The RHT concrete model is an advanced plasticity model for brittle materials developed by Riedal et al [2], [3], [4]. It is particularly useful for modeling the dynamic loading of concrete. It can also be used for other brittle materials such as rock and ceramic.

The RHT constitutive model is a combined plasticity and shear damage model in which the deviatoric stress in the material is limited by a generalized failure surface of the form:

(11–5)

This failure surface can be used to represent the following aspects of the response of geological materials

The model is modular in nature and is designed such that individual aspects of the material behavior can be turned on and off. This gives the model significant practical usefulness. Further details of how the model represents the various aspects of the material behavior are now presented.

Fracture surface

The fracture surface is represented through the expression

(11–6)

where f_c' is the cylinder strength

Additionally, there is an option to truncate the fracture surface to fit through the characteristic points that can be observed experimentally at low pressures, while retaining the flexibility to match data at high pressures. This feature is described in the figure below.

Figure 11.7: RHT Representation of Compressive Meridian

Tensile and Compressive Meridians

The RHT model can represent the difference between the compressive and tensile meridian in terms of material strength using the third invariant dependence term (R₃). This can be utilized to represent the observed reduction in strength of concrete under triaxial extension, compared with triaxial compression. The third invariant dependence term is formulated using the expression

(11–7)

The input parameter Q_2.0 defines the ratio of strength at zero pressure and the coefficient BQ defines the rate at which the fracture surface transitions from approximately triangular in form to a circular form with increasing pressure (Figure 11.8: Third invariant dependence).

Figure 11.8: Third invariant dependence

Strain Hardening

Strain hardening is represented in the model through the definition of an elastic limit surface and a "hardening" slope. The elastic limit surface is scaled down from the fracture surface by user defined ratios; (elastic strength/fc) and (elastic strength/ft). The pre-peak fracture surface is subsequently defined through interpolation between the elastic and fracture surfaces using the "hardening" slope, . This is shown in Figure 11.9: Bi-linear strain hardening function for the case of uniaxial compression.

where

Figure 11.9: Bi-linear strain hardening function

Shear Damage

Damage is assumed to accumulate due to inelastic deviatoric straining (shear induced cracking) using the relationships

(11–8)

where D₁ and D₂ are material constants used to describe the effective strain to fracture as a function of pressure. Damage accumulation can have two effects in the model

Porous Collapse Damage

The model includes the option to include a cap to limit the elastic deviatoric stress under large compressions. This effectively leads to the assumption that porous compaction results in a reduction in deviatoric strength.

The final combination of elastic, fracture and residual failure surfaces is shown schematically below in Figure 11.10: RHT Elastic, Fracture and Residual Failure Surfaces.

Figure 11.10: RHT Elastic, Fracture and Residual Failure Surfaces

Strain Rate Effects

Strain rate effects are represented through increases in fracture strength with plastic strain rate. Two different terms can be used for compression and tension with linear interpolation being used in the intermediate pressure regime.

where

= 3e-6 in tension and 30e-6 in compression.

Tensile Failure

By default, tensile failure is achieved using a hydrodynamic tensile limit. The maximum tensile pressure in the material is limited to

(11–12)

Using this option, no additional user input is required since the value of P_min is derived from f_t, which forms part of the input for the strength model.

Note that the principal tensile stress and crack softening failure properties may also be used in conjunction with this model.

Data for concrete with cube strengths of 35MPa and 140MPa are included in the distributed material library.

The model is formulated such that input can be scaled with the cube strength; fc for example. you can retrieve one of the two concretes in the library, change its cube strength to match the concrete you want to model and the remaining terms will automatically scale proportionately. The resulting data set will be approximate and we recommend validation of the material data against experimental characterization tests in all cases.

Custom results variables available for this model:

**Material status indicators (1=elastic, 2= plastic, 3 = bulk failure, 4 = bulk failure, 5= failed principal direction 1, 6= failed principal direction 2, 7 = failed principal direction 3)