Macro-Micro Relations

In the classical approach, the constitutive relation is defined by algebraic or differential relations between the stress tensor a and the strain tensor e, based on the theory of tensorial invariants. In the microplane approach, the constitutive relation is defined as a relation between the stress and strain vectors acting on a plane of arbitrary orientation in the material. The orientation of this plane, called the microplane, is characterized by the unit normal n. The basic hypothesis, which makes it possible to describe strain softening (Bazant 1984c), is that the strain vector є, on the microplane (Fig. 14.1.3a) is the projection of the macroscopic strain tensor e, that is,

ejv — ЕП (14.1.1)

The stress vector dyv on the microplane cannot be exactly equal to the projections of the macroscopic stress tensors a if the strains represent the projections of є. Thus, static equivalence or equilibrium between the macro and micro levels must be enforced only approximately, by other means. The way to enforce it is to use a variational principle, that is, the principle of virtual work. For equilibrium, it suffices that, for any variation be, the virtual work of the macrostresses within a unit sphere be equal to the virtual work of the microstresses on the surface elements of the sphere (Bazant 1984c). This condition is written as;

-—a-be — I <jjv • beГІП (14.1.2)

3 J о

where the dot represents scalar product of two vectors or two second-order tensors.

Remark: A more detailed justification of this relation may be given as follows. We consider a small repre­sentative volume of the material, given by a small cube of side Ah. A pair of two parallel sides corresponds

Macro-Micro RelationsMacro-Micro Relations

Figure 14.1.3 (a) Microplane normal and microstrain vectors, and normal and shear components of the microstrain vector, (b) Directions of microplane normals (circles) for a system of 21 – microplanes per hemisphere (after Baf. ant and Oh 1986, adapted from Bafanl, Xiang and Prat 1996).

to microplane labeled by subscript N, and the other two pairs of sides correspond to orthogonal microplanes labeled by subscripts P, and Q. The strain vectors on these microplanes may be assumed to have the meaning defined by Дии/ДЛ = єу, Айр/Ah — єр, Auq/Ah — Aeq in which Ally, Айр, and Auq are the differences in the displacement vector between the opposite sides of the cube in the directions by labeled by N, P, and Q. The equality of the incremental virtual work of stresses within the representative volume on the macrolevel and the work of stresses on the three microplanes representing the sides of the cube implies that Ah? a ■ бє — Ah2{dy ■ 6Auy 4 dp ■ 6Айр + Sq ■ 6Auq), where S denotes the variations and Ah? – area of the sides of the elementary cube. The strain vectors єу, є’р and є о include the contributions of elastic deformations as well as displacements due to cracking (and possibly also to plastic slip). The cracking or other inelastic deformation happens randomly on planes of various orientations within the material, and the macroscopic continuum must represent these strains statistically, in the average sense. Therefore,

A /t3<r • бє Ah2-— [ (cr’/V ■ 6Айу + op ■ б Айр T oq ■ б Auq) (Ю. (14.1.3)

“о Jn

in which the integral represents averaging over all spatial orientations; dQ = sin OdOdA where 0, ф = spherical angles, S3 — surface of a unit hemisphere, and Oq — 2tt — its surface area. Now, obviously, J cfw • SAuydd — Js!<fp ■ SAupdU — Jsioq ■ Ailgdll. Thus the variational equation (14.1.3) yields (14.1.2). A

Подпись: 2тг Подпись: (dy <B> n)sd(l Подпись: ■ бє — 0 Подпись: (14.1.4)

Substituting (14.1.1) into the integral in (14.1.2) and factorizing бє, we obtain

where 0 indicates tcnsorial product and superscript S for a tensor denotes the symmetric part of such tensor, i. e., Ts — (T + T2 )/2, in which T is an arbitrary second-order tensor and T2 its transpose. Since the variational equation (14.1.4) must be satisfied for any variation бє, it is not only sufficient but also necessary that the expression in parentheses vanish. This yields the following fundamental relation from which the macroscopic stress tensor is calculated:

о• — f (dy®n)sdQ (14.1.5)

27r Jn

Подпись: 3 27Г Macro-Micro Relations Подпись: (14.1.6)

To compute the integral over the unit sphere, Gaussian integration can be used, and so the cartesian stress components Oij are computed as

Macro-Micro Relations
in which

Sij = [(oyv ®n)s].. = ^(ffNiUj +aNjrii) (14.1.7)

and the last expression represents an approximate numerical evaluation of the integral over the hemisphere; subscripts /J, represent a chosen set of integration points representing orientations of discrete microplanes defined by unit vectors nf (shown by the circled points in Fig. 14.1.3b); w, t are the integration weights associated with these microplanes, normalized so that ^ = 1; and superscript /z labels the values

corresponding to these directions. While the integral over Cl represents integration over infinitely many microplanes, the numerical approximation represents summation over a finite number of suitably chosen discrete microplanes. The flow of calculation between the macro – and micro-levels is explained by Fig. 14.1.4

Formulation of an optimal numerical integration formula over the surface of a hemisphere is not a trivial matter. The problem has been studied extensively by mathematicians, and Gaussian integration formulas of various degrees of approximation have been developed. The simplest integration formulas, for which all the weights are equal, arc obtained by taking the discrete microplanes identical to the faces of a regular polyhedron (Platonic solid). В ut the regular polyhedron of the largest number of sides is the icosahedron, with 20 faces, which yields 10 microplanes per hemisphere. It has been shown that the accuracy of the corresponding integration formula is insufficient for representing the postpeak stress-strain curve of concrete (this was demonstrated by the fact that rigid-body rotations of the set of discrete microplanes can yield unacceptably large differences in stresses); see Bazant and Oh (1985, 1986). Thus, formulas based on a regular polyhedron cannot be used, which means that the discrete microplanes cannot have equal weights. Determination of the optimum weights is not a trivial matter. The weights must be determined so that the formula would exactly integrate polynomials up to the highest possible degree and that the integration error due to the next higher-degree term of the polynomial be minimized (Bazant and Oh 1986).

One sufficiently accurate formula, which consists of 28 microplanes (i. e., 28 integration points) over a hemisphere, is given by Stroud (1971). A more efficient and only slightly less accurate formula, involving 21 microplanes, was derived by Bazant and Oh (1986) (and was used in the nonlocal finite element microplane program by Ozbolt, and in the program EPIC by Adley at WES). This 21-point formula exactly integrates polynomials up to the 9lh degree. The normals to the microplanes of this formula represent the radial directions to the vertices and to the centers of the edges of a regular icosahedron (as shown in Pig. 14.1.3b). Fewerthan21 microplanes cannot give sufficient accuracy (Bazant and Oh 1985).