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Publication Number: FHWA-HRT-04-097
Date: August 2007

Measured Variability Of Southern Yellow Pine - Manual for LS-DYNA Wood Material Model 143

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1.6 HARDENING

Wood exhibits prepeak nonlinearity in compression parallel and perpendicular to the grain. Perpendicular hardening was previously demonstrated in figure 3(b) for clear pine. A choice was made between two possible approaches for modeling prepeak nonlinearity. The first approach was to model bilinear hardening behavior by specifying initial and hardening moduli. This approach simulates a sudden change in modulus. However, real data, such as that previously shown in figure 3(b), exhibits a gradual change in modulus. The second approach was to model a translating yield surface that simulates a gradual change in modulus. This is the chosen approach.

1.6.1 Model Overview

Our approach is to define initial yield surfaces that harden (translate) until they coincide with the ultimate yield surfaces, as demonstrated in figure 13 for the parallel modes. Separate formulations are modeled for the parallel and perpendicular modes. The initial location of the yield surface determines the onset of plasticity. The rate of translation determines the extent of nonlinearity.

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Figure 13.

Prepeak nonlinearity is modeled in compression with translating yield surfaces that allow user to specify the hardening response.

For each mode (parallel and perpendicular), two hardening parameters must be defined. The first parameter, N, determines the onset of nonlinearity of the load-deflection or stress-strain curves. For example, consider the case where the user wants prepeak nonlinearity to initiate at 60 percent of the peak strength. The user inputs = 0.4 so that 1 – N = 0.6. The second parameter, c, determines the rate of hardening(i.e.,how rapidly or gradually the load-deflection or stress-strain curves harden). If the user wants rapid hardening, then a large value of c is input (e.g., c = 1000). If the user wants gradual hardening, then a small value of c is input (e.g., c = 10). The value of c needed for any particular fit depends on the properties of the material being modeled. It is selected by running simulations (single-element simulations are fastest) with various values of c and comparing those simulations with data. Hardening model equations are given in section 1.6.3.

In addition to modeling prepeak nonlinearity as shown in figure 13, a separate formulation models postpeak hardening, as shown in figure 14. Instead of coinciding with the ultimate yield surface, the initial yield surface passes through the ultimate yield surface. The larger the value of the input parameter Ghard,the more pronounced the postpeak hardening. A zero value for Ghard will produce perfectly plastic behavior. The default value is zero. Currently, Ghard controls both the parallel and perpendicular behaviors simultaneously.

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Figure 14.

Postpeak hardening is modeled in compression with positive values of the parameter Ghard.

1.6.2 Default Hardening Parameters

Default hardening parameters for clear wood pine and fir are given in table 7. These hardening parameters were set by the contractor by correlating LS-DYNA simulations with compression tests conducted on 15- by 15- by 305-centimeter (cm) timbers.(2) Values of N|| = 0.5 and N^ = 0.4 were fit to the timber compression data for parallel and perpendicular behavior, respectively. These values for N|| and N^ are independent of grade, temperature, and moisture content. Values for c|| and c^ depend on the wood grade, but are independent of temperature and moisture content. Good fits to the data were obtained with the following formulas:

This figure shows two separate equations. The first equation reads Parallel hardening rate parameter equals 400 over compression quality factor superscript 2 Parallel. The second equation reads Perpendicular hardening rate parameter equals 100 over compression quality factor superscript 2 Perpendicular.

Qc = compression strength-reduction factor for graded wood discussed in section 1.12.

Table 7. Default hardening parameters for clear wood southern yellow pine and Douglas fir.
Wood Parallel Perpendicular
N|| c|| N^ c^
Southern Yellow Pine
0.5
400
0.4
100
Douglas Fir
0.5
400
0.4
100

1.6.3 Hardening Model Theory

Parallel Modes

The state variable that defines the translation of the yield surface is known as the backstress and is denoted by aij. Prepeak nonlinearity is modeled in compression, but not shear, so the only backstress required for the parallel modes is a11. The value of the backstress is a11 = 0 upon initial yield and a11 = –N||Xc at ultimate yield (in uniaxial compression). The maximum backstress occurs at ultimate yield and is equal to the total translation of the yield surface in stress space.

The hardening rule defines the growth of the backstress. Hardening rules are typically based on stress or plastic strain. Hardening is based on stress. This is accomplished by defining the incremental backstress:

This equation reads delta alpha subscript 11 equals Parallel hardening rate times parallel hardening model translation limit function times the quantity of orthotropic stress component subscript 11 minus alpha subscript 11 end quantity, times parallel scalar effective strain rate, times time-step increment.

where:
c||  = user input parameter that determines rate of translation
G||  = function that properly limits the increments
s11a11  = reduced stress that determines direction of translation (longitudinal)
effective strain-rate increment parallel to the grain  = effective strain-rate increment parallel to the grain
Dt  = time step

These terms are internally calculated by the material model and LS-DYNA, and are included to keep the hardening response independent of time step, time-step scale factor, and strain increment. The effective strain rate is a scalar value that is equal to:

This equation reads Parallel scalar effective strain-rate increment equals the square root of the quantity of strain-rate increments parallel to the grain superscript 2 plus 2 times the strain-rate increments parallel to the grain superscript 2 plus 2 times strain-rate increments parallel to the grain superscript 2, end quantity.

The function G|| restricts the motion of the yield surface so that it cannot translate outside the ultimate surface.(20) The functional form of G|| is determined from the functional form of the yield surface and the longitudinal stress definition. A brief derivation is given in appendix E. Thus, it is defined as:

This equation reads Parallel hardening model transitional limit function equals 1 minus alpha subscript 11 over parallel hardening initiation parameter orthotropic stress component superscript F equals zero.

The value of the limiting function is G||= 1 at initial yield (because a11 = 0) and G||= 0 at ultimate yield (because a11 = N|| ultimate strength in compression subscript 11 superscript Capital F). Thus, G|| limits the growth of the backstress as the ultimate surface is approached. If postpeak hardening is active, then the minimum value is maintained at G||= Ghard rather than G||= 0. The ultimate yield surface is defined from equation 13 as:

This equation reads Orthotropic stress component subscript 11 superscript F equals compression perpendicular wood strength times the square root of the quantity 1 minus isotropic material stress invariant subscript 4 over parallel shear strength superscript 2 end quantity.

For the case of uniaxial compressive stress (no shear), the ultimate yield surface reduces to ultimate strength in compression subscript 11 superscript Capital F = Xc.

Perpendicular Modes

Prepeak nonlinearity is modeled in compression, but not shear, so the backstress components required are a22 and a33. The value of the backstress sum is a22 + a33 = 0 upon initial yield and a22 + a33 = –N^Yc at ultimate yield (biaxial compression without shear). The backstress increments are defined as follows:

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where:
c^  =  user input parameter that determines rate of translation
effective strain-rate increment perpendicular to the grain  =  effective strain-rate increment perpendicular to the grain:

This equation read Perpendicular scalar effective strain-rate increment equals the square root of the quantity of strain-rate increment perpendicular to the grain superscript 2 plus strain-rate increment perpendicular to the grain superscript 2 plus 2 strain-rate increment perpendicular to the grain superscript 2, end quantity.

The functional form of G^ is determined from the functional form of the yield surface in equation 14 as:

This equation reads Perpendicular hardening model translational limit function equals 1 minus the quotient of the numerator quantity of alpha subscript 22 plus alpha subscript 33, all over perpendicular hardening initiation parameter times orthotropic stress component subscript 22 superscript F.

The value of G^ ranges from 1 at initial yield to 0 at ultimate yield. If postpeak hardening is active, then the minimum value is G^= Ghard rather than G^= 0. The ultimate yield surface is defined from equation 14 as:

This equation reads Orthotropic stress component subscript 22 superscript F equals compression perpendicular wood strength times the square root of the quantity of 1 minus the stress invariant of a transversely isotropic material subscript 3 over the perpendicular shear strength superscript 2, end quantity.

Consider the case of biaxial compressive stress (I3 = 0). Initially, a22 + a33 = 0, so G^ = 1. Ultimately, a22 + a33 = –N^Yc and I subscript 2 superscript F = s22 + s33 = Yc, so G^ = 0.

1.6.4 Implementation Aspects

The plasticity model discussed in the section 1.5 is modified to account for compressive hardening. Modifications are made to the failure surface definitions and the stress updates.

Failure Surface Definitions With Hardening

For the parallel modes, initial yielding occurs when f|| > 0, with:

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For the perpendicular modes, initial yielding occurs when f^ > 0, with:

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No modifications are necessary for the tensile modes.

Stress and Backstress Updates

Total stress is updated from the sum of the initial yield stress plus the backstress:

This equation reads Backstress tensor for hardening model superscript N plus 1 equals backstress tensor superscript N plus delta alpha backstress tensor.
This equation reads Inviscid with backstress tensor superscript N plus 1 equals inviscid stress tensor superscript N plus 1 plus backstress tensor for hardening model superscript N plus 1.

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