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Publication Number: FHWA-HRT-05-062
Date: May 2007

Users Manual for LS-DYNA Concrete Material Model 159

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Chapter 2. Theoretical Manual

This chapter begins with an overview of concrete behavior, followed by a detailed description of the formulation of concrete material model 159 in LS-DYNA. Equations are provided for each feature of the model (elasticity, plasticity, hardening, damage, and rate effects). The chapter then describes the model input properties and the basis for their default values.

CRITICAL CONCRETE BEHAVIORS

Concrete is a composite material that consists primarily of aggregate and mortar. Its response is complex, ranging from brittle in the tensile and low confining pressure regimes to ductile at high confining pressures. The critical behaviors of concrete are discussed below, particularly those in the tensile and low confining pressure regimes applicable to roadside safety analyses. Figures 1 through 14, which represent these behaviors, are reproduced from the various references cited at the end of each caption.

Stiffness. The elastic behavior of the concrete is isotropic before cracking occurs. This is because the concrete is assumed to be well mixed, vibrated, and not stratified.

Uniaxial Strength. Standard concrete has low tensile strength. The direct pull or unconfined tension strength (f 'T ) is typically 8 to 15 percent of the unconfined compression strength (f 'C ).

Multiaxial Strength. The ultimate strength of concrete depends on both the pressure and shear stresses. Concrete strength data is plotted in Figure 1 and Figure 2 in the meridian and deviatoric planes. Typical failure surfaces that may be fit to such data are represented in Figure 3 and Figure 4. Review of the reference by Chen & Han is highly recommended for a discussion of three-dimensional stress space and the meridian and deviatoric planes.(7) The general shape of a three-dimensional strength surface can be described by smooth curves in the meridian planes and by its cross-sectional shape in the deviatoric planes.

Concrete strength data is typically plotted as principal stress difference versus pressure. The principal stress difference is σx - σ r. Figure 1 is a nondimensional representation of such a plot. It is well known that concrete fails at lower values of principal stress difference for triaxial extension (TXE) tests than for triaxial compression (TXC) tests conducted at the same pressure. TXC and TXE are standard laboratory tests for measuring failure curves. These tests are typically conducted on cylindrical specimens and begin with hydrostatic compression to a desired confining pressure, i.e., the axial compressive stress, σxis equal to the radial compressive stress, σr. For TXC tests, the magnitude of the axial compressive stress is quasi-statically increased (holding σr constant) until the specimen fails. For TXE tests, the magnitude of the axial compressive stress is quasi-statically decreased until the specimen fails. TXC test data, like that shown in Figure 1, is fit the compressive meridian parameters of the concrete material model.(8,9) TXE test data is fit to the tensile meridian parameters. This behavior is schematically shown in Figure 3. The shear meridian is obtained from torsion (TOR) tests.

Concrete strength can also be plotted in the deviatoric plane, as shown in Figure 2. Here, nondimensional forms of the principal stresses (σ1, σ2, σ3) are plotted at various nondimensional pressures represented by the solid lines. In the tensile regime, strength data typically form a triangle. In the compressive regime, strength data typically transition from a triangle at low confining pressure to a circle at very high confining pressure. This behavior is schematically shown in Figure 4.

Figure 1. Graph. Example concrete data from Mills and Zimmermann plotted in the meridian plane. This plot is a reproduction from a 1988 text by Chen and Han. The Y-axis is the nondimensional ratio of the principal stress difference divided by the unconfined compressive strength, all multiplied by the square root of two-thirds. The Y-axis ranges from 0 to 6. The X-axis is the nondimensional ratio of pressure divided by unconfined compression strength. The X-axis ranges from 0 to negative 5. Two ascending, slightly nonlinear curves are shown. The top curve is the compressive meridian. The bottom curve is the tensile meridian. Each curve is fit through numerous data points taken from 1977 data by Ottosen and 1970 data by Mills et al. The point is that, for a given pressure, the triaxial compression strength is greater than the triaxial extension strength.

Figure 1. Graph. Example concrete data from Mills and Zimmermann plotted in the meridian plane.(8)

Figure 2. Graph. Example curves fit by Launay and Gachon to their concrete data and plotted in the deviatoric plane. This plot is a reproduction from a 1988 text by Chen and Han. It shows the three principal stress axes in the deviatoric plane. Each axis is in nondimensional form with the sigma 1, sigma 2, or sigma 3 stress divided by the unconfined compression strength, all multiplied by the square root of two-thirds. Seven triangular shaped curves with smooth corners are plotted. Each curve is fit through numerous data points taken from 1970 data by Launay et al.

Figure 2. Drawing. Example curves fit by Launay and Gachon to their concrete data and plotted in the deviatoric plane.(10)

Figure 3. Graph. Example plots of the failure surfaces of LS-DYNA Model 159 in the meridian plane. This plot is a reproduction from a 1988 text by Chen and Han. The Y-axis is principal stress difference multiplied by 0.8165. The X-axis is pressure times the square root of 3. It is schematic, so no units are given. It shows three ascending, slightly nonlinear curves. These are the compressive meridian, the shear meridian, and the tensile meridian. The point is that, for a given pressure, the triaxial compression strength is greater than the torsion strength, which is greater than triaxial extension strength.

psi = 145.05 MPa

Figure 3. Graph. Example plots of the failure surfaces of LS-DYNA Model 159 in the meridian plane.

Figure 4. Graph. Example plots of the failure surfaces of LS-DYNA Model 159 in the deviatoric plane. This plot is a reproduction from a 1988 text by Chen and Han. It shows the three principal stress axes in the deviatoric plane, sigma 1, sigma 2, and sigma 3. It is schematic, so no units are given. Five enclosed curves are shown surrounding the origin. The curves range from triangular closest to the origin to circular farthest away from the origin. Also shown are arrows indicating the compressive, shear, and tensile meridian directions. The compressive meridian coincides with the sigma 1 axis. The tensile meridian lays mid-angle between the sigma 1 and sigma 2 axes. The torsion meridian lays half-angle between the tensile and compressive meridians. The point is that the shape of the concrete model yield surface, in the deviatoric plane, transitions from triangular in tension, to an irregular-shaped hexagon in compression at low confining pressure, to circular at high confining pressure.

psi = 145.05 MPa

Figure 4. Drawing. Example plots of the failure surfaces of LS-DYNA Model 159 in the deviatoric plane.

Strength Degradation. Concrete softens to near zero strength in the tensile and low confining pressure regimes. This behavior is shown in Figure 5 for a variety of uniaxial strengths.(9) Concrete also softens at moderate pressures, but the concrete will exhibit a residual strength. This behavior is demonstrated in Figure 6 for concrete tested in TXC at a variety of confining pressures.

Figure 5. Graph. Softening response of concrete in uniaxial compression. Source: Comité Euro-Internacional du Béton (CEB) - Federation for Prestressing (FIP) Model Code 1990, courtesy of the International Federation for Structural Concrete (fib)). This figure is reproduced from the 1990 CEB Model Code. The Y-axis is stress in megapascals. It ranges from 0 to negative 90 megapascals. The X-axis is strain. It ranges from 0 to negative 0.01 strain. Four stress-versus-strain curves are shown. Each curve increases nonlinearly to peak stress and then softens to near zero stress. The curves are for four different strengths of concrete. These are for unconfined compression strengths of 20, 40, 60, and 80 megapascals. The higher the concrete strength, the more linear the pre-peak stress-strain curve, and the more brittle the softening. Peak strength occurs at a strain of about 0.0022 for all strengths of concrete.

psi = 145.05 MPa

Figure 5. Graph. Softening response of concrete in uniaxial compression (reprinted from the Comité Euro-Internacional du Béton (CEB) - Federation for Prestressing (FIP) Model Code 1990, courtesy of the International Federation for Structural Concrete (fib)).(11)

Figure 6. Graph. Variation of concrete softening response with confinement. Source: Joy and Moxley. This figure is reproduced from 1993 data published by Joy and Moxley. The Y-axis is principal stress difference in units of megapascals. It ranges from 0 to 300 megapascals. The X-axis is axial strain in percent. It ranges from 0 to 9 percent. Five groups of curves are shown as a function of confining pressure. The five confining pressures are 7, 14, 28, 34, and 69 megapascals. Each group of curves is nonlinear to peak strength, then each curve softens partially before unloading. The point is that the peak strength increases as the confining pressure increases, and softening becomes more ductile. The behavior is nearly perfectly plastic (very little softening) at 69 megapascals confinement.

14 MPa

psi = 145.05 MPa

Figure 6. Graph. Variation of concrete softening response with confinement.

Source: Joy and Moxley.(12)

Stiffness Degradation. As concrete softens, its stiffness also degrades. Consider the cyclic load data shown in Figure 7 through Figure 9 for uniaxial stress in both tension and compression. The unloading occurs along a different slope than the initial loading slope (elastic modulus).

Figure 7. Graph. The slope during initial loading is steeper than during subsequent loading for this uniaxial tensile stress data. Source: Reprinted with permission from Elsevier. This figure is reproduced from 1994 data published by Reinhardt and Cornelissen. The Y-axis is stress in units of megapascals. It ranges from 0 to 30 megapascals. The X-axis is displacement, called delta, in units of micrometers. It ranges from 0 to 120 micrometers. One cyclic load curve is plotted. After reaching peak strength, the curve repeatedly unloads then reloads until it softens from 26 megapascals to 2 megapascals. The slope of the unloading and reloading curves decreases with each cycle. This indicates that the modulus decreases as damage increases.

psi = 145.05 MPa

Figure 7. Graph. The slope during initial loading is steeper than during subsequent loading for this uniaxial tensile stress data.

Source: Reprinted with permission from Elsevier.(13)

Figure 8. Graph. The slope during initial loading is steeper than during subsequent loading for this uniaxial compressive stress data. Source: Reprinted with permission from Aedificatio Verlag. This figure is reproduced from 1995 data published by Lee and Willam. The Y-axis is stress in units of megapascals. It ranges from 0 to 34.45 megapascals. The X-axis is axial deformation in units of millimeters. It ranges from 0 to 15.2 millimeters for the axial curves, and 0 to 101.6 millimeters for the lateral curves. Three sets of data are plotted. These are for 137 millimeters, 91 millimeters, and 46 millimeters concrete test specimens. For each set of data, an axial deformation curve and a lateral deformation curve are plotted. After reaching peak strength, each curve repeatedly unloads then reloads until it softens from 33 megapascals to 2 megapascals. The slope of the unloading and reloading curves decreases with each cycle. This indicates that the modulus decreases as damage increases. In addition, the curves for each size test specimen lie approximately on top of each other, indicating that the stress versus displacement response is independent of specimen size. Also, effective Poisson’s ratio is greater than 0.5, because the lateral displacements become larger than the axial displacements.

Figure 8. Graph. The slope during initial loading is steeper than during subsequent loading for this uniaxial compressive stress data.

Source: Reprinted with permission from Aedificatio Verlag.(14)

Figure 9. Graph. These loading/unloading data demonstrates that concrete stiffness degrades simultaneously with strength. Source: Reprinted with permission from Aedificatio Verlag. This figure is reproduced from 1995 data published by Lee and Willam. The Y-axis is normalized stiffness and is unitless. It ranges from 0 to 1. The X-axis is normalized strength and is unitless. It ranges from 0 to 1. Numerous data points are plotted; these tend to form a straight path from position 0.0 to position 1.0. This indicates that normalized strength and stiffness degrade simultaneously.

Figure 9. Graph. These loading/unloading data demonstrate that concrete stiffness degrades simultaneously with strength.

Source: Reprinted with permission from Aedificatio Verlag.(14)

Dilation. Standard concrete exhibits volume expansion under compressive loading at low confining pressures close to pure shear and uniaxial compression. This expansion is called dilation and is as shown in Figure 10 and Figure 11 for uniaxial and biaxial compression data. The volumetric strain decreases initially, because Poisson's ratio is less than 0.5; therefore, the specimens compact in the elastic regime. Dilation initiates just before peak strength (upon initial yield) and continues throughout the softening regime. Concrete does not dilate at high confining pressures greater than about 100 MPa (14,504 psi) (not shown).

Figure 10. Graph. Concrete dilates in uniaxial compressive stress. Source: Reprinted from Defense Technical Information Center. This figure is reproduced from 1972 data from Read and Maiden. The Y-axis is stress and is unitless. The X-axis is strain and is unitless. Tensile strain is plotted negative while compressive strain is plotted positive. No magnitudes are given. Three curves are shown. These are for axial strain, volumetric strain, and transverse strain. All three curves are plotted to peak strength and become nonlinear at about two-thirds peak strength. Once nonlinearly initiates, the transverse strain begins to increase laterally, indicating that the effective Poisson’s ratio becomes large (greater than 0.5). As the transverse strain becomes large, the volumetric strain changes sign from compacting to expanding or dilating.

Figure 10. Graph. Concrete dilates in uniaxial compressive stress.

Source: Reprinted from Defense Technical Information Center.(15)

Figure 11. Graph. Concrete dilates in biaxial compression. Source: Data curves scanned from Kupfer et al., American Concrete Institute. This figure is produced by scanning and then plotting curves from 1969 data originally published by Kupfer et al. The Y-axis is normalized stress (strength divided by unconfined compression strength) and is unitless. It ranges from 0 to 1.3. The X-axis is volumetric strain and ranges from negative 0.5 in tension to 2.5 in compression. Three curves are shown. One is for uniaxial stress, which means no lateral confining stress. The second is for a lateral confining stress equal to half the axial strength. The third is for a lateral confining stress equal to the axial strength. Each curve increases to peak strength in a slightly nonlinear manner and then begins to soften. The curves demonstrate that just prior to peak strength, the incremental volumetric strain changes from compaction to expansion. Volumetric expansion decreases with confining pressure.

Figure 11. Graph. Concrete dilates in biaxial compression.
Source: Data curves scanned from Kupfer et al., American Concrete Institute (ACI).(16)

Shear Enhanced Compaction. Concrete hardens due to pore compaction. Consider the pressure-volumetric strain curves schematically shown in Figure 12. This figure demonstrates that the pressure-volumetric strain curve measured in isotropic compression tests is different from that measured in uniaxial strain tests. This difference means that the amount of compaction depends on the amount of shear stress present. This phenomenon is known as shear enhanced compaction. Slight shear enhanced compaction at low confining pressures is expected in roadside safety applications.

Figure 12. Graph. The different pressure-volumetric strain behaviors measured in isotropic compression versus uniaxial strain indicate shear enhanced compaction. Source: Data curves scanned from Joy and Moxley. These two curves were plotted from 1993 data extracted from Joy and Moxley. The Y-axis is pressure in megapascals. It ranges from 0 to 600 megapascals. The X-axis is volumetric strain, and is unitless. It ranges from 0 to 0.10. One curve is from an isotropic compression test. The other is from a uniaxial strain test. Both curves show an increase in pressure with volumetric strain. The point is that for a given strain, the pressure from the uniaxial strain test is less than the pressure from the isotropic compression test. This is because the presence of shear stress in the uniaxial strain test enhances compaction within the pore space.

psi = 145.05 MPa

Figure 12. Graph. The different pressure-volumetric strain behaviors measured in isotropic compression versus uniaxial strain indicate shear enhanced compaction.

Source: Data curves scanned from Joy and Moxley.(12)

Strain Rate Effects. The strength of concrete increases with increasing strain rate, as shown in Figure 13 and Figure 14. For roadside safety applications, strain rates in the 1 to 10 per second ( /sec) range will produce peak strength increases of about 20 to 50 percent in compression and well more than 100 percent in tension. The initial elastic modulus does not change significantly with strain rate.(17)

Figure 13. Graph. A variety of data sources indicate that the compressive strength of concrete increases with increasing strain rate. Source: Reprinted with permission from American Society of Civil Engineers. This graph is reproduced from 1995 data from Bischoff and Perry. The Y-axis is the compressive strength increase. It is nondimensional. It ranges from 0.5 to 2.5. The X-axis is strain rate in strain/second. It is plotted in log scale. It ranges from 10E-08 to 1,000. About 31 sets of concrete data are plotted. The general trend of all the data is an increase in strength with strain rate. The strength ratio ranges from 1 at a strain rate of 10E-04 per second to about 2.2 at 100 per second.

Figure 13. Graph. A variety of data sources indicates that the compressive strength of concrete increases with increasing strain rate.

Source: Reprinted with permission from American Society of Civil Engineers.(17)

Figure 14. Graph. Rate effects are more pronounced in tension than in compression. Source: Reprinted from Ross and Tedesco. This figure is reproduced from 1992 data by Ross and Tedesco. The Y-axis is the nondimensional ratio of dynamic strength divided by static strength. It ranges from negative 2 to 8. The X-axis is the logarithm of the strain rate in units of one per second. It ranges from negative 8 to 4. Two main curves are shown: one for tensile test data, and the other for compression test data. The curves are fit to numerous sets of data points. Comparison of the curves demonstrates that, for a given strain rate, the strength ratio in tension is much greater than the strength ratio in compression. For example, at a strain rate of 100 per second, the tensile strength ratio is 8 while the compression strength ratio is about 1.5.

Figure 14. Graph. Rate effects are more pronounced in tension than in compression.

Source: Reprinted from Ross and Tedesco.(18)

Overview of Model Theory

This concrete material model was developed to simulate concrete used for the National Cooperative Highway Research Program (NCHRP) 350 roadside safety hardware testing.(19) Performing roadside safety analyses with a finite element code requires a comprehensive material model for concrete, particularly for modeling strain softening in the tensile and low confining pressure regimes. Concrete material model 159 is an enhanced version of the concrete model that the developer has successfully used and progressively developed since 1990 on defense contracts to analyze dynamic loading of reinforced concrete structures. The concrete model is grouped into six formulations for ease of discussion: elastic update, plastic update, yield surface definition, damage, rate effects, and kinematic hardening. Model input parameters used in these formulations, which provide a fit of the model to data, are:

K bulk modulus
G shear modulus
α TXC surface constant term
θ TXE surface linear term
λ TXE surface nonlinear term
β TXE surface exponent
α1 TOR surface constant term
θ1 TOR surface linear term
λ1 TOR surface nonlinear term
β1 TOR surface exponent
α2 TXE surface constant term
θ2 TXE surface linear term
λ2 TXE surface nonlinear term
β2 TXE surface exponent
NH Hardening initiation
CH Hardening rate
X0 Cap initial location
R Cap aspect ratio
W Maximum plastic volume compaction
D1 Linear shape parameter
D2 Quadratic shape parameter
B Ductile shape softening parameter
Gfc Fracture energy in uniaxial stress
D Brittle shape softening parameter
Gft Fracture energy in uniaxial tension
Gfs Fracture energy in pure shear stress
pwrc Shear-to-compression transition parameter
pwrt Shear-to-tension transition parameter
pmod Modify moderate pressure softening parameter
η0c Rate effects parameter for uniaxial compressive stress
Nc Rate effects power for uniaxial compressive stress
η0t Rate effects parameter for uniaxial tensile stress
Nt Rate effects power for uniaxial tensile stress
overc Maximum overstress allowed in compression
overt Maximum overstress allowed in tension
Srate Ratio of effective shear stress to tensile stress fluidity parameters
repow Power which increases fracture energy with rate effects

Model control parameters are:

NPLOT Plotting parameter selection
INCRE Maximum strain increment for subincrementation
IRATE Option to turn rate effects on or off
ERODE Option to erode with strain at which erosion initiates
RECOV Option to recover stiffness in compression from tensile damage
IRETRC Option to retract (IRETRC = 1) or not retract (IRETRC = 0) cap
PreD Damage level for predamaged concrete

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