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Three Dimensional Fracture Growth as a Standard Dissipative System: Some General Theorems and Numerical Simulations
Full text linkCrack propagation in brittle materials has been studied by several authors exploiting its analogy with standard dissipative systems theory. In recent publications, minimum theorems were derived in terms of crack tip “quasi static velocity” for two-dimensional fracture mechanics. Following the cornerstone work of Rice on weight function theories, Leblond and coworkers proposed asymptotic expansions for stress intensity factors in three dimensions. In view of the expression of the expansions proposed by Leblond, however, symmetry of Ceradini’s theorem operators was not evident and the extension to 3D of outcomes proposed in 2D not straightforward. Following a different path of reasoning, minimum theorems have been finally derived. Moving from well-established theorems in plasticity, algorithms for crack advancing have been finally formulated. Their performance is here presented within a set of classical benchmarks
Fracture propagation in brittle materials as a standard dissipative process: General theorems and crack tracking algorithms
No full textThe present work frames the problem of three-dimensional quasi-static crack propagation in brittle materials into the theory of standard dissipative processes. Variational formulations are stated. They characterize the three dimensional crack front ``quasi-static velocity" as minimizer of constrained quadratic functionals. An implicit in time crack tracking algorithm that computationally handles the constraint via the penalty method algorithm is introduced and proof of concept provided
Dynamic multifield continualization of multilayered lattice-like metamaterials
Full text linkThis work focuses on dynamic continualization of multifield multilayered metamaterials in order to obtain energetically-consistent models able to provide an accurate description of the dispersive behavior of the corresponding discrete system. Continuum models, characterized by constitutive and inertial non-localities, have been identified through a recently proposed enhanced continualization scheme. They are identified by governing equations both of the integro-differential and higher-order gradient-type, whose regularization kernel or pseudo-differential functions accounting for shift operators are formally expanded in Taylor series. The adopted regularization kernel exhibits polar singularities at the edge of the first Brillouin zone, thus assuring the convergence of the frequency spectrum to the one of the Lagrangian system in the entire wave vector domain. The validity of the proposed approach is assessed through the investigation of multilayered discrete lattices with an antitetrachiral topology, where local resonators act as rigid links among the layers. The convergence of dispersion curves of the continuum model to the ones of the Lagrangian model is proved in the whole first Brillouin zone as the adopted continualization order increases, both considering the propagation and the spatial attenuation of Bloch waves inside the metamaterial. A low frequency continualization is also provided, leading to the identification of a first-order medium
Weight function theory and variational formulations for three-dimensional plane elastic cracks advancing
No full textThe weight function theory for three-dimensional elastic crack analysis received great attention after the
work of Rice (1985, 1989). Several applications have been considered since then, particularly in the context
of configurational stability, crack path prediction, stress intensity factor expansions, perturbation
approaches. In all cases, a specific hypothesis has been made on the variation of crack shape, in order
to formulate the problem in terms of Cauchy principal value. In the present note, such hypothesis is further
investigated and consequences discussed. A variational statement given in Salvadori and Fantoni
(2013a) is thus rephrased in terms of weight functions. Its discrete formulation shows the potential to
accurate approximation of crack front propagation
Minimum theorems in 3D incremental linear elastic fracture mechanics
No full textThe crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen in Appl Mech Rev 47, 1994, Stability and nonlinear solid mechanics. Wiley, New York, 2000; Mielke in Handbook of differential equations, evolutionary equations. Elsevier, Amsterdam, 2005; Bourdin et al. in The variational approach to fracture. Springer, Berlin, 2008). In a recent publication (Salvadori and Carini in Int J Solids Struct 48:1362–1369, 2011) minimum theorems were derived in terms of crack tip “quasi static velocity” for two-dimensional fracture mechanics. They were reminiscent of Ceradini’s theorem (Ceradini in Rendiconti Istituto Lombardo di Scienze e Lettere A99, 1965, Meccanica 1:77–82, 1966) in plasticity. Following the cornerstone work of Rice (1989) on weight function theories, Leblond et al. (Leblond in Int J Solids Struct 36:79–103, 1999; Leblond et al. in Int J Solids Struct 36:105–142, 1999) proposed asymptotic expansions for stress intensity factors in three dimensions—see also Lazarus (J Mech Phys Solids 59:121–144, 2011). As formerly in 2D, expansions can be given a Colonnetti’s decomposition (Colonnetti in Rend Accad Lincei 5, 1918, Quart Appl Math 7:353–362, 1950) interpretation. In view of the expression of the expansions proposed in Leblond (Int J Solids Struct 36:79–103, 1999), Leblond et al. (Int J Solids Struct 36:105–142, 1999) however, symmetry of Ceradini’s theorem operators was not evident and the extension of outcomes proposed in Salvadori and Carini (Int J Solids Struct 48:1362–1369, 2011) not straightforward. Following a different path of reasoning, minimum theorems have been finally derived
On a 3D crack tracking algorithm and its variational nature.
No full textThe crack propagation problem for linear elastic fracture mechanics has been studied by several authors exploiting its analogy with standard dissipative systems theory (see e.g. Nguyen (2000), Mielke (2005) and Francfort and Marigo (1998)). In recent publications Salvadori and Carini (2011) and Salvadori and Fantoni (2013) minimum theorems were derived in terms of crack tip “quasi static velocity” for two- and three-dimensional fracture mechanics. They were reminiscent of Ceradini's theorem Ceradini (1965, 1966) in plasticity.
Such an incremental picture naturally leads to explicit methods for integration in time, with well know drawbacks in terms of accuracy and stability. The present work introduces an implicit Newton–Raphson based crack tracking algorithm which is endowed with a variational formulation
High order boundary and finite elements for 3D fracture propagation in brittle materials
No full textThe quasi-static propagation of fracture in brittle materials was studied in several recent publications. A variational formulation (Salvadori, 2008; Salvadori and Carini, 2011; Salvadori and Fantoni, 2013) led to three-dimensional crack tracking strategies (Salvadori and Fantoni, 2014 [2,3]; Salvadori and Fantoni, 2016). One of the complexities of this new class of algorithms is the evaluation of high-order terms of the expansion of the crack opening and sliding. In this paper, new types of finite and boundary elements are formulated that capture the near crack-front asymptotical displacement behavior up to the order 3/2
. The use of these elements with the quasi-static crack propagation algorithms of the above references is demonstrated for a simple crack configuration
Peridynamic correspondence model for nearly-incompressible finite elasticity
No full textThis paper presents a correspondence model for use with peridynamic states in the context of nearly incompressible finite elasticity. An isochoric/volumetric decomposition is adopted, enabling the derivation of the peridynamic force state from a purely spherical, pointwise non-local deformation gradient and a deviatoric, bond-level non-local deformation gradient. This approach leads to a stable one-field, state-based peridynamic formulation that is free from zero-energy modes and capable of accurately capturing the mechanical behavior of elastic materials under large deformations, including those with low or negligible compressibility, typical of unfilled elastomers and isotropic soft biological tissues. Notably, the proposed correspondence model, based on a selective bond-associated deformation gradient, avoids the artificial stiffening commonly observed in standard displacement-based formulations near the incompressible limit. Moreover, its performance is shown to be independent of the specific compressibility ratio assumed in the hyperelastic constitutive law. The model has been successfully validated using classical polynomial strain energy functions through a series of illustrative examples involving both homogeneous and inhomogeneous finite deformations in isotropic hyperelastic solids
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