/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 | |||||||||
ρi | |||||||||
G | |||||||||
a | b | n | fc | T | |||||
c | ˙ε0 | FCUT | σ*MAX | εminf | |||||
PC | μC | PL | μL | ||||||
K1 | K2 | K3 | |||||||
D1 | D2 | IDEL | εmaxp |
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) |
|
ρi | Initial
density. (Real) |
[kgm3] |
G | Shear modulus. (Real) |
[Pa] |
A | Normalized cohesive
strength. (Real) |
|
B | Normalized pressure hardening
modulus. (Real) |
|
N | Pressure hardening
exponent. (Real) |
|
fc | Quasi-static uniaxial compressive
strength. (Real) |
[Pa] |
T | Maximum hydrostatic tensile
pressure. (Real) |
[Pa] |
C | Strain rate coefficient.
(Real) |
|
˙ε0 | Reference strain rate. Default = 1.0 (Real) |
[1s] |
FCUT | Cutoff frequency for strain rate filtering.
(Real) |
[Hz] |
σ*MAX | Maximum normalized
strength. Default = 1020 (Real) |
|
εminf | Minimum fracture strain. Default = 10-20 (Real) |
|
PC | Crushing
pressure. (Real) |
[Pa] |
μC | Crushing volumetric
strain. (Real) |
|
PL | Locking
pressure. (Real) |
[Pa] |
μL | Locking plastic volumetric
strain. (Real) |
|
K1 | Linear bulk
stiffness. (Real) |
[Pa] |
K2 | Quadratic bulk
stiffness. (Real) |
[Pa] |
K3 | Cubic bulk
stiffness. (Real) |
[Pa] |
D1 | Damage
parameter. (Real) |
|
D2 | Damage
exponent. (Real) |
|
IDEL | Element deletion flag:
(Integer) |
|
εmaxp | Critical plastic strain for element
deletion. Default = 1020 (Real) |
▸Example
Comments
- This material law is based on the theory of Johnson-Holmquist-Cook model theory (also called Johnson-Holmquist-Concrete). 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
μ
.
P={K0μifP≤PC[I](K0+(K1−K0)μpμL)(μ−μp)ifP>PC and μp≤μL[II]K1ˆμ+K2ˆμ2+K3ˆμ3for the other cases[III]withK0=PCμCˆμ=μ−μL1+μLμ=ρρ0−1
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 μp , modifying linearly the bulk modulus from K0 and K1 . When μp=μL all the cavities have been crushed and the material becomes fully dense. Then, the pressure evolution follows a polynomial equation of state.Figure 1. Hydrostatic pressure variation with respect to volumetric strain - The deviatoric behavior is defined with an elasto-plastic behavior,
where the normalized yield stress is both a yielding and a damaging limit.
Its expression is:
- If
P*>0
(compressive
loadings):σ*Y=min(σ*MAX,(A(1−D)+B(P*)N)(1+C〈ln˙ε˙ε0〉+))
Where, P*=Pfc bounded by P*=−(1−D)T* .
- If
P*≤0
(tensile loadings):σ*Y=A(1+PT)(1−D)(1+C〈ln˙ε˙ε0〉+)To get the yield stress, the normalized value is multiplied by fc . 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:
Figure 2. Yield stress evolution with respect to hydrostatic pressure To trigger the deviatoric elasto-plastic behavior, the normalized yield stress is compared to the current normalized equivalent von Mises stress:
σ*VM=σVMfcThis allows to compute the evolution of the deviatoric plastic strain denoted εp .
- If
P*>0
(compressive
loadings):
- The damage variable evolution is dependent to both volumetric and
deviatoric plastic strain. Its expression is given by:D=∑Δμp+Δεpεpf
Where, the effective strain at failure is defined by:
εpf=max(D1(P*+T*)D2,εminf)with P*=Pfc and T*=Tfc
- Time history and animation output is available using these USRi variables.
- USR1: Plastic volumetric strain μp
- USR2: Bulking pressure P
- USR3: Volumetric strain μ
- USR4: Yield stress σY
- 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 non-local regularization method can be used
(/NONLOCAL/MAT). In this case, the sum of the
deviatoric plastic strain
εp
and volumetric plastic strain
μp
is regularized and used for damage
evolution:D=∑(Δμp+Δεp)nlεpf
The regularized sum can be plotted using /ANIM/ELEM/NL_EPSP or /H3D/ELEM/NL_EPSP.