Elastic-Plastic Constitutive Law

The most general form of elastic-perfectly plastic incremental stress-strain relationship can be expressed as [1],

$$d\sigma_{ij}=C_{ijkl}^ep d\epsilon_{kl} \;\;\;\;\;\; \;\;\;\;\;\; C_{ijkl}^ep = C_{ijkl} - \frac{C_{ijmn} \displaystyl... ...sigma_{rs}}}} C_{rstu} \displaystyle{\frac{\partial f}{\partial \sigma_{tu}}} }$$ (10)

The coefficient tensor $C_{ijkl}^ep$ is the elastic-plastic tensor of tangent moduli for an elastic-perfectly plastic material. The tensor of elastic moduli, $C_{ijkl}$, can be expressed in terms of $E$ and $\nu$ as [1],

$$C_{ijkl} = \frac{E}{2(1+\nu)}\left[\frac{2\nu}{(1-\nu)}\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right]$$ (11)

By using Hooke's law and additive decomposition of strain tensors in the elastic and plastic parts [8] we obtain

$$d\sigma_{ij}=C_{ijkl}d\epsilon_{kl}-d\lambda C_{ijkl}\frac{\partial f}{\partial \sigma_{kl}}$$ (12)

By substituting Eq. (11) into Eq. (12), and with $\nu=(1/2)(3K-2G)/(3K+G)$ it follows

$$d\sigma_{ij}=2Gd\epsilon_{ij}+Kd\epsilon_{kk}\delta_{ij}-d\lambda... ...ma_{mn}}\delta_{mn}\delta_{ij}+2G\frac{\partial f}{\partial \sigma_{ij}}\right]$$ (13)

Assuming a Drucker-Prager material model, the yield function, $f$, is given by, $f(\sigma_{ij})=F(I_1,\sqrt{J_2})-K=0$ and hence it follows that

$$\frac{\partial f}{\partial \sigma_{ij}}=\frac{\partial f}{\partia... ...}\delta_{ij}+\frac{1}{2\sqrt {J_2}}\frac{\partial f}{\partial \sqrt{J_2}}S_{ij}$$ (14)

Substituting Eq. (14) into Eq. (13) we have,

$$d\sigma_{ij}=2Gd\epsilon_{ij}+Kd\epsilon_{kk}\delta_{ij}-d\lambda... ...ta_{ij}+\frac{G}{\sqrt{J_2}}\frac{\partial f}{\partial \sqrt{J_2}}S_{ij}\right)$$ (15)

where $d\lambda$ has the form,

$$d\lambda=\frac{3Kd\epsilon_{kk}\left(\frac{\partial f}{\partial I... ...{\partial I_1}\right)^2+G\left(\frac{\partial f}{\partial \sqrt{J_2}}\right)^2}$$ (16)

Now, the Drucker-Prager yield condition,f, is given by

$$f=\sqrt{J_2}+\alpha I_1-k=0$$ (17)

Combining Eqs. (15), (16) & (17), we have the most general form of Drucker-Prager elastic-perfectly plastic constitutive relationship as,

$$d\sigma_{ij} = \left[2G\delta_{im}\delta_{jn}+\left(K-\frac{2}{3}G\right)\delta_{ij}\delta_{mn} \right. -$$

$$\left. \frac{\left(\frac{G}{\sqrt{J_2}}\right)S_{ij} + 3K\alpha\d... ...ft(\frac{G}{\sqrt{J_2}}S_{mn} + 3K\alpha\delta_{mn}\right)\right]d\epsilon_{mn}$$ (18)

This relationship will be used in assessing one dimensional wave propagation characteristics of tire shred material described below.