Johnson–Holmquist damage model
Encyclopedia
In solid mechanics
, the Johnson–Holmquist damage model is used to model the mechanical behavior of damaged brittle
materials, such as ceramics, rock
s, and concrete
, over a range of strain rates. Such materials usually have high compressive strength but low tensile strength and tend to exhibit progressive damage under load due to the growth of microcracks.
There are two variations of the Johnson-Holmquist model that are used to model the impact performance of ceramics under ballistic
ally delivered loads . These models were developed by Gordon R. Johnson and Timothy J. Holmquist in the 1990s with the aim of facilitating predictive numerical simulations of ballistic armor penetration. The first version of the model is called the 1992 Johnson-Holmquist 1 (JH-1) model. This original version was developed to account for large deformations but did not take into consideration progressive damage with increasing deformation; though the multi-segment stress-strain curves in the model can be interpreted as incorporating damage implicitly. The second version, developed in 1994, incorporated a damage evolution rule and is called the Johnson-Holmquist 2 (JH-2) model or, more accurately, the Johnson-Holmquist damage material model.
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.
where
is a stress measure
,
is an equation of state
for the pressure,
is the Kronecker delta, 
is a strain measure that is energy conjugate to 
, and 
is a shear modulus. The quantity 
is frequently replaced by the hydrostatic compression 
so that the equation of state is expressed as
where
is the current mass density ans 
is the initial mass density.
The stress at the Hugoniot elastic limit is assumed to be given by a relation of the form
where
is the pressure at the Hugoniot elastic limit and 
is the stress at the Hugoniot elastic limit.
where
are material constants, 
is the time, 
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
and 
are normalized stresses and 
is a normalized tensile strength, defined as
where
are material constants.
The quantity
is a scalar variable that indicates damage accumulation.
is given by
where the strain to failure
is assumed to be
where
are material constants.
used in the Johnson–Holmquist material model is often called the Johnson–Holmquist equation of state and has the form
where
is an increment in the pressure and 
are material constants. The increment in pressure arises from the conversion of energy loss due to damage into internal energy. Frictional effects are neglected.
as * MAT_JOHNSON_HOLMQUIST_CERAMICS.
Solid mechanics
Solid mechanics is the branch of mechanics, physics, and mathematics that concerns the behavior of solid matter under external actions . It is part of a broader study known as continuum mechanics. One of the most common practical applications of solid mechanics is the Euler-Bernoulli beam equation...
, the Johnson–Holmquist damage model is used to model the mechanical behavior of damaged brittle
Brittle
A material is brittle if, when subjected to stress, it breaks without significant deformation . Brittle materials absorb relatively little energy prior to fracture, even those of high strength. Breaking is often accompanied by a snapping sound. Brittle materials include most ceramics and glasses ...
materials, such as ceramics, rock
Rock (geology)
In geology, rock or stone is a naturally occurring solid aggregate of minerals and/or mineraloids.The Earth's outer solid layer, the lithosphere, is made of rock. In general rocks are of three types, namely, igneous, sedimentary, and metamorphic...
s, and concrete
Concrete
Concrete is a composite construction material, composed of cement and other cementitious materials such as fly ash and slag cement, aggregate , water and chemical admixtures.The word concrete comes from the Latin word...
, over a range of strain rates. Such materials usually have high compressive strength but low tensile strength and tend to exhibit progressive damage under load due to the growth of microcracks.
There are two variations of the Johnson-Holmquist model that are used to model the impact performance of ceramics under ballistic
Terminal ballistics
Terminal ballistics, a sub-field of ballistics, is the study of the behavior of a projectile when it hits its target. It is often referred to as stopping power when dealing with human or other living targets. Terminal ballistics is relevant both for small caliber projectiles as well as for large...
ally delivered loads . These models were developed by Gordon R. Johnson and Timothy J. Holmquist in the 1990s with the aim of facilitating predictive numerical simulations of ballistic armor penetration. The first version of the model is called the 1992 Johnson-Holmquist 1 (JH-1) model. This original version was developed to account for large deformations but did not take into consideration progressive damage with increasing deformation; though the multi-segment stress-strain curves in the model can be interpreted as incorporating damage implicitly. The second version, developed in 1994, incorporated a damage evolution rule and is called the Johnson-Holmquist 2 (JH-2) model or, more accurately, the Johnson-Holmquist damage material model.
Johnson-Holmquist 2 (JH-2) material model
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)where
is a stress measureStress measures
The most commonly used measure of stress is the Cauchy stress. However, several other measures of stress can be defined. Some such stress measures that are widely used in continuum mechanics, particularly in the computational context, are:...
,
is an equation of stateEquation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
for the pressure,
is the Kronecker delta, is a strain measure that is energy conjugate to , and is a shear modulus. The quantity is frequently replaced by the hydrostatic compression so that the equation of state is expressed aswhere
is the current mass density ans is the initial mass density.The stress at the Hugoniot elastic limit is assumed to be given by a relation of the form
where
is the pressure at the Hugoniot elastic limit and 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 formwhere
are material constants, is the time, 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
and are normalized stresses and is a normalized tensile strength, defined asStress at complete fracture
The uniaxial stress at complete fracture is assumed to be given bywhere
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 byThe quantity
is a scalar variable that indicates damage accumulation.Damage evolution rule
The evolution of the damage variable is given bywhere the strain to failure
is assumed to bewhere
are material constants.Material parameters for some ceramics
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Johnson–Holmquist equation of state
The function used in the Johnson–Holmquist material model is often called the Johnson–Holmquist equation of state and has the formwhere
is an increment in the pressure and are material constants. The increment in pressure arises from the conversion of energy loss due to damage into internal energy. Frictional effects are neglected.Implementation in LS-DYNA
The Johnson-Holmquist material model is implemented in LS-DYNALS-DYNA
LS-DYNA is an advanced general-purpose multiphysics simulation software package developed by the Livermore Software Technology Corporation...
as * MAT_JOHNSON_HOLMQUIST_CERAMICS.
See also
- FailureFailureFailure refers to the state or condition of not meeting a desirable or intended objective, and may be viewed as the opposite of success. Product failure ranges from failure to sell the product to fracture of the product, in the worst cases leading to personal injury, the province of forensic...
- Failure theory (material)Failure theory (material)Failure theory is the science of predicting the conditions under which solid materials fail under the action of external loads. The failure of a material is usually classified into brittle failure or ductile failure . Depending on the conditions most materials can fail in a brittle or ductile...