## Abstract

The ISA-plasticity is a mathematical platform which allows to propose constitutive models for soils under a wide range of strain amplitudes. This formulation is based on a state variable, called the intergranular strain, which is related to the strain recent history. The location of the intergranular strain can be related to the strain amplitude, information which is used to improve the model for the simulation of cyclic loading. The present work proposes an ISA-plasticity-based model for the simulation of saturated clays and features the incorporation of a viscous strain rate to enable the simulation of the strain rate dependency. The work explains some aspects of the ISA-plasticity and adapts its formulation for clays. At the beginning, the formulation of the model is explained. Subsequently, some comments about its numerical implementation and parameters determination are given. Finally, some simulations are performed to evaluate the model performance with two different clays, namely a Kaolin clay and the Lower Rhine clay. The simulations include monotonic and cyclic tests under oedometric and triaxial conditions. Some of these experiments include the variation of the strain rate to evaluate the viscous component of the proposed model.

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## Appendices

### Appendix

### Short guide for the material parameter determination

The compression and swelling index, denoted with \(\lambda\) and \(\kappa\), respectively, are calibrated with isotropic or oedometric compression tests. These parameters are computed in the *e* vs. \(\log (p)\) space. As shown in Fig. 13, a compression path with constant strain rate is expected to approach asymptotically to a line with slope equal to \(\lambda\). Hence, the measurement of this slope enables us to compute parameter \(\lambda\). Parameter \(\kappa\) is calibrated instead upon an unloading path. In contrast to classical elastoplastic models, the proposed model delivers a hysteresis when simulating an unloading-reloading cycle. This implies that the slope during the cycle is not constant and is actually influenced by small strain effects. At the beginning of the unloading path, the simulation shows a slope equal to \(\kappa /m_R\), whereby \(m_R\) is the factor responsible for the stiffness increase due to reversal loading (see Table 1). Subsequently, the small strain effects vanish and the simulation exhibits a slope similar to \(\kappa\). Therefore, one may calibrate parameters \(\kappa\) and \(m_R\) with an unloading-reloading cycle.

The reference void ratio \(e_{i0}\) corresponds to the maximum void ratio at the reference pressure \(p=1\) kPa. The interpretation of \(e_{i0}\) is actually different as in other formulations considering the fact that the strain rate is decomposed into three components. According to the proposed model, the maximum void ratio curve \(e=e_i(p)\) is only reached under infinite strain rate \(\parallel \dot{{{\varvec{\varepsilon }} }}\parallel =\infty\); see Fig. 3a. A sample compressed with infinite strain rate \(\parallel \dot{{{\varvec{\varepsilon }} }}\parallel =\infty\) is not feasible from the experimental point of view, and therefore, an alternative calibration method must be followed. For this purpose, the constitutive equation for an isotropic compression test under mobilized states is examined: By performing the operation \(\dot{p}=-(\mathbf {1}:\dot{{{\varvec{\sigma }}}})/3\), the constitutive equation reduces under mobilized states to (see Table 1):

whereby \({\hbox {OCR}}=p_{i}/p\) corresponds to the overconsolidation ratio for isotropic compression (\(q=0\)) and \(\dot{\varepsilon }_v=-\mathbf {1}:\dot{{{\varvec{\varepsilon }} }}\) is the volumetric strain rate. On the other hand, an isotropic compression path with slope equal to \(\lambda\) can be also described with the non-viscous version of the model by setting \({\hbox {OCR}}=1\) and \(I_v=0\):

It is desired to find a simplified relation for compression paths under isotropic compression with the reference velocity \(\dot{\varepsilon }_v=D_{r}\). The velocity \(D_r\) can be arbitrarily selected to match a particular experiment. Similar to Niemunis [37], this particular compression path is termed “Reference isotach.” We recall that the maximum void ratio line \(e_i=e_i(p)\) presents an overconsolidation ratio of \({\hbox {OCR}}=1\), and a different value is expected for the reference isotach. Let us denote the overconsolidation ratio for the reference isotach with \({\hbox {OCR}}_\mathrm{(ri)}\). By combining Eqs. (33) and (34) and setting \(\dot{\varepsilon }_v=D_{r}\) and \(\mathrm{OCR}={\hbox {OCR}}_\mathrm{(ri)}\) yields to:

which can be numerically solved for \({\hbox {OCR}}_\mathrm{(ri)}\). After solving this equation, one may compute \(e_{i0}\) from the relation:

whereby \(e_\mathrm{(ri)}\) is the void ratio at \(p=1\) kPa at the reference isotach, which must be extrapolated from the experimental curve; see Fig. 13a. When only oedometric compression tests are performed, we suggest to use the same relations as an approximation taking advantage that the behavior is similar. The Poisson ratio \(\nu\) can be determined by measuring the shear modulus *G* for small strain amplitudes. An undrained triaxial test or a resonant column is useful for this purpose. If the shear modulus \(G=G(p,e)\) for a given mean stress *p* and void ratio *e* is known, one can compute the bulk modulus \(K=m_R \bar{K}\) according to Eq. (25) and solve for \(\nu\) from \(r=G/K=3\,(1-2\,\nu )/(2\,(1+\nu ))\). Notice that its determination depends on parameter \(m_R\), which must be previously determined.

The slope of the critical state \(M_c\) is adjusted to the critical state line CSL. It corresponds to the slope within the \(p-q\) space under triaxial compression. Points lying with vertical deformation of about \(\varepsilon _1>20\%\) are recommended for its calibration. The parameter \(f_{b0}\) controls approximately the maximum stress ratio for triaxial compression \(f_{b0}=\eta _\mathrm{max}/M_c\). It can be adjusted to highly overconsolidated samples \({\hbox {OCR}}>2\). When data are scarce, a recommended value of \(f_{b0}=1.3\) may be carefully used according to our experience with some clays. Of course, some simulations would help to check this recommendation.

The viscosity index \(I_v\) controls the intensity of the strain rate dependency. In this sense, an increasing value of \(I_v\) would return a larger creep deformation or increase the distance between the isotachs. The power relation used for the formulation of the viscous strain rate \(\dot{{{\varvec{\varepsilon }}}}^\mathrm{vis}\), see Eq. (31), has been actually examined by many authors in former works [37, 53, 61]. In particular, Niemunis [37] showed that the viscosity index \(I_v\) of this formulation can be adjusted with compression curves of different strain rates (isotachs) with the following method: Consider two different isotachs with strain rate equal to \(\Vert \varvec{\dot{\varepsilon }}_a\Vert\) and \(\Vert \varvec{\dot{\varepsilon }}_b\Vert\) and overconsolidation ratio of \({\hbox {OCR}}_a\) and \({\hbox {OCR}}_b\), respectively. Niemunis showed that for Eq. (31), the following relation, previously proposed by Leinenkugel [31], also holds:

Parameter \(n_\mathrm{ocr}\) controls the shape of the \({\hbox {OCR}}_\mathrm{3D}\) surface (see Fig. 4b) and therefore the viscous effects under stress states different than the isotropic \(q\ne 0\). As shown in Fig. 13b, an increasing value of \(n_\mathrm{ocr}\) delivers a lower excess of pore water pressure \(p_w\) upon undrained shearing, especially when approaching to the critical state line. We recommend to calibrate this parameter by trial and error given some undrained tests.

The parameter *R* corresponds to the amplitude of the threshold strain which encloses the elastic locus of the material. It can be determined from a secant shear modulus degradation curve. When this test is not available, a value \(R=10^{-4}\) is recommended to describe the elastic behavior of clays. This value has proved to provide numerical stability in finite element implementations [16]. The parameter \(\beta _h\) controls the strain amplitude required to reach the mobilized states. A relation for \(\beta\) was provided in [16] and reads:

where \(\Delta \varepsilon _s\) is the deviatoric strain amplitude and \(r_h \approx 0.99\) is a factor which defines how close tensor \(\mathbf {c}\) is to its bounding condition \(r_h =\Vert \mathbf {c}\Vert /\Vert \mathbf {c}_b\Vert\) . The parameters \(\chi _0\) and \(\chi _\mathrm{max}\) control the degradation curve shape of the secant shear modulus \(G_\mathrm{sec}\). According to Poblete et al. [43], parameter \(\chi _0\) should be calibrated on a single or a few cycles (\(N<3\)). Details of the determination of \(\chi _0\) were given in [19]. When the number of consecutive cycles under cyclic undrained triaxial test increases, the usage of curves showing the water pore pressure \(p_w\) against the number of cycles *N* is recommended in order to determine parameters \(\chi _\mathrm{max}\) and \(C_a\). Figure 14 shows the influence of these parameters in a curve fitting the Kaolin. While parameter \(\chi _\mathrm{max}\) governs the accumulation rate for a large number of consecutive cycles, \(C_a\) controls how fast the accumulation is produced during the first cycles. A trial-and-error procedure is herein recommended to calibrate simultaneously these two parameters.

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Fuentes, W., Tafili, M. & Triantafyllidis, T. An ISA-plasticity-based model for viscous and non-viscous clays.
*Acta Geotech.* **13**, 367–386 (2018). https://doi.org/10.1007/s11440-017-0548-y

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DOI: https://doi.org/10.1007/s11440-017-0548-y

### Keywords

- Clays
- Constitutive models
- Cyclic loading
- ISA-plasticity
- Viscosity