/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.
= 0 (Default)
No strain rate effect

(Real)

˙ε0 Reference strain rate.

Default = 1.0 (Real)

[1s]
FCUT Cutoff frequency for strain rate filtering.
= 0
No 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:
= 0 (Default)
No element deletion
= 1
Tensile failure when P*+T*<0
= 2
Failure when critical plastic strain is reached εp>εmaxp
= 3
Failure when σY0 (recommended)
= 4
Failure when D=1

(Integer)

εmaxp Critical plastic strain for element deletion.

Default = 1020 (Real)

Example

Comments

  1. 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.
  2. 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={K0μifPPC[I](K0+(K1K0)μpμL)(μμp)ifP>PC and μpμL[II]K1ˆμ+K2ˆμ2+K3ˆμ3for the other cases[III]withK0=PCμCˆμ=μμL1+μLμ=ρρ01

    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
  3. 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(1D)+B(P*)N)(1+Cln˙ε˙ε0+))

      Where, P*=Pfc bounded by P*=(1D)T* .

    • If P*0 (tensile loadings):
      σ*Y=A(1+PT)(1D)(1+Cln˙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=σVMfc

      This allows to compute the evolution of the deviatoric plastic strain denoted εp .

  4. 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

  5. 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
  6. Strain rate filtering can be used and activated when a cutoff frequency FCUT for filtering is defined.
  7. The damage variable can be plotted in ANIM and H3D file using the output option DAMG.
  8. 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.

1
A computational constitutive model for concrete subjected to large strains, high strain rates, and high pressure, G.R. Johnson, T.J. Holmquist, W.H. Cook,1993