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).


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.

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, \(E_{cte}\)


(r) Read-only property.

This property cannot be set by the user. Instead, it can be listed, plotted, or accessed through FISH.


  • 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)).