coordinates. The components of the stress tensor, denoted by \sigma_{ij}, represent the stress at a point in a material in the i and j directions. Therefore, there are nine components of the stress tensor, \sigma_{11}, \sigma_{12}, \sigma_{13}, \sigma_{21}, \sigma_{22}, \sigma_{23}, \sigma_{31}, \sigma_{32}, and \sigma_{33}. Each component is a function of the unit vectors \hat \lambda and \hat \mu and the scalar m, as shown in the given formulation.

To answer your question, the components of the stress tensor, \sigma_{11}, \sigma_{12}, \sigma_{13}, etc., cannot be determined based on the given formulation alone. Additional information about the material and the specific stress conditions at the point of interest is needed to calculate these components. The given formulation only provides the general form of the stress tensor.

Furthermore, the expression \hat \lambda \cdot \hat \mu represents the dot product of the two unit vectors, which is a scalar value. It is not correct to say that \hat \lambda \cdot \hat \mu = \cos 2\alpha. The dot product is a mathematical operation that results in a scalar value, whereas \cos 2\alpha is a trigonometric function that gives a ratio of two sides of a right triangle. The two are not equivalent.

In conclusion, the given formulation provides the general form of the stress tensor in Cartesian coordinates. To determine the components of the stress tensor, additional information about the material and the specific stress conditions is needed. The expression \hat \lambda \cdot \hat \mu should not be equated to \cos 2\alpha as they represent different mathematical concepts.