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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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APPENDIX E. DERIVATION OF CONSISTENCY PARAMETER FOR PLASTICITY ALGORITHM

Plasticity is modeled by enforcing separate consistency conditions for the parallel and perpendicular modes. The goal of each consistency condition is to partition the total strain increments into elastic and plastic components:

This equation reads Strain increment equals strain increment superscript e plus strain increment superscript p.

The superscripts e and p indicate the elastic and plastic components, respectively. The total strain increments (Deij) are calculated by the finite element code from the dynamic equations of motion and the time step. Once this partition is known, then the stress increments are updated from the elastic strain increments:

This equation reads viscid with damage stress tensor superscript the sum of n plus 1 equals viscid with damage stress tensor superscript n plus the product of material stiffness tensor times parenthesis strain increment subscript KL minus delta strain increment subscript kl superscript p parenthesis.

Here, n denotes the nth time step in the finite element analysis.

E.1 CHECK FOR YIELDING

The partition into elastic and plastic components requires two steps. The first step is to check for yielding. This is done by temporarily updating the stress components from the incremental strains by assuming that the entire strain increment is elastic:

This equation reads trial elastic stress tensor superscript n plus 1 equals viscid with damage stress tensor superscript n plus the product of material stiffness tensor times strain increment subscript KL.

These updated stresses are called the trial elastic stresses (trial elastic stress tensor). The trial elastic stress invariants (I subscript 1 star, I subscript 2 star, I subscript 3 star, I subscript 4 *are updated from the trial elastic stresses. The value of the yield function is evaluated from the trial elastic invariants and is denoted as f *. The new stress state is elastic if * ≤ 0 and plastic if * > 0.

E.2 CALCULATE CONSISTENCY PARAMETER

The second step is to enforce the consistency condition if *> 0. Enforcement of the consistency condition requires an assumption about the direction of plastic flow. It is assumed that the plastic strain increments are normal to the yield surface:

This equation reads Strain increment superscript p equals the product of plasticity consistency parameter divided by delta viscid with damage stress tensor evaluated at n.

where Dl is a proportionality constant known as the consistency parameter.

This assumption is known as an associated flow rule, or normality condition. Use of a potential function other than the yield function in equation 180 results in a nonassociated flow rule. Recent studies reported by Pucik(35) suggest that rate-independent models with nonassociated flow lead to spurious (non-unique) dynamic solutions, so only associated flow is proposed for the present model.

The plasticity algorithm calculates Dl by enforcing the plastic consistency condition. This condition is expressed as:

This equation reads delta yield surface function equals the difference of yield surface function superscript the sum of n plus 1 minus yield surface function superscript n which in turn equals zero.

where f is the yield surface function at time increments n to n + 1. The stress state at the beginning of the time step lies on the yield surface, thus, f n = 0. The stress state at the end of the time step is returned to the yield surface by the plasticity algorithm, thus, f n+1 = 0. Therefore, Df = 0.

The solution of the consistency condition in equation 181 determines Dl, which, in turn, determines the partitioning of the total strain rate into elastic and plastic components. The stresses are updated from the elastic strain components. Separate Dl solutions are proposed for the parallel and perpendicular modes.

Parallel Modes

The consistency condition is derived in terms of the invariants rather than the stresses, because the parallel failure criterion in equation 13 is formulated in terms of two transversely isotropic stress invariants. For the purposes of this derivation, the parallel failure criterion from equation 13 is defined as f||(I1,I4) ≥ 0, with:

This equation reads parallel yield surface function parenthesis Trial elastic stress invariant subscript 1, Trial elastic stress invariant subscript 4 parenthesis equals the quotient of Trial elastic stress invariant subscript 1 superscript 2 divided by general parallel wood strength superscript 2 plus the quotient of Trial elastic stress invariant subscript 4 divided by parallel shear strength superscript 2 minus 1.

A first-order Taylor series expansion of the consistency condition Delta parallel yield surface function equals parallel yield surface function superscript n plus 1 minus parallel yield surface function superscript n equals 0  from time increment n to n + 1 gives:

This equation reads the product of the quotient of delta parallel yield surface function divided by delta Trial elastic stress invariant subscript 1 evaluated at lowercase N times delta trial elastic stress invariant subscript 1 plus the product of the quotient delta parallel yield surface function divided by delta trial elastic stress invariant subscript 4 subscript evaluated at n times delta I subscript 4 equals zero.

Expansion of the stress invariant increments gives:

This equation reads delta Trial elastic stress invariant subscript 1 equals trial elastic increment subscript 1 plus the product of the quotient delta Trial elastic stress invariant subscript 1 divided by parallel plasticity consistency parameter evaluated at n times parallel plasticity consistency parameter.
This equation reads Trial elastic stress invariant subscript 4 equals trial elastic stress invariant subscript 4 plus the product of the quotient of delta Trial elastic stress invariant subscript 4 divided by parallel consistency parameter evaluated at n times the parallel consistency parameter.

where trial elastic stress invariant subscript 1 and trial elastic stress invariant subscript 4 are the trial elastic increments calculated with Dl|| = 0. Substitution of the updates from equations 184 and 185 into the consistency condition in equation 183 results in the following expression for Dl||:

click on the image for Section 508 compliancy text

The expression in the numerator is recognized as the first-order Taylor series expansion of parallel trial elastic yield surface function, where parallel trial elastic yield surface function equals parallel trial elastic yield surface function (I subscript 1 star, I subscript 4 star)  is the value of the failure criterion calculated from the trial elastic invariants. Therefore, the expression for Dl|| reduces to:

click on the image for Section 508 compliancy text

Perpendicular Modes

The perpendicular failure criterion in equation 14 is derived in terms of the invariants rather than the stresses. The perpendicular failure criterion is defined as f^(I2,I3) = 0, with:

This equation reads perpendicular yield surface as a function of Trial elastic stress invariant subscript 2, Trial elastic stress invariant subscript 3 equals the quotient of Trial elastic stress invariant subscript 2 superscript 2 divided by perpendicular wood strength superscript 2 plus the quotient of Trial elastic stress invariant subscript 3 divided by perpendicular shear strength superscript 2 minus 1.

Expansion of the consistency condition Df^ = 0 gives:

This equation reads the quotient of delta perpendicular yield surface function divided by delta Trial elastic stress invariant subscript 2 evaluated at n times delta Trial elastic stress invariant subscript 2 plus the quotient delta perpendicular yield surface function divided by delta Trial elastic stress invariant subscript 3 evaluated at n times Trial elastic stress invariant subscript 3 equals 0.

The stress invariant updates are:

This equation reads delta Trial elastic stress invariant subscript 2 equals trial elastic increment subscript 2 plus the product of the quotient delta Trial elastic stress invariant subscript 2 divided by perpendicular plasticity consistency parameter evaluated at n times perpendicular plasticity consistency parameter.
This equation reads delta Trial elastic stress invariant subscript 3 equals trial elastic increment subscript 3 plus the product of the quotient delta Trial elastic stress invariant subscript 3 divided by perpendicular plasticity consistency parameter evaluated at n times perpendicular plasticity consistency parameter.

Substitution of these updates into the consistency condition gives the following express for Dl^.

click on the image for Section 508 compliancy text

E.3 UPDATE STRESSES

The third step is to update the stresses. There are two options: (1) a purely elastic update and (2) an elastoplastic update. If the trial elastic stress state lies inside the failure surface (*< 0), then Dl = 0 and the stress state is purely elastic. In this case, the stress update from equations 178 and 179 is trivial:

This equation reads Viscid with damage stress tensor superscript the sum of n plus 1 equals trial elastic stress tensor superscript the sum of n plus 1.

If the trial elastic stress state lies outside the failure surface (f * > 0), then Dl ¹ 0 and the stress state is elastoplastic. In this case, equations 178, 179, and 180 combine to give:

This equation reads Viscid with damage stress tensor superscript the sum of n plus 1 equals the difference of trial elastic stress tensor superscript the sum of n plus 1 minus the product of material stiffness tensor times plasticity consistency parameter times the quotient of yield surface function divided by stress component subscript lowercase KL evaluated at n.

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