void_ratio_stress¶

Class summary¶

`AvSoilModel`(av, siga, ea)	Linear void ratio-effective stress relationship
`CcCrSoilModel`(Cc, Cr, siga, ea)	Semi-log void ratio-effective stress realationship
`FunctionSoilModel`(fn_e_from_stress, …)	Functional definition of void-ratio stress relationship
`OneDimensionalVoidRatioEffectiveStress`	Base class for defining 1D void ratio-effective stress relationships
`PwiseLinearSoilModel`(siga, ea, Cr[, xlog, ylog])	Pwise linear void ratio-effective stress realationship
`YinAndGrahamSoilModel`(lam, kap, psi, siga, …)	Yin and Graham creep void ratio-effective stress realationship

Module listing¶

Relationships between void-ratio and effective stress.

class geotecha.constitutive_models.void_ratio_stress.AvSoilModel(av, siga, ea)[source]¶

Bases: geotecha.constitutive_models.void_ratio_stress.OneDimensionalVoidRatioEffectiveStress

Linear void ratio-effective stress relationship

Parameters:	av : float Slope of compression line. siga, ea : float Effective stress and void ratio specifying point on av line.

Methods

`av_from_stress`(args, *kwargs)	Slope of void ratio from effective stress
`e_and_stress_for_plotting`(**kwargs)	Void ratio and stress values that plot the model
`e_from_stress`(estress, **kwargs)	Void ratio from effective stress
`plot_model`(**kwargs)	plot the void ratio-stress points
`stress_from_e`(e, **kwargs)	Effective stress from void ratio

av_from_stress(*args, **kwargs)[source]¶: Slope of void ratio from effective stress

e_and_stress_for_plotting(**kwargs)[source]¶

Void ratio and stress values that plot the model

Parameters:	npts : int, optional Number of points to return. Default npts=100. xmin, xmax : float, optional range of x (i.e. effective stress) values from which to return points. Default xmin=1, xmax=100
Returns:	x, y : 1d ndarray npts stress, and void ratio values between xmin and xmax.

e_from_stress(estress, **kwargs)[source]¶

Void ratio from effective stress

Parameters:	estress : float Current effective stress.
Returns:	e : float Void ratio corresponding to current stress state.

Examples

>>> a = AvSoilModel(av=1.5, siga=19, ea=4)
>>> a.e_from_stress(20)
2.5

Array inputs:

>>> a = AvSoilModel(av=1.5, siga=19, ea=4)
>>> a.e_from_stress(np.array([20, 21]))
array([2.5, 1. ])

stress_from_e(e, **kwargs)[source]¶

Effective stress from void ratio

Parameters:	e : float Current void ratio.
Returns:	estress : float Effective stress corresponding to current void ratio

Examples

>>> a = AvSoilModel(av=1.5, siga=19, ea=4)
>>> a.stress_from_e(1)
21.0

Array inputs:

>>> a = AvSoilModel(av=1.5, siga=19, ea=4)
>>> a.stress_from_e(np.array([1, 2.5]))
array([21., 20.])

class geotecha.constitutive_models.void_ratio_stress.CcCrSoilModel(Cc, Cr, siga, ea)[source]¶

Bases: geotecha.constitutive_models.void_ratio_stress.OneDimensionalVoidRatioEffectiveStress

Semi-log void ratio-effective stress realationship

Parameters:	Cc : float Compressibility index, slope of e-log(sig) line. Cr : float Recompression index, slope of e-log(sig) line. siga, ea : float Point on compression line fixing it in effective stress-void ratio space.

Methods

`av_from_stress`(estress, **kwargs)	Slope of void ratio from effective stress
`e_and_stress_for_plotting`(**kwargs)	Void ratio and stress values that plot the model
`e_from_stress`(estress, **kwargs)	Void ratio from effective stress
`plot_model`(**kwargs)	plot the void ratio-stress points
`stress_from_e`(e, **kwargs)	Effective stress from void ratio

av_from_stress(estress, **kwargs)[source]¶

Slope of void ratio from effective stress

Parameters:	estress : float Current effective stress. pstress : float, optional Preconsolidation stress. Default pstress=estress i.e. normally consolidated.
Returns:	av : float Slope of void-ratio vs effective stress plot at current stress state.

Examples

On recompression line:

>>> a = CcCrSoilModel(Cc=3.0, Cr=0.5, siga=10, ea=5)
>>> a.av_from_stress(estress=40, pstress=50)
0.00542868...

On compression line:

>>> a = CcCrSoilModel(Cc=3.0, Cr=0.5, siga=10, ea=5)
>>> a.av_from_stress(estress=60, pstress=50)
0.02171472...

Array inputs:

>>> a = CcCrSoilModel(Cc=3.0, Cr=0.5, siga=10, ea=5)
>>> a.av_from_stress(estress=np.array([40, 60.0]),
... pstress=np.array([50, 55.0]))
array([0.00542868, 0.02171472])

e_and_stress_for_plotting(**kwargs)[source]¶

Void ratio and stress values that plot the model

Parameters:	pstress : float, optional Preconsolidation stress. Default behaviour is normally consolidated. npts : int, optional Number of points to return. Default npts=100. xmin, xmax : float, optional range of x (i.e. effective stress) values from which to return points. Default xmin=1, xmax=100
Returns:	x, y : 1d ndarray npts stress, and void ratio values between xmin and xmax.

e_from_stress(estress, **kwargs)[source]¶

Void ratio from effective stress

Parameters:	estress : float Current effective stress. pstress : float, optional Preconsolidation stress. Default pstress=estress i.e. normally consolidated.
Returns:	e : float Void ratio corresponding to current stress state.

Examples

On recompression line:

>>> a = CcCrSoilModel(Cc=3, Cr=0.5, siga=10, ea=5)
>>> a.e_from_stress(estress=40, pstress=50)
2.95154...

On compression line:

>>> a = CcCrSoilModel(Cc=3, Cr=0.5, siga=10, ea=5)
>>> a.e_from_stress(estress=60, pstress=50)
2.66554...

Array inputs:

>>> a = CcCrSoilModel(Cc=3, Cr=0.5, siga=10, ea=5)
>>> a.e_from_stress(estress=np.array([40, 60]),
... pstress=np.array([50, 55]))
array([2.95154499, 2.66554625])

Normally consolidated (pstress not specified):

>>> a = CcCrSoilModel(Cc=3.0, Cr=0.5, siga=10, ea=5)
>>> a.e_from_stress(estress=10)
5.0

stress_from_e(e, **kwargs)[source]¶

Effective stress from void ratio

Parameters:	e : float Current void ratio. pstress : float, optional Preconsolidation stress. Default pstress=estress i.e. normally consolidated.
Returns:	estress : float Effective stress corresponding to current void ratio.

Examples

On recompression line:

>>> a = CcCrSoilModel(Cc=3.0, Cr=0.5, siga=10, ea=5)
>>> a.stress_from_e(e=2.95154499, pstress=50)
40...

On compression line:

>>> a = CcCrSoilModel(Cc=3, Cr=0.5, siga=10, ea=5)
>>> a.stress_from_e(e=2.66554625, pstress=50)
59.999...

Array inputs:

>>> a = CcCrSoilModel(Cc=3, Cr=0.5, siga=10, ea=5)
>>> a.stress_from_e(e=np.array([2.95154499, 2.66554625]), pstress=50)
array([40.0..., 59.99...])

Normally consolidated:

>>> a = CcCrSoilModel(Cc=3.0, Cr=0.5, siga=10, ea=5)
>>> a.stress_from_e(e=5)
10.0

class geotecha.constitutive_models.void_ratio_stress.FunctionSoilModel(fn_e_from_stress, fn_stress_from_e, fn_av_from_stress, *args, **kwargs)[source]¶

Bases: geotecha.constitutive_models.void_ratio_stress.OneDimensionalVoidRatioEffectiveStress

Functional definition of void-ratio stress relationship

User provides python functions.

Parameters:

fn_e_from_stress: callable object: Function to obtain void ratio from stress. fn_e_from_stress should be the inverse function of fn_stress_from_e.
fn_stress_from_e : callable object: Function to obtain stress from void ratio. fn_stress_from_e should be the inverse function of fn_e_from_stress.
fn_av_from_stress: callable object: Function to obtain slope of void ratio-stress relationship. fn_av_from_stress should be negative the derivative of fn_e_from_stress w.r.t. stress be the inverse function of fn_k_from_e.
*args, **kwargs : anything: Positional and keyword arguments to be passed to the fn_e_from_stress, fn_stress_from_e, fn_av_from_stress functions. Note that any additional args and kwargs passed to the functions will be appended to the args, and kwargs. You may get into a mess for example with e_from_stress because normally the first postional argument passed to such fucntions is estress. If you add your own positional arguments, then k will be after you own arguments. Just be aware that the usual way to call methods of the base object OneDimensionalVoidRatioEffectiveStress a single positonal arguement, e.g. estress, e, followed by any required keyword arguments.

Notes

Any function should be able to accept additonal keywords.

Examples

>>> def efs(s, b=2):
...     return s * b
>>> def sfe(e, b=2):
...     return e / b
>>> def afs(s, b=2):
...     return -b
>>> a = FunctionSoilModel(efs, sfe, afs, b=5)
>>> a.e_from_stress(3)
15
>>> a.stress_from_e(15)
3.0
>>> a.av_from_stress(3)
-5

Prconsolidation stress:

>>> def efs2(s, pstress=None, b=2):
...     if pstress is None:
...         pstress=8
...     return s * b + pstress
>>> a = FunctionSoilModel(efs2, sfe, afs, b=5)
>>> a.e_from_stress(3)
23

Methods

`av_from_stress`(estress, **kwargs)	Slope of void ratio from effective stress
`e_and_stress_for_plotting`(**kwargs)	Void ratio and stress values that plot the method
`e_from_stress`(estress, **kwargs)	Void ratio from effective stress
`plot_model`(**kwargs)	plot the void ratio-stress points
`stress_from_e`(e, **kwargs)	Effective stress from void ratio

class geotecha.constitutive_models.void_ratio_stress.OneDimensionalVoidRatioEffectiveStress[source]¶

Bases: object

Base class for defining 1D void ratio-effective stress relationships

Methods

`av_from_stress`(estress, **kwargs)	Slope of void ratio from effective stress
`e_and_stress_for_plotting`(**kwargs)	Void ratio and stress values that plot the method
`e_from_stress`(estress, **kwargs)	Void ratio from effective stress
`plot_model`(**kwargs)	plot the void ratio-stress points
`stress_from_e`(e, **kwargs)	Effective stress from void ratio

av_from_stress(estress, **kwargs)[source]¶: Slope of void ratio from effective stress

e_and_stress_for_plotting(**kwargs)[source]¶: Void ratio and stress values that plot the method

e_from_stress(estress, **kwargs)[source]¶: Void ratio from effective stress

plot_model(**kwargs)[source]¶: plot the void ratio-stress points

stress_from_e(e, **kwargs)[source]¶: Effective stress from void ratio

class geotecha.constitutive_models.void_ratio_stress.PwiseLinearSoilModel(siga, ea, Cr, xlog=False, ylog=False)[source]¶

Bases: geotecha.constitutive_models.void_ratio_stress.OneDimensionalVoidRatioEffectiveStress

Pwise linear void ratio-effective stress realationship

x and y data can be interpolated natural-natural, natural-log10, log10-natural, or log10, log10.

Parameters:

siga, ea : 1d array: Effective stress values and void ratio values defining a one-to-one relationship. Slope of void ratio-effectinve stress plot should never fall below Cr.
Cr : float: Precompression index, slope of void rato-effective stress line. Note that Cr is the slope in whatever scales of slog and ylog that have been chosen.
xlog, ylog : True/False, Optional: If True then interpolation on each axis is assumed to be logarithmic with base 10. Default xlog=ylog=False

Methods

`av_from_stress`(estress, **kwargs)	Slope of void ratio from effective stress
`e_and_stress_for_plotting`(**kwargs)	Void ratio and stress values that plot the model
`e_from_stress`(estress, **kwargs)	Void ratio from effective stress
`plot_model`(**kwargs)	plot the void ratio-stress points
`stress_from_e`(e, **kwargs)	Effective stress from void ratio

av_from_stress(estress, **kwargs)[source]¶

Slope of void ratio from effective stress

Parameters:	estress : float Current effective stress. pstress : float, optional Preconsolidation stress. Default pstress=estress i.e. normally consolidated.
Returns:	av : float Slope of void-ratio vs effective stress plot at current stress state.

Examples

On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.1)
>>> a.av_from_stress(estress=1.25, pstress=2.25)
0.1...

On compression line:

>>> a.av_from_stress(estress=2.25, pstress=2)
3.0...

Normally consolidated (pstress not specified):

>>> a.av_from_stress(estress=2.25)
3.0...

Array inputs:

>>> a.av_from_stress(estress=np.array([1.25, 2.25]),
... pstress=np.array([2.25, 2.]))
array([0.1, 3. ])

Logarithmic effective stress scale On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.1, xlog=True)
>>> a.av_from_stress(estress=1.25, pstress=2.25)
0.034743...

On compression line:

>>> a.av_from_stress(estress=2.25, pstress=2)
2.987...

Normally consolidated (pstress not specified):

>>> a.av_from_stress(estress=2.25)
2.987...

Array inputs:

>>> a.av_from_stress(estress=np.array([1.25, 2.25]),
... pstress=np.array([2.25, 2.]))
array([0.034743..., 2.987...])

Logarithmic void ratio scale On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.1, ylog=True)
>>> a.av_from_stress(estress=1.25, pstress=2.25)
0.76694...

On compression line:

>>> a.av_from_stress(estress=2.25, pstress=2)
2.9612...

Normally consolidated (pstress not specified):

>>> a.av_from_stress(estress=2.25)
2.9612...

Array inputs:

>>> a.av_from_stress(estress=np.array([1.25, 2.25]), pstress=2.25)
array([0.76694..., 2.9612...])

Logarithmic effective stress and void ratio scales On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.1, xlog=True, ylog=True)
>>> a.av_from_stress(estress=1.25, pstress=2.25)
0.2210045...

On compression line:

>>> a.av_from_stress(estress=2.25, pstress=2)
2.9034...

Normally consolidated (pstress not specified):

>>> a.av_from_stress(estress=2.25)
2.9034...

Array inputs:

>>> a.av_from_stress(estress=np.array([1.25, 2.25]), pstress=2.25)
array([0.2210045..., 2.9034...])

Increasing vs decreasing inputs

>>> ea = np.arange(1,10)
>>> siga = 3 * ea
>>> np.isclose(PwiseLinearSoilModel(siga, ea, Cr=0.1).av_from_stress(7.2),
... PwiseLinearSoilModel(siga[::-1], ea[::-1], Cr=0.1).av_from_stress(7.2))
True

e_and_stress_for_plotting(**kwargs)[source]¶

Void ratio and stress values that plot the model

Parameters:	pstress : float, optional Preconsolidation stress. Default behaviour is normally consolidated. npts : int, optional Number of points to return. Default npts=100 xmin, xmax : float, optional Range of x (i.e. effective stress) values from which to return points. Default minumum of model siga.
Returns:	x, y : 1d ndarray npts stress, and void ratio values between xmin and xmax.

e_from_stress(estress, **kwargs)[source]¶

Void ratio from effective stress

Parameters:	estress : float Current effective stress. etress must be within the range of the soil model points. pstress : float, optional Preconsolidation stress. Default pstress=estress i.e. normally consolidated. pstress must be in the range of the soil model points.
Returns:	e : float Void ratio corresponding to current stress state.

Examples

On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.2)
>>> a.e_from_stress(estress=1.25, pstress=2.25)
2.95...

On compression line:

>>> a.e_from_stress(estress=2.25, pstress=2)
2.75...

Normally consolidated (pstress not specified):

>>> a.e_from_stress(estress=2.25)
2.75...

Array inputs:

>>> a.e_from_stress(estress=np.array([1.25, 2.25]), pstress=2.25)
array([2.95, 2.75])

Logarithmic effective stress scale: On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.2, xlog=True)
>>> a.e_from_stress(estress=1.25, pstress=2.25)
2.75930...

On compression line:

>>> a.e_from_stress(estress=2.25, pstress=2)
2.70824...

Normally consolidated (pstress not specified):

>>> a.e_from_stress(estress=2.25)
2.70824...

Array inputs:

>>> a.e_from_stress(estress=np.array([1.25, 2.25]), pstress=2.25)
array([2.75930..., 2.70824...])

Logarithmic void ratio scale On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.1, ylog=True)
>>> a.e_from_stress(estress=1.25, pstress=2.25)
3.330803...

On compression line:

>>> a.e_from_stress(estress=2.25, pstress=2)
2.64575...

Normally consolidated (pstress not specified):

>>> a.e_from_stress(estress=2.25)
2.64575...

Array inputs:

>>> a.e_from_stress(estress=np.array([1.25, 2.25]), pstress=2.25)
array([3.330803..., 2.64575...])

Logarithmic effective stress and void ratio scales On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.1, xlog=True, ylog=True)
>>> a.e_from_stress(estress=1.25, pstress=2.25)
2.76255...

On compression line:

>>> a.e_from_stress(estress=2.25, pstress=2)
2.604857...

Normally consolidated (pstress not specified):

>>> a.e_from_stress(estress=2.25)
2.604857...

Array inputs:

>>> a.e_from_stress(estress=np.array([1.25, 2.25]), pstress=2.25)
array([2.76255..., 2.604857...])

Increasing vs decreasing inputs

>>> ea = np.arange(1,10)
>>> siga = 3 * ea
>>> np.isclose(PwiseLinearSoilModel(siga, ea,
... Cr=0.1).e_from_stress(7.2),
... PwiseLinearSoilModel(siga[::-1], ea[::-1],
... Cr=0.1).e_from_stress(7.2))
True

stress_from_e(e, **kwargs)[source]¶

Effective stress from void ratio

Parameters:	e : float Current void ratio. pstress : float, optional Preconsolidation stress. Default pstress=estress i.e. normally consolidated.
Returns:	estress : float Effective stress corresponding to current void ratio.

Examples

On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.1)
>>> a.stress_from_e(e=2.8, pstress=2.25)
1.75...

On compression line:

>>> a.stress_from_e(e=2.65, pstress=2)
2.28333...

Normally consolidated (pstress not specified): >>> a.stress_from_e(e=2.65) 2.28333…

Array inputs:

>>> a.stress_from_e(e=np.array([2.8, 2.65]),
... pstress=np.array([2.25, 2.0]))
array([1.75..., 2.28333...])

Logarithmic effective stress scale On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.1, xlog=True)
>>> a.stress_from_e(e=2.73, pstress=2.25)
1.363494...

On compression line:

>>> a.stress_from_e(e=2.73, pstress=2)
2.2427...

Normally consolidated (pstress not specified):

>>> a.stress_from_e(e=2.73)
2.2427...

Array inputs:

>>> a.stress_from_e(e=2.73, pstress=np.array([2.25, 2.]))
array([1.363494..., 2.2427...])

Logarithmic void ratio scale On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.1, ylog=True)
>>> a.stress_from_e(e=2.75, pstress=2.25)
2.082163...

On compression line:

>>> a.stress_from_e(e=2.65, pstress=2)
2.24856...

Normally consolidated (pstress not specified):

>>> a.stress_from_e(e=2.65)
2.24856...

Array inputs:

>>> a.stress_from_e(e=np.array([2.75, 2.65]),
... pstress=np.array([2.25, 2]))
array([2.082163..., 2.24856...])

Logarithmic effective stress and void ratio scales On recompression line:

>>> x, y = np.array([1,2,2.5]), np.array([4, 3.5, 2])
>>> a = PwiseLinearSoilModel(siga=x, ea=y, Cr=0.1, xlog=True, ylog=True)
>>> a.stress_from_e(e=2.65, pstress=2.25)
1.8948013...

On compression line:

>>> a.stress_from_e(e=2.65, pstress=2)
2.23463...

Normally consolidated (pstress not specified):

>>> a.stress_from_e(e=2.65)
2.23463...

Array inputs:

>>> a.stress_from_e(e=2.65, pstress=np.array([2.25, 2]))
array([1.8948013..., 2.23463...])

Increasing vs decreasing inputs

>>> ea = np.arange(1,10)
>>> siga = 3 * ea
>>> np.isclose(PwiseLinearSoilModel(siga, ea, Cr=0.1).stress_from_e(3.0, pstress=4.0),
... PwiseLinearSoilModel(siga[::-1], ea[::-1], Cr=0.1).stress_from_e(3.0, pstress=4.0))
True

class geotecha.constitutive_models.void_ratio_stress.YinAndGrahamSoilModel(lam, kap, psi, siga, ea, ta, e0=None, estress0=None, pstress0=None, t0=None)[source]¶

Bases: geotecha.constitutive_models.void_ratio_stress.OneDimensionalVoidRatioEffectiveStress

Yin and Graham creep void ratio-effective stress realationship

This model uses the integral void ratio form of the Yin and Graham (1996) [1] soil model as described by Hu et al. (2014) [2]. Basically you step through time keeping track of the stress state and a cumulatlive integral representing creep strain.

Parameters:

lam : float: Compressibility index, slope of the referece time line e-ln(sig) line.
kap : float: Recompression index, slope of e-ln(sig) line.
psi : float: Creep parameter. Slope of the void ratio-ln(time) line.
siga, ea : float: Point on reference time line fixing it in effective stress-void ratio space.
ta : float: Reference time corresponding to the reference time line.
e0 : float, optional: Initial void ratio. Default e0=None. If specifying initial conditions then only use two of the four available (e0, estress0, pstress0, t0).
estress0 : float, optional: Initial effective stress. Default estress=None. If specifying initial conditions then only use two of the four available (e0, estress0, pstress0, t0).
pstress0 : float, optional: Initial ‘preconsolidation` stress on the reference time line. Default pstress0=None. If specifying initial conditions then only use two of the four available (e0, estress0, pstress0, t0).
t0 : float, optional: Initial equivalent time at the initial conditon. Default t0=None. If specifying initial conditions then only use two of the four available (e0, estress0, pstress0, t0).

Notes

The void ratio stress relationship is:

\[e=e0-\kappa\ln\left({\frac{\sigma^{\prime}_z}{\sigma^{\prime}_{z}}}\right) -\psi\ln\left[{ \int_{0}^{t}{ \frac{1}{t_0} \left({ \frac{\sigma^{\prime}_z} {\sigma^{\prime}_{0}}}\right)^{\alpha}\,dt }+1 }\right]\]

To calculate the integral we keep track of the effective stress and the integration at time t and then add the integration increment between \(t\) and \(t+\bigtriangleup t\):

\[e=e0-\kappa\ln\left({\frac{\sigma^{\prime}_{z,t+\bigtriangleup t}}{\sigma^{\prime}_{z}}}\right) -\psi\ln\left[{ \int_{0}^{t}{ \frac{1}{t_0} \left({ \frac{\sigma^{\prime}_z} {\sigma^{\prime}_{0}}}\right)^{\alpha}\,dt } + \int_{t}^{t+\bigtriangleup t}{ \frac{1}{t_0} \left({ \frac{\sigma^{\prime}_z} {\sigma^{\prime}_{0}}}\right)^{\alpha}\,dt } +1}\right]\]

Consider a time increment of length \(\bigtriangleup t\) in which the effective stress changes from \(\sigma^{\prime}_{t}\) to \(\sigma^{\prime}_{t + \bigtriangleup t}\). Assume that the stress changes in a linear fashion over \(\bigtriangleup t\):

\[\sigma^{\prime}\left({t}\right) = \sigma^{\prime}_{t} + t \frac{\sigma^{\prime}_{t+ \bigtriangleup t} -\sigma^{\prime}_{t}} {\bigtriangleup t}\]

The time integral becomes:

\[\frac{1}{t_0}\int_{0}^{t}{ \left({ \frac{\sigma^{\prime}_z} {\sigma^{\prime}_{z0}}}\right)^{\alpha}\,dt } + \frac{1}{t_0}\int_{0}^{\bigtriangleup t} {\left({\frac{\sigma^{\prime}_{t} + t \frac{\sigma^{\prime}_{t+ \bigtriangleup t} -\sigma^{\prime}_{t}} {\bigtriangleup t}}{\sigma^{\prime}_0}}\right)^\alpha\,dt}\]

The increment that is added to the main integral is:

\[\frac{1}{t_0}\int_{0}^{\bigtriangleup t} {\left({\frac{\sigma^{\prime}_{t} + t \frac{\sigma^{\prime}_{t+ \bigtriangleup t} -\sigma^{\prime}_{t}} {\bigtriangleup t}}{\sigma^{\prime}_0}}\right)^\alpha\,dt} = \frac{1}{t_0} \frac{\sigma^{\prime}_{0} \bigtriangleup t} {\left({\alpha+1}\right) \left({\sigma^{\prime}_{t+ \bigtriangleup t} -\sigma^{\prime}_{t}}\right)} \left({ \left({\frac{\sigma^{\prime}_{t+ \bigtriangleup t}}{\sigma^{\prime}_0}}\right)^{\alpha+1}- \left({\frac{\sigma^{\prime}_{t}}{\sigma^{\prime}_0}}\right)^{\alpha+1} }\right)\]

When \(\sigma^{\prime}_{t+ \bigtriangleup t}=\sigma^{\prime}_{t}\) then the increment is:

\[\frac{1}{t_0}\int_{0}^{\bigtriangleup t} {\left({\frac{\sigma^{\prime}_{t} + t \frac{\sigma^{\prime}_{t+ \bigtriangleup t} -\sigma^{\prime}_{t}} {\bigtriangleup t}}{\sigma^{\prime}_0}}\right)^\alpha\,dt} = \frac{1}{t_0}\bigtriangleup t \left({\frac{\sigma^{\prime}_{t}}{\sigma^{\prime}_0}}\right)^{\alpha+1}\]

The value of the integral and the effective stress is stored within the YinAndGrahamSoilModel object and updated after each call to e_from_stress or stress_from_e.

e_from_stress

e_from_stress uses the above equations directly (knowing all material parameters, the value of the integral at the start of the increment, and the efffective stresses at the beginning and end of time incremetn dt, then e can be calculated directly).

stress_from_e

stress_from_e recasts the equations as estress=f(estress) and uses fixed point iteration to determine the effective stress at the end of the increment.

initial_conditions

If two of the following four parameters describing the intial state of the soil are known, then the equations below can be combined to solve for the remaining two unknowns: initial void ratio \(e_0\), initial stress \(\sigma^{\prime}_0\), pseudo initial preconsolidation stress \(\sigma^{\prime}_p\) and equivalent time corresponding to initial state \(t_0\).

On the reference time line running through point \((\sigma^{\prime}_a, e_a)\), the psuedo preconsolidation pressure \(\sigma^{\prime}_p\) corresponds to void ratio \(e_p\):

\[e_p = e_a - \lambda \ln\left({ \frac{\sigma^{\prime}_p} {\sigma^{\prime}_a} }\right)\]

The instant time line running through point \((\sigma^{\prime}_0, e_0)\) meets the reference time line at \(\sigma^{\prime}_p\) and void ratio \(e_p\):

\[e_p = e_0 - \kappa \ln\left({ \frac{\sigma^{\prime}_p} {\sigma^{\prime}_0} }\right)\]

The initial state \((\sigma^{\prime}_0, e_0)\) can be reached from the reference time line by either a) loading to \(\sigma^{\prime}_p\) and then unloading to \(\sigma^{\prime}_0\) or b) creeping from time \(t_a\) to time \(t_0\) at constant stress \(\sigma^{\prime}_0\). equating the two void ratio changes gives:

\[\left({\lambda-\kappa}\right) \ln\left({ \frac{\sigma^{\prime}_p}{\sigma^{\prime}_0} }\right) = \psi \ln\left({ \frac{t_0}{t_a} }\right)\]

YinAndGrahamSoilModel()

References

[1]	(1, 2) Yin, J.-H., and Graham, J. (1996). ‘Elastic visco-plastic modelling of one-dimensional consolidation. Geotechnique, 46(3), 515-527.

[2]	(1, 2) Hu, Y.-Y., Zhou, W.-H., and Cai, Y.-Q. (2014). ‘Large-strain elastic viscoplastic consolidation analysis of very soft clay layers with vertical drains under preloading. Canadian Geotechnical Journal, 51(2), 144-157.

Examples

Combos of initial conditions give the same results

>>> fixed_param = dict(lam=0.2, kap=0.04, psi=0.01, siga=20, ea=1, ta=1)
>>> oparam = dict(e0=0.95, estress0=18, pstress0=28.06637954,
... t0=1220.73731)
>>> a = YinAndGrahamSoilModel(e0=oparam['e0'],
... estress0=oparam['estress0'], **fixed_param)
>>> [a.e0, a.estress0, a.pstress0, a.t0]
[0.95..., 18..., 28.0..., 1220.737...]
>>> a = YinAndGrahamSoilModel(e0=oparam['e0'],
... pstress0=oparam['pstress0'], **fixed_param)
>>> [a.e0, a.estress0, a.pstress0, a.t0]
[0.95..., 18..., 28.0..., 1220.737...]
>>> a = YinAndGrahamSoilModel(e0=oparam['e0'],
... t0=oparam['t0'], **fixed_param)
>>> [a.e0, a.estress0, a.pstress0, a.t0]
[0.95..., 18..., 28.0..., 1220.737...]
>>> a = YinAndGrahamSoilModel(estress0=oparam['estress0'],
... pstress0=oparam['pstress0'], **fixed_param)
>>> [a.e0, a.estress0, a.pstress0, a.t0]
[0.95..., 18..., 28.0..., 1220.737...]
>>> a = YinAndGrahamSoilModel(estress0=oparam['estress0'],
... t0=oparam['t0'], **fixed_param)
>>> [a.e0, a.estress0, a.pstress0, a.t0]
[0.95..., 18..., 28.0..., 1220.737...]
>>> a = YinAndGrahamSoilModel(pstress0=oparam['pstress0'],
... t0=oparam['t0'], **fixed_param)
>>> [a.e0, a.estress0, a.pstress0, a.t0]
[0.95..., 18..., 28.0..., 1220.737...]

Attributes:

_alpha : float: Composite material parameter (lam - kap / psi)
_psi_zero_model : CcCrSoilModel: soil model when psi=0. i.e. CcCrSoilModel following instant time line and reference time line.
_igral : float: Current value of the material model integration w.r.t. time. _igral = integrate[1/t0 * (estress/etress0)**alpha, t,0,t]
_inc : float: Increment to add to _igral. Helper array to speed up stress_from_e. Only present after initial conditions have been initialized. _inc = integrate[1/t0 * (estress/etress0)**alpha, t,(t),(t + dt)]
_es : float: Temp effectve stress. Helper array to speed up stress_from_e calc. Only present after initial conditions have been initialized.
_estress : float: Current value of effective stress. Only present after initial conditions have been initialized. Basically when you call method e_from_stress or method stress_from_e with parameter dt, then _estress and _igral correspond to effective stress and the time integral at the start of the increment; the integration increment _inc will be calculated and then _igral and _estress will be updated to reflect values at the end of dt time increment.

Methods

`CRSN`(tvals, edot, **kwargs)	Constant rate of strain simulation
`CRSS`(tvals, estressdot, **kwargs)	Constant rate of stress simulation
`av_from_stress`(estress, **kwargs)	Slope of void ratio from effective stress
`e_and_stress_for_plotting`(**kwargs)	Void ratio and stress values that plot the method
`e_from_stress`(estress, **kwargs)	Void ratio from effective stress
`initial_conditions`([e0, estress0, pstress0, t0])	Calculate the initial conditions
`plot_model`(**kwargs)	plot the void ratio-stress points
`stress_from_e`(e, **kwargs)	Effective stress from void ratio

CRSN(tvals, edot, **kwargs)[source]¶

Constant rate of strain simulation

Note that the rate of chage of void ratio is specified. Strain can be back caluclated using strain = (e0-e) / (1 + e0).

Parameters:

tvals : 1d array of float: Time values for piecewise linear strain-rate vs time.
edot : 1d array of float: Time rate of change of void ratio corresponding to tvals. Note that you will need a negative edot for void ratio to decrease. Be careful because strain in geotech is usually positive for a decrease in void ratio.
method : [‘step’, ‘ode’] optional: Describes which method to use. ‘step’ will step through time using the stress_from_e function. ‘odeint’ will reframe the constitutive model in terms of stress-rate and use odeint to solve for effective stress. Both methods should produce the same results; the ‘odeint’ method is really just a validity check of the the ‘step’ method. Default method=’step’.
**kwargs : various: Arguments (other than method) will be passed to the subdivide_x_into_segments function. If not given then the following values will be used, dx = (tvals[-1]-tvals[0])/200, min_segments=50. You may have to fiddle around with kwargs to get meaningful results; a small dx value will give good results but could take a long time to compute.

Returns:

tvals_ : 1d array of float: Time values
estress : 1d array of float: Effective stress at tvals_.
e : 1d array of float: Void ratio at time tvals_.
edot_ : 1d array of float.: Time rate of change of void ratio at tvals_

Table Of Contents

Related Topics

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void_ratio_stress¶

Class summary¶

Module listing¶

Parameters:	e : float Current void ratio. e is assumed to be the effective stress at the end of an increment of length dt. i.e. at time t+dt. dt : float, optional Time increment. Default dt=0.
Returns:	estress : float Effective stress corresponding to current void ratio.