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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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1.5 PLASTIC FLOW

The plasticity algorithms limit the stress components once the failure criterion in equations 13 or 14 is satisfied. This is done by returning the stress state back to the yield surface.1

Our traditional approach for modeling plasticity is to partition the stress and strain tensors into elastic and plastic parts. Partitioning is done with return mapping algorithms that enforce the plastic consistency conditions. Such algorithms allow one to control plastic strain generation. In addition, return mapping algorithms with associated flow satisfy the second law of thermodynamics. Associated flow is discussed in appendix D.

1Two simple types of plasticity algorithms are those that reduce the moduli directly and those that scale back the stresses directly. Although simple to implement, such methods do not allow one to control plastic strain generation and do not necessarily satisfy the second law of thermodynamics.

1.5.1 Consistency Parameter Updates

Separate plasticity algorithms are modeled for the parallel and perpendicular modes by enforcing separate consistency conditions. Each consistency condition is derived in appendix D. The solution of each consistency condition determines the consistency parameters Dlúú and Dl^. The Dl solutions, in turn, determine the stress updates.

Parallel Modes

The parallel failure criterion from equation 13 is redefined as f|| (I1, I4) ≥ 0, with:

This equation reads Parallel yield surface as a function of stress invariant subscript 1, stress invariant subscript 4, equals stress invariant subscript 1 superscript 2 over parallel wood strength superscript 2 plus stress invariant subscript 4 over parallel shear strength superscript 2 minus 1.

where I1 and I4 are two of the five invariants of a transversely isotropic material. In this case, the expression Dl|| is:

click on the image for Section 508 compliancy text

Dl|| is calculated from specification of the total strain increments and the yield function Parrellel f star equals parrellel f (I subscript 1 star, I subscript 4 star) . The trial elastic stress invariants, I subscript 1 star and I subscript 4 star*, are calculated from the trial elastic stresses (see section 1.5.2). For the failure surface function in equation 15, the partial derivatives in equation 16 are:

This equation reads lowercase delta parallel F over lowercase delta I subscript 1 equals 2 stress invariant subscript 1 over general parallel wood strength superscript 2.
This equation reads lowercase delta parallel F over lowercase delta I subscript 4 equals 1 over parallel shear stress superscript 2.
This equation reads lowercase delta I subscript 1 over lowercase delta parallel lambda equals negative 2 times C subscript 11 times stress invariant subscript 1 over general parallel wood strength superscript 2.
This equation reads lowercase delta I subscript 4 over lowercase delta parallel lambda equals negative 4 shear moduli of an orthotropic material subscript 12 stress invariant subscript 4 over parallel shear strength superscript 2.

Perpendicular Modes

The perpendicular failure criterion from equation 14 is redefined as f^ (I3, I4) ≥ 0, with:

This equation reads Perpendicular yield surface as a function of isotropic stress invariant subscript 2, isotropic stress invariant subscript 3, equals isotropic stress invariant subscript 2 superscript 2 over perpendicular wood strength superscript 2 plus isotropic stress invariant subscript 3 over perpendicular shear strength superscript 2 minus 1.

where I2 and I3 are two of the five invariants of a transversely isotropic material. In this case, the expression for Dl^ is:

click on the image for Section 508 compliancy text

Dl^is calculated from specification of the total strain increments and the yield function F perpendicular superscript star = f perpendicular (I subscript 2 star, I subscript 3 star), where I subscript 2 star and I subscript 3 star are the trial elastic stress invariants. For the failure surface function in equation 21, the partial derivatives in equation 22 are:

This equation reads lowercase delta perpendicular F over delta I subscript 2 equals 2 stress invariant subscript 2 over general perpendicular wood strength subscript 2.
This equation reads lowercase delta perpendicular F over lowercase delta I subscript 3 equals 1 over perpendicular shear strength superscript 2.
This equation reads the quotient of lowercase delta I subscript 2 over lowercase delta perpendicular lambda equals the quantity C subscript 22 plus C subscript 23 end quantity, times stress invariant subscript 2, times the quantity of negative 4 over general perpendicular wood strength superscript 2 plus 1 over perpendicular shear strength superscript end quantity.
click on the image for Section 508 compliancy text

1.5.2 Elastoplastic Stress Updates

The stresses are readily updated from the total strain increments and the consistency parameters:

This equation reads inviscid stress tensor superscript N plus 1 equals trial elastic stress tensor superscript N plus 1 minus the product of material stiffness tensor plasticity consistency parameter times delta lambda, times lowercase delta F over lowercase delta sigma subscript KL, evaluated at N.
This equation reads Trial elastic stress tensor superscript N plus1 equals viscid with damage stress tensor superscript N plus material stiffness times delta epsilon subscript KL.

where:
n       = nth time step in the finite element analysis
s*ij   = trial elastic stress updates calculated from the total strain increments, Deij, prior to application of plasticity

Total strain increments are calculated by the finite element code from the dynamic equations of motion and the time step.

Each normal stress update depends on the consistency parameters and failure surface functions for both the parallel (Dl = Dl|| and f = f||) and perpendicular (Dl = Dl^ and f = f^) modes. Each shear stress update depends on just one consistency parameter and failure surface function. If neither parallel nor perpendicular yielding occurs F parallel superscript star is less than 0 and F perpendicular superscript star is less than 0 , then Dl = 0 and the stress update is trivial: inviscid stress tensor superscript N plus 1 equals trial elastic stress tensor superscript N plus 1.

For the stress updates in equation 27, the partial derivatives are:

This equation reads lowercase delta parallel F over orthotropic stress component subscript 11 equals 2 times orthotropic stress component subscript 11 over general parallel wood strength superscript 2.
This equation reads lowercase delta perpendicular F over lowercase delta sigma subscript 22 equals 2 stress invariant subscript 2 over general perpendicular wood strength superscript 2 minus orthotropic stress component subscript 33 over perpendicular shear strength superscript 2.
This equation reads lowercase delta perpendicular F over lowercase delta sigma subscript 33 equals 2 times the stress invariant subscript 2 over general perpendicular wood strength superscript 2, all minus orthotropic stress component subscript 22 over perpendicular shear strength superscript 2.
This equation reads lowercase delta parallel F over lowercase sigma subscript 12 equals 2 times orthotropic stress component subscript 12 over parallel shear strength superscript 2.
This equation reads lowercase delta parallel F over lowercase sigma subscript 13 equals 2 times orthotropic stress component subscript 13 over parallel shear strength superscript 2.
This equation reads lowercase delta perpendicular F over lowercase sigma subscript 23 equals 2 times orthotropic stress component subscript 23 over perpendicular shear strength superscript 2.

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