Nonlocal Damage Mechanics Research Articles

A thermodynamically consistent strain difference-based nonlocal damage mechanics approach for failure analysis of quasi-brittle materials

The nonlocal integral approaches are established methods known for their effectiveness in dealing with the challenges of ill-posed behaviour and mesh sensitivity during the strain softening process, especially within classical continuum theories. The present work extends the strain difference-based nonlocal approach to continuum damage mechanics (CDM) theories for predicting the failure behaviour of quasi-brittle materials. The strain difference approach is adopted for two reasons: (i) it maintains physical realism and indirectly incorporates nonlocal effects while computing both the damage variable and the damage energy release rate, and (ii) the constitutive model maintains a symmetry nature in constructing the nonlocal part of the stiffness matrix, reducing the computational cost in strain-softening problems. The state and damage evolution equations for the present nonlocal approach are formulated within the thermodynamic framework. Numerical examples involving 2D and 3D cases are solved: (i) to study the influence of nonlocal parameter ‘α’ on the damage distribution while capturing the post-peak behaviours, (ii) to demonstrate the effectiveness of the present methodology by obtaining mesh-independent results and accurately predicting damage propagation paths, and (iii) to compare the experimental results and numerical structural response for coarse mesh, highlighting both computational efficiency and reduction of mesh-bias dependency. The results obtained by the strain difference-based nonlocal damage model agree well with the experimental results in the literature.

Localizing Implicit Gradient Damage Based Treatment of Softening in Elasto-plasticity

As opposed to brittle fracture, the failure of ductile materials is preceded by severe plastic deformations. Microscopic mechanisms i.e., void growth and coalescence result in macroscopic property degradation causing softening, localization, and finally macroscopic crack. This contribution focuses on softening in elasto-plasticity and its mesh-objective description using an implicit gradient type of non-local damage mechanics framework. As reported in several studies Geers et al. (1998), Poh and Sun (2017), artificial widening of localization zone is observed when conventional implicit gradient type regularization is used. To circumvent this non-physical artifact, localizing implicit gradient damage (LIGD) formulation that is motivated by higher order continuum arguments, is adopted, Poh and Sun (2017). As opposed to previous remedies to artificial widening of the localization zone, LIGD proposes an internal length scale that decreases with deformation. A two-field (displacement-non-local equivalent plastic strain) hexahedra and a three-field (displacement-pressure-non-local equivalent plastic strain) tetrahedra element are formulated and implemented in commercial finite element software Abaqus through user element (UEL) subroutine. The effectiveness of the approach is demonstrated by solving two numerical examples.

A Generalized Interpolation Material Point Method for Shallow Ice Shelves. 2: Anisotropic Nonlocal Damage Mechanics and Rift Propagation.

Ice shelf fracture is responsible for roughly half of Antarctic ice mass loss in the form of calving and can weaken buttressing of upstream ice flow. Large uncertainties associated with the ice sheet response to climate variations are due to a poor understanding of these fracture processes and how to model them. Here, we address these problems by implementing an anisotropic, nonlocal integral formulation of creep damage within a large‐scale shallow‐shelf ice flow model. This model can be used to study the full evolution of fracture from initiation of crevassing to rifting that eventually causes tabular calving. While previous ice shelf fracture models have largely relied on simple expressions to estimate crevasse depths, our model parameterizes fracture as a progressive damage evolution process in three‐dimensions (3‐D). We also implement an efficient numerical framework based on the material point method, which avoids advection errors. Using an idealized marine ice sheet, we test the creep damage model and a crevasse‐depth based damage model, including a modified version of the latter that accounts for damage evolution due to necking and mass balance. We demonstrate that the creep damage model is best suited for capturing weakening and rifting over shorter (monthly/yearly) timescales, and that anisotropic damage reproduces typically observed fracture patterns better than isotropic damage. Because necking and mass balance can significantly influence damage on longer (decadal) timescales, we discuss the potential for a combined approach between models to best represent mechanical weakening and tabular calving within long‐term simulations.

On the development of direct displacement control method: application to local and nonlocal damage mechanics

The present work deals with development of method called direct displacement control (DDCM) that explicitly imposes the incremental displacement without recourse to pseudo force vector while solving nonlinear problems from solid mechanics. The limitation of the approaches, generalized displacement control method and displacement control method are overcome in DDCM approach while achieving faster inter-element force balance in finite element framework (FEF). The important application of DDCM is proposed to be elasto-plastic damage (EPD), as the damage-induced softening makes internal force balance difficult. The 1D local, and 1D and 2D nonlocal EPD formulations are developed; the derivation of 1D algorithmic tangent modulus is proposed. The applicability and correctness of DDCM approach are successfully demonstrated solving several 1D and 2D local and nonlocal EPD problems within FEF (in-house code and ABAQUS®).

A discrete nonlocal damage mechanics approach

In this paper the authors develop the governing equations for a finite element model of micro-cracking based on a novel approach, eschewing differential equations and continuum mechanics. Instead of first stating the continuum balance laws and constitutive relations followed by discretization, the body is discretized first and then the equations of equilibrium are directly stated for the discretized body. It is shown that, as a result, the balance laws and constitutive relations can be entirely stated in terms of edge forces and lengths rather than strains. Furthermore, by ensuring that microcracks always propagate along the dual mesh which represents the possible fracture microplanes in the element, the need for creating additional nodes, gap elements or cohesive zones is avoided. Finally, the notion of the survival probability of a fracture microplane is introduced and the transition probability evolution is described by using probabilistic notions from population models. Thus the resulting governing equations can be solved by a conventional elastic predictor, followed by a nonlocal fracture corrector, making this convenient to augment conventional elements with fracture abilities.

A non-local damage model for the fatigue behaviour of metallic polycrystals

ABSTRACT The development of fatigue damage in metallic materials is a complex process influenced by both intrinsic (e.g. texture, defects) and extrinsic (e.g. loading mode, frequency) factors. To better understand this process, some efforts have been made to develop microstructure-sensitive models that consider the impact of microstructural heterogeneities on the formation of fatigue cracks. An important limitation of such models is their inability to describe the transition between the nucleation and growth stages in a consistent thermodynamic framework. To circumvent this limitation, a constitutive model, which is appropriate for the description of both nucleation and early growth, is proposed. For this purpose, non-local damage mechanics is used to construct a set of constitutive relations that explicitly considers the progressive stiffness reduction due to the formation of fatigue cracks. Specifically, a damage variable is attached to each slip system, which allows considering both the anisotropic aspect of fatigue damage and closure effects. The spatial gradients of the damage variables are treated as additional state variables to account for the increase of surface energy associated with the formation of fatigue cracks. For the evolution equations to be compatible with the second law of thermodynamics, a modified form of the entropy production inequality is adopted. For illustration purposes, the non-local damage model is implemented in a spectral solver. Some numerical examples are then presented. These examples allow discussing the ability of the proposed model to describe the impact of loading conditions and pre-existing defects on the fatigue behaviour of metallic polycrystals.

Nonlocal Damage Mechanics for Quantification of Health for Piezoelectric Sensor

In this paper, a novel method to quantify the incubation of damage on piezoelectric crystal is presented. An intrinsic length scale parameter obtained from nonlocal field theory is used as a novel measure for quantification of damage precursor. Features such as amplitude decay, attenuation, frequency shifts and higher harmonics of guided waves are commonly-used damage features. Quantification of the precursors to damage by considering the mentioned features in a single framework is a difficult proposition. Therefore, a nonlocal field theory is formulated and a nonlocal damage index is proposed. The underlying idea of the paper is that inception of the damage at the micro scale manifests the evolution of damage at the macro scale. In this paper, we proposed a nonlocal field theory, which can efficiently quantify the inception of damage on piezoelectric crystals. The strength of the method is demonstrated by employing the surface acoustic waves (SAWs) and longitudinal bulk waves in Lithium Niobate (LiNbO3) single crystal. A control damage was introduced and its manifestation was expressed using the intrinsic dominant length scale. The SAWs were excited and detected using interdigital transducers (IDT) for healthy and damage state. The acoustic imaging of microscale damage in piezoelectric crystal was conducted using scanning acoustic microscopy (SAM). The intrinsic damage state was then quantified by overlaying changes in time of flight (TOF) and frequency shift on the angular dispersion relationship.

From discrete to nonlocal continuum damage mechanics: Analysis of a lattice system in bending using a continualized approach

It is shown herein that the bending problem of a discrete damage system, also called microstructured damage system or lattice damage system, can be rigorously handled by a nonlocal continuum damage mechanics approach. It has been already shown that Eringen’s nonlocal elasticity was able to capture the scale effects induced by the discreteness of a microstructured system. This paper generalizes such results for inelastic materials and first presents some results for engineering problems modelled within continuum damage mechanics. The microstructured model is composed of rigid periodic elements connected by rotational elastic damage springs (discrete damage mechanics). Such a discrete damage system can be associated with the finite difference formulation of a continuum damage mechanics problem, i.e. the Euler–Bernoulli damage beam problem. Starting from the discrete equations of this structural problem, a continualization method leads to the formulation of an Eringen’s type nonlocal model with full coupling between nonlocal elasticity and nonlocal continuum damage mechanics. Indeed, the nonlocality appears in this continualized approach both in the constitutive law and in the damage loading function. A comparison of the discrete and the continuous problems for the cantilever shows the efficiency of the new micromechanics-based nonlocal continuum damage modelling for capturing scale effects. The length scale of the nonlocal continuum damage mechanics model is rigorously calibrated from the size of the cell of the discrete repetitive damage system. The new micromechanics-based nonlocal damage mechanics model is also analysed with respect to available nonlocal damage mechanics models.

Superconvergent second derivative recovery technique and its application in a nonlocal damage mechanics model

This study proposes a nonlocal finite element technique that is based on the super-convergent patch (SCP) second derivative recovery methods. The SCP second derivative recovery methods are proven effective in evaluating the Laplacian of the equivalent strains at C0 linear elements. The element-based patches and their construction criteria are defined. Other patch-based techniques either are limited to structured meshes or only utilize nodal based patches for first derivative recoveries. The SCP technique has not previously been used to recover higher order (2nd and higher) strain derivatives. The proposed technique can deal with unstructured meshes with various element types and the patches are constructed in a fashion that greatly accelerates the evaluation of Hessian matrices. A numerical example of a strain gradient damage mechanics problem is given at the end. The results show that the proposed technique is capable of evaluating the Laplacian of the equivalent strains and has the potential for even higher order derivative recovery.

Numerical modeling of tensile fracture initiation and propagation in snow slabs using nonlocal damage mechanics

Abstract The results of numerical simulations of tensile fracture initiation and propagation in snow slabs are presented. Seventeen beam bending experiments, both notched and unnotched, were conducted using blocks of cohesive dry snow extracted from the same homogeneous layer in the natural snowpack. Material properties and fracture parameters were calculated from the experimental data using equations from beam theory and quasi-brittle fracture mechanics. These parameters were used in the nonlocal isotropic damage model to simulate the bending tests on two-dimensional finite element meshes. Using the same material parameters and boundary conditions, the model was capable of simulating the propagation of a tensile crack from an existing stress concentration as well as the initiation of a crack from a smooth boundary. Sensitivity analyses were conducted on the most uncertain model parameters. For the optimally paramaterized model, the simulated load–displacement curves agreed well with the experimental data, with the primary discrepancy related to the loss of elastic stability at peak load in the experiments. The spatial distribution of damage at peak load is shown to support a quasi-brittle interpretation of the fracture physics for both crack initiation and crack propagation problems. These results provide a foundation for future predictive modeling applications related to the tensile fractures associated with slab avalanches.

Nonlocal damage mechanics

Nonlocal damage mechanics

Nonlocal damage mechanics

ABSTRACT This paper starts with an overview of models that can provide an objective description of highly localized inelastic deformation. Basic ideas behind the integral and differential formulations of nonlocal models are explained using a simple isotropic damage model as a typical example. Regularizing effect of nonlocal enhancements is documented by one-dimensional localization analysis. The paper then focuses in detail on integral-type nonlocal damage models. The main issues addressed here include the choice of the internal variable to be averaged, the influence of boundaries, and various computational aspects ranging from efficient implementation of nonlocal averaging to adaptive techniques.

Non-local damage mechanics with application to concrete

ABSTRACT This paper explains the basic concepts of continuum damage mechanics and its nonlocal formulation. Using one- and two-dimensional examples, it is shown that a stress-strain law with softening postulated within the standard continuum theory leads to physically meaningless results and that the numerical solution suffers by a pathological sensitivity to the finite element discretization. A suitable regularization technique can be based on the non-local formulation, with damage driven by the weighted spatial average of the equivalent strain. Efficiency of the non-local simulation can be increased by mesh-adaptive techniques that adjust the finite element mesh according to the evolving strain localization pattern.

Non-local damage mechanics with application to concrete

ABSTRACT This paper explains the basic concepts of continuum damage mechanics and its nonlocal formulation. Using one- and two-dimensional examples, it is shown that a stress-strain law with softening postulated within the standard continuum theory leads to physically meaningless results and that the numerical solution suffers by a pathological sensitivity to the finite element discretization. A suitable regularization technique can be based on the non-local formulation, with damage driven by the weighted spatial average of the equivalent strain. Efficiency of the non-local simulation can be increased by mesh-adaptive techniques that adjust the finite element mesh according to the evolving strain localization pattern.

A nonlocal damage mechanics approach to high temperature fatigue crack growth

A nonlocal damage constitutive model is developed for elasto/visco-plastic materials and is used to analyze fatigue crack growth at high temperature. In this model, no kinematic hardening rule is needed to account for the subsequent yielding and strain hardening behavior of the materials. A calculation method for nonlocal damage is introduced. The fatigue crack growth tests at high temperature for pure titanium (99.5%) are carried out. By means of FEM, the mechanical fields ( σ ij , ϵ ij , D) and their redistribution due to damage effect near the crack tip are investigated. The numerical simulation results of the fatigue crack growth are shown. The validity of the presented damage model is verified by comparing the FEM numerical simulation with experimental results.