/MAT/LAW126 (JOHNSON_HOLMQUIST_CONCRETE)
Block Format Keyword This material law describes the behavior of brittle materials, more specifically dedicated to concrete.
Format
(1)  (2)  (3)  (4)  (5)  (6)  (7)  (8)  (9)  (10) 

/MAT/LAW126/mat_ID/unit_ID or /MAT/JOHNSON_HOLMQUIST_CONCRETE/mat_ID/unit_ID  
mat_title  
${\rho}_{i}$  
G  
a  b  n  ${f}_{c}$  T  
c  ${\dot{\epsilon}}_{0}$  FCUT  ${\sigma}_{MAX}^{*}$  ${\epsilon}_{f}^{\mathrm{min}}$  
P_{C}  ${\mu}_{C}$  P_{L}  ${\mu}_{L}$  
K_{1}  K_{2}  K_{3}  
D_{1}  D_{2}  IDEL  ${\epsilon}_{p}^{max}$ 
Definition
Field  Contents  SI Unit Example 

mat_ID  Material identifier. (Integer, maximum 10 digits) 

unit_ID  (Optional) Unit
identifier. (Integer, maximum 10 digits) 

mat_title  Material title. (Character, maximum 100 characters) 

${\rho}_{i}$  Initial
density. (Real) 
$\left[\frac{\text{kg}}{{\text{m}}^{\text{3}}}\right]$ 
G  Shear modulus. (Real) 
$\left[\text{Pa}\right]$ 
A  Normalized cohesive
strength. (Real) 

B  Normalized pressure hardening
modulus. (Real) 

N  Pressure hardening
exponent. (Real) 

${f}_{c}$  Quasistatic uniaxial compressive
strength. (Real) 
$\left[\text{Pa}\right]$ 
T  Maximum hydrostatic tensile
pressure. (Real) 
$\left[\text{Pa}\right]$ 
C  Strain rate coefficient.
(Real) 

${\dot{\epsilon}}_{0}$  Reference strain rate. Default = 1.0 (Real) 
$\left[\frac{\text{1}}{\text{s}}\right]$ 
FCUT  Cutoff frequency for strain rate filtering.
(Real) 
$\text{[Hz]}$ 
${\sigma}_{MAX}^{*}$  Maximum normalized
strength. Default = 10^{20} (Real) 

${\epsilon}_{f}^{\mathrm{min}}$  Minimum fracture strain. Default = 10^{20} (Real) 

P_{C}  Crushing
pressure. (Real) 
$\left[\text{Pa}\right]$ 
${\mu}_{C}$  Crushing volumetric
strain. (Real) 

P_{L}  Locking
pressure. (Real) 
$\left[\text{Pa}\right]$ 
${\mu}_{L}$  Locking plastic volumetric
strain. (Real) 

K_{1}  Linear bulk
stiffness. (Real) 
$\left[\text{Pa}\right]$ 
K_{2}  Quadratic bulk
stiffness. (Real) 
$\left[\text{Pa}\right]$ 
K_{3}  Cubic bulk
stiffness. (Real) 
$\left[\text{Pa}\right]$ 
D_{1}  Damage
parameter. (Real) 

D_{2}  Damage
exponent. (Real) 

IDEL  Element deletion flag:
(Integer) 

${\epsilon}_{p}^{max}$  Critical plastic strain for element
deletion. Default = 10^{20 } (Real) 
Example
#RADIOSS STARTER
#12345678910
/UNIT/1
Unit for material
Mg mm s
#12345678910
/MAT/LAW126/1/1
Concrete
# Init. dens.
2.440E9
# G
14860
# A B N FC T
0.79 1.60 0.61 48 4
# C EPS0 FCUT SFMAX EFMIN
0.007 1.0 10000 7 0.01
# PC MUC PL MUL
16 0.001 800 0.1
# K1 K2 K3
85000 171000 208000
# D1 D2 IDEL EPS_MAX
0.04 1.0 3 0
#12345678910
#ENDDATA
/END
#12345678910
Comments
 This material law is based on the theory of JohnsonHolmquistCook model theory (also called JohnsonHolmquistConcrete). It was proposed and designed for a concrete application. In this model, the spherical and deviatoric behavior are separated. It considers the effect of damage and strain rate sensitivity.
 The spherical behavior is described with a constitutive equation
based on hydrostatic pressure (considered positive in compression). This
behavior is divided into 3 regions (Figure 1) in the evolution hydrostatic pressure versus
volumetric strain denoted
$\mu $
.
$\begin{array}{l}P=\left\{\begin{array}{cccc}{K}_{0}\mu & \text{if}& P\le {P}_{C}& [\text{I}]\\ \left({K}_{0}+({K}_{1}{K}_{0})\frac{{\mu}_{p}}{{\mu}_{L}}\right)\left(\mu {\mu}_{p}\right)& \text{if}& P>{P}_{C}\text{and}{\mu}_{p}\le {\mu}_{L}& [\text{II}]\\ {K}_{1}\widehat{\mu}+{K}_{2}{\widehat{\mu}}^{2}+{K}_{3}{\widehat{\mu}}^{3}& & \text{fortheothercases}& [\text{III}]\end{array}\right.\\ \text{with}\\ {K}_{0}=\frac{{P}_{C}}{{\mu}_{C}}\\ \widehat{\mu}=\frac{\mu {\mu}_{L}}{1+{\mu}_{L}}\\ \mu =\frac{\rho}{{\rho}_{0}}1\end{array}$
In the first region, the pressure response is supposed linear and elastic. In the second region, the microcavities of the material are supposed to be crushed, generating a plastic volumetric strain denoted ${\mu}_{p}$ , modifying linearly the bulk modulus from ${K}_{0}$ and ${K}_{1}$ . When ${\mu}_{p}={\mu}_{L}$ all the cavities have been crushed and the material becomes fully dense. Then, the pressure evolution follows a polynomial equation of state.  The deviatoric behavior is defined with an elastoplastic behavior,
where the normalized yield stress is both a yielding and a damaging limit.
Its expression is:
 If
${P}^{*}>0$
(compressive
loadings):$${\sigma}_{Y}^{*}=\mathrm{min}\left({\sigma}_{MAX}^{*},\left(A\left(1D\right)+B{\left({P}^{*}\right)}^{N}\right)\left(1+C{\u2329\mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\u232a}_{+}\right)\right)$$
Where, ${P}^{*}=\frac{P}{{f}_{c}}$ bounded by ${P}^{*}=(1D){T}^{*}$ .
 If
${P}^{*}\le 0$
(tensile loadings):$${\sigma}_{Y}^{*}=A\left(1+\frac{P}{T}\right)\left(1D\right)\left(1+C{\u2329\mathrm{ln}\frac{\dot{\epsilon}}{{\dot{\epsilon}}_{0}}\u232a}_{+}\right)$$To get the yield stress, the normalized value is multiplied by ${f}_{c}$ . These two yield stresses shapes (for compressive and tension loadings) are plotted for a damage value of 0 (initial material) and 1 (fully fractured material) in Figure 2:
To trigger the deviatoric elastoplastic behavior, the normalized yield stress is compared to the current normalized equivalent von Mises stress:
$${\sigma}_{VM}^{*}=\frac{{\sigma}_{VM}}{{f}_{c}}$$This allows to compute the evolution of the deviatoric plastic strain denoted ${\epsilon}_{p}$ .
 If
${P}^{*}>0$
(compressive
loadings):
 The damage variable evolution is dependent to both volumetric and
deviatoric plastic strain. Its expression is given by:$$D={\displaystyle \sum \frac{\text{\Delta}{\mu}_{p}+\text{\Delta}{\epsilon}_{p}}{{\epsilon}_{f}^{p}}}$$
Where, the effective strain at failure is defined by:
$\begin{array}{l}{\epsilon}_{f}^{p}=\mathrm{max}\left({D}_{1}{\left({P}^{*}+{T}^{*}\right)}^{{D}_{2}},{\epsilon}_{f}^{\mathrm{min}}\right)\\ \text{with}{P}^{*}=\raisebox{1ex}{$P$}\!\left/ \!\raisebox{1ex}{${f}_{c}$}\right.\text{and}{T}^{*}=\raisebox{1ex}{$T$}\!\left/ \!\raisebox{1ex}{${f}_{c}$}\right.\end{array}$
 Time history and animation output is available using these USRi variables.
 USR1: Plastic volumetric strain ${\mu}_{p}$
 USR2: Bulking pressure $P$
 USR3: Volumetric strain $\mu $
 USR4: Yield stress ${\sigma}_{Y}$
 Strain rate filtering can be used and activated when a cutoff frequency FCUT for filtering is defined.
 The damage variable can be plotted in ANIM and H3D file using the output option DAMG.
 To avoid damage mesh dependency, due to mesh size or orientation,
the nonlocal regularization method can be used
(/NONLOCAL/MAT). In this case, the sum of the
deviatoric plastic strain
${\epsilon}_{p}$
and volumetric plastic strain
${\mu}_{p}$
is regularized and used for damage
evolution:$$D={\displaystyle \sum \frac{{\left(\text{\Delta}{\mu}_{p}+\text{\Delta}{\epsilon}_{p}\right)}_{nl}}{{\epsilon}_{f}^{p}}}$$
The regularized sum can be plotted using /ANIM/ELEM/NL_EPSP or /H3D/ELEM/NL_EPSP.