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High-Cycle Accumulation (HCA) model for sand

The High-Cycle Accumulation (HCA) model for sand1 can be linked to the Hypoplasticity + Intergranular Strain model (see here), Hypoplasticity + ISA model (see here), the Sanisand model (see here) or the linear elastic model, which will be used in the low-cycle phase of the HCA model. The input command takes the following form:

*Mechanical = HCA_Hypoplasticity
phi [rad], nu [-], h_s [F/A], n [-], e_d0 [-], e_c0 [-], f_ei0 [-], alpha [-]
beta [-], m_T [-], m_R [-], R [-], beta_R [-], chi [-], K^w [F/A]
C_N1 [-], C_N2 [-], C_N3 [-], C_ampl [-], C_e [-], 0, C_p [-], 0
C_Y [-], C_Pi1 [-], C_Pi2 [-], C_Pi3 [-], p_atm [-], e_ref [-], epsilon^ampl_ref [-], A [-]
a [-], n [-], nu [-], phi_c [rad]

*Mechanical = HCA-HYPO-ISA
phi [rad], h_s [F/A], n [-], e_d0 [-], e_c0 [-], e_i0 [-], alpha [-], beta [-] 
m_R [-], R [-], beta_h0 [-], chi^0 [-], chi^max [-], epsilon^acc [-], c^z [-], beta_hmax [-]
E_ph [F/A], nu_ph [-], p_{min,ph} [F/A], K^w [F/A], IM 
C_N1 [-], C_N2 [-], C_N3 [-], C_ampl [-], C_e [-], 0, C_p [-], 0
C_Y [-], C_Pi1 [-], C_Pi2 [-], C_Pi3 [-], p_atm [-], e_ref [-], epsilon^ampl_ref [-], A [-]
a [-], n [-], nu [-], phi_c [rad]

*Mechanical = HCA_Sanisand
p_atm [F/A], e_0 [-], lambda_c [-], xi [-], M_c [-], M_e [-], m [-], G_0 [-]
nu [-], h_0 [-], c_h [-], n_b[-], A_0 [-], n^d [-], z_max [-], c_z [-]
C_N1 [-], C_N2 [-], C_N3 [-], C_ampl [-], C_e [-], 0, C_p [-], 0
C_Y [-], C_Pi1 [-], C_Pi2 [-], C_Pi3 [-], p_atm [F/A], e_ref [-], epsilon^ampl_ref [-], A [-]
a [-], n [-], nu [-], phi_c [rad]

*Mechanical = HCA-Sanisand-2
p_atm [F/A], e_0 [-], lambda_c [-], xi [-], M_c [-], M_e [-], m [-], G_0 [-]
nu [-], h_0 [-], c_h [-], n_b[-], A_0 [-], n^d [-], z_max [-], c_z [-]
C_N1 [-], C_N2 [-], C_N3 [-], C_ampl [-], C_e [-], 0, C_p [-], 0
C_Y [-], C_Pi1 [-], C_Pi2 [-], C_Pi3 [-], p_atm [F/A], e_ref [-], epsilon^ampl_ref [-], A [-]
a [-], n [-], nu [-], phi_c [rad]

*Mechanical = HCA_Linear_elasticity
E [F/A], nu [-]
C_N1 [-], C_N2 [-], C_N3 [-], C_ampl [-], C_e [-], 0, C_p [-], 0
C_Y [-], C_Pi1 [-], C_Pi2 [-], C_Pi3 [-], p_atm [F/A], e_ref [-], epsilon^ampl_ref [-], A [-]
a [-], n [-], nu [-], phi_c [rad]

Properties

The material parameters of the HCA model are given in the following.

  • \(C_{N1}\), \(C_{N2}\) and \(C_{N3}\) define the accumulation rate with respect to the number of cycles.

  • \(C_{\text{ampl}}\) takes into account the influence of the strain amplitude on the accumulation rate. It can be estimated using 2 \(C_{\text{ampl}} = 1.7\).

  • \(C_{e}\) takes into account the influence of the void ratio on the accumulation rate. It can be estimated using 2 \(C_e = 0.95\cdot e_{\text{min}}\), where \(e_{\text{min}}\) is the minimum void ratio of the considered material.

  • \(C_{p}\) takes into account the influence of the mean effective stress on the accumulation rate. It can be estimated using 2 \(C_p = 0.41\cdot [1-0.34~(d_{50} [mm]-0.6)]\) where \(d_{50}\) is the mean grain size of the considered material.

  • \(C_{Y}\) takes into account the influence of the normalised stress ratio on the accumulation rate. It can be estimated using 2 \(C_Y = 2.60\cdot [1+0.12~\text{ln}(d_{50} [mm]/0.6)].\)

  • \(C_{Pi1}\), \(C_{Pi2}\) and \(C_{Pi3}\) take into account polarization changes. They are currently not used, i.e. can be chosen arbitrarily.

  • \(p_{atm}\) [F/A] is the atmospheric pressure, usually given by 100 kPa.

  • \(e_{\text{ref}}\) is the reference void ratio usually chosen identical to the maximum void ratio.

  • \(\varepsilon^\text{ampl}_{\text{ref}}\) is reference strain amplitude usually chosen to \(10^{-4}\)

  • \(A\), \(a\), \(n\) and \(\nu\) are the parameters defining the elastic stiffness of the model. They are usually chosen to \(A=1209\), \(a=1.63\), \(n=0.5\), \(\nu = 0.32\) 2.

  • \(\varphi_c\) is the friction angle used for the calculation of the flow direction and plastic strains (adopting the Matsuoka-Nakai failure surface with a 30 % higher value of \(\varphi_c\)). It is usually identical to the friction angle (but not in all cases).

Optional Parameters

*Optional mechanical parameter`
`<property 1>, <value 1>...`
`<property 2>, ...`

A list of currently supported optional parameters is given below:

  • scale_gA: A scaling factor (\(0 \leq c^{gA} \leq 1\)) for the cyclic history variable \(g^A\). At the beginning of each (new) hca phase, the cyclic history variable \(g^A\) is computed as \(g^A=c^{gA} \cdot g^A_0\), where \(g^A_0\) is the cyclic history variable at the end of the last HCA phase. By choosing \(c^{gA}=0\), \(g^A\) is reset to zero. The default value is \(c^{gA}=1\) (no scaling, a complete transfer of cyclic history to the new HCA phase).

  • dstran_rec: Required distance between two strain points during the recording step in which case a new strain point is saved. This values seldom has to be changed, except in cases where the strain path consists of values which are either much lower or larger than the default value of \(8\cdot 10^{-5}\).

  • cos_rec: Required cosine between two strain points during the recording step in which case a new strain point is saved. The default value is 0.83969.

  • proj_angle: Multiplicator for phi_c in the calculation of plastic strain using the Matsuoka-Nakai model. This multiplicator does only affect the bounding surface of the Matsuoka-Nakai model. The default value is 1.3.

  • scale_fampl_max: The maximum possible value of \(f_{ampl}\) is calculated by \(10^{\text{scale_fampl_max}\cdot C_{ampl}}\) instead of \(10^{C_{ampl}}\). The default value is 1.

  • m_factor: Determines how the critical stress ratio \(M\) is calculated in the flow rule of the HCA model. If m_factor=0, a stress-dependent non-constant value of \(M\) will be calculated. For any other value of m_factor, the critical stress ratio \(M\) for compression is used, which is the default case.

  • simple_eps_ampl: By default, the successive projection of the recorded strain path is used to calculated the strain amplitude1. If a definition based on the maximum euclidean distance is to be used, set simple_eps_ampl=1.

  • ga_definition: By default, the cyclic history state variable is calculated using

    \[ g^A = f_{\text{ampl}}C_{N1} \ln(1+C_{N2}N) \]

    If the definition derived in [@Staubach2021c] is supposed to be used, set ga_definition=1. Only in case the adaptive strain amplitude definition (see below) or update cycles are used, this influences the results. The equation used for \(g^A\) is:

    $$ gA\bigg\vert_{N=N_0}} =\int_{N_0}^{N} \dot{fN fdN\ = \ln\frac{1+C_{N2}N}{1+C_{N2}N_0} C_{N1} f_{\text{ampl}} $$ where }\(N_0\) is the number of cycles at which the strain amplitude changed (automatically calculated in the code and available as output, see end of this page).

State variables

The state variables are initialized according to the linked conventional constitutive models.

If the strain amplitude is to be given by a *initial conditions, type = state variables statement, it is possible either by using 

*initial conditions, type = state variables, default
<elset>, strain_ampl, <value>
or by using a user routine: 
*initial conditions, type = state variables, user
<elset>
In the user file, the statev(100) corresponds to the strain amplitude of the HCA model.

Additional definitions

Additionally, the name of the steps used for the low-cycle (also referred to as implicit) phase, the name of the step used for the recording of the strain and subsequent calculation of the strain amplitude and the high-cycle (also referred to as explicit) steps have to be defined. This is done by using

*Implicit hca steps
<implicit step 1>, <implicit step 2>, <implicit step 3>
*Recording hca steps
<recording step 1>, <recording step 2>
*Explicit hca steps
<HCA step 1>, <HCA step 2>

where the names of the respective steps have to be given. Lastly, the period of the individual cycles has to be defined by:

*HCA cycle time
<period HCA step 1>, <period HCA step 2>

Typically, the period is 1 for static analysis. Only for transient analyses the period matters.

Optionally, during the recording step, one may define only a certain time window for the recording of the strain path. This is defined by:

*HCA recording time
`<start time 1>, <end time 1>, <start time 2>, <end time 2>

The HCA calculations can be performed using an adaptive strain amplitude definition by using

*Adaptive strain ampl, <time =>, <frequency =>`
either <time = value 1, value 2, value 3, ...> or <frequency = value> has to be given. time defines the times at which the strain amplitude is updated during the HCA steps. frequency (\(\geq 1\)) defines the frequency in terms of increments at which the strain amplitude is updated during the HCA steps. It is strongly recommended to use the *Non-local (see here) method with the same time/frequency specifications during the HCA step and specifying statev=100 to smooth the field of updated strain amplitude.

You may define a minimum strain amplitude by using 

*Minimum eps ampl
<Value of minimum strain amplitude>
The function \(f_{\text{ampl}}\) is zero for values below the specified value.

You may define a maximum strain amplitude by using 

*Maximum eps ampl
<Value of maximum strain amplitude>

The function \(f_{\text{ampl}}\) is constant for values above the specified value.

Additional output variables

The output variables for the Hypoplasticity + Intergranular Strain (see here), the Sanisand (see here) are available.

The following additional output variables are available in the *Output command (see here):

  • Strain amplitude (only during the high-cycle step): strain_ampl

  • Accumulated strain rate: strain_acc

  • Cyclic history variable \(g^A\): cyclic_history

  • Factor \(f_{\text{ampl}}\): fampl

  • Factor \(f_{\text{e}}\): fe

  • Factor \(f_{\text{p}}\): fp

  • Factor \(f_{\text{Y}}\): fY

  • Number of recorded strain points during the recording step: Nrec_points

  • Number of cycles the strain amplitude was updated the last time (only relevant if *Adaptive strain ampl is used): N0

  • Cyclic history variable \(g^A\) the strain amplitude was updated the last time (only relevant if *Adaptive strain ampl is used): cyclic_history0

  1. A Niemunis, T Wichtmann, and T Triantafyllidis. A high-cycle accumulation model for sand. Computers and Geotechnics, 32(4):245–263, 2005. doi:https://doi.org/10.1016/j.compgeo.2005.03.002. ↩↩

  2. T Wichtmann. Soil Behaviour Under Cyclic Loading: Experimental Observations, Constitutive Description and Applications. Habilitation, Institute of Soil Mechanics and Rock Mechanics, Karlsruhe Institute of Technology, Issue No. 181, 2016. ↩↩↩↩↩