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solid mechanics Solid mechanics (also known as mechanics of solids) is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation (mechanics), deformation under the action of forces, temperature chang ...
, the Johnson–Holmquist damage model is used to model the mechanical behavior of damaged
brittle A material is brittle if, when subjected to stress, it fractures with little elastic deformation and without significant plastic deformation. Brittle materials absorb relatively little energy prior to fracture, even those of high strength. ...
materials, such as
ceramics A ceramic is any of the various hard, brittle, heat-resistant, and corrosion-resistant materials made by shaping and then firing an inorganic, nonmetallic material, such as clay, at a high temperature. Common examples are earthenware, porce ...
, rocks, and
concrete Concrete is a composite material composed of aggregate bound together with a fluid cement that cures to a solid over time. It is the second-most-used substance (after water), the most–widely used building material, and the most-manufactur ...
, over a range of strain rates. Such materials usually have high
compressive strength In mechanics, compressive strength (or compression strength) is the capacity of a material or Structural system, structure to withstand Structural load, loads tending to reduce size (Compression (physics), compression). It is opposed to ''tensil ...
but low tensile strength and tend to exhibit progressive damage under load due to the growth of microfractures. There are two variations of the Johnson-Holmquist model that are used to model the impact performance of ceramics under ballistically delivered loads.Walker, James D. ''Turning Bullets into Baseballs'', SwRI Technology Today, Spring 1998 http://www.swri.edu/3pubs/ttoday/spring98/bullet.htm 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.Johnson, G. R. and Holmquist, T. J., 1992, ''A computational constitutive model for brittle materials subjected to large strains'', Shock-wave and High Strain-rate Phenomena in Materials, ed. M. A. Meyers, L. E. Murr and K. P. Staudhammer, Marcel Dekker Inc., New York, pp. 1075-1081. 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) modelJohnson, G. R. and Holmquist, T. J., 1994, ''An improved computational constitutive model for brittle materials'', High-Pressure Science and Technology, American Institute of Physics. 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) : \sigma_ = -p(\epsilon_)~\delta_ + 2~\mu~\epsilon_ where \sigma_ is a stress measure, p(\epsilon_) is an
equation of state In physics and chemistry, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal energy. Most mo ...
for the pressure, \delta_ is the
Kronecker delta In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise: \delta_ = \begin 0 &\text i \neq j, \\ 1 &\ ...
, \epsilon_ is a strain measure that is energy conjugate to \sigma_, and \mu is a
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the Elasticity (physics), elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear s ...
. The quantity \epsilon_ is frequently replaced by the hydrostatic compression \xi so that the equation of state is expressed as : p(\xi) = p(\xi(\epsilon_)) = p\left(\cfrac-1\right) ~;~~ \xi := \cfrac-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(\rho, \mu) = p_(\rho) + \cfrac~\sigma_(\rho, \mu) where p_ is the pressure at the Hugoniot elastic limit and \sigma_ 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^_ = A~(p^* + T^*)^n~\left + C~\ln\left(\cfrac\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 ~;~ p^* = \cfrac ~;~~ T^* = \cfrac


Stress at complete fracture

The uniaxial stress at complete fracture is assumed to be given by : \sigma^_ = B~(p^*)^m~\left + C~\ln\left(\cfrac\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 In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known po ...
between the initial strength and the stress for complete failure, and is given by : \sigma^ = \sigma^_ - D~\left(\sigma^_ - \sigma^_\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 = \cfrac~\cfrac where the strain to failure \epsilon_f is assumed to be : \epsilon_f = D_1~(p^* + T^*)^ where D_1, D_2 are material constants.


Material parameters for some ceramics


Johnson–Holmquist equation of state

The function p(\xi) used in the Johnson–Holmquist material model is often called the Johnson–Holmquist equation of state and has the form : p(\xi) = \begin k_1~\xi + k_2~\xi^2 + k_3~\xi^3 + \Delta p & \qquad \text \\ k_1~\xi & \qquad \text \end where \Delta p is an increment in the pressure and k_1, k_2, k_3 are material constants. The increment in pressure arises from the conversion of energy loss due to damage into
internal energy The internal energy of a thermodynamic system is the energy of the system as a state function, measured as the quantity of energy necessary to bring the system from its standard internal state to its present internal state of interest, accoun ...
. Frictional effects are neglected.


Implementation in LS-DYNA

The Johnson-Holmquist material model is implemented in
LS-DYNA LS-DYNA is an advanced general-purpose multiphysics simulation software package developed by the former Livermore Software Technology Corporation (LSTC), which was acquired by Ansys in 2019. While the package continues to contain more and more p ...
as * MAT_JOHNSON_HOLMQUIST_CERAMICS.McIntosh, G., 1998, ''The Johnson-Holmquist ceramic model as used in the ls-DYNA2D'', Report # DREV-TM-9822:19981216029, Research and Development Branch, Department of National Defence, Canada, Valcartier, Quebec. http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADA357607&Location=U2&doc=GetTRDoc.pdf


Implementation in the IMPETUS Afea Solver

The Johnson-Holmquist material model is implemented in the IMPETUS Afea Solver as * MAT_JH_CERAMIC.


Implementation in Altair Radioss an
OpenRadioss

The Johnson-Holmquist material model is implemented in Radioss Solver a
/MAT/LAW79 (JOHN_HOLM)


Implementation in Abaqus

The Johnson-Holmquist (JH-2) material model is implemented in Abaqus a
ABQ_JH2 material name


References


See also

*
Failure Failure is the social concept of not meeting a desirable or intended objective, and is usually viewed as the opposite of success. The criteria for failure depends on context, and may be relative to a particular observer or belief system. On ...
* Material failure theory {{DEFAULTSORT:Johnson-Holmquist Damage Model Solid mechanics Equations