2.1.4. Incremental Equations

2.1.4. Incremental Equations

 *p p p = b(p - o ) + o s = b(s - o ) + o* , (10) The elastic volumetric strain increment and elastic shear strain increment can be given by:

$$d\epsilon\_v^e = \frac{dp}{K}, \, d\epsilon\_{ij}^e = \frac{ds\_{ij}}{2G} \tag{13}$$

*0 <sup>δ</sup> b = <sup>δ</sup> - <sup>δ</sup>* , (11) It is assumed that the bulk modulus and shear modulus can be obtained by:

$$\mathbf{K} = \frac{1+e\_0}{\kappa} p\_\prime \ \mathbf{G} = \frac{3\mathbf{K}(1-2\nu)}{2(1+\nu)}\tag{14}$$

*<sup>0</sup> δ* indicates the distance from the current stress point to the image stress point. The loading index is calculated by imposing the consistency condition to its corresponding bounding surface equation: ∂ ∂ *1F F <sup>Λ</sup> = L = ( dp + ds ) Kp s* , (12) in which, ν denotes Poisson's ratio. Here, an associated flow rule is used, i.e., the plastic strain increment vector is always normal to the yield surface, with and *L* coinciding. The plastic strain direction is defined by the boundary. The plastic constitutive relations can be defined as:

$$d\epsilon\_v^p = \Lambda \frac{\overline{\partial} \mathcal{F}}{\overline{\partial \overline{p}}} \,. \tag{15}$$

$$\mathrm{d}\boldsymbol{e}\_{\mathrm{ij}}^{\mathrm{p}} = \mathrm{L}\,\frac{\partial \mathcal{F}}{\partial \overline{\mathbf{s}}\_{\mathrm{ij}}} \, \mathrm{ \, \tag{16}$$

*2G* (13)

*2(1+ )* , (14)

The elastic volumetric strain increment and elastic shear strain increment can be given by:

*dp <sup>d</sup><sup>ε</sup> <sup>=</sup> <sup>K</sup>* , *ij <sup>e</sup> ij*

*de =*

*ds*

ν

ν

*e v*

It is assumed that the bulk modulus and shear modulus can be obtained by:

*<sup>0</sup> 1+e K= p*
