Johnson–Holmquist damage model

The JH-2 material assumes that the material is initially elastic and isotropic and can be described by a relation of the form (summation is implied over repeated indices)

${\displaystyle \sigma _{ij}=-p(\epsilon _{kk})~\delta _{ij}+2~\mu ~\epsilon _{ij}}$

where ${\displaystyle \sigma _{ij}}$ is a stress measure, ${\displaystyle p(\epsilon _{kk})}$ is an equation of state for the pressure, ${\displaystyle \delta _{ij}}$ is the Kronecker delta, ${\displaystyle \epsilon _{ij}}$ is a strain measure that is energy conjugate to ${\displaystyle \sigma _{ij}}$, and ${\displaystyle \mu }$ is a shear modulus. The quantity ${\displaystyle \epsilon _{kk}}$ is frequently replaced by the hydrostatic compression ${\displaystyle \xi }$ so that the equation of state is expressed as

${\displaystyle p(\xi )=p(\xi (\epsilon _{kk}))=p\left({\cfrac {\rho }{\rho _{0}}}-1\right)~;~~\xi$ :={\cfrac {\rho }{\rho _{0}}}-1}

where ${\displaystyle \rho }$ is the current mass density and ${\displaystyle \rho _{0}}$ is the initial mass density.

The stress at the Hugoniot elastic limit is assumed to be given by a relation of the form

${\displaystyle \sigma _{h}={\mathcal {H}}(\rho ,\mu )=p_{\rm {HEL}}(\rho )+{\cfrac {2}{3}}~\sigma _{\rm {HEL}}(\rho ,\mu )}$

where ${\displaystyle p_{\rm {HEL}}}$ is the pressure at the Hugoniot elastic limit and ${\displaystyle \sigma _{\rm {HEL}}}$ is the stress at the Hugoniot elastic limit.

The uniaxial failure strength of the intact material is assumed to be given by an equation of the form

${\displaystyle \sigma _{\rm {intact}}^{*}=A~(p^{*}+T^{*})^{n}~\left[1+C~\ln \left({\cfrac {d\epsilon _{p}}{dt}}\right)\right]}$

where ${\displaystyle A,C,n}$ are material constants, ${\displaystyle t}$ is the time, ${\displaystyle \epsilon _{p}}$ is the inelastic strain. The inelastic strain rate is usually normalized by a reference strain rate to remove the time dependence. The reference strain rate is generally 1/s.

The quantities ${\displaystyle \sigma ^{*}}$ and ${\displaystyle p^{*}}$ are normalized stresses and ${\displaystyle T^{*}}$ is a normalized tensile hydrostatic pressure, defined as

${\displaystyle \sigma ^{*}={\cfrac {\sigma }{\sigma _{\rm {HEL}}}}~;~p^{*}={\cfrac {p}{p_{\rm {HEL}}}}~;~~T^{*}={\cfrac {T}{p_{\rm {HEL}}}}}$

The uniaxial stress at complete fracture is assumed to be given by

${\displaystyle \sigma _{\rm {fracture}}^{*}=B~(p^{*})^{m}~\left[1+C~\ln \left({\cfrac {d\epsilon _{p}}{dt}}\right)\right]}$

where ${\displaystyle B,C,m}$ are material constants.

The uniaxial strength of the material at a given state of damage is then computed at a linear interpolation between the initial strength and the stress for complete failure, and is given by

${\displaystyle \sigma ^{*}=\sigma _{\rm {initial}}^{*}-D~\left(\sigma _{\rm {initial}}^{*}-\sigma _{\rm {fracture}}^{*}\right)}$

The quantity ${\displaystyle D}$ is a scalar variable that indicates damage accumulation.

The evolution of the damage variable ${\displaystyle D}$ is given by

${\displaystyle {\cfrac {dD}{dt}}={\cfrac {1}{\epsilon _{f}}}~{\cfrac {d\epsilon _{p}}{dt}}}$

where the strain to failure ${\displaystyle \epsilon _{f}}$ is assumed to be

${\displaystyle \epsilon _{f}=D_{1}~(p^{*}+T^{*})^{D_{2}}}$

where ${\displaystyle D_{1},D_{2}}$ are material constants.