Chapter 11:Multiscale Modeling of the Evolution of Damage 505 The similarity between (11.2)and (11.10)is sufficiently striking that one is immediately tempted to use the same modeling algorithm on both length scales (note that if the momentum terms on the right-hand side of (11.8)are not negligible at the macroscale,then a different algorithm must be used at this length scale,as will be discussed below).This indeed is the approach that will be taken herein;but it must necessarily be said that (11.10)is only exact in the limit,i.e.,(11.9)is a sufficient condition for (11.10)to be exact.However,in all real circumstances,(11.9)cannot be satisfied,so that some error must necessarily be introduced by utilizing approximate (11.10)in lieu of exact (11.10). The use of (11.10)is termed herein a"mean field theory"because the higher-order terms that are dropped from (11.6)are essentially higher area moments of the microscale stress.Thus,the macroscale analysis is per- formed only in terms of the mean stress o.Note that,in cases wherein there is localization induced by damage or large strain gradients,one or more of the higher-order terms will not be negligible.In this case,a mean field theory is no longer accurate;and a nonlocal approximation (including one or more of the higher-order terms in (11.6))or even a full field analysis performed simultaneously on all length scales may be necessary to obtain reasonable accuracy.However,the necessity for converting to this procedure may be monitored by calculating the higher-order terms in (11.6)after each time step during the local scale analysis Now consider the standard deviation of the microscale stress,given by ”=(6.-d业 (11.11) An object in which the standard deviation of all of the state variables is small compared to their respective means is termed,in this chapter, "statistically homogeneous"(this,of course,implies that the effects of any singularities are bounded when integrated over the volume).It can also be shown that,when (11.9)is satisfied,the standard deviation of the micro- scale stress,given by (11.11),goes to zero.Therefore,in many cases it is sufficient for the object to be statistically homogeneous at the microscale in order for(11.10)to be an accurate representation at the macroscale.One implication of this result is that the microcracks contained within the microscale volume must be statistically homogeneous in location and orientation.If this is not the case,then higher-order moments will neces- sarily have to be included at the macroscale [1]
The similarity between (11.2) and (11.10) is sufficiently striking that one is immediately tempted to use the same modeling algorithm on both length scales (note that if the momentum terms on the right-hand side of (11.8) are not negligible at the macroscale, then a different algorithm must be used at this length scale, as will be discussed below). This indeed is the approach that will be taken herein; but it must necessarily be said that (11.10) is only exact in the limit, i.e., (11.9) is a sufficient condition for (11.10) to be exact. However, in all real circumstances, (11.9) cannot be satisfied, so that some error must necessarily be introduced by utilizing approximate (11.10) in lieu of exact (11.10). The use of (11.10) is termed herein a “mean field theory” because the higher-order terms that are dropped from (11.6) are essentially higher area moments of the microscale stress. Thus, the macroscale analysis is performed only in terms of the mean stress σ . Note that, in cases wherein there is localization induced by damage or large strain gradients, one or more of the higher-order terms will not be negligible. In this case, a mean field theory is no longer accurate; and a nonlocal approximation (including one or more of the higher-order terms in (11.6)) or even a full field analysis performed simultaneously on all length scales may be necessary to obtain reasonable accuracy. However, the necessity for converting to this procedure may be monitored by calculating the higher-order terms in (11.6) after each time step during the local scale analysis. Now consider the standard deviation of the microscale stress, given by ( )2 SD 1 d . V V V µ µ µ µ µ σ σσ ≡ − ∫ (11.11) An object in which the standard deviation of all of the state variables is small compared to their respective means is termed, in this chapter, “statistically homogeneous” (this, of course, implies that the effects of any singularities are bounded when integrated over the volume). It can also be shown that, when (11.9) is satisfied, the standard deviation of the microscale stress, given by (11.11), goes to zero. Therefore, in many cases it is sufficient for the object to be statistically homogeneous at the microscale in order for (11.10) to be an accurate representation at the macroscale. One implication of this result is that the microcracks contained within the microscale volume must be statistically homogeneous in location and orientation. If this is not the case, then higher-order moments will necessarily have to be included at the macroscale [1]. Chapter 11: Multiscale Modeling of the Evolution of Damage 505
506 D.H.Allen and R.F.Soares Now note that,as long as any tractions on the crack faces are self- equilibrating,(11.2)may be used to show that [6,9,13] )ds. (11.12) where ri is the unit outer normal vector on the local boundary ov Note that the boundary averaged stress given in (11.12)actually is physically more palatable than the volume averaged stress given in(11.7), as it is commensurate with the original definition of stress,as defined by Cauchy [10],to act on a surface. The fact that the volume averaged stress is equivalent to the boundary averaged stress is of little importance when there are no cracks.However, when cracks grow and evolve with time,it becomes a very important aspect of the homogenization process,as will now be shown by considering the homogenization process for the strain tensor.It can be shown by careful employment of the divergence theorem that E=Em+am (11.13) where (11.14) is the mean strain at the local scale, 2,元+@n西 (11.15) is the boundary averaged strain on the initial (external)boundary of the local volume and 元+低元s (11.16) is the boundary averaged strain on the newly created (internal)boundary due to cracking ov and is called a damage parameter [17,33]. Since kinematic equation (11.15)is consistent with kinetic equation (11.12),it is reasonable to construct constitutive equations at the macro- scale in terms of these two variables,rather than in terms of volume averages.This is in striking contrast to the approach taken when there are
Now note that, as long as any tractions on the crack faces are selfequilibrating, (11.2) may be used to show that [6, 9, 13] 1 ( ) d, V nxS V µ µ µ µ µ µ σ σ ∂ = ⋅ ∫ G G (11.12) where nµ G is the unit outer normal vector on the local boundary Vµ ∂ . Note that the boundary averaged stress given in (11.12) actually is physically more palatable than the volume averaged stress given in (11.7), as it is commensurate with the original definition of stress, as defined by Cauchy [10], to act on a surface. The fact that the volume averaged stress is equivalent to the boundary averaged stress is of little importance when there are no cracks. However, when cracks grow and evolve with time, it becomes a very important aspect of the homogenization process, as will now be shown by considering the homogenization process for the strain tensor. It can be shown by careful employment of the divergence theorem that 1 1, µµ µ εε α = + + + (11.13) where 1 d V V V µ µ µ ε ε = ∫ (11.14) is the mean strain at the local scale, E T 1 1 1 [ ( ) ]d V 2 un un S V µ µ µ µ µ µ µ ε + ∂ = + ∫ G G GG (11.15) is the boundary averaged strain on the initial (external) boundary of the local volume E Vµ ∂ , and I T 1 1 1 [ ( ) ]d V 2 un un S V µ µ µ µ µ µ µ α + ∂ = + ∫ G G GG (11.16) is the boundary averaged strain on the newly created (internal) boundary due to cracking I Vµ ∂ and is called a damage parameter [17, 33]. Since kinematic equation (11.15) is consistent with kinetic equation (11.12), it is reasonable to construct constitutive equations at the macroscale in terms of these two variables, rather than in terms of volume averages. This is in striking contrast to the approach taken when there are 506 D.H. Allen and R.F. Soares