#### Installation — business terrible - 1 part

September 8th, 2015

The local constitutive law may be written in the incremental form

До – – Е{Ає – Дє") ЕАє – AS (13.3.1)

where До – and Дє are the increments of the stress and strain tensors, E is the fourth-rank tensor of elastic moduli of uncrackcd material, Дє" the inelastic strain increment tensor, and ДБ the inelastic stress increment tensor.

In a nonlocal continuum formulation, this equation is replaced by

До–.КДє-Д!3 (13.3.2)

where ДБ is the nonlocal inelastic stress increment tensor. In the phenomenological approach discussed in the previous sections, this tensor is directly obtained by a spatial averaging integral

AS(x)-=AS= f &(x,£)AS(£)dK(£)’ (13.3.3)

Jv

completely analogous to (13.1.1), in which we remember that (he weight function a is to be postulated. Following Bazant (1994b), we now describe how the equation governing the evolution of Д£> can be developed from the mechanics of crack interactions.

Consider an elastic solid that contains, at the beginning of the load step, many microcracks numbered as H = 1, …N (Fig. 13.3.1). On the macroscale, the microcracks are considered to be smeared, as required by a continuum model. Exploiting the principle of superposition, we may decompose the loading step of prescribed load or displacement increment into two substeps:

**I **In the first substep, the cracks (already opened) are imagined temporarily “frozen” (or “filled with a

glue”), that is, they can neither grow and open wider nor close and shorten. Also, no new cracks can nucleate. The stress increments, caused by strain increments Дє and transmitted across the temporarily frozen (or glued) cracks (I in Fig. 13.3.1), are then simply given by ЕАє. This is represented by the line segment 13 (Fig. 13.3,2) having the slope of the initial elastic modulus E.

**II **In the second substep, the prescribed boundary displacements and loads are held constant, the cracks

are “unfrozen” (or “unglued”), and the stresses transmitted across the cracks are relaxed. This is equivalent to applying pressures (surface tractions) on the crack faces (П in Fig. 13.3.1). In

Figure 13.3.1 Superposition method for solid with many cracks. In part i, the cracks arc closed and Acr1 = ЕАє. In part II, the stresses Apt on the crack faces generated in part I are released, either simultaneously (alternative a) or iteratively keeping all the cracks closed but one (alternative b); adapted from Bazant 1994b.

response to these pressures, the cracks are now allowed to open wider and grow (remaining critical according to the crack propagation criterion), or to close and shorten. Also, new cracks are now allowed to nucleate.

If cracks neither grew nor closed (nor new cracks nucleated), the unfreezing (or unglueing) at prescribed increments of loads or boundary displacements that cause macro-strain increment As wotdd engender the stress drop 34 down to point 4 on the secant line 01 (Fig. 13.3.2). The change of slate of the solid would then be calculated by applying the opposite of this stress drop onto the crack surfaces. However, when the cracks propagate (and new cracks nucleate), a larger stress drop defined by the local strain-softening constitutive law and represented by the segment AS = 32 in Fig. 13.3.2 takes place. Thus, the normal surface tractions

AjV ~ ng ■ AS^rtj, (13.3.4)

representing the normal component of tensor AS^, must be considered in the second substep as loads App that are applied onto the crack surfaces (Fig. 13.3.1), the unit normals of which are denoted as it,,. (Note that for mode II or III cracks, a similar equation could, in general, be written for the tangential tractions on the crack faces.)

Let us now introduce two simplifying hypotheses:

Figure 13.3.3 Details of crack interactions: (a) Actual crack pressure distribution and mean pressure; (b) mean pressure distributions generated at cracks number A and /j. by a unit uniform pressure at crack v, all other cracks being frozen. (Adapted from Bazant 1994b.)

1. Although the stress transmitted across each temporarily frozen crack varies along the crack, we consider only its average, i. e., Ais constant along each crack (Fig. 13.3.3a). This approximation, which is crucial for our formulation, was introduced by Kachanov (1985, 1987a). He discovered by numerical calculations that the error is negligible except for the rare case when the distance between two crack tips is at least an order of magnitude less than their size.

2. We consider only mode 1 crack openings, i. e., neglect the shear modes (modes II and 111). This is often justified, for instance in materials such as concrete, by a high surface roughness which prevents any significant relative slip of the microcrack faces (the mode II or III relative displacements that can occur on a macroscopic crack are mainly the result of Mode 1 openings of niicrocracks that are inclined with respect to the macrocrack).

A simple-minded kind of superposition method would be to unfreeze all the cracks, load by pressure only one crack at a time, and then superpose all the cases (Fig. 13.3.1a). In this approach, the pressure on each crack, Awould be known. But one would still have to solve a body with many cracks.

A better kind of superposition method is that adopted by Kachanov (1985, 1987a, which was also used by Datsyshin and Savruk (1973), Chudnovsky and Kachanov (1983), Chudnovsky, Dolgopolski and Kachanov (1987), and Horii and Nemat-Nasser (1985), and, in a displacement version, was introduced by Collins (1963). In this kind of superposition, all that is needed is the solution of the given body for the case of only one crack, with all the other cracks considered frozen (Fig. 13.3.1b). The cost to pay for this advantage is that the pressures to be applied at the cracks are unknown in advance and must be solved. By virtue of Kachanov’s approximation, we apply this kind of superposition only to the average crack pressures. The opening and the stress intensity factor of crack /; are approximately characterized by the uniform crack pressure Athat acts on a single crack within the given solid that has elastic moduli E and contains no other crack. This pressure is solved from the superposition relation:

where the superimposed bar indicates averaging over the crack length; Лд„ are the crack influence coefficients representing the average pressure (Fig. 13.3.3b) at the frozen crack /! caused by a unit uniform pressure applied on unfrozen crack tz, with all the other cracks being frozen (Fig. 13.3.1b); and Aw, = 0 because the summation in (13.3.5) must skip v = ц. The reason for the notation for A with a tilde instead of an overbar is that the unknown crack pressure is uniform by definition and thus its distribution over the crack area never needs to be calculated and no averaging of pressure actually needs to be carried out.

Note that the exact solution requires considering pressures App.(x’) and Ap,,(x’) that vary with coordinate Xі along each crack. In numerical analysis, the crack must then be subdivided into many intervals. This could hardly be reflected on the macroscopic continuum level, but is doubtless unimportant at that level.

Substituting (13.3.4) into (13.3.5), we obtain

_______ N ‘

A(ntl ■ SMn(I) A(n^ ■ + ^Л;і1/Д(п,/ ■ S^n,,) (13.3.6)

y=i

Now we adopt a third simplifying hypothesis. In each loading step, the influence of the. microcracks at macro-continuum point of coordinate vector £ upon the microcracks at macro-continuum point of coordinate vector x is determined only by the dominant microcrack orientation. This orientation is normal to the unit vector of the maximum principal inelastic macro-stress tensor Д5А1) at the location of the center of microcrack pt. We use the definition:

AS,?5 = Д(п,, ■ S;jn(j) = [nM ■ SMnM]new – [n/2 ■ SMn;j]oUJ (13.3.7)

The subscripts ‘new’ and ‘old’ denote the values at the beginning and end of the loading step, respectively. According to this hypothesis’, the dominant crack orientation generally rotates from one loading step to the next. Eq. (13.3.6) may now be written as:

N ____

АЩР -^ГЛ^АЗ^ = Asll} (13.3.8)

The values of Д£>(1 are graphically represented in Fig. 13.3.2 by the segment AS — .35. This segment can be smaller or larger than segment 32.

Alternatively, one might assume n/2 to approximately coincide with the direction of the maximum principal strain. Such an approximation is simpler to use in finite element programs, and it might be realistic enough, especially when the elastic strains are relatively small.

When the principal directions of the inelastic stress tensor S do not rotate, the increment operators Д can of course be moved inside each product in (13.3.6), i. e., Д(пм • S(1n,2) — n(, ■ AS,,n/t, etc. One might wonder whether this should not be done even when these directions rotate (i. e., when rt/t varies), which would correspond to crack orientations being fixed when the cracks begin to form. But according to the experience with the so-called rotating crack model, empirically verified for concrete, it is more realistic to assume that the orientation of the dominant cracks rotates with the principal direction of S.

It might seem in the foregoing equations we should have taken only the positive part of tensor AS^. But this is not necessary since the unloading criterion prevents AStl front being negative.