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In this context, we recall that the theory of brittle fracture founded in 1920 by Griffith [1,2] is based on the linear elasticity, in the framework of which the stress field is determined, and the energy criterion, which states that the energy release through the moving crack tip must be equal to the (double) surface energy of the material. It is important to note that the surface energy can be determined in an independent way. So, the physical model by itself looks impeccable. Nevertheless, the Griffith theory leaves some questions unanswered. One of them concerns the bulk-to-surface energy transition as the energy flux through the crack tip singular point (a line in the three-dimensional case). In this theory, the mechanism of the transition is hidden, the microstructure role and the transition-associated dynamic effects are not reflected.

As the next step is important for the present work, discrete models were introduced into consideration. In 1969, Novoghilov [9,10] formulated the concept of a brittle fracture, which took account of the discrete structure of the body and suggested the necessary and sufficient criterion for the estimation of the strength of an elastic body weakened by a cut. The process of destruction is treated as a loss of stability of elastic equilibrium. These two factors, the discreteness and the loss of stability during the deformation of the breaking bond, are the basis of a number of phenomena that could not be detected in the framework of the continuum mechanics [11]. One of them is the energy radiation from the crack front in each act of the rupture [12].

The crack dynamics in a lattice model, for which both these factors are inherent, was first considered analytically in 1981 [13]. In this problem, the local-to-global energy release ratio was determined. This ratio plays the role of the crack-speed-dependent and material-structure-dependent corrective coefficient in the expression of the Griffith energy criterion. For the mass-spring square lattice it appeared that, in the quasi-static crack growth, only the proportion of the macrolevel energy release is spent on the fracture itself, while the remainder is radiated with the lattice waves. As the crack speed increases, this ratio first increases non-monotonically and then monotonically decreases (it tends to zero as the crack speed approaches the long wave speed). The structure of the radiation is described in more detail in Slepyan [14].

In addition to the above-mentioned factors, the internal strain energy of self-equilibrated MS can play an important role. In addition to their effect on the crack resistance, some phenomena manifested in fracture, such as bridging, developing porosity in front of the crack and irregularities in crack growth, can be caused by MS.

Note that macrolevel residual stresses, self-equilibrated in a macrodomain were considered repeatedly. In this connection, see the series of works by Banks-Sills et al. from [15] to [16] (also see references therein) and Bebamzadeh et al. [17,18], where the role of curing residual stresses in the fracture of composites is examined. Such residual stresses manifest themselves as an additional load on the cracked body.

We here consider a mechanism of fracture under MS self-equilibrated in each cell of periodicity. A general formulation is used, where only the structure of the interface as the prospective crack path is specified. The interface is assumed to be formed by a discrete set of uniformly distributed differently stressed bonds, where compressed bonds alternate with stretched ones. For example, this may happen if the initial lengths of the elastic bonds are different. The body is assumed to be symmetric about the middle line crossing the bonds and periodic along this line (figure 1). The response of the structure to external action is reflected by means of a non-specified crack-related Green's function.

We determine the energy relations and discuss possible scenarios of the crack growth, which are defined by the bond length ratio and the ratio of the internal energy to the energy coming from the remote load. Note that the quasi-static considerations adopted in this paper allows us to determine the total energy radiated in each step of the crack growth. The dynamic effect, the effect of the radiation on the fracture development can be determined based on the dynamic formulation. The role of the dynamic factor [22] in the fracture under MS is discussed. It is found, in particular, that under MS the crack initiation energy barrier can increase, whereas the crack growth can be accompanied by irregularities and clustering. In some respect, these phenomena are similar to those found earlier for mode II crack dynamics in a triangular lattice [19] and in a lattice under the harmonic excitation [23,24].

The paper is organized as follows. First, a simple model is considered consisting of two parallel elastic strings connected by a discrete set of periodically placed bonds alternating by their initial lengths. The structure is under MS, no external forces are applied. It is assumed that under this condition only initially stretched bonds may break. In spite of the simplicity, this model demonstrates all main effects owing to MS presence. We determine the initial prestress (2.3), the tensile forces in the crack front bond in the case of a semi-infinite bridged crack (2.12) and for only one (2.16) or two bonds broken (2.19). Also, the corresponding energy relations, as the ratios of the energy of the bond to the initially stored or released energy, are presented (2.22), (2.24), (2.28) and (2.30). Here and below, the results are presented in dependence on the orthotropy parameter, α, as the ratio of the bond stiffness to that of a string between-the-bonds segment.

In a steady-state crack growth, the energy is released, so the microlevel energy release rate , where a is the between-the-bond distance. A part of this energy disappears with the breaking bond, while the other part is radiated in the form of acoustic oscillations. Owing to linearity of the problem, is proportional to Δ2, where Δ is a half of the difference in the lengths of the unstrained bonds. The analysis is given below mainly in terms of Δ. If the internal energy is given instead of the bond length initial difference, 2Δ, the results can be read based on the relation between the initial internal energy and Δ.

is less than the stored energy (2.20). In this case, in contrast to the case of the semi-infinite crack, the corresponding energy ratios of the bond energy, E, to the released energy and to the stored one are different

The energy ratios for the semi-infinite bridge crack and one and two broken bonds, and R1, respectively, are presented in figure 5 as functions of . The plots of the ratios and P1 are shown in figure 6. It is remarkable that the ratio of the fracture energy to the total released energy is practically the same for any finite and semi-infinite cracks.

The energy ratios, (2.24), R0 (2.28) and R1 (2.30), of the maximal bond energy to the stored energy as functions of the dimensionless parameter for the semi-infinite bridge crack and one and two broken bonds, respectively. (Online version in colour.)

(a,b) The energy ratios, of the maximal bond energy to the released energy as functions of the dimensionless parameter for the semi-infinite bridge crack (2.24) and one (2.28) and two (2.30) broken bonds, respectively, are presented in the left-hand side plot, which evidences that these three ratios are almost the same. Their ratios, and , are shown in the right-hand side plot. (Online version in colour.)

Recall that Ec is the critical energy of the initially stretched bond and is the internal energy per two spans stored in the initial state of the chain. The factor 1/Rj defines how much internal energy is required for the next step of the crack advance. A part of this energy goes on fracture itself, that is for the bond breakage. The remainder is radiated. The radiated acoustic energy, as the difference between the energy release and the critical energy of the bond, is equal to . The radiated oscillations, however, act on the next initially stretched bond and this can lead to the dynamic crack growth even in the case where the stored energy is below the critical level defined in (2.31) (in this connection, see [22]). Thus, the crack propagation criterion appears below the crack initiation threshold, as is usually observed.

Note that if no external force acts on the half-planes except Green's function source, the principle force acting on the upper half-plane from below is zero. It follows from the equilibrium condition that the applied unit force is completely supported by the bonds, and hence

The crack domain M it can be found, in principle, using the strength criterion for the bonds and taking into account the fracture history. In this way, it can happen that there exists a bridge crack in a part of the total crack domain, where only even bonds are broken, and a free crack face sub-domain, where the odd bonds also are broken. Note, however, that dynamic effects can play an important role in the crack development.

The normalized energy release ratio at γ=γ* (a) and the plots of γ* and (b) as functions of for the chain and lattice. The equality corresponds to the critical level of the internal energy for the body with a semi-infinite open crack. (Online version in colour.)

At least in the chain and lattice structures, a considerable part of the critical internal energy is radiated under the bond breakage (this part disappears in the framework of the quasi-static formulation). The remaining part of the released energy is the critical strain energy of the bond disappearing at fracture. The ratio of the breaking bond energy to the internal energy density for a finite crack rapidly approaches that for a semi-infinite crack (figure 5) whereas the ratio to the total released energy is practically the same for any finite and semi-infinite cracks (figure 6). 2b1af7f3a8