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)\[f(\alpha) = \left( {{\alpha - \alpha_0} \over {1 - \alpha_0}}\right)^n\]

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

(2)\[E(\alpha) = E_{cte} \cdot \left( {{\alpha - \alpha_0} \over {1 - \alpha_0}}\right)^a\]

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).

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-reference f

reference bulk modulus for α = 1, Kcte

constant-a f

material parameter, a

constant-c f

material parameter, c

hydration-minimum f

minimum hydration grade, α₀

hydration-difference-minimum f

minimum difference of (α − α₀)min

poisson f

Poisson’s ratio, ν

shear-reference f

reference shear modulus for α = 1, Gcte

young-reference f

reference Young’s modulus for α = 1, Ecte

tension-reference f

reference tensile strength for α = 1, fcte

bulk f

[read only] bulk modulus, K

cohesion-drucker f

[read only] Drucker-Prager material parameter, kφ

compression f

[read only] compressive strength limit, σc

dilation-drucker f

[read only] Drucker-Prager material parameter, qψ

friction-drucker f

[read only] Drucker-Prager material parameter, qφ

shear f

[read only] shear modulus, G

tension f

[read only] tension cut-off, σt

young f

[read only] Young’s modulus, E

Notes:

• Only one of the two options is required to define the elasticity: reference bulk modulus Kcte and reference shear modulus Gcte, or reference Young’s modulus Ecte 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 σt = min (σt(α)/3.0, kϕ(α)/qϕ(α)).

Footnote

Read only properties cannot be set by the user. However, they may be listed, plotted, or accessed through FISH.