Concrete ASR Eigenstrain

Alkali-silica reaction (ASR) swelling model

Background

Deterioration of concrete structures affected by ASR can be attributed, on the microscopic level, to the formation of a hydrophilic gel due to complex dissolution-precipitation reactions between reactive silica in aggregates, alkaline ions such as potassium and and hydroxyl ions in the cement pore solution. If water is available in concrete pores, the gel swells, creating an internal pressure in localized regions within concrete structures, which can initiate micro- and macrocracking, excessive expansion, misalignment of the structure etc. According to Ulm et al. (2000), there are two important factors that control the ASR reaction rate:

• ASR reactions are thermally activated. The higher the temperature, the faster they occur. This kinetic effect of temperature on ASR results from the thermoactivation of both the dissolution of reactive silica on aggregate-cement interface and the precipitation of gel.

• The relative humidity in concrete has a strong influence on ASR, affecting both the kinetics and magnitude of volumetric swelling. Water plays an important role as a solvent for silica dissolution, intervenes as a transport medium for diffusion of ions through the pore network, and is a necessary compound for the formation of various reaction products (gels and other mineral precipitates).

The ConcreteASREigenstrain model computes the evolution of the ASR reaction and the resulting eigenstrains as a function of the thermal, moisture, and mechanical stresses at local material points, which enables ASR to be simulated in a way that accounts for full coupling of all physics involved in this phenomenon. It is an implementation of the Saouma and Perotti (2006) ASR swelling model, which builds on the model of Ulm et al. (2000). This model has the following features:

• The ASR reaction rate is temperature dependent

• The ASR reaction can be retarded by compressive stress within concrete structures

• Both high compressive and tensile stress states inhibit ASR expansion due to the formation of micro- and macro-cracks that adsorb the expanding gel

• Triaxial compressive stress states reduce expansion

• The ASR expansion is anisotropic to account for the fact that it is constrained by compression, and is redirected into other less-constrained principal directions

Details of this model's representation of ASR reaction kinetics, volumetric strain, and anisotropic distribution of that strain are provided below. Note that throughout this discussion, the notations of Ulm et al. (2000) and Saouma and Perotti (2006) are adopted.

ASR reaction kinetics

This model employs a 1st-order ASR reaction kinetics model that is dependent on both the temperature and the first invariant of the stress tensor as: (1) in which is the ASR reaction extent ranging from 0 (not reacted) to 1 (fully reacted), is the temperature, and and are expressed as: (2) representing the latency and characteristic times of ASR reactions, respectively. Here is the reference temperature (i.e., the temperature at which stress-free ASR experiments are carried out), is the first invariant of the stress tensor, is the uniaxial compressive strength of concrete, and and are thermal activation energy constants for the latency and characteristic times, respectively, and are determined from tests conducted by Larive (1998) as

The function in Eq. (2) represents the effect of compressive stress on the ASR reaction kinetics by modifying the latency time and is defined as (3) where .

Saouma and Perotti (2006) assumed a value of 4/3 for the constant based on their analysis of experimental data from Multon and Toutlemonde (2006). When is greater than zero (i.e., tensile stress state), is equal to 1, indicating no effect of tensile stress on the reaction kinetics. When is less than zero (i.e., compressive stress state), has a value greater than 1, indicating an increase in the latency time (equivalently retardation of ASR reaction).

Eq. (1) for the ASR reaction is a nonlinear ordinary differential equation (ODE) that is solved locally at the quadrature points within each finite element by a Newton-Raphson iteration scheme, given the current temperature and the stress tensor and reaction extent at the end of the previous time step.

Stress-dependent ASR volumetric strain

The volumetric strain resulting from the ASR reaction depends on stress in this model. Once the increment of ASR reaction extent is obtained, the ASR volumetric strain increment from time to is evaluated using the following formula (4) where is the tensile strength of the concrete; is the maximum principal stress ( under tensile stress); is the crack opening displacement; is the ratio between the hydrostatic stress and compressive strength of concrete, and is the laboratory-determined maximum free volumetric expansion at the reference temperature .

The function in Eq. (4) accounts for the dependency of gel expansion on the water in concrete and takes the form: where the exponent is an empirical constant and is the relative humidity in concrete. has a value between 0 and 1.

Function in Eq. (4) accounts for the reduction of ASR expansion due to tensile cracking (e.g., gels are adsorbed into tensile macrocracks), and takes the following form: in which is the fraction of the tensile strength beyond which gel is adsorbed by cracks and is a user input parameter in simulation, and is a residual ASR expansion retention factor for ASR under tension.

Function in Eq. (4) accounts for the reduction in ASR volumetric expansion under compressive stress state, in which gel is absorbed by diffused microcracks, and takes the following form: (5) where the exponent is an empirical constant between -2 and 2, and is the ratio between the hydrostatic stress and compressive strength of concrete:

Anisotropic ASR strain distribution

In general the incremental ASR volumetric stain should not be applied isotropically because the propensity for expansion in a given direction depends on the stress in that direction. The technique of Saouma and Perotti (2006) is used here to calculate the distribution of volumetric expansion among the three directions of principal stress based on the behavior under either uniaxial, biaxial or triaxial confinement conditions.

Given the full stress tensor (in Cartesian coordinates) at a quadrature point within an element, an eigensolver is used to obtain the three principal stresses, , and , and associated eigenvectors for the directions of principal stresses, , and . These eigenvectors form a stress/strain rotational matrix that is used later to rotate the incremental ASR strain tensor expressed in principal stress/strain coordinates back into Cartesian coordinates.

Figure 1: 2D stress-space map and ASR weight interpolation quadrants (based on Saouma and Perotti (2006))

The weight allocation scheme for the principal directions starts by dividing the stress space into nine quadrants using the concrete tensile strength and compressive strength as well as a gel expansion-inhibiting compressive stress , as shown in Figure 1. is defined as the compressive stress beyond which no further gel expansion can occur. Saouma and Perotti (2006) proposed a value of -10MPa based on previous experimental studies Larive (1998). Any combination of two principal stresses will fall into one of the nine quadrants of this two-dimensional stress map, which has a total of 16 nodes.

To calculate the ASR expansion weight along the direction of a particular principal stress, using for example, the following three steps are taken:

• Identify the quadrant encompassing the other two principal stresses and using Figure 1 and the corresponding nodal numbers of that quadrant;

• Find the rows of Table 1 corresponding to the identified nodal numbers and determine the nodal weights from the last 3 columns of Table 1 in the corresponding rows using a linear interpolation of .

• Finally, compute the ASR expansion weight for the principal direction using the four nodal weights of the quadrant in which falls using the bilinear interpolation of Eq. (6)

The bilinear interpolation is performed in the same way that solution fields are interpolated in the standard finite element method: (6) where is the bilinear shape function similar to those used in finite element method and is given by

The previous steps are then repeated to calculate the ASR expansion weights along the other two principal directions, and . Note that the summation of , and equals 1. The individual incremental ASR strains along the principal directions are then obtained using these relative weights by the following formula: (7)

Finally, the full ASR expansion-induced incremental strain tensor in the Cartesian coordinate system can then be obtained by rotating into the current coordinates via (8) in which is the imposed incremental strain tensor due to ASR expansion at each quadrature point, which drives the deformation of the concrete structure.

ASR-induced Damage

The ASR-induced degradation of the mechanical properties of concrete can be simulated with a specific damage model. In this approach, the stress is computed with a ComputeDamageStress calculator as:

in which is the fourth-order stiffness tensor of the material and the ASR-induced damage.

Two options are available to compute :

• Using the ConcreteASRMicrocrackingDamage model based on Saouma and Perotti (2006), in which is a simple function of the ASR reaction extent and a material constant representing the loss of modulus when the material has fully reacted:

• Using the ConcreteExpansionMicrocrackingDamage model, in which the damage is computed as a function of the total ASR strain and the current stress. In this approach, the anisotropic effects of ASR are indirectly accounted for via the damage model rather than the eigenstrain itself, and therefore this model should be used with the expansion_type = Isotropic option.

Implementation and Usage

The ConcreteASREigenstrain model computes the ASR reaction extent , and stores it in a property named ASR_extent. It also computes an eigenstrain tensor that is used to modify the strain tensor. The loss of mechanical properties due to ASR can be represented with the ConcreteASRMicrocrackingDamage model or the ConcreteExpansionMicrocrackingDamage model, provided that the stress is computed with ComputeDamageStress.

Input Parameters

References