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Publication Number: FHWA-HRT-05-063
Date: May 2007
Evaluation of LS-DYNA Concrete Material Model 159
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Chapter 2. Single Element Simulations
Single element simulations were performed with ls-dyna to verify the implementation of the concrete model and to display the basic stress-strain behavior of the model. All simulations were conducted with cubic hex elements, 38.1 mm (1.5 inches) in each dimension. Unless otherwise specified, approximate default material properties are used for 28 MPa (4,061 lbf/inch2) concrete with 19 mm (0.75 inches) maximum aggregate size. Stresses and strains are plotted positive in tension unless otherwise noted.
Uniaxial and Triaxial Stress
Typical behavior of the concrete model in uniaxial and triaxial stress is shown in Figures 1 and 2. In uniaxial tensile stress and pure shear stress, the stress-displacement behavior is modeled linear to peak, followed by brittle softening. Although not shown, the behavior of the model in biaxial and triaxial tensile stress is nearly identical to that shown for uniaxial tensile stress, with the same peak strength and softening response.
In uniaxial compressive stress, the stress-displacement behavior is also modeled linear to peak followed by gradual softening to near zero strength. The unconfined compressive strength is approximately 10 times the unconfined tensile strength, in agreement with typical test data for concrete. The fracture energy shown in unconfined compression is approximately 100 times greater than the fracture energy in unconfined tension. The energy is the area under the softening portion of the stress versus displacement curve. The developer recommends values between 50 and 100 for the compression to tension fracture energy ratio. A value of 100 was ultimately selected at the end of the program as the default value.
The concrete elements also soften when confined at moderate pressure, although they exhibit a residual strength (plateau) rather than softening to zero. Triaxial compression refers to tests or simulations in which the lateral confining pressure is held constant around the specimen (element) while the axial compression load is quasi-statically increased. The concrete strength increases with confining pressure in the simulations, in qualitative agreement with typical test data.
Figure 1. The default behavior of the concrete model in uniaxial tensile stress and pure shear stress is linear to the peak, followed by brittle softening.
Figure 2. Strength and ductility increase with confining pressure in these triaxial compression simulations.
Standard concrete exhibits volume expansion under compressive loading at low confining pressures less than about 100 MPa (14,504 lbf/inch2). This volume expansion is called dilation and is calculated in the concrete model by the plasticity formulation. The volumetric behavior of the model is shown in Figure 3 for uniaxial compressive stress for 35 MPa (5,076 lbf/inch2) concrete. The volume initially compacts during the elastic phase, because Poisson's ratio is less than 0.5. Volume expansion commences at peak strength and continues throughout the combined plasticity-softening phase. The effective Poisson's ratio during dilation may exceed 0.5
As concrete softens, its stiffness also degrades. Stiffness degradation is shown in Figure 4 for uniaxial stress with cycling from tension to compression, and then back to tension for 45 MPa (6,527 lbf/inch2) concrete. The unloading occurs along a different slope (degraded modulus), then the initial loading slope (undamaged modulus). In addition, when the state of stress transitions from tension to compression, the undamaged modulus is recovered (recovered modulus). This is the default option obtained with recov = 0. If recov = 1, the modulus remains fully degraded when transitioning from tension to compression. If recov is set between 0 and 1, the modulus remains partially degraded.
Pressure Versus Volumetric Strain
Concrete hardens due to pore compaction. Consider the pressure versus volumetric strain behavior of the model shown in Figure 5. One simulation was conducted for isotropic compression (also called hydrostatic compression), the other for uniaxial strain. Note that the pressure versus volumetric strain curves differ. This is because the amount of compaction depends upon the amount of shear stress present. Less pressure is required to compact concrete to a given volumetric strain when shear stress is present (the uniaxial strain simulation), than when shear stress is not present (the isotropic compression simulation). This behavior is known as shear enhanced compaction. It occurs in concrete and most geological materials. Slight shear enhanced compaction at low confining pressures is expected in roadside safety applications.
Figure 3. The concrete model simulates volume expansion in uniaxial compressive stress, in agreement with typical test data (strains and stress positive in compression).
Figure 4. The modulus of concrete degrades with strength, as demonstrated by this cyclic loading simulation.
Figure 5. The difference in pressure at a given volumetric strain for these isotropic compression and uniaxial strain simulations is due to shear enhanced compaction.
The strength of concrete increases with increasing strain rate. For roadside safety applications, strain rates in the 1 to 10 per second (sec) range will produce peak strength increases of about 35 to 50 percent in compression and 150 to 300 percent in tension. The initial elastic modulus does not change significantly with strain rate.
Rate effects are computationally demonstrated in Figures 6-8 for simulations conducted in uniaxial tensile stress, pure shear stress, and uniaxial compression stress. The simulations were conducted at a high-strain rate of 100/sec. Note that the increase in strength in tension and shear is much greater than that in compression. Also note that the softening response depends on the value of repow. A value of repow = 0 minimizes the fracture energy. A value of repow = 1 increases the fracture energy in proportion to the increase in strength, and is the default value.
Figure 6. The increase in strength with strain rate is significant in tension at a strain rate of 100 per second.
Figure 7. The increase in strength with strain rate in pure shear stress is similar to that modeled in uniaxial tensile stress.
Figure 8. The increase in strength with strain rate in uniaxial compression stress is less pronounced than in uniaxial tensile stress or pure shear stress.
One optional variation on the behavior previously shown in Figure 2 is the addition of prepeak nonlinearity simulated with the kinematic hardening formulation. Such behavior is shown in Figure 9 for uniaxial stress in compression. Dilation initiates when hardening initiates, just prior to peak strength. Prepeak nonlinearity is more pronounced in compression than in tension or shear. This formulation is optional and not default because it is not essential to good performance of the model. The developer has found that accurate modeling of the damage fracture energies and rate effects are the key behaviors needed for good correlations with most dynamic roadside safety test data.
Figure 9. Application of kinematic hardening simulates prepeak nonlinearity accompanied by plastic volume expansion.