# Numerical Aspects

In finite element programs, a system of at least 21 microplanes must be associated with each integration point of each finite element (Bazant and Oh 1985, 1986; used in Bazant and Ozbolt 1990). Their number, however, can be reduced for the symmetries of plane stress, plane strain, axisymmetric behavior, and uniaxial stress.

For a given microplane, the normal and shear components of the microstrain vector are conveniently handled by defining an orthonormal base (Fig. 14.1.6). Since the selection of m and t is

arbitrary, we may, for example, choose vector m,- to be normal to the global axis xj, in which case the cartesian components of rn in the global coordinate system are m ni{n + пХ)" І/,2,?П2 = —П (n + n|)“!/2, ГП] = 0 but mi — I and rn.2 — тт, — 0 if щ = щ — 0. To get a vector пщ normal to axis X| or axis X2, we carry out permutations 123 –> 231 —> 312 of the indices in the preceding equations. (To minimize directional bias, the procedure of generating vectors rn,- should be such that if for one microplane m, is normal to Xi, for the next numbered microplane it is normal to a. g, for the next to xj, for the next again to xi, etc.) The other coordinate vector within the microplane is obtained as vector product, і — m x n.

Once the components of the base vectors for the microplane are obtained, the determination of the components of the microstrain vector given the macrostrain tensor immediately follow as:

£дг — Л/jjCjy, Ем ~ MijEij, c/; — LijEij (14.1.28)

where Em and £/, are the components of the shear microstrain vector (i. e., E-p = єд/m EiJ), and the projection tensors N, M, and L arc given in component form by

Nij—riirij, Mi] — + rtijUi) , Lij — ~(ІіП]-i-ijiij) (14.1.29)

To write an efficient finite element program, the values of N’j, Mshould be calculated, for all the discrete microplanes (labeled here by superscript fi), in advance of finite element analysis and stored in memory. The values of nf and of the weights uy. in the integration formula (14.1.6) must also be stored in advance.

In each loading step, an explicit computational algorithm can be formulated as follows. First, the new values of macro-strains £.;y are calculated at each integration point from the new (incremented) values of nodal displacements. Then, for each integration point, the new values of £n, sm, and are calculated for all the microplanes from (14.1.28) and the volumetric and deviatoric components £y and Єц are determined from the first of (14.1.9) and (14.1.13). Using these values, the following new stress values

Gy — Gy + Ьу(єу Єу) J GCd — alD + Е[)(єо — ££>) , G% ~ Gy ■{- GC} (14.1.30)

ам = а’м + ®г(£м – єгм), <?i ^ a +- ET(eh – є), a’v = Max[cry, – Fy(ey)} (14.1.31) = °y + МіП{Мах[ст^,-^((-Єс>))],.р£((єо))} _ (14.1.32)

aN — Min[o’Jv. i;V((£Jv),CTV/),(Ti/] (14.1.33)

Superscripts і denote the previously calculated initial values at the beginning of the loading step, and superscripts e denote the new stress values based on elastically calculated increments; (x) — Max(a;, 0) — positive part of x (this symbol, called the Macauley bracket, is used so that functions Fn, . ..Ft could be defined for only the positive values of strain arguments), and ay a. but if the load step is iterated, it helps accuracy to take ay as the value of ay obtained in the previous iteration. Alter sweeping through all the microplancs /х — 1,… Nm, one must calculate

Nm

Gy – (14.1.34)

tl=]

Then, for each microplane one can calculate

ay — M’m(a’v, ay) (14.1.35)

for є у — є у > 0 :

а’р – Міп{а? Г,гГа? Г,рг((єт),ау)]}, ат = Мах[а! Г,-Fr({ ~єт), ау)} (14.1.36)

forey – є, < 0 ;

а’т = Мах{с4,2′[с4, – Fr((-eT), ау)]}, ат ~ Міп[а’г, РГ({єт),ау)} (14.1.37)

After sweeping again through all the microplanes, all the new values of the microplane stresses at the end of the loading step are known, and the macrostresses can then be calculated from (14.1.6) where the expression for the components of sy in terms of the components of the microslress vector is easily seen to be;

•Sy — a pi Nij – b aMMtj T apL^j (14.1.38)

The inelastic parts of the new macrostresses must subsequently be modified according to a suitable nonlocal formulation. This subject is discussed later in Section 14.3.

Note that the foregoing algorithm gives the new stresses as explicit functions of the new strains. No equations need to be solved. This is important for computational efficiency.

14.1.5 Constitutive Characterization of Material on Microplane Level

By fitting of various types of test data for concrete, the following functions, characterizing the constitutive properties of the material, have been identified (Bazant, Xiang and Prat 1996): Fv(—£y) — fy exp y~ff7~J ’ fv = EkxkA, (any єу) Fd{-sd) = f°D (l – , /& = Я*,С4, (ев<0)       fM? d) = ъ/Ъ (l + , (£D>0) Fr(aiv) — (Ekfc2 — k3a.)

in which k….. k5 are adjustable empirical constants, which take different values for different types of concretes, while C[,…, c.\$ are fixed empirical constants that can be kept the same for all normal concretes. They have the values C 5,02 = 6,03 — 50, c* = 130, and cs — 6 (parameter cj affects almost only the standard triaxial tests at very high pressures). It has been recommended that, in absence of sufficient test data, the adjustable parameters may be taken with the following reference values k — 72 x 10 6, kj — 0.1, k) — 0.05, A4 — 15, and As — 150. The value of Poisson’s ratio may be considered as и — 0.18. Except for E. all llic parameters are dimensionless.

The macroscopic Young’s modu las is a parameter whose change causes a vertical sealing transformation (affinity transformation) of all the response stress-strain curves. If this parameter is changed from E to some other value E’, all the stresses are multiplied by the ratio EJ/Е at no change of strains (Fig. 14.1,7a). Parameter ki describes radial scaling (affinity transformation) with respect to the origin. If this parameter is changed from кt to some other value Aj, all the stresses and all the strains are multiplied by the ratio A-;/A, (Fig. 14.1.7b).

The aforementioned reference values of material parameters along with E =58000 MPa yield the uniaxial compression strength /’ =42.4 MPa, as calculated by simulating the uniaxial compression test by incremental loading. The strain corresponding to the stress peak has been found to be єр 0.0022. Now, if the user needs a microplane model that yields the uniaxial compressive strength f* and the corresponding strain at peak є*, one needs to modify the reference values of only two parameters as follows:  (14.1.44)

Table 14.1.1 shows the values of /’, f[ for some typical values of material parameters (“R” in Table

14.1.1 refers to the reference values stated above). It also gives the corresponding ductility r = evE j/’, representing the ratio 03/04 in Fig. 14.1.7c. The smaller r, the steeper the postpeak softening. The transformations according to (14.1.44) do not change the ratio r.

The aforementioned reference values of material parameters have been selected so that the ratio of tensile to compressive uniaxial strengths be approximately f[jfc = 0.082; the ratio of equitriaxal to uniaxial compression strength fbcj/’ = 1.17; the ratio of the strength in pure shear to the uniaxial compressive strength approximately /*//’ = 0.069; the ratio of residual stress for very large uniaxial compressive strain to the uniaxial compression strength approximately oy//’ = 0.07; and the ratio of residual stress for very large shear strain to the shear strength approximately ту/f‘ = 0.3. The transformations according to (14.1.44) do not change these ratios. These ratios can be changed only by adjusting material parameters other than E and k.

Parameters A4 and k\$ can be determined exclusively from the data on hydrostatic compression tests. Taking the logarithm of (14.1.43), the equation can be reduced to a linear regression plot, and thus parameters A4 and As can be obtained by fitting the data on the hydrostatic compression test, separate from all other parameters (because the value of ey for hydrostatic compression is the same for all microplanes and єд = єм — £i — 0 for all microplanes). The softening tail in uniaxial compression can be lengthened by increasing C2 while reducing k a little, and for tension by reducing A3 while reducing k

Table 14.1.1 Strength, ductility, and typical material parameters

 Tests E k кг h /u4 bffc5 fe(io-6) I’c Vc fl rt Hognestad 5900 TIT к R R R R 5.18 1.9(T 0.4 6 1.78 van Mier 29000 R R R R R R 40.0 1.98 3.75 1.80 Petersson 26000 R R R 0.4 R R 32.1 2.43 3.62 1.65 Bazant 6000 112 R R R 12 175 7.55 1.91 0.69 1.73 Green 5100 R R R R R 125 7.39 1.93 0.66 1.78 Balmcr 3500 90 20 R 0.5 18 R 3.05 2.29 0.33 1.71 Bresler 5100 R R 0.2 R R R 6.49 2.36 0.40 2.44 Kupfer 5100 R 40 0.3 .06 R R 4.86 4.41 0.62 2.22 Launay 5100 R R R 0.3 R R 4.87 2.93 0.77 1.73 Sinha 3200 113 R R R R R 4.00 2.07 0.36 1.79

“R” means reference values; rc = £? C/fL rt — et b’//t

a little. The ratio of the tensile-to-compressive strength can be increased by reducing c4 or k-). The ratio of the strength in pure shear to the uniaxial compressive strength can be increased by increasing кг while reducing c4 or fcr a little.