Dilatancy of granular materials in a strain space multiple mechanism model

Iai, Susumu; Tobita, Tetsuo; Ozutsumi, Osamu; Ueda, Kyohei

doi:10.1002/nag.899

Cited by 92 publications

(70 citation statements)

References 38 publications

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“…Furthermore, the necessity and advantages of introducing the second mode with the parameter a 2 in the expression are examined in Appendix A through comparison with a DEM simulation. If the anisotropic parameters are ignored and thus the function F * ( θ ) is independent of direction θ (ie, F * ( θ ) = 1) as shown in the circle in Figure , the second‐order fabric tensor in Equation is reduced to its original form without any inherent‐anisotropy effects …”

Section: Integrated Form Of the Strain Space Multiple Mechanism Modelmentioning

confidence: 99%

“…The incremental form of the constitutive model is derived by taking the time derivative of both sides of Equation as

{\dot{boldσ}}^{'} = - {\dot{p}}^{'} boldI + \int_{0}^{π} ({}, F ((), ω - ω_{0}) {\dot{q}}_{Iso} + {\dot{q}}_{Aniso}) (〈〉, boldt \otimes boldn) normald ω,

where

{\dot{p}}^{'} = \frac{normald p^{'}}{normald ε^{'}} {\dot{ε}}^{'};

{\dot{q}}_{Iso} = \frac{\partial q_{Iso}}{\partial γ} \overset{γ}{true} + \frac{\partial q_{Iso}}{\partial ε^{'}} {\dot{ε}}^{'} + \frac{\partial q_{Iso}}{\partial ε^{″}} {\dot{ε}}^{″};

{\dot{q}}_{Aniso} = \frac{normald q_{Aniso}}{normald p^{'}} {\dot{p}}^{'} = \frac{normald q_{Aniso}}{normald p^{'}} \frac{normald p^{'}}{normald ε^{'}} {\dot{ε}}^{'} .

A volumetric strain rate due to contractive dilatancy is formulated using a parameter,

M_{v}^{*}

, determined by the phase transformation angle through a second‐order fabric tensor,

I_{d}^{c}

, as follows:

…”

Section: Incremental Form Of the Strain Space Multiple Mechanism Modementioning

confidence: 99%

“…By ignoring the terms associated with inherent anisotropy in Equation , the fourth‐order fabric tensor, ℂ, is reduced to the following expression, which has already been established for isotropic granular materials without inherent anisotropy:

\begin{matrix} lefttrue \\ ℂ = K_{L / U} I \otimes I + \int_{0}^{2 π} G_{L / U} 〈n \otimes n〉 \otimes 〈n \otimes n〉 d ω - K_{L / U} I \otimes {boldI}_{normald} \\ + \int_{0}^{2 π} H_{Iso} 〈n \otimes n〉 \otimes (I - {boldI}_{normald}) d ω + \int_{0}^{2 π} L 〈n \otimes n〉 \otimes (I - {boldI}_{normald}^{normalc}) d ω . \end{matrix}

…”

Section: Incremental Form Of the Strain Space Multiple Mechanism Modementioning

confidence: 99%

“…As a prelude to the formulation of dilatancy in the anisotropic strain space multiple mechanism model, the relationship between microscopic and macroscopic strain energies is identified following Iai et al First, the rate of strain energy,

\overset{W}{true}

, in the constitutive model is computed through Equations , , and as

\overset{W}{false} = σ^{'} : \overset{ε}{false} = {\dot{W}}_{p} + \int_{0}^{π} ({}, F ((), ω - ω_{0}) {\dot{W}}_{q_{Iso}} + {\dot{W}}_{q_{Aniso}}) normald ω,

where

{\dot{W}}_{p} = - p^{'} \overset{ε}{false},

{\dot{W}}_{q_{Iso}} = q_{Iso} \overset{γ}{false},

{\dot{W}}_{q_{Aniso}} = q_{Aniso} \overset{γ}{false} = - \frac{1}{π} a_{1} cos ((), ω - ω_{0}) p^{'} \overset{γ}{false} .

Equation defines the fundamental relationship between the macroscopic strain‐energy rate

\overset{W}{true}

and the microscopic strain‐energy rates,

{\dot{W}}_{q_{Iso}}

and

{\dot{W}}_{q_{Aniso}}

given by the individual virtual simple shear mechanism that constitutes the anisotropic strain space multiple mechanism model. This relationship will play an important role in formulating dilatancy as described later.…”

Section: Strain Energy and The Dilative Component Of Dilatancymentioning

confidence: 99%

“…When the shear strain is very large such that γ xy = ∞, Equation with Equation leads to the macroscopic shear strength considering inherent anisotropy,

τ_{m}^{Aniso}

, as follows:

τ_{m}^{Aniso} = {τ_{xy}|}_{γ_{xy} = \infty} = \int_{0}^{π} ([], ({}, 1 + a_{2} cos 2 ((), ω - ω_{0})) q_{v} - \frac{1}{π} a_{1} cos ((), ω - ω_{0}) p^{'}) sin ω normald ω .

Here, the macroscopic shear strength without inherent anisotropy is given as

τ_{m}^{Iso} = q_{v} \int_{0}^{π} sin ω normald ω = 2 q_{v} .

Substitution of Equation into Equation yields

τ_{m}^{Aniso} = ((), 1 - \frac{1}{3} a_{2} cos 2 ω_{0}) τ_{m}^{Iso} - \frac{1}{2} a_{1} p^{'} sin ω_{0};

by normalizing Equation with the isotropic stress p ' , the internal friction angle with inherent anisotropy,

ϕ_{f}^{Aniso}

, can be related to that without inherent anisotropy,

ϕ_{f}^{Iso}

, using the anisotropy parameters (ie, a 1 , a 2 , and ω 0 ) as follows:

sin ϕ_{f}^{Aniso} = (()...)

…”

Section: Model Parametersmentioning

confidence: 99%

See 4 more Smart Citations

Constitutive modeling of inherent anisotropy in a strain space multiple mechanism model for granular materials

Ueda

Iai

2018

Num Anal Meth Geomechanics

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Summary Consideration of fabric anisotropy is crucial to gaining an improved understanding of the behavior of granular materials. This paper presents a constitutive model to describe the sand behavior associated with fabric anisotropy within a framework of a strain space multiple mechanism model. In the proposed model, a second‐order fabric tensor is extended by incorporating a new function that represents the effect of inherent (or initial fabric) anisotropy, along with three additional parameters: two of them, a1 and a2, control the degree of anisotropy, and the second mode of inherent anisotropy can be expressed by introducing the parameter a2 as well as the first mode by the parameter a1. The third parameter, θ0, expresses the principal direction of inherent anisotropy (eg, the normal vector direction of bedding planes relative to horizontal axis). The formulation of the dilative component of dilatancy (ie, positive dilatancy) is also extended to consider the effect of inherent anisotropy based on the interlocking mechanism. Experimental data on the complex anisotropic responses of Fraser River sand and Toyoura sand under monotonic loading is used to validate this model. The proposed model is shown to successfully capture anisotropic responses, which become contractive or dilative depending on different principal‐stress directions, with a single set of anisotropy parameters; thus, the model is considered to possess the capability to simulate the anisotropic behaviors of granular materials. In addition to different loadings on the same fabric, the effects of different fabric anisotropies upon the sand behavior under the same loadings are also investigated.

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Section: Integrated Form Of the Strain Space Multiple Mechanism Modelmentioning

confidence: 99%

“…The incremental form of the constitutive model is derived by taking the time derivative of both sides of Equation as

{\dot{boldσ}}^{'} = - {\dot{p}}^{'} boldI + \int_{0}^{π} ({}, F ((), ω - ω_{0}) {\dot{q}}_{Iso} + {\dot{q}}_{Aniso}) (〈〉, boldt \otimes boldn) normald ω,

where

{\dot{p}}^{'} = \frac{normald p^{'}}{normald ε^{'}} {\dot{ε}}^{'};

{\dot{q}}_{Iso} = \frac{\partial q_{Iso}}{\partial γ} \overset{γ}{true} + \frac{\partial q_{Iso}}{\partial ε^{'}} {\dot{ε}}^{'} + \frac{\partial q_{Iso}}{\partial ε^{″}} {\dot{ε}}^{″};

{\dot{q}}_{Aniso} = \frac{normald q_{Aniso}}{normald p^{'}} {\dot{p}}^{'} = \frac{normald q_{Aniso}}{normald p^{'}} \frac{normald p^{'}}{normald ε^{'}} {\dot{ε}}^{'} .

A volumetric strain rate due to contractive dilatancy is formulated using a parameter,

M_{v}^{*}

, determined by the phase transformation angle through a second‐order fabric tensor,

I_{d}^{c}

, as follows:

…”

Section: Incremental Form Of the Strain Space Multiple Mechanism Modementioning

confidence: 99%

\begin{matrix} lefttrue \\ ℂ = K_{L / U} I \otimes I + \int_{0}^{2 π} G_{L / U} 〈n \otimes n〉 \otimes 〈n \otimes n〉 d ω - K_{L / U} I \otimes {boldI}_{normald} \\ + \int_{0}^{2 π} H_{Iso} 〈n \otimes n〉 \otimes (I - {boldI}_{normald}) d ω + \int_{0}^{2 π} L 〈n \otimes n〉 \otimes (I - {boldI}_{normald}^{normalc}) d ω . \end{matrix}

…”

Section: Incremental Form Of the Strain Space Multiple Mechanism Modementioning

confidence: 99%

\overset{W}{true}

, in the constitutive model is computed through Equations , , and as

\overset{W}{false} = σ^{'} : \overset{ε}{false} = {\dot{W}}_{p} + \int_{0}^{π} ({}, F ((), ω - ω_{0}) {\dot{W}}_{q_{Iso}} + {\dot{W}}_{q_{Aniso}}) normald ω,

where

{\dot{W}}_{p} = - p^{'} \overset{ε}{false},

{\dot{W}}_{q_{Iso}} = q_{Iso} \overset{γ}{false},

{\dot{W}}_{q_{Aniso}} = q_{Aniso} \overset{γ}{false} = - \frac{1}{π} a_{1} cos ((), ω - ω_{0}) p^{'} \overset{γ}{false} .

Equation defines the fundamental relationship between the macroscopic strain‐energy rate

\overset{W}{true}

and the microscopic strain‐energy rates,

{\dot{W}}_{q_{Iso}}

and

{\dot{W}}_{q_{Aniso}}

Section: Strain Energy and The Dilative Component Of Dilatancymentioning

confidence: 99%

“…When the shear strain is very large such that γ xy = ∞, Equation with Equation leads to the macroscopic shear strength considering inherent anisotropy,

τ_{m}^{Aniso}

, as follows:

τ_{m}^{Aniso} = {τ_{xy}|}_{γ_{xy} = \infty} = \int_{0}^{π} ([], ({}, 1 + a_{2} cos 2 ((), ω - ω_{0})) q_{v} - \frac{1}{π} a_{1} cos ((), ω - ω_{0}) p^{'}) sin ω normald ω .

Here, the macroscopic shear strength without inherent anisotropy is given as

τ_{m}^{Iso} = q_{v} \int_{0}^{π} sin ω normald ω = 2 q_{v} .

Substitution of Equation into Equation yields

τ_{m}^{Aniso} = ((), 1 - \frac{1}{3} a_{2} cos 2 ω_{0}) τ_{m}^{Iso} - \frac{1}{2} a_{1} p^{'} sin ω_{0};

by normalizing Equation with the isotropic stress p ' , the internal friction angle with inherent anisotropy,

ϕ_{f}^{Aniso}

, can be related to that without inherent anisotropy,

ϕ_{f}^{Iso}

, using the anisotropy parameters (ie, a 1 , a 2 , and ω 0 ) as follows:

sin ϕ_{f}^{Aniso} = (()...)

…”

Section: Model Parametersmentioning

confidence: 99%

See 3 more Smart Citations

Constitutive modeling of inherent anisotropy in a strain space multiple mechanism model for granular materials

Ueda

Iai

2018

Num Anal Meth Geomechanics

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Reduced Plastic Dilatancy in Polymer Glasses

Peng

Deng

Utz

2020

Macro Theory & Simulations

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Amorphous solids in general exhibit a volume change during plastic deformation due to microstructure change during plastic relaxation. Here the deformation dilatancy of alkane polymer glasses upon shearing is investigated using molecular static simulations at zero temperature and pressure. The dilatancy of linear alkane chains has been quantified as a function of strain and chain length. It is found that the system densities decrease linearly with respect to strain after yield point. In addition, dilatability decreases considerably with increasing chain length, suggesting enhanced cooperation of different deformation mechanisms. An analytic model is introduced for dilatability based on the atomistic study. The entanglement chain length is predicted as 43 for alkane polymers from the model, agreeing well with experiments. The study provides insights of correlations of the physical properties and chain length of polymers which might be useful in material design and applications of structural polymers.

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Induced fabric under cyclic and rotational loads in a strain space multiple mechanism model for granular materials

Iai

Tobita

Ozutsumi³

2011

Num Anal Meth Geomechanics

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SUMMARYThe strain space multiple mechanism model idealizes the behavior of granular materials on the basis of a multitude of virtual simple shear mechanisms oriented in arbitrary directions. Within this modeling framework, the virtual simple shear stress is defined as a quantity dependent on the contact distribution function as well as the normal and tangential components of interparticle contact forces, which evolve independently during the loading process. In other terms, the virtual simple shear stress is an intermediate quantity in the upscaling process from the microscopic level (characterized by contact distribution and interparticle contact forces) to the macroscopic stress. The stress space fabric produces macroscopic stress through the tensorial average. Thus, the stress space fabric characterizes the fundamental and higher modes of anisotropy induced in granular materials. Herein, the induced fabric is associated with monotonic and cyclic loadings, loading with the rotation of the principal stress, and general loading. Upon loading with the rotation of the principal stress axis, some of the virtual simple shear mechanisms undergo loading whereas others undergo unloading. This process of fabric evolution is the primary cause of noncoaxiality between the axes of principal stresses and strains. Although cyclic behavior and behavior under the rotation of the principal stress axis seem to originate from two distinct mechanisms, the strain space multiple mechanism model demonstrates that these behaviors are closely related through the hysteretic damping factor.

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Dilatancy of granular materials in a strain space multiple mechanism model

Cited by 92 publications

References 38 publications

Constitutive modeling of inherent anisotropy in a strain space multiple mechanism model for granular materials

Constitutive modeling of inherent anisotropy in a strain space multiple mechanism model for granular materials

Reduced Plastic Dilatancy in Polymer Glasses

Induced fabric under cyclic and rotational loads in a strain space multiple mechanism model for granular materials

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