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Shrimali and Lopez-Pamies (2023, “The ‘Pure-Shear’ Fracture Test for Viscoelastic Elastomers and Its Revelation on Griffth Fracture,” Extreme Mech. Lett., 58, p. 101944) have recently shown that the Griffith criticality condition that governs crack growth in viscoelastic elastomers can be reduced to a fundamental form that involves exclusively the intrinsic fracture energy Gc of the elastomer, and, in so doing, they have brought resolution to the complete description of the historically elusive notion of critical tearing energy Tc. The purpose of this article—which can be viewed as the third installment of a series—is to make use of this fundamental form to explain one of the most popular fracture tests for probing the growth of cracks in viscoelastic elastomers, the trousers test.

 

It is now a well-established fact that even simple topology variations can drastically change the fracture response of structures. With the objective of gaining quantitative insight into this phenomenon, this paper puts forth a density-based topology optimization framework for the fracture response of structures subjected to quasistatic mechanical loads. One of the two key features of the proposed framework is that it makes use of a complete phase-field fracture theory that has been recently shown capable of accurately describing the nucleation and propagation of brittle fracture in a wide range of nominally elastic materials under a wide range of loading conditions. The other key feature is that the framework is based on a multi-objective function that allows optimizing in a weighted manner: ( ) the initial stiffness of the structure, ( ) the first instance at which fracture nucleates, and ( ) the energy dissipated by fracture propagation once fracture nucleation has occurred. The focus is on the basic case of structures made of a single homogeneous material featuring an isotropic linear elastic behavior alongside an isotropic strength surface and toughness. Novel interpolation rules are proposed for each of these three types of material properties. As a first effort to gain quantitative insight, the framework is deployed to optimize the fracture response of 2D structures wherein the fracture is bound to nucleate in three different types of regions: within the bulk, from geometric singularities (pre-existing cracks and sharp corners), and from smooth parts of the boundary. The obtained optimized structures are shown to exhibit significantly enhanced fracture behaviors compared to those of structures that are optimized according to conventional stiffness maximization. Furthermore, the results serve to reveal a variety of strengthening and toughening mechanisms. These include the promotion of highly porous structures, the formation of tension-compression asymmetric regions, and the removal of cracks and sharp corners. The particular mechanism that is preferred by a given structure, not surprisingly, correlates directly to the elastic, strength, and toughness properties of the material that is made of. 

A numerical and analytical study is made of the macroscopic or homogenized mechanical response of a random isotropic suspension of liquid n -spherical inclusions ( n = 2, 3), each having identical initial radius A , in an elastomer subjected to small quasistatic deformations. Attention is restricted to the basic case when the elastomer is an isotropic incompressible linear elastic solid, the liquid making up the inclusions is an incompressible linear elastic fluid, and the interfaces separating the solid elastomer from the liquid inclusions feature a constant initial surface tension γ . For such a class of suspensions, it has been recently established that the homogenized mechanical response is that of a standard linear elastic solid and hence, for the specific type of isotropic incompressible suspension of interest here, one that can be characterized solely by an effective shear modulus  n in terms of the shear modulus μ of the elastomer, the initial elasto-capillary number eCa = γ /2 μA , the volume fraction c of inclusions, and the space dimension n . This paper presents numerical solutions—generated by means of a recently introduced finite-element scheme—for  n over a wide range of elasto-capillary numbers eCa and volume fractions of inclusions c . Complementary to these, a formula is also introduced for  n that is in quantitative agreement with all the numerical solutions, as well as with the asymptotic results for  n in the limit of dilute volume fraction of inclusions and at percolation . The proposed formula has the added theoretical merit of being an iterated-homogenization solution.