The Perzyna option has the following form:

(4–95)

where:

= equivalent plastic strain rate

m = strain rate hardening parameter (input as C1 via TBDATA command)

γ = material viscosity parameter (input as C2 via TBDATA command)

σ = equivalent stress

σo = static yield stress of material (defined via TB,BISO, or TB,NLISO commands)

As γ tends to , or m tends to zero or tends to zero, the solution converges to the static (rate-independent) solution. However, for this material option when m is very small (< 0.1), the solution shows difficulties in convergence (Peric and Owen([298])).

The Peirce option has the following form:

(4–96)

Similar to the Perzyna model, the solution converges to the static (rate-independent) solution, as γ tends to , or m tends to zero, or tends to zero. For small value of m, this option shows much better convergency than PERZYNA option (Peric and Owen([298])).

For information about this option, see Exponential Visco-Hardening (EVH) Option in the Material Reference.

Metal under elevated temperature, such as the hot-metal-working problems, the material physical behaviors become very sensitive to strain rate, temperature, history of strain rate and temperature, and strain hardening and softening. The systematical effect of all these complex factors can be taken into account and modeled via Anand viscoplasticity([160], [148]). The Anand option is categorized into the group of the unified plasticity models where the inelastic deformation refers to all irreversible deformation that can not be simply or specifically decomposed into the plastic deformation derived from the rate-independent plasticity theories and the part resulted from the creep effect. Compare to the traditional creep approach, the Anand option introduces a single scalar internal variable "s", called the deformation resistance, which is used to represent the isotropic resistance to inelastic flow of the material.

Although the Anand option was originally developed for the metal forming application ([160], [148]), it is however applicable for general applications involving strain and temperature effect, including but not limited to such as solder join analysis, high temperature creep etc.

The inelastic strain rate is described by the flow equation as follows:

(4–97)

where:

= inelastic strain rate tensor

= rate of accumulated equivalent plastic strain

S, the deviator of the Cauchy stress tensor, is:

(4–98)

and q, equivalent stress, is:

(4–99)

where:

The rate of accumulated equivalent plastic strain, , is defined as follows:

(4–100)

The equivalent plastic strain rate is associated with equivalent stress, q, and deformation resistance, s, by:

(4–101)

Equation 4–101 implies that the inelastic strain occurs at any level of stress (more precisely, deviation stress). This theory is different from other plastic theories with yielding functions where the plastic strain develops only at a certain stress level above yielding stress.

The evolution of the deformation resistance is dependent of the rate of the equivalent plastic strain and the current deformation resistance. It is:

(4–102)

where:

The sign, , is determined by:

(4–103)

The deformation resistance saturation s* is controlled by the equivalent plastic strain rate as follows:

(4–104)

where:

= constant with stress unit

n = dimensionless constant

Because of the , Equation 4–102 is able to account for both strain hardening and strain softening. The strain softening refers to the reduction on the deformation resistance. The strain softening process occurs when the strain rate decreases or the temperature increases. Such changes cause a great reduction on the saturation s* so that the current value of the deformation resistance s may exceed the saturation.

The material constants and their units specified for the Anand option are listed in Table 4.3: Material Parameter Units for the Anand Option. All constants must be positive, except constant "a", which must be 1.0 or greater. The inelastic strain rate in Anand's definition of material is temperature and stress dependent as well as dependent on the rate of loading. Determination of the material parameters is performed by curve-fitting a series of the stress-strain data at various temperatures and strain rates as in Anand([160]) or Brown et al.([148]).

Table 4.3: Material Parameter Units for the Anand Option

TBDATA Constant	Parameter	Meaning	Units
1	so	Initial value of deformation resistance	stress, e.g. psi, MPa
2	Q/R	Q = activation energy	energy / volume, e.g. kJ / mole
		R = universal gas constant	energy / (volume temperature), e.g. kJ / (mole - K
3	A	pre-exponential factor	1 / time e.g. 1 / second
4	ξ	multiplier of stress	dimensionless
5	m	strain rate sensitivity of stress	dimensionless
6	ho	hardening/softening constant	stress e.g. psi, MPa
7		coefficient for deformation resistance saturation value	stress e.g. psi, MPa
8	n	strain rate sensitivity of saturation (deformation resistance) value	dimensionless
9	a	strain rate sensitivity of hardening or softening	dimensionless

where:

If h₀ is set to zero, the deformation resistance goes away and the Anand option reduces to the traditional creep model.

We first assume that the material point yields so that both plastic deformation and creep deformation occur. Figure 4.16: Material Point in Yielding Condition Elastically Predicted illustrates such a stress state. We next decompose the inelastic strain rate as follows:

(4–105)

where:

The plastic strain rate is further defined as follows:

(4–106)

where:

= plastic multiplier

Q = flow function that has been previously defined in Equation 4–64, Equation 4–65, and Equation 4–66 in Extended Drucker-Prager Model

Here we also apply these plastic flow functions to the creep strain rate as follows:

(4–107)

where:

= creep multiplier

Figure 4.16: Material Point in Yielding Condition Elastically Predicted

As material yields, the real stress should always be on the yielding surface, implying that:

(4–108)

where:

It is assumed that material hardening is related to plasticity only and not to creep. The yield stress, , evolves as a function of the hardening variable, :

where the hardening variable is the plastic work and is calculated from the summation of:

The creep behaviors could be measured through a few simple tests such as the uniaxial compression, uniaxial tension, and shear tests. We here assume that the creep is measured through the uniaxial compression test described in Figure 4.17: Uniaxial Compression Test.

Figure 4.17: Uniaxial Compression Test

The measurements in the test are the vertical stress and vertical creep strain at temperature T. The creep test is targeted to be able to describe material creep behaviors in a general implicit rate format as follows:

(4–109)

We define the equivalent creep strain and the equivalent creep stress through the equal creep work as follows:

(4–110)

where:

and = equivalent creep strain and equivalent creep stress to be defined.

For this particular uniaxial compression test, the stress and creep strain are:

(4–111)

Inserting (Equation 4–111) into (Equation 4–110) , we conclude that for this special test case the equivalent creep strain and the equivalent creep stress just recover the corresponding test measurements. Therefore, we are able to simply replace the two test measurements in (Equation 4–109) with two variables of the equivalent creep strain and the equivalent creep stress as follows:

(4–112)

Once the equivalent creep stress for any arbitrary stress state is obtained, we can insert it into (Equation 4–112) to compute the material creep rate at this stress state. We next focus on the derivation of the equivalent creep stress for any arbitrary stress state.

We first introduce the creep isosurface concept. Figure 4.18: Creep Isosurface shows any two material points A and B at yielding but they are on the same yielding surface. We say that the creep behaviors of point A and point B can be measured by the same equivalent creep stress if any and the yielding surface is called the creep isosurface. We now set point B to a specific point, the intersection between the yielding curve and the straight line indicating the uniaxial compression test. From previous creep measurement discussion, we know that point B has for the coordinate p and for the coordinate q. Point B is now also on the yielding surface, which immediately implies:

(4–113)

It is interpreted from (Equation 4–113) that the yielding stress σ _Y is the function of the equivalent creep stress . Therefore, we have:

(4–114)

We now insert (Equation 4–114) into the yielding condition (Equation 4–108) again:

(4–115)

We then solve (Equation 4–115) for the equivalent creep stress for material point A on the isosurface but with any arbitrary coordinates (p,q). (Equation 4–115) is, in general, a nonlinear equation and the iteration procedure must be followed for searching its root. In the local material iterations, for a material stress point not on the yielding surface but out of the yielding surface like the one shown in Figure 4.16: Material Point in Yielding Condition Elastically Predicted, (Equation 4–115) is also valid and the equivalent creep stress solved is always positive.

Figure 4.18: Creep Isosurface

When the loading is at a low level or the unloading occurs, the material doesn’t yield and is at an elastic state from the point view of plasticity. However, the inelastic deformation may still exist fully due to material creep. In this situation, the equivalent creep stress obtained from (Equation 4–115) may be negative in some area. If this is the case, (Equation 4–115) is not valid any more. To solve this difficulty, the stress-projection method shown in Figure 4.19: Stress Projection is used, where the real stress σ is multiplied by an unknown scalar β so that the projected stress σ* = βσ is on the yielding surface. The parameter β can be obtained by solving the following equation:

(4–116)

Again, Equation 4–116 is a nonlinear equation except in the linear Drucker-Prager model. Because the projected stress σ* is on the yielding surface, the equivalent creep stress denoted as and calculated by inserting σ* into (Equation 4–115), as follows:

(4–117)

is always positive. The real equivalent creep stress is obtained by simply rescaling as follows:

(4–118)

For creep flow in this situation, (Equation 4–107) can be simply modified as follows:

(4–119)

For stress in a particular continuous domain indicated by the shaded area in Figure 4.19: Stress Projection, the stresses are not able to be projected on the yielding surface (that is, Equation 4–116) has no positive value of solution for β). For stresses in this area, no creep is assumed. This assumption makes some sense partially because this area is pressure-dominated and the EDP models are shear-dominated.

Having Equation 4–105,Equation 4–106, Equation 4–107, or Equation 4–119, Equation 4–108, Equation 4–110, and Equation 4–112, the EDP creep model is a mathematically well-posed problem.

Figure 4.19: Stress Projection

The following assumptions and restrictions apply to the cap creep model:

The following yielding equation applies of the cap portion of the cap creep model:

(4–120)

This equation represents the flow function of the cap creep model:

(4–121)

The equations are similar to those used for yielding and flow function (respectively) in the cap (rate-independent plasticity) model.

The tension cap portion for the cap creep model is nearly identical to the cap (rate-independent plasticity) yielding function except that the J₃ (Lode angle) effect is ignored. The notations and conventions remain the same.

Equation 4–120 is also used to calculate the equivalent creep stresses, and Equation 4–121 is used to calculate the creep strain.

For the sake of convenience, the cap domain shown in the following figure is divided into six zones (labeled 1, 2, 3, 10, 20, and 30):

Figure 4.20: Cap Zoning

Following is a brief description of each zone:

Creep in zones 2, 20, 3, and 30 is measured via uniaxial compression testing. Those zones do not experience compaction creep contributed from the compaction cap.

Zones 1 and 10 may also experience shear creep because a certain amount of shear creep exists in all zones. It is assumed that stress points below line CD are not subject to shear creep.

The shear and expansion zones share the same uniaxial compression for creep measurement. The derivation for the equivalent creep stress for the two zones is similar to that used for extended Drucker-Prager creep stress.

In coordinate system shown in the following figure, the coordinates of the stress points from the uniaxial compression test are , where is the measured stress or the equivalent creep stress.

Figure 4.21: Uniaxial Compression Test and Creep Isosurface for Shear and Expansion Zones

The point also on the shear yielding surface should satisfy the corresponding yielding function, as follows:

(4–122)

or

(4–123)

Equation 4–123 implies , which is used in the yielding condition

resulting in:

(4–124)

Equation 4–124 is then solved for the equivalent creep stress for stresses at either the shear zone or the expansion zone. As shown in the EDP creep model, the measured creep strain component in the axial direction is the equivalent creep strain; therefore, the creep rate function for both shear and expansion zones is:

where is the equivalent creep strain for the shear or expansion zone and T is the temperature.

 

4.3.4.4.1. Equivalent Creep Stress in the Compaction Zone

The equivalent creep stress for compaction cap is assumed to be independent of J₂ and only related to the first invariant of stress:

The creep test measurement for the compaction cap is regulated via hydrostatic compression testing. The measurement and record are

• where p is the hydrostatic pressure, and

• where is the volume creep strain.

Thus, the creep rate function for the compaction cap is:

(4–125)

In zones 30, 20, 10, and 1 (where the material does not yield at the shear envelope criterion), loading is light; it also possible that unloading occurs. Inelastic deformation may still exist, however, due exclusively to material shear creep. In such a case, the equivalent creep stress obtained from the previous derivation may be negative. To avoid a negative equivalent creep stress, the stress-projection method proposed in EDP Elastic Creep and Stress Projection is used.

To model cap creep, use the input from the cap (rate-independent plasticity) model. The cap creep model ignores any tension-to-compression-strength ratio values and sets them to 1.

Creep data is input via the TB,CREEP command. Because the cap creep model requires two independent creep function inputs, an additional material table data command, TBEO, allows shear and compaction creep data to be defined separately.