crack_growth_curve#

The py_fatigue.material.crack_growth_curve module contains the py_fatigue.ParisCurve and py_fatigue.WalkerCurve classes, together with the helpers used to evaluate crack-growth curves.

class py_fatigue.ParisCurve(slope: int | float | list | ndarray, intercept: int | float | list | ndarray, threshold: int | float = 0, critical: int | float = inf, load_ratio: int | float = 0, walker_exponent: int | float = 0, environment: str | None = None, curve: str | None = None, norm: str | None = None, unit_string: str = 'MPa √mm', color: str | None = None)#

Define a crack growth curve inthe form of Paris’ law that can have an arbitrary number of slopes/intercepts as well as a threshold and critical stress intensity factor. For example:

>>> #
>>> #         ^                log da/dN - log ΔK
>>> #         │                                *
>>> #         │                                *
>>> #         │                                *
>>> #         │                                *
>>> #         │                             *  .
>>> #         │                          *  │  .
>>> #         │                       *     │ m2
>>> #         │                    *        │  .
>>> #         │                 *-----------┘  .
>>> #         │              *                 .
>>> #         │           .*│                  .
>>> #         │        . *  │ m1               .
>>> #         │     .  *    │                  .
>>> #         │  .   *------┘                  .
>>> # (da/dN)1+    *                           .
>>> #         │  . *                           .
>>> #         │.   *                           .
>>> # (da/dN)2+----|---------------------------|--------------->
>>> #             ΔK0                         ΔKc
>>> #                          Number of cycles

The slope-intercept couples ((m1, log_a1), (m2, log_a2)), and the endurance value (Ne) are the parameters necessary to fully describe this trilinear Paris’ law. If the endurance is not set, it defaults to Inf.

Note

The curve can account for mean stress effect according to Walker’s model. The walker_exponent is set to 0 by default, meaning no correction is applied, i.e. the base curve is valid for R=0.

Example

>>> from py_fatigue import ParisCurve
>>> import numpy as np

>>> # Define a Paris' law with 5 slopes and 5 interceps
>>> SIF = np.linspace(1,2500, 300)
>>> SLOPE_5 = np.array([2.88, 5.1, 8.16, 5.1, 2.88])
>>> INTERCEPT_5 = np.array([1E-16, 1E-20, 1E-27, 1E-19, 1E-13])
>>> THRESHOLD = 20
>>> CRITICAL = 2000

>>> pc = ParisCurve(slope=SLOPE_5, intercept=INTERCEPT_5,
                  threshold=THRESHOLD, critical=CRITICAL, norm="The norm",
                  environment="Environment", curve="nr. 4")
>>> pc_.get_knee_sif()
array([ 63.35804993, 193.9017369 , 411.50876026, 504.31594872])

Define crack growth rate curve (Paris’ law). See class docstring for more information.

Parameters:

  • slope (Union[int, float, list, np.ndarray]) – Paris curve slope

  • intercept (Union[int, float, list, np.ndarray]) – crack growth rate axis intercept

  • threshold (Union[int, float]) – propagation threshold, below which crack growth rate is null

  • critical (Union[int, float]) – critical propagation stress intensity factor, imminent failure

  • load_ratio (Union[int, float]) – load ratio. Used for mean stress-corrected Paris’ law

  • walker_exponent (Union[int, float]) – Walker’s exponent. Used for mean stress-corrected Paris’ law

  • environment (str, optional) – Paris’ law envirnoment, by default None

  • curve (str, optional) – Paris’ law category, by default None

  • norm (str, optional) – Paris’ law norm, by default None

  • unit_string (str, optional) – units, by default “MPa √mm”

  • color (str, optional) – RGBS or HEX string for color, by default None

__init__(slope: int | float | list | ndarray, intercept: int | float | list | ndarray, threshold: int | float = 0, critical: int | float = inf, load_ratio: int | float = 0, walker_exponent: int | float = 0, environment: str | None = None, curve: str | None = None, norm: str | None = None, unit_string: str = 'MPa √mm', color: str | None = None) → None#

Define crack growth rate curve (Paris’ law). See class docstring for more information.

Parameters:

  • slope (Union[int, float, list, np.ndarray]) – Paris curve slope

  • intercept (Union[int, float, list, np.ndarray]) – crack growth rate axis intercept

  • threshold (Union[int, float]) – propagation threshold, below which crack growth rate is null

  • critical (Union[int, float]) – critical propagation stress intensity factor, imminent failure

  • load_ratio (Union[int, float]) – load ratio. Used for mean stress-corrected Paris’ law

  • walker_exponent (Union[int, float]) – Walker’s exponent. Used for mean stress-corrected Paris’ law

  • environment (str, optional) – Paris’ law envirnoment, by default None

  • curve (str, optional) – Paris’ law category, by default None

  • norm (str, optional) – Paris’ law norm, by default None

  • unit_string (str, optional) – units, by default “MPa √mm”

  • color (str, optional) – RGBS or HEX string for color, by default None

classmethod from_knee_points(knee_sif: ndarray | List[float], knee_growth_rate: ndarray | List[float], environment: str | None = None, curve: str | None = None, norm: str | None = None, unit_string: str = 'MPa √mm', color: str | None = None) → ParisCurve#

Create a Paris curve from a set of knee points, i.e. build slopes and intercepts from pairs of SIF and crack growth rate. The first and last knee points are used to set the threshold and critical SIF values.

Parameters:

  • knee_sif (Union[np.ndarray, List[float]]) – SIF values at the knee

  • knee_growth_rate (Union[np.ndarray, List[float]]) – crack growth rate at the knee

  • environment (str, optional) – Paris’ law environment, by default None

  • curve (str, optional) – Paris’ law category, by default None

  • norm (str, optional) – Paris’ law norm, by default None

  • unit_string (str, optional) – units, by default “MPa”

  • color (str, optional) – RGBS or HEX string for color, by default None

Returns:

Paris curve instance

Return type:

ParisCurve

format_name(html_format: bool = False) → None#

Reformat ParisCurve name.

Parameters:

html_format (bool, optional) – Choose whether the name format shall be HTML or not, by default True

property load_ratio: int | float#

Preventing attribute modification outside of constructor

Returns:

Load ratio

Return type:

Union[int, float]

property walker_exponent: int | float#

Preventing attribute modification outside of constructor

Returns:

Walker’s exponent

Return type:

Union[int, float]

property walker_intercept: ndarray#

Walker-corrected intercept.

\[a_w = \frac{a}{(1 - R)^{\frac{m}{1 - \gamma}}}\]

Preventing attribute modification outside of constructor

Returns:

Corrected intercept

Return type:

np.ndarray

property threshold: int | float#

Returns: Union[int, float]: Threshold SIF

property critical: int | float#

Returns: Union[int, float]: Critical SIF

property threshold_growth_rate: float#

Calculates the crack growth rate at threshold, if threshold SIF is defined.

Returns:

threshold crack growth rate

Return type:

float

property critical_growth_rate: float#

Calculates the crack growth rate at critical, if critical SIF is defined.

Returns:

critical crack growth rate

Return type:

float

get_knee_growth_rate(check_knee: Collection | None = None, significant_digits: int = 2) → ndarray#

Calculates the crack growth rate at the knee, if the Paris’ law is more than linear.

Parameters:

  • check_knee (Iterable, optional) – Iterable of SIF values to check for the knee, by default None

  • significant_digits (int, optional) – Number of significant digits to round the knee to, by default 2

Returns:

knee crack growth rate

Return type:

np.ndarray

get_knee_sif(check_knee: Collection | None = None, significant_digits: int = 2) → ndarray#

Calculates the SIF at the knee, if the Paris’ law is more than linear.

Parameters:

  • check_knee (iterable, optional) – SIF values to check the knee SIF against, by default None

  • significant_digits (int, optional) – number of significant digits to check the knee SIF against, by default 2

Returns:

knee SIF

Return type:

np.ndarray

get_growth_rate(sif_range: int | float | list | ndarray) → float | ndarray#

Return cycles value(s) for stress range(s).

Parameters:

sif_range (int or float or list or 1darray) – SIF range(s) to find the corresponding crack growth rate(s) for.

Returns:

crack growth rate(s) for the SIF range(s) indicating sif_range.

Return type:

float or 1darray

get_sif(growth_rate: int | float | list | ndarray) → float | ndarray#

Return SIF range(s) for the endurance(s) N.

Parameters:

cycles (int or float or list or 1darray) – crack growth rate(s) to find the corresponding SIF range(s) for.

Returns:

SIF range(s) for corresponding crack growth rate(s).

Return type:

float or 1darray

plot(sif: list | None = None, growth_rate: list | None = None, dataset_name: str | None = None, dataset_color: str = '#000', fig: Figure | None = None, ax: Axes | None = None, **kwargs: Any) → Tuple[Figure, Axes]#

Use matplotlib to plot the Paris’ law and a da/dN vs ΔK history dataset.

Example

Use matplotlib to plot the Paris’ law and a da/dN vs ΔK history

>>> fig, ax = pc.plot()
Parameters:

  • sif (list, optional) – SIF value(s) in the dataset, by default None

  • growth_rate (list, optional) – Crack growth rate(s) in the dataset, by default None

  • dataset_name (str, optional) – history dataset name, by default None

  • dataset_color (str, optional) – history dataset color, by default “#000”

  • fig (matplotlib.figure.Figure, optional) – figure object, by default None

  • ax (matplotlib.axes.Axes, optional) – axis object, by default None

  • **kwargs (Any, optional) – additional keyword arguments

Returns:

The figure and axes.

Return type:

matplotlib.figure.Figure, matplotlib.axes.Axes

class py_fatigue.WalkerCurve(slope: int | float | list | ndarray, intercept: int | float | list | ndarray, threshold: int | float = 0, critical: int | float = inf, load_ratio: int | float = 0, walker_exponent: int | float = 0, environment: str | None = None, curve: str | None = None, norm: str | None = None, unit_string: str = 'MPa √mm', color: str | None = None)#

Define a crack growth curve inthe form of Paris’ law that can have an arbitrary number of slopes/intercepts as well as a threshold and critical stress intensity factor. For example:

>>> #
>>> #         ^                log da/dN - log ΔK
>>> #         │                                *
>>> #         │                                *
>>> #         │                                *
>>> #         │                                *
>>> #         │                             *  .
>>> #         │                          *  │  .
>>> #         │                       *     │ m2
>>> #         │                    *        │  .
>>> #         │                 *-----------┘  .
>>> #         │              *                 .
>>> #         │           .*│                  .
>>> #         │        . *  │ m1               .
>>> #         │     .  *    │                  .
>>> #         │  .   *------┘                  .
>>> # (da/dN)1+    *                           .
>>> #         │  . *                           .
>>> #         │.   *                           .
>>> # (da/dN)2+----|---------------------------|--------------->
>>> #             ΔK0                         ΔKc
>>> #                          Number of cycles

The slope-intercept couples ((m1, log_a1), (m2, log_a2)), and the endurance value (Ne) are the parameters necessary to fully describe this trilinear Paris’ law. If the endurance is not set, it defaults to Inf.

Note

The curve can account for mean stress effect according to Walker’s model. The walker_exponent is set to 0 by default, meaning no correction is applied, i.e. the base curve is valid for R=0.

Example

>>> from py_fatigue import ParisCurve
>>> import numpy as np

>>> # Define a Paris' law with 5 slopes and 5 interceps
>>> SIF = np.linspace(1,2500, 300)
>>> SLOPE_5 = np.array([2.88, 5.1, 8.16, 5.1, 2.88])
>>> INTERCEPT_5 = np.array([1E-16, 1E-20, 1E-27, 1E-19, 1E-13])
>>> THRESHOLD = 20
>>> CRITICAL = 2000

>>> pc = ParisCurve(slope=SLOPE_5, intercept=INTERCEPT_5,
                  threshold=THRESHOLD, critical=CRITICAL, norm="The norm",
                  environment="Environment", curve="nr. 4")
>>> pc_.get_knee_sif()
array([ 63.35804993, 193.9017369 , 411.50876026, 504.31594872])

Define crack growth rate curve (Paris’ law). See class docstring for more information.

Parameters:

  • slope (Union[int, float, list, np.ndarray]) – Paris curve slope

  • intercept (Union[int, float, list, np.ndarray]) – crack growth rate axis intercept

  • threshold (Union[int, float]) – propagation threshold, below which crack growth rate is null

  • critical (Union[int, float]) – critical propagation stress intensity factor, imminent failure

  • load_ratio (Union[int, float]) – load ratio. Used for mean stress-corrected Paris’ law

  • walker_exponent (Union[int, float]) – Walker’s exponent. Used for mean stress-corrected Paris’ law

  • environment (str, optional) – Paris’ law envirnoment, by default None

  • curve (str, optional) – Paris’ law category, by default None

  • norm (str, optional) – Paris’ law norm, by default None

  • unit_string (str, optional) – units, by default “MPa √mm”

  • color (str, optional) – RGBS or HEX string for color, by default None

__init__(slope: int | float | list | ndarray, intercept: int | float | list | ndarray, threshold: int | float = 0, critical: int | float = inf, load_ratio: int | float = 0, walker_exponent: int | float = 0, environment: str | None = None, curve: str | None = None, norm: str | None = None, unit_string: str = 'MPa √mm', color: str | None = None) → None#

Define crack growth rate curve (Paris’ law). See class docstring for more information.

Parameters:

  • slope (Union[int, float, list, np.ndarray]) – Paris curve slope

  • intercept (Union[int, float, list, np.ndarray]) – crack growth rate axis intercept

  • threshold (Union[int, float]) – propagation threshold, below which crack growth rate is null

  • critical (Union[int, float]) – critical propagation stress intensity factor, imminent failure

  • load_ratio (Union[int, float]) – load ratio. Used for mean stress-corrected Paris’ law

  • walker_exponent (Union[int, float]) – Walker’s exponent. Used for mean stress-corrected Paris’ law

  • environment (str, optional) – Paris’ law envirnoment, by default None

  • curve (str, optional) – Paris’ law category, by default None

  • norm (str, optional) – Paris’ law norm, by default None

  • unit_string (str, optional) – units, by default “MPa √mm”

  • color (str, optional) – RGBS or HEX string for color, by default None

property load_ratio: int | float#

Preventing attribute modification outside of constructor

Returns:

Load ratio

Return type:

Union[int, float]

property walker_exponent: int | float#

Preventing attribute modification outside of constructor

Returns:

Walker’s exponent

Return type:

Union[int, float]

property walker_correction: bool#

Check if walker correction is applied.

Returns:

True if walker correction is applied

Return type:

bool

property walker_intercept: ndarray#

Walker-corrected intercept.

\[a_w = \frac{a}{(1 - R)^{\frac{m}{1 - \gamma}}}\]

Preventing attribute modification outside of constructor

Returns:

Corrected intercept

Return type:

np.ndarray