Unified Formulation for a Triaxial Elastoplastic Constitutive Law for Concrete
2. Triaxial Constitutive Formulation
2.1. Yield Surface Criterion
- is the mean normal stress or hydrostatic pressure expressed by:
- ρ is the deviatoric stress defined by:
- θ is the polar angle that determines the direction of the octahedral shear stress and locates the stress state relative to the meridians of tension and compression around the hydrostatic axis. The angle θ is defined as follows:
2.2. Isotropic Loading Surfaces in Pre- and Post-Peak
2.2.1. Isotropic Hardening
2.2.2. Isotropic Softening
3. Plastic Potential Function
4. Hardening and Softening Parameter Functions
5. Algorithmic Formulation
5.1. Evaluation of Convenient Stress for Plastic Potential
5.2. Resolution Scheme
- Calculating the first elastic prediction:
- From the stresses at point B (Figure 7), calculate the value of F and the gradient .
- In the presence of non-associated flow, identify a particular value of ρ for and calculate the gradient .
- Compute and the stresses at point C: , where is the elastic test point; is elasticity tensor; and is effective plastic modulus (g is generic variable, for hardening and for softening).
- Update the equivalent plastic strain in hardening (softening) and the strength parameter during the hardening(softening).
- Beginning the implicit backward-Euler method:
- Calculate F and at the current point C.
- Minimize the potential for and calculate the gradient .
- Calculate the residual .
- Compute the change of the plastic multiplier:
- Update the stresses at the point C: , then calculate the changes in plastic multiplier at point B (Figure 7):
- Update the equivalent plastic strain and the strength parameter during the hardening(softening).
- Repeat the procedure from step 5 until and F are below a certain tolerance.
- It should promote a positive change in volumetric plastic in the region of positive pressure related to the mode of crack opening.
- It should promote a change in plastic form in the region of negative pressure related to the mode of cracking or splitting in shear compression.
7. Numerical Experiments for Various Loading Scenarios
- A five parameters loading surface, which was adapted and calibrated by a simple procedure.
- Uncoupled hardening and softening functions following the accumulation of plastic strain and ductility evolution.
- A new ductility function was proposed and fitted experimentally.
- A new nonlinear plastic potential function was developed and calibrated using database of test results (uniaxial compression).
- As the failure criterion and plastic potential do not undergo the same stress states, a projection procedure has been adopted and applied to the concrete case. The calculation of normal is accurate and verified through numerical simulations.
Conflicts of Interest
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