# Energy Criterion for Crack Bands with Sudden Cracking

As just described, if the cracks are assumed to form suddenly, i. e., the stress to drop suddenly to zero, a spurious mesh sensitivity and lack of objectivity appears because of the dependence of the apparent energy dissipation on the element size. This effect is eliminated by the previous equivalent strength method, but can also be eliminated by directly applying an energy criterion analogous to linear elastic fracture mechanics. The proper form of the energy criterion, which was obtained by Bazant and Cedolin (1979, 1980) by generalization of Rice’s (1968b) energy analysis of the extension of a notch, can be formulated as follows. The crack band extension by length Да into volume ДЕ (of the next finite element, Fig. 8.6.6) may be decomposed for calculation purposes into two stages.

Stage I. Smeared cracks are created in concrete inside volume ДЕ of the element ahead of the crack in the direction of tensile principal stress (Fig. 8.6.6b), while at the same time, the deformations and stresses in the rest of the body are imagined to remain fixed (frozen). This means that one must introduce surface tractions At0 applied on the boundary AS of volume ДЕ, and distributed forces Af° applied at the concrete-steel interface, such that they replace the action of concrete that has cracked upon the remaining volume E — ДЕ and upon the reinforcement within ДЕ.

Stage II. Next, forces At0 and Af° (Fig. 8.6.6c) are released (unfrozen) by gradually applying the opposite forces —At0 and — Дґ°, reaching in this way the final state.

Let u° and £° be the displacement vector and strain tensor before the crack band advance, and let u and є be the same quantities after the crack band advance. For the purpose of analysis, the reinforcement may be imagined to be smeared in a separate layer coupled in parallel and undergoing the same strains as concrete in the crack band. The interface forces between steel and concrete, /f°, then appear as volume forces applied on the concrete layer.

Upon passing from the initial to the intermediate state (Stage I), the strains are kept unchanged while cracking goes on. Thus, the corresponding stress changes within concrete in ДЕ are given by Дсгц — erf і — Е’є®! (gf, + t/g?2)i?’/(l – г/2) — b”gf,; Д<т22 = <т®2; Дег^ ~ 0 (cracks are assumed to propagate in the principal stress direction). Here, denote the components of stress carried before cracking by the concrete alone (they are defined as the forces in concrete per unit area of the steel-concrete composite); and E and и are the Young’s modulus and Poisson’s ratio of concrete. The values E’ — E and и’ = и apply to plane stress and E’ = E/{ 1 — a2) and t/ — v/( — v) to plane strain. The change in strain energy of the system during Stage I in Fig. 8.6.6b is given by the elastic energy initially stored in AE and released by cracking, i. e.,

AU — ~ f “(°-° • e° ~ E’£°n2)dV. (8.6.17)

J&V 2

The change in strain energy during Stage II is given by the work done by the forces At0 and Af° while they are being released, i. e.,

Coefficients 1 /2 must be used because forces t and f at the end of Stage II are reduced to zero.

If the concrete is reinforced, part of the energy is consumed by the bond slip of reinforcing bars during cracking within volume AV. This part may be expressed as AW;, = Js Fbfads, in which &ь represents the relative slip between the bars and the concrete which is required to accommodate fracture advance; h’b is the average bond force during displacement &ь per unit length of the reinforcing bar (force during the slip); and s is the length of the bar segment within the actual fracture process zone of width hc (and not within volume Д V, since the energy consumed by bond slip would then depend on the clement size).

The energy criterion for the crack band extension may now. be expressed as follows:

AU — U’Aa = GjAa AU[ — AU2 — AWj, > 0 stable, no propagation

= 0 equilibrium propagation (8.6.19) < 0 unstable

Here AU is the energy that must be externally supplied to the structure to extend the crack band of width h by length Да. (AU = total energy in the case of rapid, or adiabatic fracture, and Helmholtz free energy in the case of slow, isothermal fracture.) If AU >0, then no crack extension can occur without supplying energy to the structure, and so the crack band is stable and cannot propagate. If AU < 0, crack band extension causes a spontaneous energy release by the structure, which is an unstable situation, and so the crack extension must happen dynamically, the excess energy — AU being transformed into kinetic energy. If AU — 0, no energy needs to be supplied and none is released, and so the crack band can extend in a static manner.

For practical calculations, the volume integral in Hq. (8.6.17) needs to be expressed in terms of nodal displacements using the distribution functions of the finite element. The boundary integral in Eq. (8.6.18) is evaluated from the change of nodal forces acting on volume AV from the outside. The energy AUi released from the surrounding body into A V may also be alternatively calculated as the difference between the total strain energy contained in all the finite elements of the structure before and after the crack band advance. According to the principle of virtual work, the result is exactly the same as that from Eq. (8.6.18). This calculation, however, is possible only if the structure is perfectly elastic whereas Eq. (8.6.18) is correct even for inelastic behavior outside the process zone, providing A a is so small that t and f vary almost linearly during Stage I.

It may also be noted that Pan, Marchcrtas and Kennedy (1683) calculated AU2 in their crack band finite element program by means of the J-integral, keeping the integration contour the same for various crack lengths. This calculation must yield the same AU2 if the integration contour passes through only the clastic part of the structure, except for crossing the crack band behind the fracture process zone where the stresses are already reduced to zero.

Under general loading, the crack band may propagate through a mesh of finite elements in an inclined direction, in which case the band has a zig-zag shape. This means that the crack length increment during the breakage of the next element is not well defined, and an effective crack extension Aae must then be used for the element. This crack extension is easily determined based on the effective bandwidth of the element, by writing that (for two dimensions) the area of the element AAc must be identical to the effective bandwidth he times the effective crack extension Aae, and thus (8.6.20)

where, if no’ further analysis is available, it may be assumed that the effective bandwidth is the projected element size hp. For rectangular meshes this reduces to the formula proposed by Bazant (1985a).

The ability of the energy balance and equivalent strength methods to describe the fracture processes in large structures was demonstrated in a series of papers by Bazant and Cedolin (1979, 1980, 1983) and by Bazant and Oh (1983a). As an example of their results, we consider here the problem of a plain concrete panel with a center crack, as depicted in Fig. 8.6.7a. Bazant and Cedolin (1980) analyzed the results for Figure 8.6.7 (a) Center-cracked panel analyzed by Bazant and Cedolin (1980). (b) Mesh-dependent results derived from constant-strength formulations, (c) Mesh independent results based on the energy formulation, (d) Comparison of the finite element results with the exact LEFM predictions.

three finite element meshes with element sizes in the relation 1:2:4, as sketched in Fig. 8.6.7a. If the classical tensile strength criterion (i. e., constant tensile strength) with sudden drop is used, the results shown in Fig. 8.6.7b are found, where it appears that the effect of the element size is tremendous. The strength for each crack length is seen to be smaller, the smaller the element size. This result may be expected because the results must converge to LEFM with an apparent fracture energy equal to the elastic energy density at fracture f’t/2E times the element width h. This means that O’/.’apparent oc h —» 0 for h -> 0, and the strength tends to zero for infinite mesh refinement, which is obviously wrong.

On the contrary, Fig. 8.6.7c shows the result obtained following the energy balance method, which shows very little influence of the mesh size. The curves for the equivalent strength method closely follow the results of the energy method (Bazant and Cedolin 1980). To check that the results of the crack band analysis are not only mesh independent, but also accurate, it suffices to compare them with the prediction deduced from LEFM analysis, which can be obtained in closed form for this case (using the solutions for the center cracked pane] and K/c = fEGp ). The comparison in Fig. 8.6.7d shows that the correspondence is excellent.

From the foregoing, we can conclude that if the mesh refinement is feasible so that h < 2£сь, and if each element displays progressive softening, the classical finite element analysis suffices to get consistent results. For larger elements, either the equivalent strength approximation in Section 8.6.4 or the energy balance method just described, will give mesh-independent and accurate results.

Comparison of Crack Band and Cohesive Crack Approaches Exercises

8.33 Consider a rectangular uniform mesh, with elements of dimensions hx and’/ij, in the horizontal and vertical directions, respectively, and a crack band extending at an angle в from the horizontal. Show that the mean or effective width for these elements is hb — hx sin 0 + hy cos 0. Show that for square meshes of size h, this reduces to hs/l cos(45° – 0) (Bazant 1985a).

8.34 Consider a rectangular uniform mesh, with elements of dimensions hx and hy in the horizontal and vertical directions, respectively, and a crack band extending at an angle 0 from the horizontal. Show that the effective crack extension Aac|-f when a crack band extends by one element is given by Дас|у — hxhy/(hx sin в – і – hy cos 0). Show that for square meshes of size h, this reduces to h/{/2 cos(45° – 0)} (Bazant 1985a).

8.35 To make a simple and fast analysis of a concrete gravity dam, a bidimensional model is generated, having approximately square elements with 3 m sides. The elastic modulus, strength, and fracture energy are estimated to be E — 19 GPa, r_ 21 MPa and Gy ■ 92 N/m. Determine the stress-strain curve with sudden strength drop that should be used.