/MAT/LAW24 (CONC)

Block Format Keyword This material law models brittle elastic-plastic behavior of concrete. A yield surface is deduced from an Ottosen triaxial failure surface. Orthotropic damage is modeled and cracks can open and close.

An optional embedded model allows to take into account steel rebars into a homogenized model. Otherwise, rebars are usually meshed with 1D or 3D elements.

This material law is compatible with solid elements only.

Format

(1)	(3)	(5)	(7)	(9)
/MAT/LAW24/`mat_ID`/`unit_ID` or /MAT/CONC/`mat_ID`/`unit_ID`
`mat_title`
$ρ_{i}$
`E`_c	$ν$	`I`_cap
`f`_c	`f`_t`/f`_c	`f`_b/`f`_c	`f`₂`/f`_c	`s`₀`/f`_c
`H`_t	`D`_sup	$ε_{\max}$
`k`_y	$ρ_{t}$	$ρ_{c}$	`H`_bp	`Etc`
$α_{y}$	$α_{f}$	`v`_max
`f`_k	`f`₀	`H`_v0	`Eps`₀	`h`_fac
`E`	$σ_{y}$	`E`_t
$α_{1}$	$α_{2}$	$α_{3}$

Definition

Field	Contents	SI Unit Example
`mat_ID`	Material identifier. (Integer, maximum 10 digits)
`unit_ID`	Unit Identifier. (Integer, maximum 10 digits)
`mat_title`	Material title. (Character, maximum 100 characters)
$ρ_{i}$	Initial density. (Real)	$[\frac{kg}{m^{3}}]$
`E`_c	Concrete elasticity Young's modulus. (Real)	$[Pa]$
$ν$	Poisson's ratio. (Real)
`I`_cap	Cap formulation flag. 8 = 0 (Default) Improved cap formulation. = 1 Original cap formulation (initial formulation used in Radioss 14.0 and older). = 2 Cap model with updated multi-axial and hydrostatic load representation (recommended). (Integer)
`f`_c	Concrete uniaxial compression strength. (Real)	$[Pa]$
`f`_t/`f`_c	Concrete tensile strength ratio. Default = 0.10 (Real)
`f`_b/`f`_c	Concrete biaxial strength ratio. Default = 1.20 (Real)
`f`₂/`f`_c	Concrete confined strength ratio. Default = 4.00, if `I`_cap = 0 or 1 Default = 7.00 if `I`_cap = 2 (Real)
`s`₀`/f`_c	Concrete confining stress ratio. Default = 1.25 (Real)
`H`_t	Concrete tensile tangent modulus. Default = -E_c (Real)	$[Pa]$
`D`_sup	Concrete maximum damage. Default = 0.99999 (Real)
$ε_{\max}$	Concrete data total failure strain. Default = 10²⁰ (Real)
`k`_y	Concrete plasticity initial value of hardening parameter (first part). Default = 0.5 (Real)
$ρ_{t}$	Concrete plasticity failure/plastic transition pressure (first part). Default = 0.0 (Real)	$[Pa]$
$ρ_{c}$	Concrete plasticity proportional yield transition pressure (first part). Default = `-f`_c/3 (Real)	$[Pa]$
`H`_bp	Concrete plasticity base plastic modulus (first part). 3 Default is computed by Starter (Real)	$[Pa]$
`Etc`	Concrete plastic modulus. 3 (Real)	$[Pa]$
$α_{y}$	Concrete plasticity dilatancy factor at yield (second part). Default = -0.2, if `I`_cap= 0 or 2 Default = 0.0, if `I`_cap= 1 (Real)
$α_{f}$	Concrete plasticity dilatancy factor at failure (second part). Default = 0.0 (Real)
`v`_max	Concrete plasticity maximum volumetric compaction ( < 0 ) (second part). Default = -0.35 (Real)
`f`_k	Initial beginning of cap. 7 Default = `-f`_c/3 (Real)	$[Pa]$
`f`₀	Initial end of cap. 7 Default = -0.8 f_c, if `I`_cap = 0 or 1 Default = -2 f_c, if `I`_cap = 2 (Real)	$[Pa]$
`H`_v0	Initial triaxial plastic modulus. Default = 0.2 E_c (Real)	$[Pa]$
`Eps`₀	Reference plastic strain for plastic hardening (`I`_cap = 2 only). Default = 0.02 (Real)
`h`_fac	Reduction factor for plastic hardening default (`I`_cap = 2 only). Default = 0.1 (Real)
`E`	Steel properties Young's modulus. (Real)	$[Pa]$
$σ_{y}$	Yield strength. (Real)	$[Pa]$
`E`_t	Tangent modulus. (Real)	$[Pa]$
$α_{1}$	Rebar section fraction of reinforcement in direction 1. (Real)
$α_{2}$	Rebar section fraction of reinforcement in direction 2. (Real)
$α_{3}$	Rebar section fraction of reinforcement in direction 3. (Real)

Example (Concrete)

#RADIOSS STARTER
/UNIT/1
unit for mat
                   g                  mm                  ms
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#-  2. MATERIALS:
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
/MAT/CONC/1/1
concrete - air dry
#              RHO_I
               .0024  
#                E_c                  NU      Icap
               41200                  .2         0
#                 fc            ft_on_fc            fb_on_fc            f2_on_fc            s0_on_fc
                  44                   0                   0                   0                   0
#                H_t               D_sup             EPS_max
                   0                   0                   0
#                k_y                 r_t                 r_c                H_bp
                   0                   0                   0                   0
#            ALPHA_y             ALPHA_F               V_max
                -0.2                -0.1                   0
#                f_k                 f_0                H_v0                eps0               h_fac
                   0                   0                   0
#                  E             sigma_y                 E_t
                   0                   0                   0
#             ALPHA1              ALPHA2              ALPHA3
                   0                   0                   0
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|
#ENDDATA
/END
#---1----|----2----|----3----|----4----|----5----|----6----|----7----|----8----|----9----|---10----|

Comments

This material law can be used with only four parameters: $ρ_{i}, E_{c}, ν$ and $f_{c}$ . Values $f_{t}, f_{b}, f_{2}, s_{0}$ are input as ratios of $f_{c}$ and are based on the failure of a typical concrete material.
The parameters for the damage model are:

Figure 1. Stress Strain Curve for LAW24 Damage Model. meridians of failure and yield surfaces

Where, $f_{c}$ is the uniaxial compression strength.
The default value of the plastic modulus in compression is defined as:(1)
$E_{t c} = \frac{(1 - k_{y}) E_{c} f_{c}}{{2.10}^{- 3} E_{c} - k_{y} f_{c}}$

The base plastic modulus is then calculated:(2)
$H_{p}^{b} = \frac{E_{c} E_{t c}}{(E_{c} - E_{t c})}$

Figure 2.
The yield envelope is derived from the failure envelope using a scale factor $k (σ_{m}, k_{0})$ .(3)
$f = r - k (σ_{m}, k_{0}) \cdot r_{f} = 0$
$k_{0}$ is the hardening parameter, it characterizes the limit of the current elastic domain. The initial limit of the elastic domain is set at $k_{0} = k_{y}$ and the initial failure surface at $k_{0} = 1$ .
The scale factor $k (σ_{m}, k_{0})$ models the hardening effect depending on the mean stress $σ_{m}$ .

In tension (when $σ_{m} \geq ρ_{t}$ ), $k (σ_{m}, k_{0}) = 1$ and the yield envelope and failure envelope are superimposed.

In compression (when $ρ_{c} \geq σ_{m}$ ), the scale factor $k (σ_{m}, k_{0})$ depends on I_cap value (0, 1 or 2).

Figure 3. Scale factor $k (σ_{m}, k_{0})$ shapes the yield surface from the failure surface
- For I_cap =0 or 1: $k_{y} \leq k (σ_{m}, k_{0}) \leq 1$
  
  Figure 4.
- For I_cap =2 (with cap): $0 \leq k (σ_{m}, k_{0}) \leq 1$
  
  Figure 5.
Where, $r = \sqrt{2 J_{2}} = \sqrt{\frac{2}{3}} σ_{V M}$ and $σ_{m} = \frac{I_{1}}{3}$ is mean stress (pressure), $I_{1}$ and $J_{2}$ are the first and second stress invariant. The factor $k$ can be output in time history file under VK keyword.
There is no cap effect when $α_{y} = α_{f} = 0$ because there is no compaction. The default values of dilatancy parameters $α_{y}$ and $α_{f}$ were modified for I_cap = 0 or 2 in Radioss 2017. These parameters should be negative with recommended values of -0.2 and -0.1, respectively. The default cap formulation was updated in Radioss 2017. To enable the original cap formulation used in Radioss 14.0 and older, set I_cap = 1. The I_cap =2 formulation is more accurate for triaxial and hydrostatic loadings. The cap hardening is a function of the compaction coefficient. The shear hardening modulus is reduced in the transition region which assures better stability. I_cap = 2 must be used to output plastic strain using /ANIM/BRICK/EPSP or /H3D/SOLID/EPSP.
The embedded rebar model is optional. It uses an elastic-plastic with hardening material model. When defining rebar section fractions, a homogenized behavior is assumed for each element. The element should be large enough to stand for a Representative Elementary Volume (REV). This homogenized model is mostly used with large structures and a coarse mesh. Otherwise, rebar can be modeled with trusses, springs, beams, or even brick elements. the section ratio of rebars must be provided $α_{1}$ , $α_{2}$ , and $α_{3}$ :

Figure 6.
The rebar directions must be defined in the orthotropic solid property /PROP/TYPE6. Otherwise, the local element coordinate r, s, and t are taken respectively as directions 1, 2, and 3; unless I_solid = 1 or 2 is used with I_frame = 2; in which case the orthotropic directions 1, 2 and 3 are defined with the local co-rotating element coordinate r, s, and t, when time = 0.
In an axisymmetrical analysis, direction 3 is the $θ$ direction.
The 10 node tetrahedron elements are compatible with this law.