Isotropic Kelvin Material

As proposed by Kelvin [24] material strain $D\hackscore{ij}=\frac{1}{2}(v\hackscore{i,j}+v\hackscore{j,i})$ can be decomposed into an elastic part $D\hackscore{ij}^{el}$ and visco-plastic part $D\hackscore{ij}^{vp}$ :

$$D\hackscore{ij}=D\hackscore{ij}^{el}+D\hackscore{ij}^{vp}$$ (156)

with the elastic strain given as

$$D\hackscore{ij}^{el'}=\frac{1}{2 \mu} \dot{\sigma}\hackscore{ij}'$$ (157)

where $\sigma'\hackscore{ij}$ is the deviatoric stress (Notice that $\sigma'\hackscore{ii}=0$ ). If the material is composed by materials $q$ the visco-plastic strain can be decomposed as

$$D\hackscore{ij}^{vp'}=\sum\hackscore{q} D\hackscore{ij}^{q'}$$ (158)

where $D\hackscore{ij}^{q}$ is the strain in material $q$ given as

$$D\hackscore{ij}^{q'}=\frac{1}{2 \eta^{q}} \sigma'\hackscore{ij}$$ (159)

where $\eta^{q}$ is the viscosity of material $q$ . We assume the following betwee the the strain in material $q$

$$\eta^{q}=\eta^{q}\hackscore{N} \left(\frac{\tau}{\tau\hackscore{t}^q}\right)^{1-n^{q}} \tau=\sqrt{\frac{1}{2}\sigma'\hackscore{ij} \sigma'\hackscore{ij}}$$ (160)

for a given power law coefficients $n^{q}\ge1$ and transition stresses $\tau\hackscore{t}^q$ , see [24]. Notice that $n^{q}=1$ gives a constant viscosity. After inserting equation 6.59 into equation 6.58 one gets:

$$D\hackscore{ij}'^{vp}=\frac{1}{2 \eta^{vp}} \sigma'\hackscore{ij} \frac{1}{\eta^{vp}} = \sum\hackscore{q} \frac{1}{\eta^{q}} \;.$$ (161)

and finally with 6.56

$$D\hackscore{ij}'=\frac{1}{2 \eta^{vp}} \sigma'\hackscore{ij}+\frac{1}{2 \mu} \dot{\sigma}\hackscore{ij}'$$ (162)

The total stress $\tau$ needs to fullfill the yield condition

$$\tau \le \tau\hackscore{Y} + \beta \; p$$ (163)

with the Drucker-Prager cohesion factor $\tau\hackscore{Y}$ , Drucker-Prager friction $\beta$ and total pressure $p$ . The deviatoric stress needs to fullfill the equilibrion equation

$$-\sigma'\hackscore{ij,j}+p\hackscore{,i}=F\hackscore{i}$$ (164)

where $F\hackscore{j}$ is a given external fource. We assume an incompressible media:

$$-v\hackscore{i,i}=0$$ (165)

Natural boundary conditions are taken in the form

$$\sigma'\hackscore{ij}n\hackscore{j}-n\hackscore{i}p=f$$ (166)

which can be overwritten by a constraint

$$v\hackscore{i}(x)=0$$ (167)

where the index $i$ may depend on the location $x$ on the bondary.