Appendix II. Evaluation of the reference stress tensor in 2D

In this section, we describe briefly the evaluation of the two factors that are present in the expression (4.20) that defines the reference stress tensor ˇσ, which is repeated here for completeness:

σˇ =ξ¡

is a suitably chosen interpolation function, withtandβ given in Appendix III of chapter2.

To evaluate the factor αm, it is necessary to provide an estimate for the effective behavior of the porous material in purely hydrostatic loading. In this regard, we recall that the effective stress potentialUe of a porous material consisting of cylindrical voids, defined by (2.195), is expressed as

Ue(σ;f, w) =ε˙oeσw

For the estimation of effective flow stressσew, we make use of the approximation proposed in relation (2.199), such that Here, the notationw→ ∞has been used to indicate that the void has a cylindrical shape, in accor-dance with the general definitions introduced in subsection (2.2.1). The quantityeσw→∞corresponds to the flow stress of a porous medium containing cylindrical voids with circular cross-section subjected to purely hydrostatic pressure. On the other hand,σe_w→∞^var andσe^var_w are the effective flow stresses of a

porous material consisting of cylindrical voids with circular and elliptical cross-section, respectively, as predicted by the “variational” method. The effective flow stress eσ_w^var is given by (2.201) in the general case of ellipsoidal voids. However, for cylindrical voids, it can be shown to reduce to

e

σ_w^var =σe^var_w→∞

µ 2w w²+ 1

¶ⁿ⁺¹

2n

. (4.47)

Then, the factorαmis determined numerically by solving the following condition Uesom→Ue as |XΣ| → ∞ ⇒ αm=αm

¡S, f, w, n¢

, (4.48)

where Uesom is the effective stress potential of the porous material given by (4.14), and needs to be computed in the hydrostatic limit. On the other hand, for the computation of the second factorαeq, we make use of the analytical relation (2.207), which holds also for the cylindrical microstructures.

Porous media with an ideally-plastic matrix phase. For the computation of αm and αeq

in the limit of ideal-plasticity, we just need to consider the limitn→ ∞in the previous expressions.

Thus, one finds in this limit that e

σw→∞=

√3 2 log 1

f, and eσ_w→∞^var σo

=

√3 2

1−f

√f , (4.49)

and

e

σ_w^var=σe^var_w→∞

r 2w

w²+ 1. (4.50)

By making use of relation (4.29) (see also (2.290)), as well as of expression (4.49), the equation for the factorαmbecomes

e

σ^som_w =σew, as |XΣ| → ∞, (4.51) witheσw given by (4.45). On the other hand, the factorαeq, is determined in the ideally-plastic limit from relation (2.297), which is also valid in the case of cylindrical microstructures.

Chapter 5

Evolution of microstructure: cylindrical voids

The main objective of this chapter is to make use of the results of the previous one in order to estimate the evolution of microstructure for porous materials with cylindrical voids subjected to plane-strain loading conditions. For completeness and validation of the “second-order” model (SOM), presented in section 2.6, we study also the evolution of microstructure as predicted by the Lee and Mear (1999) (LM) method (see subsection 3.2.3), the “variational” method (V AR) (see section2.5) and unit-cell finite element (F EM) calculations (see section 3.5). As already discussed in chapter 3, the two homogenization methods used in this work consider that the porous material consists of a random distribution of cylindrical voids aligned in the 3−direction. On the other hand, the unit-cell calculation requires periodic boundary conditions. In this regard, the two problems, random and periodic, are not equivalent unless a sufficiently small value for the porosity is used. Note, however, that the computation of the fields inF EM becomes too cumbersome for very small porosities, since a large number of degrees of freedom needs to be used. Thus, for our problem, a porosity offo= 0.01%

(sufficiently small) is used to perform the F EM calculations. Furthermore, it is noted that in the case of plane-strain loading, the problem reduces in estimating the in-plane effective behavior of the composite.

5.1 Evolution laws in two-dimensions

In particular, we study the problem of porous materials consisting of cylindrical voids with initially circular cross-section, subjected to plane-strain loading conditions. Due to the finite deformations, the initially circular voids evolve into elliptical ones with certain orientation in the plane. Thus, the relevant microstructural variables are the porosity f, the in-plane aspect ratio w and the in-plane orientation angleψ. The general evolution equations presented in section2.7, reduce here to:

Porosity. By making use of the incompressibility of the matrix phase, the evolution law for the porosity is obtained from the kinematical relations

f˙= (1−f)Dαα, α= 1,2, (5.1)

ψ

e

e n

( )²

n

( )²

e

( )³

e

( )¹

e

α2

Figure 5.1: Representative ellipsoidal void in the case of cylindrical microstructures.

where D is evaluated from relation (4.8) for the “variational” method and (4.22) for the “second-order” method.

Aspect ratio. The evolution of the in-plane aspect ratio of the void is defined by

˙ w=w³

n⁽²⁾·D⁽²⁾n⁽²⁾−n⁽¹⁾·D⁽²⁾n⁽¹⁾´

=w³

n⁽²⁾⊗n⁽²⁾−n⁽¹⁾⊗n⁽¹⁾´

·D⁽²⁾. (5.2) The average strain-rate in the void D⁽²⁾ is computed by relation (4.9) for the “variational” method and (4.27) for the “second-order” method.

Orientation vectors. The evolution of the orientation vectorsn⁽ⁱ⁾(withi= 1,2) is determined by the spin of the in-plane principal axes of the void, or microstructural spinω, via

˙

n⁽ⁱ⁾=ωn⁽ⁱ⁾, i= 1,2. (5.3)

Because of the two-dimensional character of the problem, the unit-vectorsn⁽ⁱ⁾are defined completely by the in-plane angleψ, such that

n⁽¹⁾= cosψe⁽¹⁾+ sinψe⁽²⁾, n⁽²⁾=−sinψe⁽¹⁾+ cosψe⁽²⁾. (5.4) Then, by making use of the standard notation

ω=ω(e⁽¹⁾⊗e⁽²⁾−e⁽²⁾⊗e⁽¹⁾), (5.5) then the evolution law (5.3) can be replaced by the scalar expression

ψ˙ =−ω. (5.6)

Now, the microstructural spinωis related to the average spin in the void, Ω⁽²⁾, and the average strain-rate in the void,D⁽²⁾, by the well-known kinematical relation (2.224) (Hill, 1978; Ogden, 1984), which takes the following form in two-dimensional problems

ω=Ω⁽²⁾−1 2

1 +w² 1−w²

³

n⁽¹⁾⊗n⁽²⁾+n⁽²⁾⊗n⁽¹⁾´

·D⁽²⁾, w 6= 1. (5.7) The average spin tensorΩ⁽²⁾ in the pore phase is given by (4.9) for the “variational” method and by (4.28) for the “second-order” method.

Following definition (2.228) and expression (2.229) for the plastic spin (i.e., Ω^p =Ω−ω), the Jaumann rate of the orientation vectorsn⁽ⁱ⁾is defined by

nO⁽ⁱ⁾=−Ω^pn⁽ⁱ⁾ i= 1,2,3. (5.8)

The above definition is helpful for the computation of the Jaumann hardening rate in the context of ideal-plasticity. Note that the above evolution laws are valid for both the “variational” and the

“second-order” method, while the computation of the phase average fields in the composite is the one that brings about the difference between the “variational” and the “second-order” estimates.

In the following, we study the evolution of microstructure in dilute porous media consisting of cylindrical voids with initially circular or elliptical cross-section. The Lee and Mear (1999) (LM) predictions are used as a comparison for the corresponding “second-order” (SOM) estimates. Next, we study the evolution of microstructure in porous materials subjected to loading conditions that do not induce a rotation of the principal axes of the void. In this case, we discuss the effect of the porosity and the aspect ratio on the effective response of the porous material. The results obtained by theSOM and the “variational” (V AR) methods are compared with corresponding estimates by F EM unit-cell calculations. Finally, a simple shear loading is applied. In this case, the effect of the orientation angle on the effective response of the material is studied.