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Fabric Tensors

Type of Fabric Tensors

The following fabric tensors are supported in the fabric parameter in the type parameter in the Model.setAnisotropicParameters method and Model.setMaterialAnisotropicParameters method.

Name Equation Parameters References
chang1990 \(\mathbf{F}_{ij}=\begin{bmatrix} a_{20} & 0 & 0 \\ 0 & -\dfrac{a_{20}}{2}+3a_{22} & 3b_{22} \\ 0 & 3b_{22} & -\dfrac{a_{20}}{2}-3a_{22} \end{bmatrix}\) \(a_{20},a_{22},b_{22}\) (a20,a22,b22) Chang and Misra (1990) 1
chang1990-ext \(\mathbf{F}_{ij}=\begin{bmatrix} a_{20} & -\dfrac{3}{2}a_{21} & 6b_{21} \\ -\dfrac{3}{2}a_{21} & -\dfrac{a_{20}}{2}+3a_{22} & 3b_{22} \\ 6b_{21} & 3b_{22} & -\dfrac{a_{20}}{2}-3a_{22} \end{bmatrix}\) \(a_{20},a_{22},b_{22},a_{21},b_{21}\) (a20,a22,b22,a21,b21)
general \(\mathbf{F}_{ij}=\begin{bmatrix} F_{11} & F_{12} & F_{13} \\ F_{12} & F_{22} & F_{23} \\ F_{13} & F_{23} & -F_{11}-F_{22} \end{bmatrix}\) \(F_{11},F_{22},F_{12},F_{13},F_{23}\) (F11,F22,F12,F13,F23)
general-cross-anisotropic \(\mathbf{F}_{ij}(\text{major_axes}=3)=\begin{bmatrix} -F/2 & Fhh & Fvh \\ Fhh & -F/2 & Fvh \\ Fvh & Fvh & F \end{bmatrix}\) \(F,F_{hh},F_{vh}\) and major material axes (1/2/3) (F,Fhh,Fvh,major_axes)

Note

You can specify r1, r2, and r3 (in degrees) to rotate the fabric tensor along the x1, x2, and x3 axes, respectively.

Fabric Evolution

The following fabric evolution models are supported in the evolution parameter in the Model.setAnisotropicParameters method and Model.setMaterialAnisotropicParameters method.

Name Equation Parameters References
wan2004 \(\dot{\mathbf{F}}=\chi\dot{\boldsymbol{\eta}}=\chi\left(\dfrac{\dot{\mathbf{s}}}{p}-\dfrac{\dot{p}}{p^2}\mathbf{s}\right)\) \(\chi\) (chi) Wan and Guo (2004) 2
sun2011 \(\dot{\mathbf{F}}=s_1\dot{\boldsymbol{\epsilon}}+s_2\|\dot{\boldsymbol{\epsilon}}\|\mathbf{F}+s_3\left(\mathbf{F}:\dot{\boldsymbol{\epsilon}}_q\right)\mathbf{F}\) \(s_1,s_2,s_3\) (s1,s2,s3) Sun and Sundaresan (2011) 3
li2012 \(\dot{\mathbf{F}}=c(\mathbf{n}_l-r\mathbf{F})\langle\lambda\rangle\) \(c,r\) (c,r) Li and Dafalias (2012) 4
yang2014 \(\dot{\mathbf{F}}=c_1(1+c_2\Vert\boldsymbol{\eta}\Vert)\dot{\boldsymbol{\eta}}\) \(c_1,c_2\) (c1,c2) Yang (2014) 5
hu2015 \(\dot{\mathbf{F}}=c_1(1+c_2\Vert\boldsymbol{\eta}\Vert)\dot{\boldsymbol{\eta}}+c_3(c_4\boldsymbol{\eta}-\mathbf{F})\Vert\dot{\mathbf{\epsilon}}_q\Vert\) \(c_1,c_2,c_3,c_4\) (c1,c2,c3,c4) Hu (2015) 6
zhao2020 \(\begin{aligned} &\dot{\mathbf{F}}=h\left\{F_c-\mathbf{F}:\dfrac{\dot{\boldsymbol{\epsilon}}_q^p}{\Vert\dot{\boldsymbol{\epsilon}}_q^p\Vert}\exp\left[m(M-\eta)+n\psi\right]\right\}\dot{\boldsymbol{\epsilon}}_q^p\\ &M=g_s(\theta_s)M_c,F_c=g_F(\theta_F)F_{c0},\psi=e-e_c(p)\\ &g_s(\theta)=\dfrac{2c}{(1+c)-(1-c)\cos(3\theta)},c=M_e/M_c\\ &g_F(\theta)=\dfrac{1}{g_s(\theta)} \end{aligned}\) \(M_c,M_e,F_{c0},h,m\) (Mc,Me,Fc0,h,m,n) Zhao and Kruyt (2020) 7
wang2020 \(\dot{\mathbf{F}}=\vert\varepsilon_q\vert c_c\left[\dfrac{\exp\left[-\beta(1+A_p)\right]^m}{1+c_{cd}D}\mathbf{n}-\mathbf{F}\right]\) \(\beta,m,c_c,c_{cd}\) (beta,m,cc,ccd) Wang et al. (2020) 8
wang2024 \(\dot{\mathbf{F}}=c_1(1+c_2\Vert\dot{\boldsymbol{\eta}}\Vert)\dot{\boldsymbol{\eta}}\) \(c_1,c_2\) (c1,c2) This work

Note

You can use the + operator to combine multiple fabric evolution models as long as there are no conflicting parameters. For example, wan2004+sun2011 will combine the wan2004 and sun2011 fabric evolution models, in which the fabric tensor increment is calculated as the sum of the fabric tensor increments calculated by both models.

References

  1. Ching S. Chang and Anil Misra. Packing Structure and Mechanical Properties of Granulates. Journal of Engineering Mechanics, 116(5):1077–1093, May 1990. doi:10.1061/(ASCE)0733-9399(1990)116:5(1077). ↩

  2. Richard G. Wan and Pei J. Guo. Stress Dilatancy and Fabric Dependencies on Sand Behavior. Journal of Engineering Mechanics, 130(6):635–645, June 2004. doi:10.1061/(ASCE)0733-9399(2004)130:6(635). ↩

  3. Jin Sun and Sankaran Sundaresan. A constitutive model with microstructure evolution for flow of rate-independent granular materials. Journal of Fluid Mechanics, 682:590–616, September 2011. doi:10.1017/jfm.2011.251. ↩

  4. Xiang Song Li and Yannis F. Dafalias. Anisotropic Critical State Theory: Role of Fabric. Journal of Engineering Mechanics, 138(3):263–275, March 2012. doi:10.1061/(ASCE)EM.1943-7889.0000324. ↩

  5. Dunshun Yang. Microscopic Study of Granular Material Behaviours under General Stress Paths. PhD thesis, University of Nottingham, 2014. URL: http://eprints.nottingham.ac.uk/14251/. ↩

  6. Nian Hu. On Fabric Tensor-Based Constitutive Modelling of Granular Materials: Theory and Numerical Implementation. PhD thesis, University of Nottingham, 2015. ↩

  7. Chao-Fa Zhao and Niels P. Kruyt. An evolution law for fabric anisotropy and its application in micromechanical modelling of granular materials. International Journal of Solids and Structures, 196-197:53–66, July 2020. doi:10.1016/j.ijsolstr.2020.04.007. ↩

  8. Rui Wang, Yannis F. Dafalias, Pengcheng Fu, and Jian-Min Zhang. Fabric evolution and dilatancy within anisotropic critical state theory guided and validated by DEM. International Journal of Solids and Structures, 188-189:210–222, April 2020. doi:10.1016/j.ijsolstr.2019.10.013. ↩