Feeding and dissipative waves in fracture and phase transition. I: Some 1D structures and a square-cell lattice. II: Phase-transition waves.

*(English)*Zbl 1003.74007These two parts are devoted to a common approach to the description of feeding and dissipative waves in fracture and phase transition, with statement of a boundary between external (macrolevel) and internal (microlevel) problems. A zero feeding wave number corresponds to a macrolevel-associated solution with the classical homogeneous-material solution as its long-wave approximation. A non-zero wavenumber corresponds to a genuine microlevel solution which has no analogue on the macrolevel. Dissipative waves appear in both macrolevel-associated and microlevel soilutions.

In the first part the author presents a complete set of steady-state solutions for mode III crack propagation in a square-cell lattice. For this, at first, three simplified one-dimensional structures (an inclined thread falling on a rigid foundation, a periodic set of oscillators, and a string on an elastic foundation under a moving load) are studied, allowing to retrace qualitatively the main phenomena in the lattice. In particular, these structures demonstrate the structure- and speed-dependent radiation and different types of feeding and dissipative waves. Then, the analytical technique for mode III crack propagaton is described in detail. The crack propagation is considered as a sequence of bond breakings caused by feeding waves carrying energy to the crack front and accompanied by dissipative waves carrying a part of this energy away from the front. The final expressions are presented for all possible feeding waves, and fast and slow decreasing dissipative waves are shown.

Analogously, in the second part the author classifies the micro- and macrolevel solutions according to the type of feeding wave delivering energy to the propagating phase transition front. Both the discrete chain consisting of particles connected by massless bonds and the higher-order-derivate (HOD) model are considered. A steady-state phase-transition wave is considered for the discrete chain. As in the first part, a general solution is derived using Fourier transform and Wiener-Hopf techniques. In terms of Fourier transform, a long-wave approximation of the solution coincides with that for a homogeneous body, while nonzero real singular points correspond to the microlevel feeding and dissipative waves. So, the macrolevel-associated and microlevel solutions with the corresponding dissipative waves are analyzed. The uniqueness of macrolevel solution is achieved by using an expression for speed-dependent total dissipation obtained as a result of the macrolevel-associated formulation.

Then, the author considers a model where the strain energy is represented as quadratic form including first- and higher-order derivatives, while the corresponding moduli are different for both phases. For the homogeneous model it is shown that the contradiction between the limiting stress and energy criteria is eliminated if and only if the phase transition does not concern the highest-order modulus. For the fourth-order partial differential equation, the author shows the existence of the Maxwell type, dissipation-free, subsonic phase-transition wave. In this case the microlevel plays the role of a catalyst. The obtained results show that the HOD model and discrete chain possess both the macrolevel-associated and microlevel types of solutions. However, there are some distinctions between these models. In particular, in the HOD model, in contrast to the discrete chain, the manifestation of the dynamic amplification factor with its influence on the phase-transition wave-speed cannot be revealed.

In the first part the author presents a complete set of steady-state solutions for mode III crack propagation in a square-cell lattice. For this, at first, three simplified one-dimensional structures (an inclined thread falling on a rigid foundation, a periodic set of oscillators, and a string on an elastic foundation under a moving load) are studied, allowing to retrace qualitatively the main phenomena in the lattice. In particular, these structures demonstrate the structure- and speed-dependent radiation and different types of feeding and dissipative waves. Then, the analytical technique for mode III crack propagaton is described in detail. The crack propagation is considered as a sequence of bond breakings caused by feeding waves carrying energy to the crack front and accompanied by dissipative waves carrying a part of this energy away from the front. The final expressions are presented for all possible feeding waves, and fast and slow decreasing dissipative waves are shown.

Analogously, in the second part the author classifies the micro- and macrolevel solutions according to the type of feeding wave delivering energy to the propagating phase transition front. Both the discrete chain consisting of particles connected by massless bonds and the higher-order-derivate (HOD) model are considered. A steady-state phase-transition wave is considered for the discrete chain. As in the first part, a general solution is derived using Fourier transform and Wiener-Hopf techniques. In terms of Fourier transform, a long-wave approximation of the solution coincides with that for a homogeneous body, while nonzero real singular points correspond to the microlevel feeding and dissipative waves. So, the macrolevel-associated and microlevel solutions with the corresponding dissipative waves are analyzed. The uniqueness of macrolevel solution is achieved by using an expression for speed-dependent total dissipation obtained as a result of the macrolevel-associated formulation.

Then, the author considers a model where the strain energy is represented as quadratic form including first- and higher-order derivatives, while the corresponding moduli are different for both phases. For the homogeneous model it is shown that the contradiction between the limiting stress and energy criteria is eliminated if and only if the phase transition does not concern the highest-order modulus. For the fourth-order partial differential equation, the author shows the existence of the Maxwell type, dissipation-free, subsonic phase-transition wave. In this case the microlevel plays the role of a catalyst. The obtained results show that the HOD model and discrete chain possess both the macrolevel-associated and microlevel types of solutions. However, there are some distinctions between these models. In particular, in the HOD model, in contrast to the discrete chain, the manifestation of the dynamic amplification factor with its influence on the phase-transition wave-speed cannot be revealed.

Reviewer: I.A.Parinov (Rostov-na-Donu)

##### MSC:

74A45 | Theories of fracture and damage |

74N20 | Dynamics of phase boundaries in solids |

74J99 | Waves in solid mechanics |

##### Keywords:

fracture; phase transition; dissipative waves; Fourier transform; Wiener-Hopf techniques; feeding waves; energy dissipation; square-cell lattice; macrolevel solution; microlevel solution; mode III crack propagation; higher-order derivate model; phase-transition wave; long-wave approximation; uniqueness; strain energy; fourth-order partial differential equation##### References:

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