Concrete slump test

The material properties reported by Famiglietti and Prevost are used in this example.

A Young's modulus of 2.25 and a Poisson's ratio of 0.125 define the elastic response of the concrete. A density of $\rho =$ 0.1 is used.

It is assumed that the inelastic behavior is governed by the cohesion or shear strength and by the friction angle of the material. A cohesion of $c=$ 0.0011547 is used, and the responses at four different friction angles ($\varphi =$ 0°, 5°, 20°, and 35°) are compared. Perfect plasticity is assumed. Since these parameters are provided for a Mohr-Coulomb plasticity model, they must be converted to linear Drucker-Prager parameters. Extended Drucker-Prager Models describes a method for converting Mohr-Coulomb parameters to equivalent linear Drucker-Prager parameters. Plane strain deformation and an associated plastic flow rule, where the dilation angle $\psi$ is equal to the material friction angle $\beta$, are assumed for the purpose of this conversion. The corresponding linear Drucker-Prager parameters, $\beta$ and d, are given in Table 1. The values are obtained using the expressions given in Extended Drucker-Prager Models.

Reducing the hyperbolic yield function into a linear form requires that ${p_t|}_0=d/\tan \beta .$ Reducing the exponent yield function into a linear form requires that $b=$ 1.0 and that $a=$ ($\tan \beta$)⁻¹. The material parameters for the exponential and hyperbolic yield criteria that create equivalent linear models are given in Table 1. Neither the hyperbolic nor the exponential yield criteria can be reduced to a linear model where $\beta =$ 0° (Mises yield surface).

The hyperbolic and exponential yield criteria both use a hyperbolic flow potential in the meridional stress plane. This flow potential, which is continuous and smooth, ensures that the flow direction is well-defined. The function asymptotically approaches the straight-line Drucker-Prager flow potential at high confining pressure stress but intersects the hydrostatic pressure axis at an angle of 90°. This function is, therefore, preferred as a flow potential for the Drucker-Prager model over the straight-line potential, which has a vertex on the hydrostatic pressure axis.

To match the hyperbolic flow potential as closely as possible to the straight-line Drucker-Prager flow potential, the parameter $ϵ$ must be set to a small value. The default value for the exponent model, $ϵ=$ 0.1, is assumed in this example. This value ensures that the results obtained with this model will not deviate substantially from an equivalent straight-line flow potential, except for a small region in the meridional plane around the triaxial extension point. The size of this region diminishes as $ϵ$ decreases. This parameter rarely needs to be modified for problems where a linear flow potential is desired for modeling the inelastic deformation. Reducing $ϵ$ to a smaller value may cause convergence problems.

The inelastic material properties are specified using the extended Drucker-Prager plasticity model with hardening.

The loading is a gravity load, $g=$ 0.666, applied to the entire model. In Abaqus/Standard the load is increased linearly from zero at the beginning of the step to its maximum value at the end of the step. In Abaqus/Explicit the load is ramped up using a smooth step amplitude definition. This amplitude definition provides a smooth loading rate, which is desirable in quasi-static or steady-state simulations.

The base of the concrete cone is held fixed in the vertical (2) direction but is free to move in the radial (1) direction. Thus, friction between the concrete and the support is not considered in this example.

This step accounts for finite strains and large displacements.

The models with the hyperbolic and exponential yield criteria use the default values for solution controls. However, for the linear Drucker-Prager model field equation tolerances are used to override the automatic calculation of the average forces to decrease the computational time required for the analysis. The convergence criteria is set to 1%, and the average force is set to 5.0 × 10⁻⁵. The convergence check for the maximum allowable correction in displacement during an increment is also disabled. In addition, the time incrementation parameters are set automatically for this model to avoid premature cutbacks of the automatic time incrementation scheme. This is done because the linear flow potential used with this model creates a discontinuity in the solution when a material point reaches the vertex of the yield surface on the hydrostatic pressure axis. The error introduced in the solution by these relaxed tolerances is not large but results in a substantial reduction in computational time.

The maximum time increment is limited in the models such that no more than 2.0% of the total load is applied in any given increment. This is done so that the point of initial yield and the shape of the inelastic response are captured accurately during the analyses (see Figure 4 and Figure 5).

The unsymmetric solver is activated for the exponential and hyperbolic yield models. This is needed because the hyperbolic flow potential used with the linear yield criteria causes nonassociated inelastic flow that results in an unsymmetric system of equations.