# 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).

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)).