# Hydration-Drucker-Prager Model

The mechanical aspects of hydration in FLAC3D are handled by a modified Drucker-Prager constitutive model where elastic and strength properties depend on the hydration grade, $$\alpha$$ (Hinze 1987). [*]

Due to a dormant phase, the evolution of strength and stiffness starts with some delay. This is taken into account by the minimum degree of hydration, $$\alpha_0$$. This value marks the transition between the suspension and solid-state behavior. Beyond $$\alpha_0$$, strength and stiffness do not always depend linearly on hydration grade. Thus, a relationship is introduced, based on the idea of a multiplicative split of the final values of material properties and the degree of hydration, including the minimum degree of hydration according to the power law in Equation (1):

(1)$f(\alpha) = \max \left(1 \times 10^{-4}, \left( {{\alpha - \alpha_0} \over {1 - \alpha_0}}\right)^a \right)$

During evolving, $$(\alpha - \alpha_0)$$ may be less than zero when $$\alpha < \alpha_0$$. This is avoided by enforcing $$(\alpha - \alpha_0) \ge (\alpha - \alpha_0)_{min}$$, where $$(\alpha - \alpha_0)_{min}$$ is an input with a default value 1e-6.

With this formulation, the actual (and initial) Young’s modulus, $$E$$, during the hydration process is

(2)$E(\alpha) = f(\alpha) \cdot E_{cte}$

where $$E_{cte}$$ is the Young’s modulus (stress unit) after complete hydration, and $$a$$ is the power exponent (no unit).

The actual uniaxial compressive strength $$\sigma_c$$ and the uniaxial strength $$\sigma_t$$ also depend on the function in Equation (1).

(3)$\sigma_c (\alpha) = 0.85 \cdot {f_{cte} \over c} \cdot \left({{\alpha - \alpha_0} \over {1 - \alpha_0}} \right)^{3/2}$
(4)$\sigma_t (\alpha) = f_{cte} \cdot \left({{\alpha - \alpha_0} \over {1 - \alpha_0}} \right)$

where $$f_{cte}$$ is the uniaxial strength (stress unit) after total completion of the hydration process, and $$c$$ is a material parameter (no unit). In the above two equations, $$(\alpha - \alpha_0) \ge (\alpha - \alpha_0)_{min}$$ is enforced as well.

The yield criterion in the Drucker-Prager model is

(5)$0 = \tau + q \cdot \sigma - k$

where $$q$$ and $$k$$ are material parameters, and $$\tau$$ and $$\sigma$$ are stress invariants. $$q$$ and $$k$$ can be derived from the actual uniaxial compressive and tensile strengths, $$\sigma_c$$ and $$\sigma_t$$.

(6)$q = {{\sqrt{3}(\sigma_c - \sigma_t)} \over {\sigma_c + \sigma_t}}$
(7)$k = {{2 \cdot \sigma_c \cdot \sigma_t} \over {\sqrt{3} (\sigma_c + \sigma_t)}}$

In this model, the compression strength is assumed no less than one third of $$k/q$$ ($$\sigma_c \ge k/q$$) and greater than the extension strength ($$\sigma_c \ge 1.001 \times \sigma_t$$ is used).

 [*] During the hydration process, the values of elastic material parameters can vary over several orders of magnitude. Accordingly, the gridpoint masses have to be adjusted for numerical stability in both small-strain mode and large-strain mode. The frequency of the update can be set by the user with the zone geometry-update command.

Reference

Hinze, D. “Zur Beurteilung des phsikalischen nicht-linearen Betonverhaltens bei mehrachsigem Spannungszustand mit Hilfe differenzeiller Stoffgesetze unter Anwendung der FEM,” Thesis, Hochschule für Architektur und Bauwesen, Weimar (1987).

Hydration-Drucker-Prager model properties

Use the following keywords with the zone property command to set these properties of the Hydration-Drucker-Prager model.

hydration-drucker-prager
bulk f

bulk modulus, $$K$$

bulk-reference f

reference bulk modulus for $$α$$ = 1, $$K_{cte}$$

cohesion-drucker f

Drucker-Prager material parameter, $$k_φ$$

compression f

compressive strength limit, $$σ_c$$

constant-a f

material parameter, $$a$$

constant-c f

material parameter, $$c$$

dilation-drucker f

Drucker-Prager material parameter, $$q_ψ$$

friction-drucker f

Drucker-Prager material parameter, $$q_φ$$

hydration-minimum f

minimum hydration grade, $$α$$0

hydration-difference-minimum f

minimum difference of $$(α - α_0)_{min}$$

poisson f

Poisson’s ratio, $$ν$$

shear f

shear modulus, $$G$$

shear-reference f

reference shear modulus for $$α$$ = 1, $$G_{cte}$$

tension f

tension cut-off, $$σ^t$$

tension-reference f

reference tensile strength for $$α$$ = 1, $$f_{cte}$$

young f

Young’s modulus, $$E$$

young-reference f

reference Young’s modulus for $$α$$ = 1, $$E_{cte}$$

Key

• Only one of the two options is required to define the elasticity: reference bulk modulus $$K_{cte}$$ and reference shear modulus $$G_{cte}$$, or reference Young’s modulus $$E_{cte}$$ and Poisson’s ratio $$v$$. When choosing the latter, reference Young’s modulus $$E$$ must be assigned in advance of Poisson’s ratio $$v$$.
• The tension cut-off is $$σ^t$$ = min ($$σ_t$$(α)/3.0, $$k_ϕ$$(α)/$$q_ϕ$$(α)).