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Fracture of materials with rate-dependent mechanical behaviour, e.g. polymers, is a highly complex process. For an adequate modelling, the coupling between rate-dependent stiffness, dissipative mechanisms present in the bulk material and crack driving force has to be accounted for in an appropriate manner. In addition, the resistance against crack propagation can depend on rate of deformation. In this contribution, an energetic phase-field model of rate-dependent fracture at finite deformation is presented. For the deformation of the bulk material, a formulation of finite viscoelasticity is adopted with strain energy densities of Ogden type assumed. The unified formulation allows to study different expressions for the fracture driving force. Furthermore, a possibly rate-dependent toughness is incorporated. The model is calibrated using experimental results from the literature for an elastomer and predictions are qualitatively and quantitatively validated against experimental data. Predictive capabilities of the model are studied for monotonic loads as well as creep fracture. Symmetrical and asymmetrical crack patterns are discussed and the influence of a dissipative fracture driving force contribution is analysed. It is shown that, different from ductile fracture of metals, such a driving force is not required for an adequate simulation of experimentally observable crack paths and is not favourable for the description of failure in viscoelastic rubbery polymers. Furthermore, the influence of a rate-dependent toughness is discussed by means of a numerical study. From a phenomenological point of view, it is demonstrated that rate-dependency of resistance against crack propagation can be an essential ingredient for the model when specific effects such as rate-dependent brittle-to-ductile transitions shall be described. [ABSTRACT FROM AUTHOR]

In this work, the geometry-constraint-based non-ordinary state-based peridynamic (GC-NOSBPD) model is further generalized to study the mechanical responses and rate-dependent fracture behaviors of viscoelastic materials. The viscoelasticity is generated by directly incorporating the classical viscoelastic constitutive relation in the form of the Generalized Maxwell Model, which is expressed as Prony-series type expansion, into the GC-NOSBPD framework. A strain-based scalar damage variable only driven by elastic energy, whose evolution follows a rate-independent law, is proposed to describe the progressive growth of the mode-I crack. Combining the rate-independent damage variable and viscoelasticity results in a rate-dependent damage model in the hypothesis that the fracture of viscoelastic material is driven by both elastic and viscous components of the energy. The fidelity of this model is established in the absence of damage by comparison with the benchmark solution for the viscoelastic response of a 3D bar subjected to various load histories. Further, the proposed model is employed to simulate two quasi-static fracture tests under various loading rates: a three-point bend fracture test for concrete and a double-cantilever-beam delamination test for rubber interface. These numerical results agree well with experimental tests, showing that the proposed model can effectively capture rate-dependent crack propagation. [ABSTRACT FROM AUTHOR]

Investigation of dynamic fracture of elastomers can still be considered to be a relatively open area. When a sheet of elastomer is stretched in a tensile machine and a crack is introduced, the crack propagates at a speed that depends on the initial stretch level. There are instances where this speed is noted to exceed the shear wave speed based on the elastic modulus under high imposed stretches. Such cracks are called transonic cracks. It was usually hypothesized that either the hyperelastic or viscoelastic stiffening of the bulk material raises the wave speeds resulting in crack speeds entering the transonic regime. This article revisits the experiments performed on Polyurethane elastomers in Corre et al. (Int J Fract 224(1):83–100, 2020) to study the implications of both these hypotheses. Crack propagation has not been explicitly modeled, but the crack speeds are implicitly imposed on the geometry using the boundary conditions extracted from the experimental data. It has been determined that the viscoelasticity in the bulk is needed to describe and understand the transonic cracks in polyurethane elastomer. The inclusion of viscoelasticity results in the notions of 'rubbery' and 'glassy' wave speeds and hence, the transonic regime is defined considering the rubbery wave speed. [ABSTRACT FROM AUTHOR]

The aim of this chapter is to revisit the historical works, mechanisms, and modeling approaches available in the field of fatigue crack growth resistance and rupture properties. After introducing the methodology developed to evaluate these properties, the impact of testing parameters such as temperature, loading speed, and pre-deformation will be highlighted. We will then review the influence of some material characteristics on rupture and crack propagation and the local mechanisms involved. Finally, a theoretical framework primarily dedicated to the description of crack propagation under static load will be discussed that aims to underline the connection between resistance to crack growth and the ability of a material to dissipate energy. [ABSTRACT FROM AUTHOR]

The paper reviews recent investigations of fatigue crack propagation in filler-reinforced elastomer blends based on recipes referring to truck tire tread compounds. One focus lies on viscoelastic energy dissipation effects that explain the well-known power law behavior of the crack growth rate vs. tearing energy. It is demonstrated that the crack growth rate fulfills the time-temperature superposition principle by constructing master curves with shifting factors from viscoelastic data. In addition, energy dissipation mechanisms due to stress softening in vicinity of the crack tip are evaluated under quasi-static conditions by calculating the J-integral, which decreases significantly while approaching the crack tip. Therefore, we use combined experimental and theoretical techniques consisting of digital image correlation measurements of the strain field around the crack and calculations of the corresponding energy densities and stress fields based on a physically motivated stress-softening model of filled rubbers. A second focus lies on the role of phase morphology of carbon black (CB) filled NR/BR and NR/SBR blends in fatigue crack propagation. Therefore, the filler distribution is evaluated by an established technique referring to the variation of viscoelastic loss peaks in the glass transition regime. It is found that in NR/BR blends, almost all CB is located in the NR phase, while in NR/SBR blends, the majority of CB is located in the SBR phase. The effect of batching on the phase morphology is also considered, by mixing the NR first with CB and then blending with the second rubber phase. This leads to a more homogeneous distribution of CB in NR/SBR blends. Finally, the obtained fatigue crack growth rates of these blend systems, measured under pulsed harmonic excitations in analogy to rolling tires, are discussed on the basis of the evaluated phase morphology and filler distribution. Here, also the influence of the polymerfiller interphase on the filler distribution is considered. [ABSTRACT FROM AUTHOR]

The viscoelastic and thermodynamic properties of four stock indices, namely, DJI, Nasdaq-100, Nasdaq-Composite, and S&P were analyzed for a period of 30 years from 1986 to 2015. The asset values (or index) can be placed into Aristotelian 'potentiality-actuality' framework by using scattering diagram. Thus, the index values can be transformed into vectorial forms in a scattering diagram, and each vector can be split into its horizontal and vertical components. According to viscoelastic theory, the horizontal component represents the conservative, and the vertical component represents the dissipative behavior. The related storage and the loss modulus of these components are determined and then work-like and heat-like terms are calculated. It is found that the change of storage and loss modulus with Wiener noise ( W) exhibit interesting patterns. The loss modulus shows a featherlike pattern, whereas the storage modulus shows figurative man-like pattern. These patterns are formed due to branchings in the system and imply that stock indices do have a kind of 'fine-order' which can be detected when the change of modulus values are plotted with respect to Wiener noise. In theoretical calculations it is shown that the tips of the featherlike patterns stay at negative W values, but get closer to W = 0 as the drift in the system increases. The shift of the tip point from W = 0 indicates that the price change involves higher number of positive Wiener number corrections than the negative Wiener. The work-like and heat-like terms also exhibit patterns but with different appearance than modulus patterns. The decisional changes of people are reflected as the arrows in the scattering diagram and the propagation path of these vectors resemble the path of crack propagation. The distribution of the angle between two subsequent vectors shows a peak at 90°, indicating that the path mostly obeys the crack path occurring in hard objects. Entropy mimics the Wiener noise in the evolution of stock index value although they describe different properties. Entropy fluctuates at fast increase and fast fall of index value, and fluctuation becomes very high at minimum values of the index. The curvature of a circle passing from the two ends of the vector and the point of intersection of its horizontal and vertical components designates the reactivity involved in the market and the radius of circle behaves somehow similar to entropy and Wiener noise. The change of entropy and Wiener noise with radius exhibits patterns with four branches. [ABSTRACT FROM AUTHOR]

In this work we develop an adhesive zone model for polymeric interfaces, which describes the kinetics of dissociation and association of polymer chains bridging the interface. Compared with previous works on interfacial bond rupture, our adhesive zone model includes two novel features: possibility of bond reforming and a highly nonlinear force-extension relationship for the polymer chain motivated by previous experimental measurements. The absence of these two features was demonstrated in an earlier work to cause unphysical crack propagation under zero load as well as overextension of the chains beyond their full contour length. Using the rate dependent crack propagation in a double cantilever beam, which may be elastic or viscoelastic, as an example, the new adhesive zone model is shown to correct the unphysical results obtained earlier. Specifically, it leads to a significantly increased adhesive fracture energy, i.e., the energy per unit area required to rupture the chains on the interface, for slow crack propagation, owing to the ability to achieve a dynamic equilibrium of bond dissociation and association. Furthermore, the nonlinear chain model predicts a near-catastrophic decrease in chain density as the finite extensibility limit is approached. This results in an adhesive fracture energy which is orders of magnitude smaller than that predicted by the linear chain model for fast crack propagation. Although the adhesive zone model has only been applied to a double cantilever beam in this work, it is a generic model for polymeric interface, and can be implemented in finite element models to simulate fracture in bulk polymers in general. [ABSTRACT FROM AUTHOR]