Spheroidal voids using spheroidal coordinates

This section deals with the work of Lee and Mear (1992c), who extended the previous study for cylindrical voids to porous materials consisting of spheroidal voids subjected to axisymmetric load-ing conditions aligned with the pore symmetry axis by makload-ing use of prolate or oblate spheroidal coordinates. In this connection, it is convenient to use the notation introduced in subsection2.2.1for

the description of the microstructural variables. Thus, for spheroidal voids, the corresponding aspect ratios are defined byw1=a3/a1 andw2=a3/a1, witha1=a2. This implies that the voids can be either prolate, with aspect ratiosw1=w2>1, or oblatew1=w2<1. The effective stress potential is given by the same expression (3.20) as in the two-dimensional case, such that

Ue(σ) =U(σ) +f δUe where V2 and V1 are the regions occupied by the void and the matrix phase, while U denotes the stress potential of the matrix phase and is given by relation (2.23)1. Similar to the two-dimensional case presented previously, expression (3.31) implies that the effective stress potentialUe of the porous solid has a linear correction in the concentration of the voids, i.e., the porosity f. However, we will see in the following subsection that such an expansion may have a very small range of validity for very nonlinear solids and particularly for ideally-plastic solids where such an expansion may not be valid at all (see Duva and Hutchinson, 1984). In any case, the above correction termδU /δfe can be directly related to the minimum principle, defined previously in the context of relation (3.4), through the relation (3.21) (Duva and Hutchinson, 1984; Lee and Mear, 1992b), which takes the form

δUe

δf =W(D^∞)−Fmin

V2

, (3.32)

for the case of spheroidal voids.

The next step in this procedure involves the definition of a stream function in terms of spheroidal coordinates (η,θ,φ). Then, the physical components of the velocity fieldev, defined by relation (3.5), are given for prolate voids by

e In turn, for oblate voids, the components ofve are chosen as

e α is a scaling factor related to the distance from the origin to the foci of the spheroidal void, Pk

are the associated Legendre functions of order one, Qm are the associated Legendre functions (of the second kind) of order one, i =√

−1 is the imaginary unity, and the constants{A, Bkm} are the

amplitude factors with respect to which F (see relation (3.4)) must be minimized. It is remarked at this point that because of the symmetry of the microstructure and the axisymmetric loading, the velocity fields ve do not depend on the third coordinate φ. Note that if the void was ellipsoidal or misaligned with the principal loading directions, the above procedure would become too complicated, and so far there are no results for those more general cases. This is because this procedure is based on the construction of a stream function, which is a technique that is used mainly in two-dimensional problems, or three-dimensional problems with certain symmetries.

In the following, we recall the boundary conditions used by Lee and Mear (1992c) to solve the above described problem. The loading is axisymmetric about the 3−axis, such that the only non-zero components of the stress tensor are

σ^∞₁₁=σ₁₁^∞=T^∞, σ^∞₃₃=S^∞. (3.38) The stress triaxiality may be defined by

XΣ= σm

σeq, (3.39)

whereσm= (S^∞+ 2T^∞)/3 andσeq=|S^∞−T^∞|.

In this regard, it is essential to analyze the relevant dilatation and deviatoric quantities introduced by Lee and Mear (1992c). In particular, the Lee and Mear (LM) effective stress potential can be written similarly to the two-dimensional case, so that it takes the form

UeLM(σ) = (1 +f h(XΣ))ε˙oσo

n+ 1 µσeq

σo

¶_n+1

. (3.40)

By comparing the last expression for the effective stress potential with the corresponding expression provided by the homogenization methods, in relation (2.26), it is easily deduced that

h(XΣ) = 1

f(bh(XΣ)−1). (3.41)

Then, Lee and Mear defined a scalar function P(XΣ), which is directly related to the dilatational rate of the void, defined by

P = 1 n+ 1

∂h

∂XΣ = Dii

f D₃₃^∞, i= 1,2,3. (3.42)

Here, D^∞₃₃ is the axial component of the remote strain-rate field in the absence of voids. In addition to the dilatational part of the macroscopic strain-rate, it is useful to study, as well, the deviatoric part ofD. For this reason, Lee and Mear have defined the scalar function Q(XΣ), which is directly related to the axial component of the deviatoric macroscopic strain-rateD₃₃⁰ , such that

Q=h−XΣP= D₃₃⁰ −D^∞₃₃

f D^∞₃₃ , (3.43)

where the prime is used to denote the deviatoric part ofD. Making use of the last two expressions, we are able to compare the estimates obtained by the “second-order” and the “variational” method with

“exact” numerical results in the dilute limit. Note that the above quantitiesP andQdepend also on the microstructural variables, such as the aspect ratiosw1=w2. Relevant comparisons between the

homogenization methods and the numerical results of this section will also be made in a following chapter.

In addition to the aforementioned analysis associated with the initial response of a dilute porous solid subjected to axisymmetric loading conditions, Lee and Mear (1994) have also studied the evo-lution of microstructure in dilute porous viscoplastic composites subjected to the loading conditions described previously by relation (3.38). Thus, it is also relevant to compare the corresponding ho-mogenization estimates, described in the previous chapter, with the corresponding Lee and Mear predictions for the microstructure evolution. However, in order to determine the evolution of the pore shape, Lee and Mear have idealized the nonlinear problem by allowing the voids to remain spheroidal during the deformation process. As already explained in the previous subsection, this ap-proximation is different than the one described in the context of the homogenization theories, where the objective is to compute anaverage void shape by making use of the average fields in the vacuous phase. As we will see in a following chapter, this idealization proposed by Lee and Mear could lead to inaccurate estimates for the evolution of the microstructural variables, including the evolution of porosity.

In summary, Lee and Mear (1994) have proposed a velocity field based on a stream function by making use of spheroidal coordinates. In the limit that the void becomes spherical this procedure is identical to the one followed by Budiansky et al. (1982) in the context of spherical voids. However, it is emphasized that this procedure is limited to spheroidal voids and axisymmetric loading conditions aligned with the pore symmetry axis. Moreover, it is based on the assumption that the effective stress potential has a linear correction in the porosity in the dilute limit. An attempt to comment on the validity of this expansion is performed in the following subsection.