The Johnson-Holmquist material model (JH-2), with damage, is useful when modeling brittle materials, such as ceramics, subjected to large pressures, shear strain and high strain rates. The model attempts to include the phenomena encountered when brittle materials are subjected to load and damage, and is one of the most widely used models when dealing with ballistic impact on ceramics. The model simulates the increase in strength shown by ceramics subjected to hydrostatic pressure as well as the reduction in strength shown by damaged ceramics. This is done by basing the model on two sets of curves that plot the yield stress against the pressure. The first set of curves accounts for the intact material, while the second one accounts for the failed material. Each curve set depends on the plastic strain and plastic strain rate. A damage variable D accounts for the level of fracture.
Intact elastic behavior 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) : \sigma_{ij} = -p(\epsilon_{kk})~\delta_{ij} + 2~\mu~\epsilon_{ij} where \sigma_{ij} is a
stress measure, p(\epsilon_{kk}) is an
equation of state for the pressure, \delta_{ij} is the
Kronecker delta, \epsilon_{ij} is a
strain measure that is energy conjugate to \sigma_{ij}, and \mu is a
shear modulus. The quantity \epsilon_{kk} is frequently replaced by the hydrostatic compression \xi so that the equation of state is expressed as : p(\xi) = p(\xi(\epsilon_{kk})) = p\left(\cfrac{\rho}{\rho_0}-1\right) ~;~~ \xi := \cfrac{\rho}{\rho_0}-1 where \rho is the current mass density and \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 : \sigma_h = \mathcal{H}(\rho, \mu) = p_{\rm HEL}(\rho) + \cfrac{2}{3}~\sigma_{\rm HEL}(\rho, \mu) where p_{\rm HEL} is the pressure at the Hugoniot elastic limit and \sigma_{\rm HEL} is the stress at the Hugoniot elastic limit.
Intact material strength The uniaxial failure strength of the intact material is assumed to be given by an equation of the form : \sigma^{*}_{\rm intact} = A~(p^* + T^*)^n~\left[1 + C~\ln\left(\cfrac{d\epsilon_p}{dt}\right)\right] where A, C, n are material constants, t is the time, \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 \sigma^{*} and p^* are normalized stresses and T^* is a normalized tensile hydrostatic pressure, defined as : \sigma^* = \cfrac{\sigma}{\sigma_{\rm HEL}} ~;~ p^* = \cfrac{p}{p_{\rm HEL}} ~;~~ T^* = \cfrac{T}{p_{\rm HEL}}
Stress at complete fracture The uniaxial stress at complete fracture is assumed to be given by : \sigma^{*}_{\rm fracture} = B~(p^*)^m~\left[1 + C~\ln\left(\cfrac{d\epsilon_p}{dt}\right)\right] where B, C, m are material constants.
Current material strength 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 : \sigma^{*} = \sigma^{*}_{\rm initial} - D~\left(\sigma^{*}_{\rm initial} - \sigma^{*}_{\rm fracture}\right) The quantity D is a scalar variable that indicates damage accumulation.
Damage evolution rule The evolution of the damage variable D is given by : \cfrac{dD}{dt} = \cfrac{1}{\epsilon_f}~\cfrac{d\epsilon_p}{dt} where the strain to failure \epsilon_f is assumed to be : \epsilon_f = D_1~(p^* + T^*)^{D_2} where D_1, D_2 are material constants.
Material parameters for some ceramics == Johnson–Holmquist equation of state ==