Brittle Damage: Reinforced Concrete Material (LAW24)

The model is a continuum, plasticity-based, damage model for concrete. It assumes that the main two failure mechanisms are tensile cracking and compressive crushing of the concrete material.

The material law will enable to formulate the brittle elastic - plastic behavior of the reinforced concrete.

The input data for concrete are:

E _c: Young's modulus; 32000 MPa
V_c: Poisson's ratio; 0.2
f_c: Uniaxial compressive strength; 32 $[MPa]$
f_i/f_c: Tensile strength ratio; Default = 0.1
f_b/f_c: Biaxial strength ratio; Default = 1.2
f₂/f_c: Confined strength ratio; Default = 4.0
s₀/f_c: Confining stress ratio; Default = 1.25

Experimental results enable to determine the material parameters. This can be done by in-plane unidirectional and bi-axial tests as shown in Figure 2. The expression of the failure surface is in a general form as:(1)

f = r - k (σ_{m}, k_{0}) \cdot r_{f} = 0

Where,

$J_{2}$: Second invariant of stress; Where, $r = \sqrt{2 J_{2}} = \sqrt{\frac{2}{3}} σ_{V M}$
$σ_{m} = \frac{I_{1}}{3}$: Mean stress

A schematic representation of the failure surface in the principal stress space is given in Figure 2. The yield surface is derived from the failure envelope by introducing a scale factor $k (σ_{m}, k_{0})$ . The meridian planes are presented in Figure 3.

The steel directions are defined identically to material LAW14 by a TYPE6 property set. If a property set is not given in the element input data, r, s, $θ$ are taken respectively as direction 1, 2, 3. For quad elements, direction 3 is taken as the $θ$ direction.

Steel data properties are:

E: Young's modulus
$σ_{y}$: Yield strength
E _t: Tangent modulus
$α_{1}$: Ratio of reinforcement in direction 1
$α_{2}$: Ratio of reinforcement in direction 2
$α_{3}$: Ratio of reinforcement in direction 3