Evolving discontinuities and cohesive fracture

Multi-scale methods provide a new paradigm in many branches of sciences, including applied mechanics. However, at lower scales continuum mechanics can become less applicable, and more phenomena enter which involve discon- tinuities. The two main approaches to the modelling of discontinuities are briefly reviewed, followed by an in-depth discussion of cohesive models for fracture. In this discussion emphasis is put on a novel approach to incorporate triaxi- ality into cohesive-zone models, which enables for instance the modelling of crazing in polymers, or of splitting cracks in shear-critical concrete beams. This is followed by a discussion on the representation of cohesive crack models in a continuum format, where phase-field models seem promising

Homogenization of cohesive fracture in masonry structures

We derive a homogenized mechanical model of a masonry-type structure constituted by a periodic assemblage of blocks with interposed mortar joints. The energy functionals in the model under investigation consist in (i) a linear elastic contribution within the blocks, (ii) a Barenblatt's cohesive contribution at contact surfaces between blocks and (iii) a suitable unilateral condition on the strain across contact surfaces, and are governed by a small parameter representing the typical ratio between the length of the blocks and the dimension of the structure. Using the terminology of Gamma-convergence and within the functional setting supplied by the functions of bounded deformation, we analyze the asymptotic behavior of such energy functionals when the parameter tends to zero, and derive a simple homogenization formula for the limit energy. Furthermore, we highlight the main mathematical and mechanical properties of the homogenized energy, including its non-standard growth conditions under tension or compression. The key point in the limit process is the definition of macroscopic tensile and compressive stresses, which are determined by the unilateral conditions on contact surfaces and the geometry of the blocks

Phase field approximation of cohesive fracture models

We obtain a cohesive fracture model as a $\Gamma$-limit of scalar damage models in which the elastic coefficient is computed from the damage variable $v$ through a function $f_k$ of the form $f_k(v)=min\{1,\varepsilon_k^{1/2} f(v)\}$, with $f$ diverging for $v$ close to the value describing undamaged material. The resulting fracture energy can be determined by solving a one-dimensional vectorial optimal profile problem. It is linear in the opening $s$ at small values of $s$ and has a finite limit as $s\to\infty$. If the function $f$ is allowed to depend on the index $k$, for specific choices we recover in the limit Dugdale's and Griffith's fracture models, and models with surface energy density having a power-law growth at small openings

A phase-field model for cohesive fracture

In this paper, a phase-field model for cohesive fracture is developed. After casting the cohesive zone approach in an energetic framework, which is suitable for incorporation in phase-field approaches, the phase-field approach to brittle fracture is recapitulated. The approximation to the Dirac function is discussed with particular emphasis on the Dirichlet boundary conditions that arise in the phase-field approximation. The accuracy of the discretisation of the phase field, including the sensitivity to the parameter that balances the field and the boundary contributions, is assessed at the hand of a simple example. The relation to gradient-enhanced damage models is highlighted, and some comments on the similarities and the differences between phase-field approaches to fracture and gradient-damage models are made. A phase-field representation for cohesive fracture is elaborated, starting from the aforementioned energetic framework. The strong as well as the weak formats are presented, the latter being the starting point for the ensuing finite element discretisation, which involves three fields: the displacement field, an auxiliary field that represents the jump in the displacement across the crack, and the phase field. Compared to phase-field approaches for brittle fracture, the modelling of the jump of the displacement across the crack is a complication, and the current work provides evidence that an additional constraint has to be provided in the sense that the auxiliary field must be constant in the direction orthogonal to the crack. The sensitivity of the results with respect to the numerical parameter needed to enforce this constraint is investigated, as well as how the results depend on the orders of the discretisation of the three fields. Finally, examples are given that demonstrate grid insensitivity for adhesive and for cohesive failure, the latter example being somewhat limited because only straight crack propagation is considered

Numerical modelling of sandstone uniaxial compression test using a mix-mode cohesive fracture model

A mix-mode cohesive fracture model considering tension, compression and shear material behaviour is presented, which has wide applications to geotechnical problems. The model considers both elastic and inelastic displacements. Inelastic displacement comprises fracture and plastic displacements. The norm of inelastic displacement is used to control the fracture behaviour. Meantime, a failure function describing the fracture strength is proposed. Using the internal programming FISH, the cohesive fracture model is programmed into a hybrid distinct element algorithm as encoded in Universal Distinct Element Code (UDEC). The model is verified through uniaxial tension and direct shear tests. The developed model is then applied to model the behaviour of a uniaxial compression test on Gosford sandstone. The modelling results indicate that the proposed cohesive fracture model is capable of simulating combined failure behaviour applicable to rock

Phase-field approximation for a class of cohesive fracture energies with an activation threshold

We study the $\Gamma$-limit of Ambrosio-Tortorelli-type functionals $D_\varepsilon(u,v)$, whose dependence on the symmetrised gradient $e(u)$ is different in $\mathbb{A} u$ and in $e(u)-\mathbb{A} u$, for a $\mathbb{C}$-elliptic symmetric operator $\mathbb{A}$, in terms of the prefactor depending on the phase-field variable $v$. This is intermediate between an approximation for the Griffith brittle fracture energy and the one for a cohesive energy by Focardi and Iurlano. In particular we prove that $G(S)BD$ functions with bounded $\mathbb{A}$-variation are $(S)BD$

Phase-field models for brittle and cohesive fracture

In this paper we first recapitulate some basic notions of brittle and cohesive fracture models, as well as the phase-field approximation to fracture. Next, a critical assessment is made of the sensitivity of the phase-field approach to brittle fracture, in particular the degradation function, and the use of monolithic versus partitioned solution schemes. The last part of the paper makes extensions to a recently developed phase-field model for cohesive fracture, in particular for propagating cracks. Using some simple examples the current state of the cohesive phase-field model is shown

Porous LSCF/Dense 3YSZ Interface Fracture Toughness Measured by Single Cantilever Beam Wedge Test

Sandwich specimens were prepared by firing a thin inter-layer of porous La0.6Sr0.4Co0.2Fe0.8O3-d (LSCF) to bond a thin tetragonal yttria-stabilised zirconia (YSZ) beam to a thick YSZ substrate. Fracture of the joint was evaluated by introducing a wedge between the two YSZ adherands so that the stored energy in the thin YSZ cantilever beam drives a stable crack in the adhesive bond and allows the critical energy release rate for crack extension (fracture toughness) to be measured. The crack path in most specimens showed a mixture of adhesive failure (at the YSZ-LSCF interface) and cohesive failure (within the LSCF). It was found that the extent of adhesive fracture increased with firing temperature and decreased with LSCF layer thickness. The adhesive failures were mainly at the interface between the LSCF and the thin YSZ beam and FEM modelling revealed that this is due to asymmetric stresses in the LSCF. Within the firing temperature range of 1000-1150C, the bonding fracture toughness appears to have a strong dependence on firing temperature. However, the intrinsic adhesive fracture toughness of the LSCF/YSZ interface was estimated to be 11 Jm2 and was not firing temperature dependent within the temperature range investigated.Comment: 13 figures, 1 table, journal paper publishe

Effect of airborne particle abrasion on microtensile bond strength of total-etch adhesives to human dentin

Aim of this study was to investigate a specific airborne particle abrasion pretreatment on dentin and its effects on microtensile bond strengths of four commercial total-etch adhesives. Midcoronal occlusal dentin of extracted human molars was used. Teeth were randomly assigned to 4 groups according to the adhesive system used: OptiBond FL (FL), OptiBond Solo Plus (SO), Prime & Bond (PB), and Riva Bond LC (RB). Specimens from each group were further divided into two subgroups: control specimens were treated with adhesive procedures; abraded specimens were pretreated with airborne particle abrasion using 50 mu m Al2O3 before adhesion. After bonding procedures, composite crowns were incrementally built up. Specimens were sectioned perpendicular to adhesive interface to producemultiple beams, which were tested under tension until failure. Data were statistically analysed. Failure mode analysis was performed. Overall comparison showed significant increase in bond strength (p < 0.001) between abraded and no-abraded specimens, independently of brand. Intrabrand comparison showed statistical increase when abraded specimens were tested compared to no-abraded ones, with the exception of PB that did not show such difference. Distribution of failure mode was relatively uniform among all subgroups. Surface treatment by airborne particle abrasion with Al2O3 particles can increase the bond strength of total-etch adhesive