Spalling failure is one of the main techniques for studying processes that occur in solids in dynamic tension [1–۵]. Consideration of wave processes that are typical of the spalling problem makes it possible to determine a stress momentum that leads to failure [1, 2]. As a rule, the so-called acoustic approximation, in which only elastic stresses that act in the spalling zone are taken into account, is used to estimate the spalling momentum parameters. Sample failure may occur sometime after the maximum of tensile stress has been reached in the spalling section. This implies that the failure moment falls onto the descending or constant segment of effective local stress. This phenomenon was coined as “failure delay,” and pulses that correspond to this failure type are, as a rule, close to threshold ones. Threshold pulses are understood to be pulses of given duration and shape that have the minimum failure-producing amplitude. Failure delay has been registered in some spalling experiments, for example, in [6, 7]. However, any detailed discussion of this phenomenon is practically absent from the literature. In a situation like this, strength is often linked, purely formally, to the period of action of tensile stresses in the spalling section [6]. In this article, spalling failure delay is explained using the criterion of an incubation time. The notion of the structural–temporal criterion is based on the following two material characteristics: quasistatic material strength and failure incubation time. These quantities are strength-related parameters of a material; they are independent of experimental conditions and can be applied to different types of loading. This approach proved its effectiveness when determining conditions of emergence of various transients such as brittle fracture [8, 9], electrical breakdown [10], and cavitation in fluids [11].

ONE-DIMENSIONAL WAVE PROBLEM

Spalling failure occurs as follows. A shock pulse creates a compression wave in a sample. The wave propagates along the sample axis until it reaches a free surface. Upon reflection from the surface, the compression wave reverses sign and propagates backwards as a tensile wave. Since the tensile strength of a material is usually considerably lower than its compressive strength, a failure may occur in a certain sample section. Under threshold loads, the failure may reveal itself as microcrack nucleation. With more intense actions, the fraction of the material is completely separated from the sample in the form of the so-called spalling plate. Solving the one-dimensional spalling problem in its elastic formulation shows that the time profile of the wave of compressive stresses coincides, bar a multiplier, with V(t), the velocity of motion of particles on the free surface. Thus, the time dependence of stresses in the spalling section with a coordinate x can be repwhere ρ is the material density and a is the propagation speed of a longitudinal stress wave. It should be noted that the temporal profiles of the compressive and tensile waves coincide completely, with the tensile wave lagged behind by Δt = 2x/a, which is the doubled travel time of the elastic wave through the spalling section.