Bounding Surface Elasto-Viscoplasticity: A General Constitutive Framework for Rate-Dependent Geomaterials

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INTRODUCTION
Similar to classical plasticity, the plastic strain rate is computed according to a plastic flow rule, 131 which can be written as follows: where the symbol hi indicates Macauly brackets such that hxi = x if x 0 and hxi = 0 if x < 0, 134 R i j denotes the direction of the plastic flow, and ⇤ is a plastic multiplier. To evaluate the plastic 135 multiplier for an incremental loading path, a bounding surface is defined in stress space: 136 F(¯ i j , q n ) = 0 (4) 137 where q n represents the plastic internal variables (PIVs) (Dafalias and Popov 1976) and the image 138 stress¯ i j is the radial projection of the current stress i j onto F = 0 ( Fig. 1(a)). This radial mapping 139 rule can be analytically expressed as where ↵ i j is the projection center with respect to which the current stress state is mapped to the 142 bounding surface. The loading direction at the current state is assumed to coincide with the gradient 143 of the bounding surface at the image stress (L i j in Fig. 1(a)): The combination of this assumption with the radial mapping rule implies a loading surface ( f = 0 146 in Fig. 1(a)), which passes through the current stress and is homothetic to the bounding surface 147 with the projection center as the center of homothecy. The variable b in Eq. (5) can be further 148 interpreted as the similarity ratio between the bounding surface and the loading surface. It varies ⇤ = 1 K p L i j € i j (7) 153 where K p , the plastic modulus at the current stress, is a function of the plastic modulus at the image 154 stress,K p , and the similarity ratio b. A convenient example of such an equation linking K p toK p 155 and b was suggested by Dafalias (1986): stress space is again homothetic to the bounding surface with reference to the projection center, and 161 when s = 1, the elastic nucleus degenerates to the projection point, thus modeling materials with 162 vanishing elastic range. This "elastic nucleus" was used by Dafalias (1982) to define the overstress 163 for calculating the viscoplastic strain rate in the BS-EP/VP framework discussed previously. The 164 variableK p in Eq. (8) is evaluated by enforcing the consistency condition at the bounding surface: 165K p = @F @q nq n (9) 166 whereq n denotes the direction of the rate of PIVs and is specified by certain evolution rules, which 167 can be written as follows: 168 € q n = h⇤iq n (10) 169 The evolution of ↵ i j can be expressed in a similar form, thus treating the projection center as a 170 particular PIV:

GENERAL FORMULATION OF BOUNDING SURFACE ELASTO-VISCOPLASTICITY
A general formulation of bounding surface elasto-viscoplasticity (BS-EVP) will be presented 174 here, which is applicable to incorporate time/rate e ects into inviscid bounding surface models.

175
As stressed before, a fundamental assumption of the proposed BS-EVP framework is that the strain 176 rate can be additively decomposed into elastic and viscoplastic parts, as follows: where the superscript vp stands for viscoplastic. Note that the elastic response is still governed by 179 Eq.
(2). In accordance with Perzyna's overstress theory, the viscoplastic strain rate and the rate of 180 change of the internal variables can be related to a viscous nucleus function, as follows: where R i j ,q n and↵ i j retain the same definitions previously provided with reference to the rate-183 independent framework. The viscous nucleus is a function of the overstress, y, that satisfies the 184 following requirements: This work employs a unique viscous nucleus expression to control the evolution of both vis-187 coplastic strains and internal variables. In principle, however, di erent expressions of could be 188 hypothesized for each of these variables.

Static Loading Surface and Overstress
This work introduces a static loading surface ( f s = 0 in Fig. 1(b)) and uses the departure of the 191 current stress i j from f s = 0 to define the overstress, y, such that Note that the static loading surface resembles the loading surface in the rate-independent framework, implies that the static loading surface will tend to approach the current stress, when the viscous 199 nucleus is not null, accompanied by delayed plasticity until when the current stress lies on the 200 static loading surface. As a result, at low loading rates (i.e., when su cient time is allowed for 201 viscoplastic strains to develop), the current stress will tend to remain on the static loading surface.

202
Consequently, to ensure convergence to the underlying bounding surface rate-independent behavior 203 at low loading rates, the static loading surface ( f s = 0) and the loading surface ( f = 0) in the 204 underlying rate-independent models have to share the same analytical expression and evolution 205 rules with plastic strains. This condition is here referred to as the identity condition. The analytical 206 expression of f s is obtained by inserting the radial mapping of Eq. (5) into F = 0 (i.e., Eq. (4)): where the variable b s replaces b in Eq. (5), denoting the homothetic ratio between the bounding 209 surface and the static loading surface. In contrast to the variable b in rate-independent BS models, 210 b s has to be treated as an independent PIV, because the current stress is not required to lie on 211 the static loading surface. Similar to other PIVs, the evolution of b s can be related to the viscous 212 nucleus, as follows: where the expression ofb s has to be defined in agreement with the underlying bounding surface 215 formulation (see next section). The variable i j,s in Eq. (16) denotes stress states laying on the 216 static loading surface. A particular instance of i j,s (see Fig. 1(b)), which is here referred to as the 217 static stress, is the radial projection of the current stress onto f s = 0 by using ↵ i j as the projection 218 center. As will be discussed in next section, this stress variable will replace the appearance of the 219 current stress in the expression ofb s .

220
Similar to the static loading surface, a dynamic loading surface (i.e., f d = 0 in Fig. 1(b)) exists, 221 which always passes through the current stress i j and is homothetic to the bounding surface, with 222 ↵ i j being the homothetic center. The dynamic loading surface can be written as follows: Eq. (5) with b d .

227
As the static stress and the current stress lie along the same radial projection ( Fig. 1(b)), inserting 228 b s and the static stress i j,s or b d and the current stress i j into Eq. (5) yields the same image stress.

229
Accordingly, s,i j can be related to the current stress i j by knowing the values of b s and b d : where b d /b s can be further interpreted as the similarity ratio of the static loading surface over the 232 dynamic loading surface, through which a normalized overstress can be defined as: This overstress function satisfies the requirement in Eq. (15), in that when the current stress lies outside the static loading surface (i.e., the dynamic loading surface encloses the static loading 236 surface), the ratio b s /b d > 1 and consequently y > 0.

238
To fulfill the identity condition, the evolution of the static loading surface with plastic strains 239 has to be identical to that of the loading surface in the rate-independent models, and thusb s ,q n and 240↵ i j should be defined consistently with the hardening rules employed in inviscid baseline models 241 (i.e., Eqs. (10) and (11)).

242
Since the value of b in rate-independent models is directly computed from the current stress i j , 243 an explicit definition of its evolution is not required. This rule, which also governs the evolution of : Using the chain rule and the radial mapping of Eq. (5), the partial derivatives in Eq. (21) can 248 be expressed as By substituting Eq. (22) into Eq. (21), and taking Eq. (7) into account, the following relation is Considering the rate equations of q n and ↵ i j given in Eq. (10) and (11), respectively, and the 254 definition ofK p given in Eq. (9), the evolution functionb controlling the rate of the similarity ratio 255 b (i.e., € b = h⇤ib) can be expressed as:  The plastic modulus at the current stress state K p is related to the plastic modulus at the image 325 stressK p by The variable R p is the volumetric component of the plastic flow direction, which is given by the 336 following flow rule: in which⌘ is the image stress ratio (i.e.,⌘ =q/p). Equation (31) implies that an associative flow 341 rule is employed, i.e., the loading direction coincides with the plastic flow direction.

342
Finally, the elastic response is governed by a hypoelastic law often used in critical state models 343 for clay (Roscoe and Burland 1968; Schofield and Wroth 1968): where the elastic bulk moduli K and shear moduli G are given by In Eq. (33), ⌫ is the Poisson's ratio.

349
The rate equation of the viscoplastic strain is defined by: where R p and R q have been specified in Eq. (31). Here a simple linear viscous nucleus is used: where v is a material parameter related to viscosity. To compute y from Eq. that a plastic equilibrium state is achieved once delayed plasticity has fully developed.

417
In contrast to classical plasticity models augmented by the overstress theory (e.g., see Adachi  For NC clays (Fig. 9), the model satisfactorily replicates the increase of undrained strength 445 as well as the simultaneous decrease of excess pore pressure generated by increasing strain rates.