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)\[f(\alpha) = \max (1 \times 10^{-4}, A^a)\]

where

(2)\[A = {{\alpha - \alpha_0} \over {1 - \alpha_0}}\]

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

(3)\[E(\alpha) = f(\alpha) \times E_{cte}\]

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).

(4)\[\sigma_c (\alpha) = 0.85 \cdot {f_{cte} \over c} A^{3/2}\]

(5)\[\sigma_t (\alpha) = A f_{cte}\]

where \(f_{cte}\) is the uniaxial strength (stress unit) after total completion of the hydration process, and \(c\) is a material parameter (no unit). Dividing the two equations shows what \(c\) controls the ratio of the two strengths \(\sigma_c / \sigma_t = (0.85/c) \sqrt{A}\), which at complete hydration is simply \(0.85/c\). The default \(c\) = 0.1 therefore gives a compressive strength 8.5 times the tensile strength. Users can customize the \(c\) value, for example, \(c\) = 0.085 will make the ratio of the two strengths \(\sigma_c / \sigma_t\) = 10. In the above two equations, \((\alpha - \alpha_0) \ge (\alpha - \alpha_0)_{min}\) is enforced as well.

Lower Bound on the Compressive Strength

Early in the hydration process, Equation (4) can return a uniaxial compressive strength smaller than the uniaxial tensile strength of Equation (5). Comparing the two, this is the case while

(6)\[A \le \left({c \over 0.85}\right)^2\]

Such a state is not admissible, because it makes the friction parameter \(q\) of Equation (9) negative. Whenever the condition above holds, the compressive strength is therefore raised to just above the tensile strength:

(7)\[\sigma_c (\alpha) = 1.001 \times \sigma_t (\alpha)\]

Over that range the material is very nearly frictionless, with \(q \approx 8.7 \times 10^{-4}\) and \(k \approx 0.578 \sigma_t\). The width of the range grows with the square of \(c\). With the default \(c\) = 0.1 the bound applies only until \(A\) exceeds 0.014.

Because \(c\) divides the compressive strength in Equation (4), it must be positive whenever a reference tensile strength \(f_{cte}\) is given.

The yield criterion in the Drucker-Prager model is

(8)\[0 = \tau + q \sigma - k\]

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 \(\sigma_t\).

(9)\[q = {{\sqrt{3}(\sigma_c - \sigma_t)} \over {\sigma_c + \sigma_t}}\]

(10)\[k = {{2 \sigma_c \sigma_t} \over {\sqrt{3} (\sigma_c + \sigma_t)}}\]

[*]

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, \(k_φ\)

constant-a f

material parameter, \(a\)

constant-c f

material parameter, \(c\), setting the ratio of the uniaxial compressive to tensile strength; at complete hydration \(σ_c\)/\(σ_t\) = 0.85/\(c\). Must be positive whenever a reference tensile strength \(f_{cte}\) is given. The default is 0.1.

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 \((α - α_0)_{min}\)

shear f

shear modulus, \(G\)

shear-reference f

reference shear modulus for \(α\) = 1, \(G_{cte}\)

tension-drucker f

tensile strength, \(σ^t\)

tension-reference f

reference tensile strength for \(α\) = 1, \(f_{cte}\)

compression-uniaxial f (r)

current uniaxial compressive strength, \(σ_c\)

tension-uniaxial f (r)

current uniaxial tensile strength, \(σ_t\)

poisson f (r)

Poisson’s ratio, \(ν\)

young f (r)

Young’s modulus, \(E\)

young-reference f (r)

reference Young’s modulus for \(α\) = 1, \(E_{cte}\)

Key

(r) Read-only property.

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

Notes

• The effective Drucker-Prager tension cut-off is \(\sigma_{cut}^t = \min(\sigma_{DP}^t, k_\phi\ (\alpha)/q_\phi\ (\alpha))\)