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2025 | OriginalPaper | Chapter

A High Cycle Accumulation Model for Polychromatic and Stochastic Loading

Authors : Andrzej Niemunis, Hans Henning Stutz

Published in: Recent Developments of Soil Mechanics and Geotechnics in Theory and Practice

Publisher: Springer Nature Switzerland

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Abstract

This paper introduces a novel high-cycle accumulation model for sand, referred to as the NHCA model. While many empirical formulas for deformation accumulation rates (such as stress or density dependence) remain consistent with the HCA model proposed in 2005 (referred to as the OHCA), the NHCA model brings essential novelty through the definition of strain amplitude and the cycle counting method.

The definition of strain amplitude is rooted in kinematic hardening principles, commonly found in multi-surface plasticity models. Notably, for polychromatic cycles (comprising oscillations with different frequencies), the amplitude lacked clear definition within the OHCA framework. This paper specifically explores the ovality and polarization of strain loops, as well as stochastic and polychromatic multiaxial cycles, as key points of interest. Additionally, experimental results and model calibration procedures are presented to support the NHCA model.

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Readers familiar with OHCA may notice that the chunk is a new amplitude factor rather than the amplitude itself.

The polychromatic cyclic loading was proposed to be decomposed into several oscillations with individual frequencies, analogously to the Fourier transform. Unfortunately, the accumulation rates due to the individual oscillations turned out to be non-additive, as demonstrated experimentally by M. Poblete [11].

This adaptation was introduced in the OHCA [13] and denoted as g there.

According to experimental results with millions of cycles the empirical formula (5) underestimates slightly the observed cyclic compaction rate and hence a small constant $C_{N3}$ was added Its effect becomes significant for $N > 10^6$, see [13]. The effect of $ C_{N3}$ is disregarded in the NHCA for simplicity. It may reappear due to the decay of G described in Sect. 6.

In the special case of $ f_\textrm{ampl}= \textrm{const}$ one may integrate

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-031-71896-0_7/MediaObjects/558583_1_En_7_Figw_HTML.gif

to $ g_A = f_\textrm{ampl}f_g $ and substitute $f_g = g_A/f_\textrm{ampl}$ into (5) for $f_g^\textrm{old}$. This results in $ 1 + C_{N2} N = \exp ( \frac{ g_A}{ f_\textrm{ampl}C_{N1}} ) $. Replacing $1 + C_{N2} N$ by $ \exp ( \frac{ g_A}{ f_\textrm{ampl}C_{N1}} )$ in (6)$_2$ one obtains

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-031-71896-0_7/MediaObjects/558583_1_En_7_Figx_HTML.gif

and further These equations generalize (5) in the sense that they are claimed to hold also for $f_\textrm{ampl}\ne \textrm{const}.$.

This relationship is used for the calibration of the NHCA from the OHCA material constants.

In the geomechanical context, the fatigue damage accumulation must be replaced by the adaptation variable for sand.

Contributions to G from individual simultaneously applied oscillations of polychromatic cyclic loading cannot be added. Section 5.1 explains how the NHCA handles such oscillations.

For small or moderate

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-031-71896-0_7/MediaObjects/558583_1_En_7_Figac_HTML.gif

amplitudes

https://static-content.springer.com/image/chp%3A10.1007%2F978-3-031-71896-0_7/MediaObjects/558583_1_En_7_Figad_HTML.gif

this part is assumed to be absent.

The following fourth order tensor can be examined for this purpose

$$ \mathfrak {f}_\textrm{Ampl}= R_1 \int _{t_0}^{t_0 + T} \left( w_0 \, \displaystyle \frac{\dot{\textbf{e}} \,\dot{\textbf{e}}}{ \sqrt{ \dot{\textbf{e}} : \dot{\textbf{e}}} } + \sum _{i=1}^n w_i \, \displaystyle \frac{ \dot{\textbf{a}}_i \, \dot{\textbf{a}}_i}{ \sqrt{\dot{\textbf{a}}_i : \dot{\textbf{a}}_i} } \right) \textrm{d}t \ \text { with}\ f_\textrm{Ampl}= \Vert \mathfrak {f}_\textrm{Ampl}\Vert $$

wherein $\sqrt{ \textbf{a}\, \textbf{a}} = \textbf{a}\, \textbf{a}/ \sqrt{\textbf{a}:\textbf{a}} $ holds.

Also the OHCA amplitude factor increases over-proportionally with the amplitude, namely $f_\textrm{ampl}= (\epsilon ^\textrm{ampl}/ \epsilon ^\textrm{ampl}_\textrm{ref})^{n_\textrm{ampl}} $ where $\epsilon ^\textrm{ampl}_\textrm{ref}= 10^{-4}$ and $n_\textrm{ampl}\approx 1.5$.

It just has to be roughly representative for the given cyclic loading.

The Prager condition is known from elasto-plastic hardening.

The proportionality $\dot{\boldsymbol{\alpha }}^i \sim \textbf{d}$ follows from the minimum of $\dot{\boldsymbol{\alpha }}^i : \dot{\boldsymbol{\alpha }}^i - \lambda (\textbf{d}: \dot{\epsilon } - \textbf{d}^i: \dot{\boldsymbol{\alpha }}^i ) $ in which the constraint (30) appears with the Lagrange multiplier $\lambda $. The partial derivative with respect to $\dot{\boldsymbol{\alpha }}^i$ leads to $2 \boldsymbol{\alpha }^i - \lambda \textbf{d}^i = 0$.

The state after cycles with slowly dwindling amplitude.

Six conditioning (irregular) chunks = cycles are used in the further examples.

A deformation commenced from the shakedown $\epsilon =0$ and $\alpha _i = 0$, see Fig. 12(a), and continuing to $\epsilon = r$ would not be a representative quarter of the cycle. The half-cycle from the initial position of the back strains shown in Fig. 12(b) to the same position (but mirrored) is reached at the end of the half-cycle in Fig. 12(d) is representative because the lengths of all back-strain paths are identical in each half-cycle.

If the back strain $i+1$ is activated the inequality becomes the equality.

Usually we deal with large slow cycles overlaid by small cycles of higher frequency.

Such decomposition was proposed in [10] using multidimensional Fourier analysis.

Preloading with (57) repeated over five periods.

In spite of $G=\textrm{const}$, the rate $\dot{\epsilon }^\textrm{acc}$ may slowly decrease due to changes in the stress and in the void ratio which reduce $f_\textrm{all}$.

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Knittel, L., Lamparter, A., Niemunis, A., Stutz: The high-cyclic model for sand tested beyond the usual ranges of application. Acta Geotechnica (2023), https://doi.org/10.1007/s11440-023-02031-y

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Title: A High Cycle Accumulation Model for Polychromatic and Stochastic Loading
Authors: Andrzej Niemunis
Hans Henning Stutz
Publisher: Springer Nature Switzerland
Book: Recent Developments of Soil Mechanics and Geotechnics in Theory and Practice
Print ISBN: 978-3-031-71895-3

Electronic ISBN: 978-3-031-71896-0

Copyright Year: 2025
DOI: https://doi.org/10.1007/978-3-031-71896-0_7

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