# 15.6. Karagozian and Case Concrete Model

    BEGIN PARAMETERS FOR MODEL KC_CONCRETE
      #
      # Elastic constants
      #
      YOUNGS MODULUS = <real>
      POISSONS RATIO = <real>
      SHEAR MODULUS  = <real>
      BULK MODULUS   = <real>
      LAMBDA         = <real>
      TWO MU         = <real>
      #
      #
      #
      COMPRESSIVE STRENGTH = <real>compressive_strength
      FRACTIONAL DILATANCY = <real>omega
      HARDEN-SOFTEN FUNCTION = <string>harden_soften_function_name
      LAMBDAM = <real>lambda_m
      LAMBDAZ = <real>lambda_z
      MAXIMUM AGGREGATE SIZE = <real>max_aggregate_size
      ONE INCH = <real>one_inch
      PRESSURE FUNCTION = <string>pressure_function_name
      RATE SENSITIVITY FUNCTION = <string>rate_function_name
      SINGLE RATE ENHANCEMENT = <enum>TRUE|FALSE
      TENSILE STRENGTH = <real>tensile_strength
      UNLOAD BULK MODULUS FUNCTION = <string>bulk_function_name
    END PARAMETERS FOR MODEL KC_CONCRETE

The Karagozian and Case (or K&C) concrete model is an inelasticity model appropriate for approximating the constitutive behavior of concrete. Coupled with appropriate elements for capturing the embedded deformation of reinforcing steel, the K&C concrete model can be used effectively for simulating the mechanical response of reinforced concrete structures. The K&C model has several useful characteristics for estimating concrete response, including strain-softening capabilities, some degree of tensile response, and a nonlinear stress-strain characterization that robustly simulates the behavior of plain concrete. This model is described in detail in [[1]].

In the above command blocks:

- Consult the Sierra/SM User Manual for more information on elastic constants input.

- The compressive strength for a uniaxial compression test is defined with the `COMPRESSIVE STRENGTH` command line.

- The tensile strength for the uniaxial tension test is defined with the `TENSILE STRENGTH` command line.

- The abscissa of the hardening/softening curve where this curve takes on the value of one is termed Lambda-M, and it is defined with the `LAMBDAM` command line ([[1]], pg. B-3).

- The abscissa of the hardening/softening curve where this curve takes on the value of zero after its peak value has been attained is termed Lambda-Z, and it is defined with the `LAMBDAZ` command line. This parameter should satisfy \(\lambda_z > \lambda_m\) ([[1]], pg. B-3). This input is Sierra-specific, and differs from the previous PRONTO3D definitions.

- The `SINGLE RATE ENHANCEMENT` parameter indicates whether the rate enhancement of the model should be independent of the sign of the deformation. If this parameter is set to `TRUE`, the same enhancement function is used for both compression and tension. If it is set to `FALSE`, the enhancement function must assign values for both positive and negative values of strain rate ([[1]], pg. B-5). This parameter is also Sierra-specific, and is different from the previous PRONTO3D definitions.

- The `FRACTIONAL DILATANCY` is an estimate of the size of the plastic volume strain increment relative to that corresponding to straining in the hydrostatic plane. This value normally ranges from 0.3 to 0.7, and a value of one-half is commonly used in practice.

- The `MAXIMUM AGGREGATE SIZE` parameter provides an estimate of the largest length dimension for the aggregate component of the concrete mix. The American Concrete Institute code [[2]] includes specifications for maximum aggregate size that are based on member depth and clear spacing between adjacent reinforcement elements. This parameter is also useful in modifying the post-peak fall-off of both compressive and tensile behaviors. A large aggregate size (e.g. 2.5 inches) may result in a rapid drop after reaching peak stress. A small aggregate size (e.g. 0.5 inches) may result in a gradual decline after reaching peak stress.

Warning

Simulations utilizing the K&C model can be subject to damage-failure waves wherein the initiation of element damage can cause a rapid chain reaction of damage that propagates through the entire simulation within a few time steps. This damage-failure wave could be caused by a too-sudden drop in element strength after failure. One method to stabilize the simulation is to lower the value used for `MAXIMUM AGGREGATE SIZE`, allowing for a more gradual post-peak element strength decline.

- The parameter `ONE INCH` provides for conversion to units other than the pounds/inch system commonly used in U.S. concrete venues. This parameter should be set to the equivalent length in the system used for analysis. If centimeters are to be used, for example, then `ONE INCH = 2.54`.

The following functions describe the evolution of material coefficients in this model:

- The function characterizing the enhancement of strength with strain rate is described via the `RATE SENSITIVITY FUNCTION` ([[1]], pg. B-3).

Warning

The `RATE SENSITIVITY FUNCTION` command should be used with caution. The implementation appears to overestimate concrete strength in tension, and users are cautioned to provide rate sensitivity function values that have the value of 1.0 for positive (tensile) values of strain rate. These values correspond to no additional strength in tension due to strain rate, and are both physically realistic and conservative.

- The function describing the relationship between pressure and volumetric strain is described via the `PRESSURE FUNCTION`.

- The function characterizing the relationship between bulk modulus and volumetric strain during unloading is described via the `UNLOAD BULK MODULUS FUNCTION`.

- The function describing the hardening and softening functions function eta as a function of the material parameters lambda (see `LAMBDAM` and `LAMBDAZ`) is defined via the `HARDEN-SOFTEN FUNCTION` (see [[1]], pg. B-3). The `HARDEN-SOFTEN FUNCTION` dictates damage accrual. `ETA` is a function of `LAMBDA`. At `LAMBDA` = `ETA` = 0, the material is undamaged. At `ETA(LAMBDA)` = 1, damage = 1; the concrete has reached maximum stress and cannot support more. At `ETA(LAMBDA)` = 0 (after `ETA(LAMBDA)` = 1), damage is approaching 2; the concrete is mostly rubble/cracked. At damage = 2, the concrete has fully become rubble/cracked.

The above-listed functions are calculated as follows. The `HARDEN-SOFTEN FUNCTION` is referenced in [[1]], pg. B-3.

    RATE SENSITIVITY FUNCTION CALCULATIONS
      #
      # Equation Constants
      #
      delta   = 1/(1+8*fpc/1450) fpc in psi
      beta    = 10**(6*delta-2)
      difmax  = beta*(300/1e-6)**(1/3)
      alpha   = 1/(5+9*fpc/1450) fpc in psi
      gamma   = 10**(6.156*alpha-2)
      difmaxc = gamma*(300/(30e-6)**(1/3))
      #
      # Value Calculations
      #
      -30.e5     difmax
      -3.0e2     difmax
      -100.0     beta*(100/1e-6)**(1/3)
      -10.00     beta*(10/1e-6)**(1/3)
      -1.000     (1/1e-6)**delta
      -0.100     (0.1/1e-6)**delta
      -0.010     (0.01/1e-6)**delta
       0.000     1.0
       0.010     (0.01/30e-6)**(1.026*alpha)
       0.100     (0.1/30e-6)**(1.026*alpha)
       1.000     (1/30e-6)**(1.026*alpha)
       30.00     gamma*(30/30e-6)**(1/3)
       100.0     gamma*(100/30e-6)**(1/3)
       300.0     difmaxc
       30.e5     difmaxc

    PRESSURE FUNCTION CALCULATIONS
      #
      # Equation Constants
      #
      bulk0 = MATERIAL BULK MODULUS
      p2    = bulk0*0.15e-2
      #
      # Value Calculations
      # volume strain vs pressure
      0.0        0.0
      0.15e-2    p2
      0.27e-2    p2*1.53
      0.43e-2    p2*2.18
      0.6e-2     p2*2.74
      0.8e-2     p2*3.13
      0.197e-1   p2*5.13
      0.89e-1    p2*21.7
      0.1e1      p2*221.9
      0.1e4      p2*221.9

    UNLOAD BULK MODULUS FUNCTION CALCULATIONS
      #
      # Equation Constants
      #
      bulk0 = MATERIAL BULK MODULUS
      #
      # Value Calculations
      # volume strain vs bulk modulus
      0.0        bulk0
      0.15e-2    bulk0
      0.27e-2    min(10*bulk0,
                  bulk0*(1+10*(0.27e-2 - 0.15e-2)))
      0.43e-2    min(10*bulk0,
                  bulk0*(1+10*(0.43e-2 - 0.15e-2)))
      0.60e-2    min(10*bulk0,
                  bulk0*(1+10*(0.6e-2 - 0.15e-2)))
      0.80e-2    min(10*bulk0,
                  bulk0*(1+10*(0.8e-2 - 0.15e-2)))
      0.197e-1   min(10*bulk0,
                  bulk0*(1+10*(0.197e-1 - 0.15e-2)))
      0.89e-1    min(10*bulk0,
                  bulk0*(1+10*(0.89e-1 - 0.15e-2)))
      0.1e1      min(10*bulk0,
                  bulk0*(1+10*(0.1e1 - 0.15e-2)))
      0.1e4      min(10*bulk0,
                  bulk0*(1+10*(0.1e1 - 0.15e-2)))

The following are sample values for a concrete with a compressive strength of 7000 psi (base units inch-pounds):

    RATE SENSITIVITY FUNCTION SAMPLE VALUES
      -30.e5     9.4873
      -3.0e2     9.4873
      -100.0     6.5781
      -10.00     3.0533
      -1.000     1.0201
      -0.100     1.0190
      -0.010     1.0179
       0.000     1.0
       0.010     1.1310
       0.100     1.1874
       1.000     1.2468
       30.00     1.3399
       100.0     2.0015
       300.0     2.8867
       30.e5     2.8867

    PRESSURE FUNCTION SAMPLE VALUES
      0.0        0.0
      0.15e-2    3974
      0.27e-2    6080
      0.43e-2    8664
      0.6e-2     10889
      0.8e-2     12439
      0.197e-1   20387
      0.89e-1    86239
      0.1e1      881861
      0.1e4      881861

    UNLOAD BULK MODULUS FUNCTION SAMPLE VALUES
      0.0        2649423
      .15e-2     2649423
      .27e-2     2681216
      .43e-2     2723607
      .60e-2     2768647
      .80e-2     2821636
      .197e-1    3131618
      .89e-1     4967669
      .1e1       26494234
      .1e4       26494234

    HARDEN-SOFTEN FUNCTION SAMPLE VALUES
    (for most concrete strengths)
    (damage parameter d also shown for reference)
      0          0.0     # --> d=0
      8e-06      0.85    # --> d=0.25
      2.4e-05    0.97    # --> d=0.6
      4e-05      0.99    # --> d=0.8333333333
      5.6e-05    1.0     # --> d=1
      7.2e-05    0.99    # --> d=1.125
      8.8e-05    0.97    # --> d=1.222222222
      .00032     0.5     # --> d=1.70212766
      .00052     0.1     # --> d=1.805555556
      .00057     0.0     # --> d=1.821086262
      1.00056    0.0     # --> d=1.999888013
      10.00056   0.0     # --> d=1.9999888
      1e+10      0.0     # --> d=2

[1] (1,2,3,4,5,6,7)

S.W. Attaway, R.V. Matallucci, S.W. Key, K.B. Morrill, L.J. Malvar, and J.E. Crawford. Enhancements to Pronto3D to predict structural response to blast. Technical Report SAND2000-1017, Sandia National Laboratories, Albuquerque, NM, 2000. URL: http://infoserve.sandia.gov/sand_doc/1972/720883.pdf.

[2]

ACI318-08: building code requirements for structural concrete and commentary. Farmington Hills, MI. American Concrete Institute. 2008.