PROCESO DE REGISTRO Y DOCUMENTACIÓN DE LA PRODUCCIÓN

In this appendix, we discuss the numerical algorithms used in computing the LCC model predictions of this chapter for the macroscopic response and texture evolution of semi-crystalline polymers. As already mentioned in the main body of the text, the computation of the LCC model for any given finite deformation loading process is performed incrementally from the initial to the final state of the body. Therefore, for the purpose of the present discussion it suffices to restrict our consideration to a generic time increment [𝑡𝑚, 𝑡𝑚+ Δ𝑡], with𝑚being a nonnegative integer, during an arbitrary finite strain loading history. The increment Δ𝑡may be chosen to be different for different values of

𝑚. Note that the special case𝑚= 0 corresponds to the reference configuration of the body (Figs.

4.1(a) and4.1(b)). Within the increment [𝑡𝑚, 𝑡𝑚+ Δ𝑡], we consider the numerical treatment of two problems: (𝑖) the computation of the instantaneous effective constitutive relation (4.80) at

𝑡=𝑡𝑚and (𝑖𝑖) the integration of the evolution laws of section4.3for the internal variables (4.32) over the increment [𝑡𝑚, 𝑡𝑚+ Δ𝑡]. The later problem allows updating the state of the underlying sub-structure at the end of the increment (𝑡 = 𝑡𝑚+ Δ𝑡), which is in turn used as input in the corresponding calculations at the beginning of the next increment, i.e., at time 𝑡 = 𝑡𝑚+1. In this connection, it is remarked that the sub-structure at 𝑡 = 𝑡0, corresponding to the reference configuration (Figs. 4.1(a) and 4.1(b)), is assumed to be known (see subsection 4.5.1). Hence, the values of the internal variables (4.32) at 𝑡 =𝑡𝑚 are assumed to be known in the context of problem (𝑖) and to provide the initial conditions in the context of problem (𝑖𝑖). At this point, it should be emphasized that both problems (𝑖) and (𝑖𝑖) are highly nonlinear and, therefore, they must be treated numerically. In passing, it is remarked that the numerical procedures described below determine also the effective stress-potential (4.52) of the semi-crystalline polymer at time

𝑡 =𝑡𝑚. Furthermore, appropriately modified versions of these procedures have been used in the calculations of the LCC estimates of chapter5for two-scale polycrystalline metals.

Instantaneous effective response at time 𝑡 =𝑡𝑚. Recall that the LCC relation (4.80) de- termines any combination of five independent components of the macroscopic deformation rate tensor D and the deviatoric stress tensor 𝝈𝑑 of the semi-crystalline polymer in terms of the re- manning five components of these tensors, which are prescribed through the boundary conditions. In turn, the use of the LCC relation (4.80) requires the computation of the local and effective properties of the associated two-scale LCC. Specifically, the quantities that have to be determined

in the context of the LCC model (4.80) are: (𝑖) the scalar phase-moduli 𝛼_𝑘(𝑟) and 𝜆(𝑟)_{, (𝑖𝑖) the} stress-phase-average tensors𝝈(𝑟,𝑐)and𝝈(𝑟,𝑎), (𝑖𝑖𝑖) the stress-phase-fluctuation variables ˆ𝜏_𝑘(𝑟,𝑐)and ˆ

𝜏(𝑟,𝑎)_{, (𝑖𝑣) the stress-grain-average tensors𝝈}(𝑟)_{, (𝑣) the effective-grain properties M}(𝑟)_{, 𝜶}(𝑟)_and

𝜙(𝑟)_{and (𝑣𝑖) the overall properties˜M,𝜶˜ and ˜𝜙of the two-scale LCC. The quantities ˆ𝜏}(𝑟,𝑐)

𝑘 , ˆ𝜏(𝑟,𝑎), M(𝑟)_,𝜶(𝑟)_,𝜙(𝑟)_,

˜

𝜶 and ˜𝜙are given explicitly by means of the expressions (4.54), (4.55), (4.101), (4.82), (4.83), (4.81)1 and (4.81)2, respectively, in terms of the variables

{𝛼(_𝑘𝑟), 𝜆(𝑟)_,𝝈(𝑟,𝑐)_,𝝈(𝑟,𝑎)_,𝝈(𝑟)_{,M˜}, (4.117)} which are, therefore, chosen as the principal unknowns of the problem. These variables are deter- mined by means of the system of equations

{(4.44),(4.45),(4.94),(4.95),(4.96),(4.53)2,(4.87)}, (4.118) which, as already mentioned, is nonlinear and must be treated numerically.

The system of equations (4.118) is solved in this work by means of an algorithm consisting of four nested iterative procedures. In these procedures, the values of the variables (4.117) at 𝑡=𝑡𝑚−1, with𝑚≥1—obtained by solving equations (4.118) for𝑡=𝑡𝑚−1in the previous increment—is used as an initial guess for the calculations at 𝑡=𝑡𝑚. The prescription of the initial guess for𝑡 =𝑡0, i.e., in the reference configuration, is discussed further below. In the innermost procedure, the moduli of the crystalline𝛼(𝑘𝑟)and the amorphous𝜆(𝑟)phase in each grain𝑟= 1, ..., 𝑁 of the LCC are held fixed and the system of equations (4.94), (4.95) and (4.96) is solved for the corresponding stress-phase-average tensors 𝝈(𝑟,𝑐)and𝝈(𝑟,𝑎). From the relevant discussion of subsection4.4.2, it is recalled that the equations (4.94), (4.95) and (4.96) may be reduced to the system of three scalar equations (4.96) for any three independent components of the stress-phase-average tensors 𝝈(_𝑑𝑟,𝑐)

and𝝈(_𝑑𝑟,𝑎), with the equations (4.94) and (4.95) providing explicit expressions for the computation of the remanning unknown components of 𝝈(𝑟,𝑐) and 𝝈(𝑟,𝑎). The system of equations (4.96) is nonlinear, but it can be easily solved numerically by means of Newton’s method.

The two intermediate iterative procedures compute successively the effective modulus tensor

˜

M and the stress-grain-average tensors 𝝈(𝑟). These calculations are performed by means of the “viscoplastic self-consistent” (VPSC) software, which is a Fortran code developed by C.N. Tom´e and R.A. Lebensohn at the “Los Alamos National Laboratory” and it has been provided to us by R.A. Lebensohn. The core component of the VPSC code is a quite efficient solver which, making use of the fixed-point method, computes iteratively the self-consistent estimate (4.87) for the effective modulus tensor ˜M of a granular composite with arbitrary but fixed values of the associated grain modulus tensorsM(𝑟)_{. For a given value of˜M, the VPSC code then employs the} fixed-point method once again in computing iteratively the estimate (4.53)2 for the stress-grain- average tensors𝝈(𝑟).

Finally, the outermost iterative procedure applies the fixed-point method to the equations (4.44) and (4.45), and—using the results of the previous procedures—computes the phase-moduli variables𝛼(_𝑘𝑟)and𝜆(𝑟)_{. It should be remarked that this procedure involves the computation of the} second-moment tensors⟨𝝈⊗𝝈⟩(𝑟,𝑝)_{in the amorphous (𝑝=𝑎) and crystalline (𝑝=𝑐) phases of the} LCC, given by (4.84), which in turn requires the calculation of the second-moment tensors⟨𝝈⊗𝝈⟩(𝑟) in the grains. The computation of the later quantities is quite involved and it is performed by employing an appropriate subroutine that is available in the VPSC package, following a procedure proposed by Liu and Ponte Casta˜neda [86].

As already mentioned, in the above algorithm the initial guess for the variables (4.117) at time

𝑡=𝑡𝑚for𝑚≥1 is chosen to be the solution of the equations (4.118) at time𝑡=𝑡𝑚−1. However, in the case of the first increment (𝑡=𝑡0), corresponding to the reference configuration, this option is not available. Given the fact that the system of equations (4.118) are highly nonlinear, it is very difficult to make a good initial guess for this case. In order to obtain the required initial estimate for the variables (4.117) in the reference configuration, we solve the system of equations (4.118), as discussed above, for increasing values of the nonlinearity exponent𝑛from𝑛= 1 to 𝑛= 9 with a step Δ𝑛. This approach has the advantage that the case 𝑛= 1 corresponds to a linear elastic composite for which an initial guess can be easily found, while for any higher value of𝑛= 1+𝑚Δ𝑛, with𝑚being a positive integer, the solution for𝑛= 1 + (𝑚−1)Δ𝑛can be used as an initial guess. Specifically, for 𝑛 = 1 the phase-moduli are given by𝛼(_𝑘𝑟) =𝛾0/𝜏0𝑘 and 𝜆(𝑟) =𝜏0/(2𝛾0) for all

𝑟 = 1, ..., 𝑁, where 𝛾0 is the reference strain rate while 𝜏0𝑘 and 𝜏0 are the associated reference stresses, given in subsection 4.5.1. Hence, the corresponding phase modulus tensors M(𝑟,𝑐) _and

M(𝑟,𝑎)_{are constant and the associated effective-grain tensorsM}(𝑟)_{may be readily obtained from} (4.101). The Taylor estimateM˜= [∑𝑁_𝑟=1𝑐𝑟_(M(𝑟)₎−1_]−1 _{turns out to be a good initial guess for} the effective modulus tensor for𝑛= 1. Making use of this estimate forM˜, corresponding estimates for the stress-grain-average tensors𝝈(𝑟) and the stress-phase-average tensors𝝈(𝑟,𝑝)(𝑝=𝑎, 𝑐) may be readily obtained from the expressions (4.53)2and (4.53)1, respectively.

Evolution of the internal variables (4.32) over the time increment [𝑡𝑚, 𝑡𝑚+ Δ𝑡]. It is recalled once again that the evolution laws of section4.3are highly nonlinear and, therefore, their integration over the time increment [𝑡𝑚, 𝑡𝑚+ Δ𝑡] can only be performed numerically.

Direct integration of the evolution equations (4.61) for the aspect ratios𝑤𝛼 (𝛼= 1,2) of the shape tensorZ by employing a forward Euler scheme yields the result

𝑤𝛼(𝑡𝑚+1) =𝑤𝛼(𝑡𝑚) +[𝑤𝛼(z3⊗z3−z𝛼⊗z𝛼)⋅D]_{𝑡=𝑡𝑚}Δ𝑡, (4.119) where the notation 𝑡𝑚+1 = 𝑡𝑚+ Δ𝑡 has been introduced and it is recalled thatz𝑖 (𝑖 = 1,2,3) denote the principal vectors ofZ.

The forward Euler scheme may also be used in integrating the evolution laws (4.62) for the principal vectorsz𝑖 ofZ. However, such an approximation has the disadvantage that the resulting vector z𝑖(𝑡𝑚+1) does not have unit length (see, e.g., [5]). For this reason, following Aravas and Ponte Casta˜neda [5], we approximate the evolution equation (4.62) in the time interval 𝑡𝑚≤𝑡≤ 𝑡𝑚+ Δ𝑡by

˙

z𝑖(𝑡) =𝜔z(𝑡𝑚)z𝑖(𝑡), (4.120) where the spin tensor𝜔z(𝑡𝑚) may be readily obtained from (4.63) by making use of the results for the effective instantaneous response of the composite at time𝑡=𝑡𝑚, discussed earlier. The above equation may be integrated exactly, yielding the result

z𝑖(𝑡𝑚+1) = exp [𝜔z(𝑡_𝑚)Δ𝑡]z_𝑖(𝑡_𝑚), (4.121) where it should be recalled that the exponential of an anti-symmetric tensorAis given by

exp [A] =I+sin𝑎

𝑎 A+

1−cos𝑎 𝑎2 A

2_{, (4.122)}

with 𝑎 = √A⋅A/2 standing for the magnitude of A. Making use of (4.122), it can be easily shown thatz𝑖(𝑡𝑚+1) given by (4.121) is indeed a unit vector.

The evolution laws (4.67)1and (4.69) for the lamellar normalsn(𝑟)and the lattice vectorsa(𝑟),

b(𝑟)_,c(𝑟)_{, respectively, are integrated by following the same approach as in the case of the vectors}

z𝑖. The results read as follows

n(𝑟)(𝑡𝑚+1) = exp

[

𝜔(n𝑟)(𝑡𝑚)Δ𝑡

]

n(𝑟)(𝑡𝑚), (4.123) where the spin𝜔(n𝑟)(𝑡𝑚) of the lamellar grains is given by (4.67)2, and

a(𝑟)(𝑡𝑚+1) = exp [ 𝜔(c𝑟)(𝑡𝑚)Δ𝑡 ] a(𝑟)(𝑡𝑚), b(𝑟)_(𝑡 𝑚+1) = exp [ 𝜔(𝑟) c (𝑡𝑚)Δ𝑡 ] b(𝑟)_(𝑡 𝑚), c(𝑟)(𝑡𝑚+1) = exp [ 𝜔(c𝑟)(𝑡𝑚)Δ𝑡 ] c(𝑟)(𝑡𝑚), (4.124) where the lattice spin 𝜔(c𝑟)(𝑡𝑚) is obtained from (4.70).

The evolution equations (4.74) for the back-stress tensorsT(_𝑑𝑟)are integrated by means of the forward Euler scheme, leading to

T(_𝑑𝑟)(𝑡𝑚+1) =T(𝑑𝑟)(𝑡𝑚) +{ 𝑔(_𝐼𝐼𝑟)T(_𝑑𝑟)⋅D(𝑟,𝑎) (𝑔(_𝐼𝑟))2 T (𝑟) 𝑑 + 2 3 [( 2𝐼(𝑟)𝑔(_𝐼𝑟))D(𝑟,𝑎)−(T(_𝑑𝑟)⋅D(𝑟,𝑎))I] +(D(𝑟,𝑎)T(_𝑑𝑟)+T_𝑑(𝑟)D(𝑟,𝑎))+(W(𝑟,𝑎)T(_𝑑𝑟)−T(_𝑑𝑟)W(𝑟,𝑎))}𝑡=𝑡𝑚Δ𝑡, (4.125) where it is recalled that 𝑔_𝐼(𝑟)(𝑡𝑚) and 𝑔𝐼𝐼(𝑟)(𝑡𝑚) denote respectively the first and second derivative of the stored-energy function (4.16) with respect to the invariant𝐼, evaluated at𝑡=𝑡𝑚, and that the variable𝐼(𝑟)_(𝑡

𝑚),D

(𝑟,𝑎)

Finally, recall that the critical resolved shear stresses (CRSSs)𝜏₀(𝑟,𝑐_𝑘 )(𝑡𝑚+1) in the slip systems of the crystalline phase within grain 𝑟 are completely determined in terms of the associated de- formation measure Γ(𝑟,𝑐)(𝑡𝑚+1) by means of the expression (4.77). In turn, given Γ

(𝑟,𝑐)

(𝑡𝑚), the computation of the quantity Γ(𝑟,𝑐)(𝑡𝑚+1) requires the numerical evaluation of the integral (4.126). Assuming that the average shear rates ˙𝜸(𝑘𝑟,𝑐)(𝑡) in the corresponding phase of the LCC are constant during the time increment 𝑡𝑚 ≤𝑡 ≤𝑡𝑚+ Δ𝑡 and equal to its value at 𝑡=𝑡𝑚, equation (4.126) yields the result

Γ(𝑟,𝑐)(𝑡𝑚+1) = Γ (𝑟,𝑐) (𝑡𝑚) + 𝐾 ∑ 𝑘=1 ˙ 𝛾(𝑘𝑟,𝑐)(𝑡𝑚)Δ𝑡. (4.126)