A unified potential-based cohesive model of mixed-mode fracture

Mathematics and Mechanics of Solids

The objective of this contribution is to elaborate on the notion of “traction continuity” across an interface at finite deformations. The term interface corresponds to a zero-thickness model representing the interphase between different constituents in a material. Commonly accepted interface models are the cohesive interface model and the elastic interface model. Both the cohesive and elastic interface models are the limit cases of a generalized interface model. This contribution aims to rigorously analyze the concept of the traction jump for the general interface model. The governing equations of the general interface model in the material as well as spatial configurations are derived and the traction jump across the interface for each configuration is highlighted. It is clearly shown that the elastic interface model undergoes a traction jump in both the material and spatial configurations according to a generalized Young–Laplace equation. For the cohesive interface model, however,...

Computational Mechanics

The interface between constituents of a multiphase material exhibits properties different from those of the bulk and can lead to major alternation of the material response. Interface effects are particularly important for multiphase nano-materials where the area-to-volume ratio is significantly large. In this contribution, we study the influence of a degrading general interface. That is, we allow for the initiation and accumulation of damage on a generalized interface accounting for both jumps of the displacement and the traction across the interface. The applicability of the proposed framework is demonstrated through several numerical examples. We present a parametric study on the influence of a broad range of interface material parameters on the overall behavior of various microstructures subject to volumetric loading and unloading. The numerical results illustrate that the resistance along the interface plays a key role in the resulting damage mechanism and could potentially prevent the detachment of the inclusion from the matrix regardless of the resistance across the interface or bulk material parameters. This behavior is observed and shown for both two-and three-dimensional examples. Moreover, the size-effect due to the general interface model is examined and compared against other interface models. Finally, the influence of the boundary conditions on the effective response and damage initiation of several microstructures is studied.

International Journal for Numerical and Analytical Methods in Geomechanics

We propose a numerical method that couples a cohesive zone model (CZM) and a finite elementbased continuum damage mechanics (CDM) model. The CZM represents a mode II macrofracture, and CDM finite elements (FE) represent the damage zone of the CZM. The coupled CZM/CDM model can capture the flow of energy that takes place between the bulk material that forms the matrix and the macroscopic fracture surfaces. The CDM model, which does not account for micro-crack interaction, is calibrated against triaxial compression tests performed on Bakken shale, so as to reproduce the stress/strain curve before the failure peak. Based on a comparison with Kachanov's micro-mechanical model, we confirm that the critical micro-crack density value equal to 0.3 reflects the point at which crack interaction cannot be neglected. The CZM is assigned a pure mode II cohesive law that accounts for the dependence of the shear strength and energy release rate on confining pressure. The cohesive shear strength of the CZM is calibrated by calculating the shear stress necessary to reach a CDM damage of 0.3 during a direct shear test. We find that the shear cohesive strength of the CZM depends linearly on the confining pressure. Triaxial compression tests are simulated, in which the shale sample is modeled as an FE CDM continuum that contains a predefined thin cohesive zone representing the idealized shear fracture plane. The shear energy release rate of the CZM is fitted in order to match to the post-peak stress/strain curves obtained during experimental tests performed on Bakken shale. We find that the energy release rate depends linearly on the shear cohesive strength. We then use the calibrated shale rheology to simulate the propagation of a meter-scale mode II fracture. Under low confining pressure, the macroscopic crack (CZM) and its damaged zone (CDM) propagate simultaneously (i.e., during the same loading increments). Under high confining pressure, the fracture propagates in slip-friction, that is, the debonding of the cohesive zone alternates with the propagation of continuum damage. The computational method is applicable to a range of geological injection problems including hydraulic fracturing and fluid storage and should be further enhanced by the addition of mode I and mixed mode (I+II+III) propagation.

Applied Sciences

In brittle fracture applications, failure paths, regions where the failure occurs and damage statistics, are some of the key quantities of interest (QoI). High-fidelity models for brittle failure that accurately predict these QoI exist but are highly computationally intensive, making them infeasible to incorporate in upscaling and uncertainty quantification frameworks. The goal of this paper is to provide a fast heuristic to reasonably estimate quantities such as failure path and damage in the process of brittle failure. Towards this goal, we first present a method to predict failure paths under tensile loading conditions and low-strain rates. The method uses a k-nearest neighbors algorithm built on fracture process zone theory, and identifies the set of all possible pre-existing cracks that are likely to join early to form a large crack. The method then identifies zone of failure and failure paths using weighted graphs algorithms. We compare these failure paths to those computed wi...

Nanomaterials

In this study, the mechanical behavior of epoxy/carbon nanotubes (CNTs) nanocomposite is predicated by a two-scale modeling approach. At the nanoscale, a CNT, the interface between the CNT and the matrix and a layer of the matrix around the CNT are modeled and the elastic behavior of the equivalent fiber (EF) has been identified. The CNT/epoxy interface behavior is modeled by the Park–Paulino–Roesler (PPR) potential. At the microscale, the EFs are embedded in the matrix with the extracted elastic properties from the nanoscale model. The random pattern has been used for the dispersing of EFs in the representative volume element (RVE). The effect of CNTs agglomeration in the epoxy matrix has also been investigated at the micro level. The Young’s modulus of the nanocomposite was extracted from simulation of the RVE. CNT/epoxy nanocomposites at four different volume fractions were manufactured and the modeling results were validated by tensile tests. The results of the numerical models ...

Engineering Fracture Mechanics, 2021

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Meccanica, 2017

In the present paper the non-linear behaviour a solid body with embedded cohesive interfaces is examined in a nite displacements context. The principal target is the formulation of a two dimensional interface nite element which is referred to a local reference frame, dened by normal and tangential unit vectors to the interface middle surface. All the geometric operators, such as the interface elongation and the reference frame, are computed as function of the actual nodal displacements. The constitutive cohesive law is dened in terms of Helmholtz free energy for unit of undeformed interface surface and, in order to obtain the same nodal force vector and stiness matrix by the two integration schemes, the cohesive law in the deformed conguration is dened in terms of Cauchy traction, as a function of separation displacement and of interface elongation. Explicit expression of the nodal force vector is integrated either over the reference conguration or over the current conguration, which is shown to produce the same analytical nite element operators. No dierences between the integration carried out in the reference and in the current conguration are shown, provided that elongation of the interface is taken in to account.

Metals, 2021

This review paper discusses the basic problems related to the use of cohesive models to simulate the initiation and development of failure in various types of engineering issues. The most commonly used cohesive zone models (CZMs) are described. Recent achievements in the field of cohesive modeling are characterized, with particular emphasis on the problem of mixed mode loading, the influence of the strain rate, the stress state triaxiality, and fatigue. A separate chapter of the work is devoted to the identification of cohesive parameters. Examples of the use of CZMs for the analysis of the fracture and failure process in various applications, both on the macro and microscopic scale, are given. The directions of CZMs development were indicated as well as the issues that are currently under particularly intensive development.

Experimental Mechanics, 2020

The mechanical and fracture properties of refractory ceramics are determined by means of an inverse identification procedure between experimental data and numerical simulations. An experimental setup is proposed to perform Wedge Splitting Tests (WST) at elevated temperature with Digital Image Correlation (DIC) to assess the crack propagation. The ceramic Young's Modulus, fracture energy and strength are determined by indirect confrontation to Finite Element simulations of crack propagation in WST specimens employing Cohesive Zone Modeling (CZM). The variations of the force and crack length are used to set an inverse problem for estimating the material parameters for various temperatures. The method, illustrated through the analysis of an industrial refractory ceramic from 25 • C to 1200 • C, combines experimental and numerical approaches to understand and optimize the fracture behaviour of refractories in application. Keywords Cohesive Zone Model • Refractories • Fracture • Digital Image Correlation • Wedge Splitting Test • High Temperature 1 Introduction Refractory ceramics are widely employed in parts that are used in high temperature environments such as, e.g., glass furnaces, steel ladles or blast furnaces. These parts are subjected to severe thermal gradients that may provoke the initiation and propagation of cracks. The presence of such cracks is critical since they may weaken the whole structure and locally enhance the corrosion effects, leading to a lifetime reduction of the furnaces. A major challenge thus consists in predicting the initiation and propagation of such cracks. To this end, modeling the temperature distribution within the part as well as providing realistic description of failure can help in the definition of a robust design. These models require some essential parameters such as the refractory fracture properties (critical energy release rate and strength) and also their variations from room temperature to really high temperature (up to 1500 • C).

International Journal of Fracture, 2011

We propose a hybrid technique to extract cohesive fracture properties of a quasi-brittle (not exhibiting bulk plasticity) material using an inverse numerical analysis and experimentation based on the optical technique of digital image correlation (DIC). Two options for the inverse analysis were used-a shape optimization approach, and a parameter optimization for a potential-based cohesive constitutive model, the so-called PPR (Park-Paulino-Roesler) model. The unconstrained, derivative free Nelder-Mead algorithm was used for optimization in the inverse analysis. The two proposed schemes were verified for realistic cases of varying initial guesses, and different synthetic and noisy displacement field data. As proof of concept, both schemes were applied to a Polymethyl-methacrylate (PMMA) quasi-static crack growth experiment where the near tip displacement field was obtained experimentally by DIC and was used as input to the optimization schemes. The technique

Journal of Applied Mechanics, 2013

The size effect in the failure of a hybrid adhesive joint of a metal with a fiber-polymer composite, which has been experimentally demonstrated and analytically formulated in preceding two papers, is here investigated numerically. Cohesive finite elements with a mixed-mode fracture criterion are adopted to model the adhesive layer in the metal-composite interface. A linear traction-separation softening law is assumed to describe the damage evolution at debonding in the adhesive layer. The results of simulations agree with the previously measured load-displacement curves of geometrically similar hybrid joints of various sizes, with the size ratio of 1:4:12. The effective size of the fracture process zone is identified from the numerically simulated cohesive stress profile at the peak load. The fracture energy previously identified analytically by fitting the experimentally observed size effect curves agrees well with the fracture energy of the cohesive crack model obtained numericall...

Engineering Fracture Mechanics, 2021

A new complex-variable version of a cohesive element is presented that provides highly accurate first order derivatives of the nodal displacements with respect to the cohesive fracture parameters. These sensitivities are provided as a byproduct of the analysis using the complex Taylor series expansion method. This information is useful for inversely determining the cohesive fracture parameters from experimental or synthetic data using a finite element-based approach. In particular, the PPR cohesive element (Park et al., 2009), was extended using complex variables as a user element for the well-known commercial finite element program, Abaqus. The source code for the element is provided as an educational resource. The advantage of having accurate first order derivatives on both accuracy and efficiency is demonstrated through numerical examples.

Journal of Central South University, 2016

A new test method was proposed to evaluate the cohesive strength of composite laminates. Cohesive strength and the critical strain energy for Mode-II interlamiar fracture of E-glass/epoxy woven fabrication were determined from the single lap joint (SLJ) and end notch flexure (ENF) test, respectively. In order to verify their adequacy, a cohesive zone model simulation based on interface finite elements was performed. A closed form solution for determination of the penalty stiffness parameter was proposed. Modified form of Park-Paulino-Roesler traction-separation law was provided and conducted altogether with trapezoidal and bilinear mixed-mode damage models to simulate damage using Abaqus cohesive elements. It was observed that accurate damage prediction and numerical convergence were obtained using the proposed penalty stiffness. Comparison between three damage models reveals that good simulation of fracture process zone and delamination prediction were obtained using the modified PPR model as damage model. Cohesive zone length as a material property was determined. To ensure the sufficient dissipation of energy, it was recommended that at least 4 elements should span cohesive zone length.

International Journal of Damage Mechanics, 2016

A cohesive zone model implemented in an augmented Lagrangian functional is used for simulation of meso-scale fracture problems in this paper. The method originally developed by Lorentz is first presented in a rigorous variational framework. The equivalence between the stationary point of the one-field problem and the saddle point of the mixed formulation is proved by solving the double inequality of the mixed functional. An adaptation to simulate fracture phenomena in the meso-scale via mesh modification is also presented as an algorithm to insert zero-thickness interface elements based on Lagrange multipliers, boarding the non-trivial task of the field interpolation for different crack paths (plain and tortuous). A suitable tool to study the matrix fracture and debonding phenomena in composites with strongly different component stiffnesses that avoids ill-conditioning matrices associated with intrinsic cohesive zone models is obtained. The method stability is discussed using a simp...

Materials

Structural adhesives have shown significant improvements in their behavior over the past few decades [...]

Fatigue & Fracture of Engineering Materials & Structures, 2011

In the recent years, Cohesive Zone Models (CZMs) have gained increasing popularity for modelling the fracture process [1] and also in other applications like composite delamination [2] solder failures [3] in circuits, etc. This can be attributed to the ability of the CZM to adapt to the nonlinearities in the process it represents by adjusting the model parameters. These parameters that are selected to represent the material behaviour in the vicinity of the crack or a damage zone are non-deterministic in nature resulting in random fracture strength estimates. Currently, there are no standardized tests for measuring the CZM parameters and their random scatter. Numerous researchers in the literature suggest values for the CZM parameters based on their experience from limited test data. Traditionally, fracture toughness is determined through coupon tests for any material system that is being analysed using Linear Elastic Fracture Mechanics to determine the fracture strength of a specimen. Since data for fracture toughness are available, this research is aimed at determining the probability density functions (PDFs) for the cohesive zone parameters that would give the same scatter in fracture strength as that obtained from the test statistics. Correlations between the model parameters were introduced to improve the accuracy of the fracture strength PDF. A finite width cracked plate was selected as a test case to demonstrate the process. This paper also presents evidence that material scatter can be isolated from the geometric effects to determine a normalized PDF of fracture strength for a given material. This normalized PDF can then be scaled, using mean fracture strength, to any crack configuration to develop a nomograph that can be used to rapidly assess risk without the need for a probabilistic fracture analysis.

Journal of the European Ceramic Society

This paper aims to evaluate cohesive properties for an alumina refractory with mullite-zirconia aggregates from wedge splitting tests (WST), and to assess their sensitivity to sintering and testing temperature. Five experiments were analyzed of which four were performed at 600°C. The sought parameters were determined via weighted finite element model updating. The cohesive strength and the fracture energy were successfully calibrated and resulted in simulated data close to their experimental counterparts (i.e., between 4 and 11 times the measurement uncertainty). Increasing the sintering temperature from 1400°C to 1450°C enhanced the cohesion between the mullite-zirconia aggregates and the alumina matrix (20% increase of the fracture energy and of the fracture process zone length). When the WSTs were performed at 600°C, the cohesive strength was 10% smaller while the fracture energy was 70% higher than that at room temperature.