# Nonlinear Microplane Behavior and the Concept of Stress-Strain Boundaries

In the original microplane model for compressive failure (Bazant and Prat 1988b), the stress-strain relations for the microplancs were smooth curves. However, difficulties arose in the handling of the transition from reloading to virgin inelastic loading in the quadrants of negative stress-strain ratio, and complicated rules had to be devised (Hasegawa and Bazant 1993; andOz. bolt and Bazant 1992). Also, the modeling of cyclic loading was difficult. These difficulties can be circumvented with the concept of stress-strain boundaries. However, the main reason for introducing this concept is the modeling of triaxial behavior in tension.

The condition that the response must not exceed a specified boundary curve ox = Fx{sx) —where X indicates the appropriate microplane component— makes it easy to ensure continuity at the transition from elastic behavior, which is defined separately for volumetric and deviatoric components, to the strain­softening damage behavior in tension, which is defined without the volumctric-deviatoric split (Bazant 1993c). It seems next to impossible to devise explicit algebraic stress-strain relations that would describe such transitions without any discontinuity.

The stress-strain boundaries, shown in Fig. 14.1.5, are defined as (Bazant, Xiang and Prat 1996):

ff/v — Fn(£n),vv – —Fv{ £v), з — —Fd[—£u),ctd — Ер(єц). or = Fr(ox) (14.1.25)

in which от stand for either ом or 07, (the components of shear stress vector on two arbitrarily assigned orthogonal axes M and L within the microplane). It might seem that, from the viewpoint of rotational invariance in the microplane, the shear stress vector or = (ом, or) should be considered parallel to £7, i. e., отІ5т — є’х/£Т. Such a formulation (Bazant and Prat 1988b), however, did not perform very well for complex loading paths, ft appeared preferable and simpler to consider that or in (14.1.25) stands either for ом or or, i. e., ом — Fp(ox) and or — fr(ayv), thus allowing or and dp to be, in general, nonparallcl. Of course, this implies a directional bias for the chosen orientations of axes M and L on each microplane. However, due to averaging on the macroscale, such a bias becomes negligible on the macroscale if the orientations of M and L on various microplanes are chosen with nearly equal probability (or frequency) for various possible orientations, and if there are many integration points of finite elements within the representative volume of material.

Function Ft defines only the boundary for positive stresses. The other, for negative stresses, is symmetric. The reason for writing the minus signs in (14.1.25) is that functions F, v, Fy, T)> are defined as positive-valued functions of positive arguments. Function Fp defines only the boundary for the magnitudes of the shear stresses (Fig. 14.1.5d). The dependence of о-p on o, characterizes friction on the microplane, as well as the fact that a widely opened rough crack offers less resistance to shear than a narrow rough crack.

The response anywhere within the boundaries may be simply assumed to be clastic, as given in the rate Figure 14.1.5 Stress-strain boundaries. General form for the deviatoric (a), volumetric (b), normal (c)} and shear (d) components. For a classical macroscopic formulation, the boundary would be an arbitrary surface in the 12-dimensional Oij-Sij as indicated by the thick curve in (e). (Adapted from Bazant, Xiang and Prat 1996.)

form by equations similar to (14.1.18):

ay = Eyiy, ap — ЕрЄр, and ap — Epep (14.1.26)

This simple assumption, of course, implies the stress-strain path for the microplane to exhibit a sudden change of slope when the elastic response arrives to the boundary curve. 1 lowever, such changes of slope on the macroscale are not so abrupt because different microplanes reach the boundary at different times. Nevertheless, the response can be made smoother by the formulation in the following remark.

Remark: The response on the microscale can be made smooth by introducing a transition curve between the elastic straight line and the boundary curve. The transition curve, however, cannot be defined as a simple function of strains because the elastic lines and boundary curves are functions of different components. A helpful idea is to define the transition implicitly, in terms of (i) the elastic stress value ac and (ii) the boundary curve value ob, both of them corresponding to the same strain є. When ob ac > 0, the transition curve must nearly coincide with ae, and when a*’ — ab, it must lie farthest below both curves. These required properties can be achieved by the following formula for the transition curve (Bazant, Xiang and Prat 1996):

T{ae, ab) = —±-2І±£і _ (14.1.27)

2 у 2c() J

where ae — cr’jf, o”j;. 0ГСТ7.; ab — obn, oS, or oj; and Si, So are constants, which have been chosen as <Si = 0.10/osign(cr) and So ~ 0.24 foaign(crb) with /о = /с,/ло or/)1-. For the volumetric boundary, no transition curve is introduced because the slope change is mild.

For Si — 0 the transition curve would approach the elastic curve and the boundary curve asymptotically at ±00 (this may be easily checked by noting that, for large [x|,2cosha: = expjxj). But the response near the origin of stress-strain space must be exactly clastic. Therefore, the left-side asymptote of the transition curve is shifted up by distance її. This causes the transition to intersect the elastic curve. By choosing a small enough 61, the slope change at the intersection is small and acceptable.

The transition curve (14.1.27) with Si = 0 approaches the elastic line and the boundary curve exponentially, i. e., very rapidly. Another formula of similar properties was also explored: T(cr’ , a’1) -■■■ {ab + ae + S1 – [(ct6 — <7e — <5|)2 + Syy/2. This formula would be faster to execute computationally (which matters somewhat because it is evaluated a great many limes). However, for Si = 0, it approaches the elastic and boundary curves too slowly, much slower than (14.1.27), which is therefore preferable, Л

The stress-strain boundary may be regarded as a strain-dependent yield limit. Such an idea could hardly be introduced in the classical macroscopic invariant approach to plasticity, because the boundary would be a surface in a 12-dimensional space of all сту – and єу components. The micruplanc concept makes the idea of strain-dependent yield limit feasible, in fact simple, because there arc only a few components on the microplane level. The strain-dependent yield limit may be illustrated by the curve in Fig. 14.1.5e. The classical (stress space) plasticity is in this figure represented by the horizontal line for the yield limit. Now  Figure 14.1.6 (a) Orthonormal base associated to a microplane, (b) Microstrain components.

note that plastic metals and fracturing materials (Dougill 1976) have also been satisfactorily described by strain-space plasticity, which corresponds to the vertical line in this figure. Obviously, a general curve should allow a better description because it is a combination of stress-space and strain-space plasticity theories.