generate.inclined_plane¶

Inclined plane stress field simulation.

This module provides functions for simulating the photoelastic response of a material under inclined plane stress conditions (rectangular mass with gravity acting at an angle).

`inclined_plane` ¶

Functions¶

`inclined_stress_cartesian(X, Y, rho, g=9.81, theta_deg=0.0, K0=0.5)` ¶

Stress field for an inclined plane (rectangular mass with inclined gravity).

This computes the stress field in a rectangular domain with gravity acting at an angle theta from vertical. The resulting stress field includes both normal stresses that vary with depth and shear stresses due to the inclined gravity.

Parameters:

Name	Type	Description	Default
`X`	`array - like`	X coordinates (horizontal position).	required
`Y`	`array - like`	Y coordinates (depth, positive downward from surface).	required
`rho`	`float`	Density (kg/m^3).	required
`g`	`float`	Gravitational acceleration (m/s^2).	`9.81`
`theta_deg`	`float`	Inclination angle of gravity from vertical (degrees). 0 degrees = vertical (standard lithostatic) Positive angles tilt gravity towards +x direction	`0.0`
`K0`	`float`	Coefficient of lateral earth pressure at rest. Used to relate stresses perpendicular to the inclined direction.	`0.5`

Returns:

Name	Type	Description
`sigma_xx`	`array - like`	Normal stress in x direction (Pa).
`sigma_yy`	`array - like`	Normal stress in y direction (Pa).
`tau_xy`	`array - like`	Shear stress (Pa).

Notes

For an inclined gravity field, the body forces are: - f_x = rho * g * sin(theta) (horizontal component) - f_y = rho * g * cos(theta) (vertical component)

The stress field must satisfy equilibrium: - dσ_xx/dx + dτ_xy/dy + f_x = 0 - dτ_xy/dx + dσ_yy/dy + f_y = 0

For a uniform rectangular mass with no boundaries except at the top, we assume a solution where stresses increase linearly with depth in the direction of gravity, and include shear stress from the inclined component.

Source code in photoelastimetry/generate/inclined_plane.py

def inclined_stress_cartesian(X, Y, rho, g=9.81, theta_deg=0.0, K0=0.5):
    """
    Stress field for an inclined plane (rectangular mass with inclined gravity).

    This computes the stress field in a rectangular domain with gravity
    acting at an angle theta from vertical. The resulting stress field
    includes both normal stresses that vary with depth and shear stresses
    due to the inclined gravity.

    Parameters
    ----------
    X : array-like
        X coordinates (horizontal position).
    Y : array-like
        Y coordinates (depth, positive downward from surface).
    rho : float
        Density (kg/m^3).
    g : float
        Gravitational acceleration (m/s^2).
    theta_deg : float
        Inclination angle of gravity from vertical (degrees).
        0 degrees = vertical (standard lithostatic)
        Positive angles tilt gravity towards +x direction
    K0 : float
        Coefficient of lateral earth pressure at rest.
        Used to relate stresses perpendicular to the inclined direction.

    Returns
    -------
    sigma_xx : array-like
        Normal stress in x direction (Pa).
    sigma_yy : array-like
        Normal stress in y direction (Pa).
    tau_xy : array-like
        Shear stress (Pa).

    Notes
    -----
    For an inclined gravity field, the body forces are:
    - f_x = rho * g * sin(theta)  (horizontal component)
    - f_y = rho * g * cos(theta)  (vertical component)

    The stress field must satisfy equilibrium:
    - dσ_xx/dx + dτ_xy/dy + f_x = 0
    - dτ_xy/dx + dσ_yy/dy + f_y = 0

    For a uniform rectangular mass with no boundaries except at the top,
    we assume a solution where stresses increase linearly with depth
    in the direction of gravity, and include shear stress from the
    inclined component.
    """
    theta_rad = np.deg2rad(theta_deg)

    # Components of gravity
    g_x = g * np.sin(theta_rad)  # Horizontal component
    g_y = g * np.cos(theta_rad)  # Vertical component

    # For a simple inclined plane solution, we assume:
    # 1. Vertical stress component increases with depth due to g_y
    # 2. Horizontal stress is related by K0
    # 3. Shear stress arises from the horizontal gravity component

    # Stress in the direction of gravity (normal to inclined layers)
    # For simplicity, we compute stress as if in a rotated coordinate system
    # then transform back to x-y coordinates

    # Normal stress perpendicular to gravity direction
    sigma_normal = rho * g * Y  # Total weight effect with depth

    # In the inclined case, we decompose this into x-y components
    # The vertical stress component
    sigma_yy = rho * g_y * Y

    # The horizontal stress has contributions from both K0 effect and inclination
    sigma_xx = K0 * sigma_yy

    # Shear stress arises from the inclined gravity
    # In equilibrium, tau_xy must balance the horizontal body force
    # For a linear variation: dtau_xy/dy = -rho * g_x
    # Integrating: tau_xy = -rho * g_x * Y + C
    # With free surface at Y=0: tau_xy(Y=0) = 0, so C = 0
    tau_xy = rho * g_x * Y

    # Ensure non-negative normal stresses (no tension)
    sigma_yy = np.maximum(0, sigma_yy)
    sigma_xx = np.maximum(0, sigma_xx)

    return sigma_xx, sigma_yy, tau_xy

`generate_synthetic_inclined_plane(X, Y, rho, g, theta_deg, K0, S_i_hat, wavelengths_nm, thickness, C)` ¶

Generate synthetic inclined plane stress data.