Note on the Theoretical Background of 3D Stress Transform

Note on the Theoretical Background of 3D Stress Transformation

1. State of Stress in 3D

In three dimensions, the state of stress at a point is represented by a stress tensor in matrix form:

$$ \mathbf{\sigma} = \begin{bmatrix} \sigma_x & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_y & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_z \end{bmatrix} $$

where:

$\sigma_x, \sigma_y, \sigma_z$ are the normal stresses in the x, y, and z directions.
$\tau_{xy}, \tau_{xz}, \tau_{yz}$ are the shear stresses.
Due to the complementary property of shear stress: $\tau_{xy} = \tau_{yx}, \tau_{xz} = \tau_{zx}, \tau_{yz} = \tau_{zy}$.

Fig.1. State of stress in 3D

Since the stress tensor is symmetric (due to the complementary property of shear stress), it has six independent components:

Three normal stress components: $\sigma_x, \sigma_y, \sigma_z$
Three shear stress components: $\tau_{xy}, \tau_{xz}, \tau_{yz}$

Sign Convention

Normal stresses ($\sigma_x, \sigma_y, \sigma_z$) are positive when they cause tension and negative when they cause compression.
Shear stresses ($\tau$) are positive if they act in the positive coordinate direction on a face whose outward normal is in the positive coordinate direction.
Shear stresses are negative if they act in the negative coordinate direction on a face whose outward normal is in the positive coordinate direction.

2. Transformation of Stress in 3D

To determine the normal and shear stresses acting on a plane with a unit normal vector $ \mathbf{n} = (n_x, n_y, n_z) $, we proceed as follows.

Geometric Interpretation of $ n_x, n_y, n_z $

Let the plane intersect the x, y, and z axes at points A, B, and C, respectively, with the origin as O.

Fig.2. The plane with a normal of $ \mathbf{n} = (n_x, n_y, n_z) $

The components of the unit normal vector can be expressed in terms of area ratios:

$$ n_x = \frac{A_{\triangle BOC}}{A_{\triangle ABC}}, \quad n_y = \frac{A_{\triangle AOC}}{A_{\triangle ABC}}, \quad n_z = \frac{A_{\triangle ABO}}{A_{\triangle ABC}} $$

These values satisfy the fundamental condition:

$$ n_x^2 + n_y^2 + n_z^2 = 1 $$

Since this equation involves three unknowns, specifying two angles is sufficient to fully determine the normal vector $ \mathbf{n} $.

The angles $\alpha$, $\beta$, and $\theta$ are the angles between the normal vector $\mathbf{n}$ and the coordinate axes:

$\alpha$ is the angle between $\mathbf{n}$ and the x-axis.
$\beta$ is the angle between $\mathbf{n}$ and the y-axis.
$\theta$ is the angle between $\mathbf{n}$ and the z-axis.

Fig.3. Definition of Angles $\alpha, \beta, \theta$ in 3D Stress Transformation

The normal components can be expressed in terms of these angles as:

$$ n_x = \cos(\alpha), \quad n_y = \cos(\beta), \quad n_z = \cos(\theta) $$

Stress Vector on the Inclined Plane

The stress vector $ \boldsymbol{\sigma_n} $ acting on the plane with normal $ \mathbf{n} $ is obtained by:

$$ \boldsymbol{\sigma_n} = \boldsymbol{\sigma} \cdot \mathbf{n} $$

Expanding in component form:

$$ \boldsymbol{\sigma_n} = \begin{bmatrix} \sigma_x n_x + \tau_{xy} n_y + \tau_{xz} n_z \\ \tau_{xy} n_x + \sigma_y n_y + \tau_{yz} n_z \\ \tau_{xz} n_x + \tau_{yz} n_y + \sigma_z n_z \end{bmatrix} = \begin{bmatrix} \sigma_{n_x} \\ \sigma_{n_y} \\ \sigma_{n_z} \end{bmatrix} $$

The maginute of $ \boldsymbol{\sigma_n} $ can be given as follows:

$$ \sigma_n = \sqrt{\sigma_{n_x}^2 + \sigma_{n_y}^2 + \sigma_{n_z}^2} $$

Normal and Shear Stress Components

The normal stress $ \sigma $ is the projection of $ \boldsymbol{\sigma_n} $ onto $ \mathbf{n} $:

$$ \sigma = \boldsymbol{\sigma_n} \cdot \mathbf{n} $$

Expanding:

$$ \sigma = \sigma_{n_x} n_x + \sigma_{n_y} n_y + \sigma_{n_z} n_z $$ $$ \sigma = \sigma_x n_x^2 + \sigma_y n_y^2 + \sigma_z n_z^2 + 2(\tau_{xy} n_x n_y + \tau_{xz} n_x n_z + \tau_{yz} n_y n_z) $$

This scalar value represents the normal stress acting in the direction of $ \mathbf{n} $.

The remaining component, the shear stress $ \tau $, is the magnitude of the stress vector perpendicular to $ \mathbf{n} $:

$$ \tau = \sqrt{\sigma_n^2 - \sigma^2} $$

3. Principal Stresses

The principal stresses $ \sigma_1, \sigma_2, \sigma_3 $ are the eigenvalues of the stress tensor. They are found by solving the characteristic equation:

$$ \det(\mathbf{\sigma} - \lambda I) = \begin{vmatrix} \sigma_x - \lambda & \tau_{xy} & \tau_{xz} \\ \tau_{yx} & \sigma_y - \lambda & \tau_{yz} \\ \tau_{zx} & \tau_{zy} & \sigma_z - \lambda \end{vmatrix} = 0 $$

Expanding the determinant, we obtain the characteristic equation:

$$ -\lambda^3 + (\sigma_x + \sigma_y + \sigma_z)\lambda^2 $$ $$ - (\sigma_x\sigma_y + \sigma_y\sigma_z + \sigma_z\sigma_x - \tau_{xy}^2 - \tau_{yz}^2 - \tau_{zx}^2)\lambda $$ $$ + (\sigma_x\sigma_y\sigma_z + 2\tau_{xy}\tau_{yz}\tau_{zx} - \sigma_x\tau_{yz}^2 - \sigma_y\tau_{zx}^2 - \sigma_z\tau_{xy}^2) = 0 $$

Using stress invariants, the characteristic equation can be rewritten as:

$$ \lambda^3 - I_1 \lambda^2 + I_2 \lambda - I_3 = 0 $$

where the stress invariants are:

First invariant: $ I_1 = \sigma_x + \sigma_y + \sigma_z $ (Sum of normal stresses)
Second invariant: $ I_2 = \sigma_x\sigma_y + \sigma_y\sigma_z + \sigma_z\sigma_x - \tau_{xy}^2 - \tau_{yz}^2 - \tau_{zx}^2 $
Third invariant: $ I_3 = \sigma_x\sigma_y\sigma_z + 2\tau_{xy}\tau_{yz}\tau_{zx} - \sigma_x\tau_{yz}^2 - \sigma_y\tau_{zx}^2 - \sigma_z\tau_{xy}^2 $ (Determinant of stress tensor)

Principal Directions

For each principal stress $ \sigma_i $, the corresponding principal direction can be determined in two ways:

i) Using Eigenvector Equation

The principal directions correspond to the eigenvectors $ \mathbf{n} = (n_x, n_y, n_z) $ of the stress tensor, which satisfy the eigenvalue problem:

$$ (\mathbf{\sigma} - \sigma_i I) \mathbf{n} = 0 $$

Here, the eigenvalues $ \sigma_i $ are the principal stresses, as mentioned earlier, and the corresponding eigenvectors define the principal directions for each principal stress. This forms a system of three homogeneous equations, which can be solved to obtain $ n_x, n_y, n_z $.

ii) Solving the Characteristic Equations

Alternatively, by substituting a principal stress $ \sigma_i $ into the system:

$$ (\sigma_x - \sigma_i)n_x + \tau_{xy} n_y + \tau_{xz} n_z = 0 $$ $$ \tau_{yx} n_x + (\sigma_y - \sigma_i)n_y + \tau_{yz} n_z = 0 $$ $$ \tau_{zx} n_x + \tau_{zy} n_y + (\sigma_z - \sigma_i)n_z = 0 $$

Solving this system gives $ n_x, n_y, n_z $, which are proportional values.

Since these are direction cosines, they must satisfy:

$$ n_x^2 + n_y^2 + n_z^2 = 1 $$

This constraint ensures the correct unit direction vector for each principal stress.

4. Mohr’s Circle for 3D Stress

Mohr’s Circle provides a graphical representation of the state of stress at a point in 3D. Unlike the 2D case, three principal stresses exist, and we construct three Mohr’s Circles to visualize the normal and shear stresses on different planes.

i) Steps to Draw Mohr’s Circles

To construct Mohr’s Circles for 3D stress, follow these steps:

First, determine the three principal stresses $ \sigma_1, \sigma_2, \sigma_3 $ by solving the characteristic equation.
Plot these principal stresses on the horizontal axis of the coordinate system.
Draw three circles using these principal stresses as follows:
- Largest circle: Center at $ (\sigma_1 + \sigma_3)/2 $, radius $ (\sigma_1 - \sigma_3)/2 $
- Middle circle: Center at $ (\sigma_1 + \sigma_2)/2 $, radius $ (\sigma_1 - \sigma_2)/2 $
- Smallest circle: Center at $ (\sigma_2 + \sigma_3)/2 $, radius $ (\sigma_2 - \sigma_3)/2 $

Fig.4. Mohr's Circle in 3D

ii) Interpretation of the Mohr’s Circles

Each point on Mohr’s Circles represents a state of stress on a particular plane. Largest circle ($ \sigma_1, \sigma_3 $) represents the absolute maximum shear stress.

The center of the largest Mohr’s Circle is the average normal stress:

$$ \sigma_{avg} = \frac{\sigma_1 + \sigma_3}{2} $$

The maximum shear stress in the system is given by:

$$ \tau_{max} = \frac{\sigma_1 - \sigma_3}{2} $$

iii) Summary

Mohr’s Circle in 3D stress analysis is a useful tool to visualize the stress state, determine maximum shear stress, and analyze stress transformation on arbitrary planes.