Anisotropy state variable based on phase transformation for generalized plasticity constitutive model

Abstract

This study presents a novel formulation for incorporating anisotropy into the generalized plasticity constitutive model. Generalized plasticity is a hierarchical framework allowing for extensibility, in order to encompass new phenomena and improve its predictive capabilities. Anisotropy formulation is based experimentally on the phase transformation state and considers explicitly the direction of the maximum principal stress and the magnitude of the intermediate principal stress, through an anisotropy state variable that contributes to the state parameter. Additionally, the model incorporates the fabric using an evolving fabric variable that reflects initial fabric due to sample preparation method for granular soils. The formulation is simple and introduces three constitutive parameters, allowing for straightforward implementation into the constitutive model and direct application in finite element analysis. The model is validated with undrained triaxial tests conducted on Toyoura sand, covering a wide range of initial conditions with a unique set of constitutive parameters, and yielding overall satisfactory results despite some limitations.

Appendices

Appendix 1

Formulation for generalized plasticity model according to [37] and [26] is outlined in this section. Notation coincides mostly with the latter. Bold figures render second-order tensors and double-struck capitals (${\mathbb{C}}, {\mathbb{D}})$ fourth-order tensors. Stress–strain relation, also known as constitutive relation, is expressed as per Eq. (11).

$${\varvec{d}}{{\varvec{\sigma}}}^{\boldsymbol{^{\prime}}}={\mathbb{D}}^{ep}: {\varvec{d}}{\varvec{\varepsilon}}$$

(11)

where ${\varvec{d}}{{\varvec{\sigma}}}^{\boldsymbol{^{\prime}}}$ and ${\varvec{d}}{\varvec{\varepsilon}}$ are the differential of strains and co-rotational stresses, respectively, and ${\mathbb{D}}^{ep}$ is the elasto-plastic stiffness tensor, see [37]. Inverse relation of the constitutive equation is formulated in Eq. (12).

$${\varvec{d}}{\varvec{\varepsilon}}={\mathbb{C}}^{ep}: {\varvec{d}}{{\varvec{\sigma}}}^{\boldsymbol{^{\prime}}}$$

(12)

where ${\mathbb{C}}^{ep}$, denoted the (elasto-plastic) compliance tensor, corresponds to the inverse of the stiffness tensor. The implementation of the Generalized Plasticity can be summarised in three steps. Firstly, formulation is elaborated in the compliance variant (Eq. 12), and the compliance tensor is obtained. Secondly, the stiffness tensor is determined, not directly from inversion of the compliance tensor, but following a proper procedure. Finally, the constitutive relation in its stiffness variant, Eq. (11), is solved. This last equation allows reproducing strain softening, which is a key feature of soil behaviour.

The stress tensor is expressed in the invariant base $\left\{p\left({I}_{1}\right),q\left({J}_{2}\right),\theta \left({J}_{2},{J}_{3}\right)\right\}$, see Eqs. (13) to (17). The mean effective stress, p', is related to the first stress invariant, I₁, Eq. (13). The deviatoric stress, q, depends on the second deviatoric stress invariant (J₂), which depends on the deviatoric stress tensor (s), see Eqs. (14) and (15). Lode angle, $\theta$, a function of the 2nd and 3rd deviatoric stress invariants, acts as the third invariant of the defining base, Eq. (16). Definition of ${J}_{2}$ and ${J}_{3}$ is shown in Eqs. (17), refer to [56].

$${p}^{\prime}=\frac{1}{3}{I}_{1}=\frac{1}{3}tr({{\varvec{\sigma}}}^{\boldsymbol{^{\prime}}})$$

(13)

$$q=\sqrt{3{J}_{2}}$$

(14)

$${\varvec{s}}={{\varvec{\sigma}}}^{\boldsymbol{^{\prime}}}-\frac{1}{3}{I}_{1}{\varvec{I}}$$

(15)

$$\theta =-\frac{1}{3}{{\text{sin}}}^{-1}\left(\frac{3\sqrt{3}}{2}\frac{{J}_{3}}{{J}_{2}^{3/2}}\right)$$

(16)

$$J_{2} = \frac{1}{2}tr(s^{2} );J_{3} = \frac{1}{3}tr(s^{3} )$$

(17)

The elasto-plastic constitutive (compliance) tensor, ${\mathbb{C}}^{ep}$, is defined according to Eq. (18), where ${\mathbb{C}}^{e}$ is the elastic constitutive tensor with elastic constants, G and K, defined as per Eq. (19). G₀ and K₀ are adimensional elastic shear and bulk moduli and ${p}_{a}^{\prime}$ is the atmospheric pressure. Alternatively, G₀ and $\nu$ can be specified, where Poisson modulus, $\nu$, relates G and K, Eq. (20). Subsequent improvement of shear and bulk moduli was attained by introducing their dependence on void ratio, as per Eqs. (21) and (22), see [42].

$${\mathbb{C}}^{ep}={\mathbb{C}}^{e}+\frac{1}{H}{{\varvec{n}}}_{{\varvec{g}}}\otimes {\varvec{n}}$$

(18)

$$G={{\text{G}}}_{0}\frac{{{\text{p}}}^{\prime}}{{p}_{a}^{\prime}};K={K}_{0}\frac{{{\text{p}}}^{\prime}}{{p}_{a}^{\prime}}$$

(19)

$$K=\frac{2}{3}G\frac{(1+\nu )}{(1-2\nu )}$$

(20)

$$G={G}_{0}\frac{{(2.97-{\text{e}})}^{2}}{(1+e)}\sqrt{{p}^{\prime}{p}_{a}^{\prime}}$$

(21)

$$K={K}_{0}\frac{{(2.97-{\text{e}})}^{2}}{(1+e)}\sqrt{{p}^{\prime}{p}_{a}^{\prime}}$$

(22)

Director vectors of plastic flow, ${{\varvec{n}}}_{{\varvec{g}}}$, and loading, ${\varvec{n}}$, are defined according to Eqs. (23) and (24).

$${{\varvec{n}}}_{{\varvec{g}}}={\left(\frac{{d}_{g}}{\sqrt{1+{d}_{g}^{2}}}, \frac{1}{\sqrt{1+{d}_{g}^{2}}}\right)}^{T}$$

(23)

$${\varvec{n}}={\left(\frac{{d}_{f}}{\sqrt{1+{d}_{f}^{2}}}, \frac{1}{\sqrt{1+{d}_{f}^{2}}}\right)}^{T}$$

(24)

where the loading direction vector, ${\varvec{n}}$, allows discriminating between states of loading (L) and unloading (U), according to Eq. (25), being ${\varvec{d}}{{\varvec{\sigma}}}^{{\varvec{e}}}$ the elastic stress infinitesimal increment. This loading criteria considers the elastic part of the differential of stresses in order to account for softening of the material (for which H < 0), which can not be reproduced with ${\varvec{d}}{{\varvec{\sigma}}}^{\boldsymbol{^{\prime}}}$, refer to [37].

$${\varvec{n}}\boldsymbol{ }\boldsymbol{ }: {\varvec{d}}{{\varvec{\sigma}}}^{{\varvec{e}}}\left\{\begin{array}{c}>0\to Loading\\ =0\to Neutral\\ <0\to Unloading\end{array}\right.$$

(25)

For the case of unloading, the direction of ${{\varvec{n}}}_{{\varvec{g}}}$ is denoted ${{\varvec{n}}}_{{\varvec{g}}{\varvec{U}}}$ and, for triaxial conditions, is specified as per Eq. (26).

$${{\varvec{n}}}_{{\varvec{g}}{\varvec{U}}}= \left( {{n_{{gU,v}}},{{n_{{gU,s}} }}} \right)^{T} = \left( { - n_{{g,v}} },{{n_{{g,s}} }} \right)^{T}$$

(26)

Dilatancy law, ${d}_{g}$, from Li and Dafalias [17], is defined in Eq. (27), whereas function ${d}_{f}$, Eq. (28), is defined similarly to dilatancy. The stress ratio is $\eta =q/p^{\prime}$ and m and ${d}_{0}$ are two material parameters.

$${d}_{g}={d}_{0}\left({e}^{m\psi }-\frac{\eta }{{M}_{g}}\right)$$

(27)

$${d}_{f}={d}_{0}\left({e}^{m\psi }-\frac{\eta }{{M}_{f}}\right)$$

(28)

The state parameter $\psi$ is defined according to [3], Eq. (29). The void ratio at CS, ${e}_{c}$, is assumed from Li [16] as per Eq. (30), being ${e}_{\Gamma }, \lambda , {\zeta }_{c}$ material parameters. The remaining CS parameter is ${M}_{g}$, which represents the stress ratio at CS, ${M}_{g}={\left(q/p^{\prime}\right)}_{CS}$, and is a function of the Lode angle (Zienkiewicz and Pande, [57]), Eq. (31), and the internal friction angle, Eq. (32). ${M}_{gc}$ stands for the value of ${M}_{g}$ at triaxial compression state ($\theta =30^\circ )$.

$$\psi =e-{e}_{c}$$

(29)

$${e}_{c}={e}_{\Gamma }-\lambda {\left(\frac{{{\text{p}}}{^\prime}}{{p}_{a}{^\prime}}\right)}^{{\zeta }_{c}}$$

(30)

$${M}_{g}\left(\theta ,{M}_{gc}\right)=\frac{6 {M}_{gc}}{6+{M}_{gc}(1-{\text{sin}}3\theta )}$$

(31)

$${M}_{gc}=\frac{6 {\text{sin}}\varphi }{3-{\text{sin}}\varphi }$$

(32)

And ${M}_{f}$ and ${M}_{g}$ are related by Eq. (33). Therefore, ${M}_{f}$ also depends on $\theta$ as per Eq. (31). Associative behaviour corresponds to ${M}_{f}={M}_{g}$, i.e., ${h}_{1}=1$ and ${h}_{2}=0$; behaviour being non-associative otherwise; ${h}_{1}, {h}_{2}$ and $\beta$ are material parameters.

$$\frac{{M}_{f}}{{M}_{g}}={h}_{1}-{h}_{2}{\left(\frac{e}{{e}_{c}}\right)}^{\beta }$$

(33)

Finally, H is the plastic modulus, which, as described in Sect. 3, comprises several factors. For the loading case, plastic modulus, denoted ${H}_{L}$, is formulated as per Eqs. (34) and (35).

$${H}_{L}={H}_{0} \sqrt{{p}^{\prime}{p}_{a}^{\prime}}{H}_{DM} f\left(\eta ,\psi \right)$$

(34)

$$f\left(\eta ,\psi \right)=\left\{\begin{array}{c}1\; for \;\eta =0\\ {H}_{f}\left({H}_{\upsilon }+{H}_{s}\right) \;for\; \eta \ne 0\end{array}\right.$$

(35)

These factors are related to: initial plastic strains (H₀), Eq. (36), where ${H}_{0}^{\prime}$, ${\beta }_{0}^{\prime}$ are material parameters; discrete memory (H_DM), which allows considering history to reproduce reloading, Eq. (37), where α and γ are material parameters and the function $\zeta$, Eq. (38), accounts for mobilization of deviatoric plastic strains; H_f stands for a limitation of admissible states, Eqs. (39) and (40), being µ = 4 (default) a material parameter; volumetric strain hardening (H_V), Eq. (41), where ${H}_{\nu 0}$, ${\beta }_{\nu }$ are material parameters; and deviatoric strain hardening (H_S), Eq. (42), ${\beta }_{1}$ and ${\beta }_{0}$ are model parameters and ${\xi }_{dev}$ represents the accumulated deviatoric plastic strain.

$$H_{0} = {H}_{0}^{\prime}{e}^{ - {\beta }_{0 }^{\prime}(\frac{e}{e_{c}})^\beta}$$

(36)

$${H}_{DM}= {\left(\frac{{\zeta }_{max}}{\zeta }\right)}^{\gamma }$$

(37)

$$\zeta ={p}^{\mathrm{^{\prime}}}{\left[1-\left(\frac{\alpha }{1+\alpha }\right)\frac{\eta }{{M}_{f}}\right]}^{-\frac{1}{\alpha }}$$

(38)

$${H}_{f}={\left(1-\frac{\eta }{{\eta }_{f}}\right)}^{\mu }$$

(39)

$${\eta }_{f}=\left(1+\frac{1}{\alpha }\right){M}_{f}$$

(40)

$${H}_{\nu }={H}_{\nu 0} \left[{M}_{g }{e}^{-{\beta }_{\nu }\psi }-\eta \right]$$

(41)

$${H}_{s}={\beta }_{1} {e}^{-{\beta }_{0}{\xi }_{dev}}$$

(42)

Regarding the unloading case, the plastic modulus, denoted ${H}_{U}$, is formulated as per Eq. (43).

$${H}_{U}=\left\{\begin{array}{c}{H}_{U0}{\left(\frac{{M}_{g}}{{\eta }_{u}}\right)}^{{\gamma }_{u}} for \left|\frac{{M}_{g}}{{\eta }_{u}}\right|>1 \\ {H}_{U0} for \left|\frac{{M}_{g}}{{\eta }_{u}}\right|\le 1\end{array}\right.$$

(43)

where ${\eta }_{u}$ is the stress ratio at the beginning of unloading and ${H}_{U0}$ and ${\gamma }_{u}$ are constitutive parameters.

Note that under neutral loading, both loading and unloading formulation yield the same strain increment, thus avoiding non-uniqueness of solutions, see [37]. The detailed calibration procedure to determine the constitutive parameters can be found in [24] and Cuomo et al. [8].

Once the constitutive model is formulated, a proper procedure for the determination of the stiffness tensor is considered, as the plastic modulus can reach a null value that renders the direct inversion of the compliance tensor (${\mathbb{C}}^{ep}$) inadequate, see Eq. (21). This procedure was first developed by Zienkiewicz and Mróz [57], and elaborated by [37], refer to the Appendices of both references. Equation (44) expresses the elastoplastic stiffness tensor as a function of: the elastic stiffness tensor, ${\mathbb{D}}^{e}$, which is the inverse of ${\mathbb{C}}^{e}$; the plastic modulus, $H$; and the director vectors ${\varvec{n}}$ (${{\varvec{n}}}^{{\varvec{T}}}$ denoting its transpose) and ${{\varvec{n}}}_{{\varvec{g}}}$. These are the four components of the constitutive model to be determined.

$${\mathbb{D}}^{ep}={\mathbb{D}}^{e}- \frac{{\mathbb{D}}^{e}{{\varvec{n}}}_{{\varvec{g}}}{{\varvec{n}}}^{{\varvec{T}}}{\mathbb{D}}^{e}}{H+{{\varvec{n}}}^{{\varvec{T}}}{\mathbb{D}}^{e}{{\varvec{n}}}_{{\varvec{g}}}}$$

(44)

Appendix 2

Experimental tests considered in this work are shown in Table 2: first 18 rows correspond to Yoshimine et al. [53] and last 15 rows to Nakata et al. [32]. The first set includes 4 TC tests, 4 TE tests, 5 constant-α tests and 5 constant-b tests. The second set includes 15 constant-α tests, 5 for each relative density: D_r = 30%, 60% and 90%.

Table 2 Summary of undrained tests on Toyoura sand (data from [53] and [32])

Appendix 3

This Appendix includes some aspects of the variation of the stress ratio at CS, M_g, which should be considered to adequately reproduce anisotropy and interpret results.

3.1 Variation of M _g with α

The magnitude of M_g from experimental tests is estimated from undrained tests in p'-q space [24], as the slope ($\eta =q/{p}^{\prime}$) at CS. Strictly, CS is not observed in analyzed tests from Yoshimine et al. [53] and Nakata et al. [32]. However, the $q/{p}^{\prime}$ at the end of the tests are considered for this purpose.

Experimentally, considering constant-α tests, a variation of M_g with α is noticed. These values are depicted in Fig. 16 for D_r = 40%, 60% and 90%. Note that in the case of tests from [32], for which b = 0.5, deviatoric stress is given by Eqs. (45). Thus, to show the same levels of deviatoric stress, values from [53], corresponding to D_r = 40%, are divided by factor $\sqrt{3}/2$ only to compare both magnitudes.

$${q}_{b}=\sqrt{\frac{1}{2}\left[{\left({\sigma }_{1}-{\sigma }_{2}\right)}^{2}+{\left({\sigma }_{2}-{\sigma }_{3}\right)}^{2}+{\left({\sigma }_{3}-{\sigma }_{1}\right)}^{2}\right]}$$

(45.1)

$$b=0.5\to {\sigma }_{2}=\frac{1}{2}{\left({\sigma }_{1}+{\sigma }_{3}\right)\to q}_{b=0.5}=\frac{\sqrt{3}}{2}{(\sigma }_{1}-{\sigma }_{3})= \frac{\sqrt{3}}{2}q$$

(45.2)

For all densities, a tendency can be remarked with a cosine shape, which is in line with Dong et al. [9] and Gao et al. [11]. This experimental tendency is implemented into the constitutive model according to Eq. (46). Both simulation curves are shown in Fig. 16.

$${M}_{g}\left(\alpha \right)={M}_{g,a}+k{\text{cos}}\left(2\alpha \right)$$

(46)

where ${M}_{g,a}=1.15$ for b = 0 and 1.03 for b = 0.5; $k=0.1$ for both cases.

Fig. 16

Experimental values of Mg for D_r = 40%, 60% and 90% (data [53] and [32]) and simulation functions for b = 0 and b = 0.5

3.2 Variation of M _g with b

From a constitutive point of view, the stress-invariant base of the model is $\left\{p\left({I}_{1}\right),q\left({J}_{2}\right),\theta \left({J}_{2},{J}_{3}\right)\right\}.$ A variation in ${\sigma }_{2}$, or in b, modifies the invariants and the model response. In particular, it is relevant to note that the third invariant (Lode’s angle), and variables that depend on it, are affected. One of these variables is the stress ratio at critical state (M_g). From Eqs. (18), (19) and (31), M_g as a function of b via $\theta$ is calculated [57], as shown in Fig. 17. Introducing the dependency of M_g on $\theta$ allows the model reproducing all shearing modes (α, b), and not only the triaxial space.

A divergence between theoretical values and the experimental ones from [53] is noticed, which are depicted in Fig. 17. Note that experimental values corresponding to b ≠ 0.5 are obtained from Petalas et al. [40].

As can be seen, experimental values are significantly higher than theoretical ones. This is the reason of the observed offset between tests and simulations.

Fig. 17

Mg as a function of b: theoretical formula from [57], ZP77, and experimental values (data from [53] after [40])