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19.47

1

1p

C

εβ = + (19.24.2)

where ε is the strain rate.

II. For complete generality a load curve, defining β , which scales the yield stress may be input instead. In this curve the scale factor versus strain rate is defined.

III. If different stress versus strain curves can be provided for various strain rates, the option using the reference to a table definition can be used. See Figure 19.24.1.

A fully viscoplastic formulation is optional which incorporates the different options above within the yield surface. An additional cost is incurred over the simple scaling but the improvement is results can be dramatic.

If a table ID is specified a curve ID is given for each strain rate, see Section 23. Intermediate values are found by interpolating between curves. Effective plastic strain versus yield stress is expected. If the strain rate values fall out of range, extrapolation is not used; rather, either the first or last curve determines the yield stress depending on whether the rate is low or high, respectively.

Material Model 25: Kinematic Hardening Cap Model The implementation of an extended two invariant cap model, suggested by Stojko [1990], is based on the formulations of Simo, et al. [1988, 1990] and Sandler and Rubin [1979]. In this model, the two invariant cap theory is extended to include nonlinear kinematic hardening as suggested by Isenberg, Vaughn, and Sandler [1978]. A brief discussion of the extended cap model and its parameters is given below.

The cap model is formulated in terms of the invariants of the stress tensor. The square

root of the second invariant of the deviatoric stress tensor, 2DJ is found from the deviatoric

stresses s as

2

1

2D ij ijJ s s≡

and is the objective scalar measure of the distortional or shearing stress. The first invariant of the stress, 1J , is the trace of the stress tensor.

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Figure 19.25.1. Rate effects may be accounted for by defining a table of curves.

Figure 19.25.2. The yield surface of the two-invariant cap model in pressure 2 1DJ J− space

Surface 1f is the failure envelope, 2f is the cap surface, and 3f is the tension

cutoff.

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The cap model consists of three surfaces in 2 1DJ J− space, as shown in Figure 19.25.1.

First, there is a failure envelope surface, denoted 1f in the figure. The functional form of 1f is

( )( )1 2 1min ,D e misesf J F J T= − ,

where eF is given by

( ) ( )1 1 1expeF J J Jα γ β θ≡ − − +

and ( ) ( )mises n nT X Lκ κ≡ − . This failure envelope surface is fixed in 2 1DJ J− space, and

therefore does not harden unless kinematic hardening is present. Next, there is a cap surface, denoted 2f in the figure, with 2f given by

( )2 2 1,D cf J F J κ= −

where cF is defined by

( ) ( ) ( ) ( )2 2

1 1

1, ,cF J X L J L

Rκ κ κ κ≡ − − −

( )X κ is the intersection of the cap surface with the 1J axis

( ) ( ) ,eX RFκ κ κ= +

and ( )L κ is defined by

( ) 0

0 0

if L

if

κ κκ

κ>

≡≤

.

The hardening parameter κ is related to the plastic volume change pvε through the hardening

law

( )( ){ }01 exppv W D X Xε κ= − − −

Geometrically, κ is seen in the figure as the 1J coordinate of the intersection of the cap surface

and the failure surface. Finally, there is the tension cutoff surface, denoted 3f in the figure. The

function 3f is given by

3 1,f T J+ −

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where T is the input material parameter which specifies the maximum hydrostatic tension

sustainable by the material. The elastic domain in 2 1DJ J− space is then bounded by the

failure envelope surface above, the tension cutoff surface on the left, and the cap surface on the right. An additive decomposition of the strain into elastic and plastic parts is assumed:

Pε ε ε= + ,

where eε is the elastic strain and pε is the plastic strain. Stress is found from the elastic strain using Hooke’s law,

( )PCσ ε ε= − ,

where σ is the stress and C is the elastic constitutive tensor. The yield condition may be written

( )( )( )

1

2

3

0

, 0

0

f

f

f

σσ κσ

≤

≤

≤

and the plastic consistency condition requires that

0

1,2,3

0

k k

k

f

k

λ

λ

==

≥

where kλ is the plastic consistency parameter for surface k . If kf < 0 , then 0kλ = and the

response is elastic. If kf 0> , then surface k is active and kλ is found from the requirement that

0kf = .

Associated plastic flow is assumed, so using Koiter’s flow rule the plastic strain rate is given as the sum of contribution from all of the active surfaces,

3

1

p kk

k=

f

s

∂ε λ∂

=

One of the major advantages of the cap model over other classical pressure-dependent plasticity models is the ability to control the amount of dilatency produced under shear loading. Dilatency is produced under shear loading as a result of the yield surface having a positive slope

in 2 1DJ J− space, so the assumption of plastic flow in the direction normal to the yield surface

produces a plastic strain rate vector that has a component in the volumetric (hydrostatic)

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direction (see Figure 19.25.1). In models such as the Drucker-Prager and Mohr-Coulomb, this dilatency continues as long as shear loads are applied, and in many cases produces far more dilatency than is experimentally observed in material tests. In the cap model, when the failure surface is active, dilatency is produced just as with the Drucker-Prager and Mohr-Columb models. However, the hardening law permits the cap surface to contract until the cap intersects the failure envelope at the stress point, and the cap remains at that point. The local normal to the yield surface is now vertical, and therefore the normality rule assures that no further plastic volumetric strain (dilatency) is created. Adjustment of the parameters that control the rate of cap contractions permits experimentally observed amounts of dilatency to be incorporated into the cap model, thus producing a constitutive law which better represents the physics to be modeled. Another advantage of the cap model over other models such as the Drucker-Prager and Mohr-Coulomb is the ability to model plastic compaction. In these models all purely volumetric response is elastic. In the cap model, volumetric response is elastic until the stress point hits the cap surface. Therefore, plastic volumetric strain (compaction) is generated at a rate controlled by the hardening law. Thus, in addition to controlling the amount of dilatency, the introduction of the cap surface adds another experimentally observed response characteristic of geological material into the model. The inclusion of kinematic hardening results in hysteretic energy dissipation under cyclic loading conditions. Following the approach of Isenberg, et al., [1978] a nonlinear kinematic hardening law is used for the failure envelope surface when nonzero values of and N are specified. In this case, the failure envelope surface is replaced by a family of yield surfaces bounded by an initial yield surface and a limiting failure envelope surface. Thus, the shape of the yield surfaces described above remains unchanged, but they may translate in a plane orthogonal to the J axis. Translation of the yield surfaces is permitted through the introduction of a “back stress” tensor, α . The formulation including kinematic hardening is obtained by replacing the stress σ with the translated stress tensor η σ α≡ − in all of the above equation. The history tensor αis assumed deviatoric, and therefore has only 5 unique components. The evolution of the back stress tensor is governed by the nonlinear hardening law

( ), pcFα σ α ε=

where c is a constant, F is a scalar function of σ and α and pε is the rate of deviator plastic strain. The constant may be estimated from the slope of the shear stress - plastic shear strain curve at low levels of shear stress. The function F is defined as

( )( )1

max 0,12 e

FNF J

σ α α− •≡ −

where N is a constant defining the size of the yield surface. The value of N may be interpreted as the radial distant between the outside of the initial yield surface and the inside of the limit surface. In order for the limit surface of the kinematic hardening cap model to correspond with

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the failure envelope surface of the standard cap model, the scalar parameter a must be replaced Nα − in the definition eF .

The cap model contains a number of parameters which must be chosen to represent a particular material, and are generally based on experimental data. The parameters , , ,α β θ and γ are usually evaluated by fitting a curve through failure data taken from a set of triaxial

compression tests. The parameters W, D, and 0X define the cap hardening law. The value W

represents the void fraction of the uncompressed sample and D governs the slope of the initial loading curve in hydrostatic compression. The value of R is the ration of major to minor axes of the quarter ellipse defining the cap surface. Additional details and guidelines for fitting the cap model to experimental data are found in [Chen and Baladi, 1985].

Material Model 26: Crushable Foam This orthotropic material model does the stress update in the local material system

denoted by the subscripts, a, b, and c. The material model requires the following input parameters:

• E, Young’s modulus for the fully compacted material; • ν , Poisson’s ratio for the compacted material;

• yσ , yield stress for fully compacted honeycomb;

• LCA, load curve number for sigma-aa versus either relative volume or volumetric strain (see Figure 19.26.1.);

• LCB, load curve number for sigma-bb versus either relative volume or volumetric strain (default: LCB = LCA);

• LCC, the load curve number for sigma-cc versus either relative volume or volumetric strain (default: LCC = LCA);

• LCS, the load curve number for shear stress versus either relative volume or volumetric strain (default LCS = LCA);

• fV, relative volume at which the honeycomb is fully compacted;

• aauE , elastic modulus in the uncompressed configuration;

• bbuE , elastic modulus in the uncompressed configuration;

• ccuE , elastic modulus in the uncompressed configuration;

• abuG , elastic shear modulus in the uncompressed configuration;

• bcuG , elastic shear modulus in the uncompressed configuration;

• cauG , elastic shear modulus in the uncompressed configuration; • LCAB, load curve number for sigma-ab versus either relative volume or volumetric

strain (default: LCAB = LCS); • LCBC, load curve number for sigma-bc versus either relative volume or volumetric

strain default: LCBC = LCS); • LCCA, load curve number for sigma-ca versus either relative volume or volumetric

strain (default: LCCA = LCS); • LCSR, optional load curve number for strain rate effects.