Nonlocal Adaptation of Microplane Model or Other Constitutive Models
In unconfined straining, the microplane model displays softening. Therefore, localization limiters of some kind must be used to avoid spurious localization and nresh sensitivity, as for all other models with strain softening. This can be easily implemented using a nonlocal adaptation of the microplane model in which the inelastic stress increment is made nonlocal following the theory of microcrack interactions presented in the previous chapter (Bazant 1994b; see §13.3). This approach affects the flow of calculation only partially and a general finite element scheme can be used. Fig. 14.1.1c shows the basic calculation flowchart for this approach in which the nonlocal adaptation is implemented just after the microplane stresses get computed; the flow bifurcates and the inelastic incremental stress is computed following the nonlocal theory with microcrack interactions.
The microplane model as presented, or for that matter any constitutive model for damage, gives a prescription to calculate the stress tensor or as some tensor-valued function It of the strain tensor є (and of some further parameters depending on the loading history, e. g., on whether there is loading or unloading). So, o’ = Іі(є). The most robust (although not always the most accurate) method of structural analysis is to base the solution of a loading step or lime step on the incremental elastic stress-strain relation with inelastic strain involving the initial elastic moduli tensor E, as explained in Section 13.3.1. Then, for a local formulation, the inelastic stress increment tensor AS defined in (13.3.1) can be computed as
AS – E( ^IlCW £old) — ІЦЄпет/) + ЩЄоІсі) (14.3.1)
in which subscripts old and new label the old and new value of the variables at the beginning and end of the loading step (or time step); and S is the inelastic stress tensor due to nonlinear behavior. This stress-strain relation is used for both dynamic explicit analysis and static implicit analysis (as the iterative initial stiffness method).
A possible simple approach to introduce nonlocal effects is similar to the isotropic scalar nonlocal approach (Pijaudier-Cabot and Bazant 1987), which was applied to the microplane model by Bazant and Ozbolt (1990, 1992) andOz. bolt and Bazant (1992). In this approach, the elastic parts of stress increments are calculated locally. The inelastic parts of the increments of S must be calculated nonlocally. This is accomplished by first determining, at each integration point of each finite, element, the average (or nonlocal) strains є, and then calculating nonlocal AS from these, i. e.,
Ac — ЕАє – AS; AS = E(WM – £„c„) – R(eIK. w) + К(є0м) (14.3.2)
The only modification required in a local finite element program is to insert the spatial averaging subroutine just before the calculation of AS.
A better approach is to introduce the crack interaction concept explained in Section 13.3 and write the incremental elastic stress-strain relation as
Дог =■ ЕАє – AS (14.3.3)
in which AS is given by Eq. (13.3.9). The spatially averaged strains are not calculated in this approach. The nonlocal part of the analysis proceeds in the following steps:
1. First, AS is calculated (in the local form) from (14.3.1) according to the microplane model. Then one calculates at each integration point of each finite element the maximum principal direction vectors n(4 [і = 1,2, 3) of strain tensor e, for which the value of e0id may be used as an approximation.
Figure 14.3.1 Local representative volume: orientation and size (adapted from Ozbolt and Bazant 1996).
2. Then one starts a loop on principal strain directions n‘ (і — 1,2,3) of tensor є and evaluates
the inelastic stress changes in the directions that is, ASM = ■ ASn^ or A=
ASijn, j’ For those principal directions for which Дб’М < 0, the nonlocal calculations are skipped because the inelastic strain is not due to cracking, i. e., one jumps directly to the end of this loop; here we make the assumption that, on the microscale (but not on the macroscale), there is no softening in compression, which is true for the microplane model.
3. The values of ASW for the integration points of finite elements are then spatially averaged:
A“4l) = ~ it Д5<4Ч„Д V„ (14.3.4)
‘ V— 1
where V’fj — Yl*-=i AV„ = normalizing factor, n = number of all the integration points inside the averaging volume, and afiu = given weight coefficients, whose distribution is suitably chosen with a bell shape in both x and у directions, described by a polynomial of the fourth degree. The bell shape, which is similar to that in the nonlocal damage approach (Chapter 13, Eq. (13.1.5)) is reasonable in that it gives larger contributions to the sum from points that lie closer. Because the spacing of major cracks in concrete is approximately the same as the spacing of the largest aggregate pieces, the size of the averaging volume may be assumed to be approximately proportional to the maximum aggregate size, da. For two-dimensional analysis, the region of averaging should probably be taken as a rectangle with its longer side in the direction normal to (Fig. 14.3.1)
4. The values of the nonlocal principal inelastic stress increments Amust then be solved from the system of linear equations (13.3.31) based on the crack influence function However, as discussed before, exact solution is normally not needed. Depending on the type of program, one of two approximate methods can be used:
(a) In programs in which the loading step is iterated, these equations may be solved iteratively within the same iteration loop as that used to solve the nonlinear constitutive relation, using the following equation:
ASfnew AS-W I — ДКЛ^Л5»оШ (м – 1,2, …N) (14.3.5)
in which A’ = crack influence matrix defined in Eq. (13.3.43), which must, however, be adjusted with factor kb for finite elements close to the boundary of concrete (Section 13.3.9).
(b) In explicit finite element programs without iteration, one may calculate from (14.3.5) only the first iterate (r — 1), which represents one explicit calculation, requiring only the values of Aand (Д5,?). The premise of this approximation is that the repetitions of similar calculations (for r — 1) in the next loading step (or time step) effectively serve as the subsequent iterations (for r = 2, 3,4,…) because the loading steps in the explicit programs are very small. This of course means that the correct value of А6’|,^ gets established with a delay of several steps or time intervals (in other words, the computer program is using nonlocal inelastic stress increments that are several steps old; the nonlocal interactions expressed by the crack influence function are delayed by several steps).
We recall from Section 13.3.9 that the adjustment by factor кь must ensure that, even if part of the influencing volume protrudes beyond the boundary, the condition А’/ш 0 be met. Because
this condition may be written as ^interior ^ /^boundary ~ 0>lllc following adjustment is needed for the integration points of the elements adjoining the boundary;
= ki, A,lL,-, кь — – Y, AfW / Y, Л,,„ (14.3.6)
interior a ‘ boundary ;/
For the remaining integration points in the interior, no adjustment is done, i. e., А!,ш — Л.
5. At each integration point of each finite element, the nonlocal inelastic stress increment tensor is then constituted from its principal values according to the following equation:
ASkl = ]T AS^n^n^ or AS = Y 5(!>n(,) Jo n(0 (14.3.7)
І~. 1 І–1
based on the spectral decomposition theorem of a tensor.
Note that if, at some integration point, all the principal values of tensor AS arc nonpositive, then the foregoing nonlocal procedure may be skipped for that point.