#### Installation — business terrible - 1 part

September 8th, 2015

It is well known in continuum mechanics that for many purposes it is useful to decompose the strain tensor into its hydrostatic and dcviatoric parts, by writing s = (1 /3)tr є 1 + є’, where 1 is the unit tensor and e’ the dcviatoric strain tensor. When applied to (14.1.1), the following decomposition of the microstrain vector follows

in which £y is called the volumetric strain and £p the deviatoric strain vector acting on the microplane; they are defined as

1 .

£y — “tre and єр~єп (14.1.9)

The deviatoric strain vector is further decomposed into its normal component Ep that we call deviatoric strain for short, and its component tangential to the microplane that we call the shear strain vector £p:

єр = ep ■ n — s’n ■ n and £т—Єр~£рп (14.1.10)

The microplane strain vector can, thus, be written as

s’n — єуп + єріі + єр (14.1.11)

Analogous components ay, ap, and dp are defined for the microstress vector, and so we write

(7/v " ay ft T apfi – ap (14.1.12)

Note that both the volumetric and deviatoric components contribute to the normal component at the microplane. We can thus define the total normal microstrain and microstress £n and cr/v as

£n = £V + £u і atn –ay–ap (14.1.13)

Based on the foregoing definitions, a particular microplane constitutive law consists in a set of rules specifying how the microstress components ay, ap, and dp change as £y,£p, and ip evolve. The simplest case to be solved is the linear elastic case that we analyze next.

In the elastic regime we must have a linear relationship between <T/v and є for every ii; therefore, we must seek a relationship of the form

арі = L(e, n) (14.1.14)

where the function L{e, n) is linear in є. Moreover, isotropy requires that if the microplane (and its normal vector) and the macrostrain tensor are both rotated through any orthogonal tensor Q, the resulting microstress must be correspondingly rotated, i. e.,

I^QeQ1, Qfi) — Q3n = QL(e, n) (14.1.15)

which indicates that the function /.(e’/v, n) is an isotropic vector-valued function of a second-order tensor and a vector. The most general function of this type that is linear in e can be written as

Е(єх, п) — aitr є n 4- Ьі(єії ■ fi)n + с,єп (14.1.16)

where a,b{, and C| arc scalar constanls. This can readily be rewritten in terms of the volumetric, deviatoric, and shear components of the microstrain:

dfj — ЕуЄу n + Ep£p n – I Epip ‘ (14.1.17)

where Ey, Ep, and Ep are microplane elastic moduli corresponding to volumetric, deviatoric, and shear straining. In view of (14.1.12), the foregoing expression can be split into the following three relations:

ay – Eyey, ap=Ep£p and ap = Epip (14.1.18)

The microplane elastic moduli can be determined in terms of the macroscopic elastic moduli by identifying the macroscopic stress-strain response predicted by the microplane model with the classical Hooke equations. The macroscopic response is obtained by substituting Sfj from (14.1.17) into (14.1.5) and

(lien into the expressions for the microplane strain components in terms of the macrostrain tensor. The resulting macroscopic relationship is

or = “tr £ A – f (Ає’ + є’A) – I – (Ep – Ет)Вє’ (14.1.19)

where A and В are, respectively, the following second – and fourth-order tensors:

A — — [ n® ті (Ю. and В = — f n®n&n<F)ndil (14.1.20)

^ Jn 2тг Jn

These two tensors can be computed with relative ease using various methods (see the exercises at the end of this section). The result is simple:

A–1 and SDijki = 6ij6kl F 6ik&ji + 6n8jg (14.1.21)

where Btjki arc the rectangular cartesian components of B. Substituting these expressions into (14.1.19) we get the linal expression for the macrostress tensor as:

where /( — Erf Eq is a free parameter which may be chosen.

Parameter /л can be optimized so as to best match the given test data. Bazant and Prat (1988b), who gave relations equivalent to (14.1.24) but in terms of parameter rj ~ Ep/Ey instead of/г, found the range of ‘//-values giving the optimum (its of test data for concrete. This range corresponds to /./-values close to 1. Therefore, the value fi= 1 has subsequently been used in all the data lilting that we cite later in this section. Note also that the inverse of (14.1.24) yields E and v in terms of Ey, Ер, and /л.

As revealed by the study of Carol, Bazant and Prat (1991), the value /it = 1 is also conceptually advantageous because it makes it possible to characterize damage, in the sense of continuum damage mechanics, by a fourth-rank tensor that is independent of the material stiffness properties. This will be discussed in more depth later.

It is interesting to note that for the choice /t — (1 — 4i/)/(l + u), one has Ey — Ер. Then one can set с/v — Ерєл’, where £/v = Ey — Eq – Sb, in that case there is no volumctric-deviatoric split. But that would not be realistic for concrete.

One reason that the normal strain on the microplane must be split into the volumetric and deviatoric normal components is that a general model ought to be capable of giving (for any /./) any thermodynamically admissible value of Poisson’s ratio, that is, —1 < v < 0.5. That this is indeed so can be checked by eliminating /і from (14.1.24) and solving for u, which yields и = (5Ey — 2Ер — ЪЕр) j(QEy T 2Ер + 3E-p). This relation also shows that, for the case of no split (which corresponds to the case Ey — Ep = Ep), one would have v — (Ер — Ет)/(4Ер + Ет), and so the Poisson ratio would be restricted to the range — 1 < v < 0.25. Although this range would suffice for concrete, the microplane model, in principle, could not be fully realistic if it were restricted to Poisson’s ratios less than 0.25.

it may also be noted that if the shear stiffness were neglected (Er — 0 or /t — 0), then any Poisson ratio between —I and 0.5 could still be obtained, provided that the volumetric and deviatoric normal microplane strains would be split. However, if they were not (i. e., rt/v — ЕрЄр), which was implied in the initial model of Bazant and Oh (1983b, 1985) for tensile fracturing only, then Poisson’s ratio would be restricted to the value v — 0.25. Such a restriction is not realistic, and besides, the shear stiffness on the microplane level appears to be important for correct modeling of the effect of confining pressure on compression failure.

The main reason for the volumctric-deviatoric split with independent moduli Ev and Epj (Bgzant and Prat 1988b) is the absence of a peak and of postpeak strain softening for hydrostatie compression test and uniaxial strain compression test (see the tests of Bazant, Bishop and Chang 1986), while at the same time the loading by uniaxial compressive stress or other compressive loadings with uninhibited volume expansion exhibits stress peak followed by postpeak strain softening. Without the aforementioned split, compressive loading with restricted volume expansion (hydrostatic compression and uniaxial strain) would also, incorrectly, exhibit a peak stress and postpeak strain softening.

In the initial proposal of microplane model with strain softening (Bazant 1984e), the stress-strain relation for the normal and shear components of stresses and strains of the microplanes had the form of incremental plasticity, based on subsequent yield surfaces and loading potentials for the microplane. However, subsequent studies have shown that this was unnecessarily complicated. As it turned out (Bazant and Oh 1985; Bazant and Prat 1988b), one can assume a total algebraic stress-strain relation for these components for the case of virgin loading, that is, о у, о d, and d-p can be assumed to be functions of £у,£д, and є-p. Further it turned out that each stress component can be considered to depend only on the associated strain component, with the exception of shear stress d-p, which is considered to depend on (Tfj to express internal friction (and, at high pressures, plasticity). Without the frictional aspect, it is not possible to model standard triaxial tests at high confining pressures.