Continuum Damage Mechanics Approach Research Articles

Comparison of local, gradient-enhanced and integral form of continuum damage approaches to strain localization in fiber reinforced composites

Structures often have inserts and notches that cause stress concentration and consequent reduction in the load bearing capacity. The literature on numerical implementation of continuum damage mechanics (CDM) has generally focused on using local theories of damage. However, their use to analyze deformations of notched laminates that are prone to strain localization can cause numerical convergence issues that can be alleviated by employing a non-local theory. There are a few studies using a non-local damage theory for fiber-reinforced polymeric composites (FRPCs) and even fewer for strain localization in FRPCs. The non-local theories employ an equivalent (or effective or von Mises) strain often used in metal plasticity while most theories for studying failure of FRPCs use individual strain/stress components. Here, we compare predictions from local and nonlocal CDM approaches that are effective in studying strain localization in FRPCs. For the local approach, we employ a rate-dependent evolution relation. For the non-local approach, we either compute damage at a point as the weighted sum of damage at its neighbors that introduces a length scale into the problem, or employ a gradient-enhanced approach that also introduces a length scale. It is shown that numerical predictions from both the rate-dependent and the non-local theory alleviate to different degrees sensitivity of results to the finite element mesh used and the strain localization issues. Of these, a local rate-type approach is more convenient and cost effective.

Numerical prediction of fretting fatigue life for cold-drawn eutectoid carbon steel wire using continuum damage mechanics and effective slip amplitude

The effective and safe use of wire ropes in various applications relies on a comprehensive understanding of the fretting fatigue phenomena that occur between internal wires. Estimating the fatigue life of steel wires subjected to fretting loading through numerical simulation techniques requires significant computational capacity and time, especially when considering wear process. In this study, a more straightforward and practical approach for predicting fretting fatigue life is developed using Continuum Damage Mechanics (CDM). This involves introducing damage stresses that account for relative slip amplitude and modified maximum principal stresses into the CDM framework. Additionally, the Effective Slip Amplitude (ESA) approach is utilized, incorporating relative slip amplitude and Walker effective stress, to evaluate fretting fatigue lifetime. A finite element model is replicated to obtain the relative slippages and stresses at the contact region. The accuracy of the CDM approach and the ESA approach is individually validated and verified by comparing them to experimental data and simulation results including the wear process from published literature. The results indicate that the CDM and ESA approaches, which combine relative slip amplitude and Walker effective stress, effectively predict the fretting fatigue lifetime of cold-drawn eutectoid carbon steel wires.

ANALYSIS OF CORD–RUBBER COMPOSITE RESPONSE BY USING THE NONLINEAR CONTINUUM DAMAGE MECHANICS APPROACH

ABSTRACT The phenomenon of damage observed in cord–rubber composite laminates is the result of deformation, heat, chemical damage, and fracture. The micro cracks and the initial voids, present before any load is applied, grow through the mechanism of coalescence and generate permanent macroscopic cracks. A damage approach is proposed to describe the cumulative effects and damage evolution under cyclic loading and thermal and chemical impact. The approach parallels the continuum damage mechanics approach advocated by Kachanov and Rabotnov.1,2 It is a phenomenological model that depends on laboratory testing to describe the evolution of damage and contains one scalar parameter to describe the collective effect of material damage. The following analysis is based on the premise that the cyclic interlaminar shear strain, coupled with the running temperature at the free edge, is the primary cause of damage. The model constants were derived from an alternating stress (S) against the number of cycles to failure (N) (i.e., S-N) curve at room temperature. The temperature effect on the material damage was accounted for by an Arrhenius shift function of the S-N curve. Numerical simulation of a composite laminate was conducted using the user subroutine UMAT in ABAQUS. The results presented reflect the accuracy of the proposed methodology to predict the location of the ensuing damage and the path of the damage propagation.

A 3D objective material model for elastic–plastic damage behavior of fiber reinforced polymer composites

To simulate the plastic deformation and the progressive failure process of fiber reinforced polymer composites under 3D stress conditions, a new elastic–plastic damage model is proposed to satisfy the objectivity of 3D stress conditions. A new two-parameter 3D plasticity criterion and a simple non-associated flow potential, expressed by stress invariants of a transversely isotropic unidirectional (UD) lamina, are developed to characterize plastic deformation in an objective way. Moreover, the material degradation model and damage evolution laws are constructed on a matrix fracture angle-dependent coordinate system to ensure the objectivity of the continuum damage mechanics (CDM) approach. In addition, a new linear law alleviating mesh dependence is derived to govern the damage variable evolution in the elastic–plastic damage process. ​The effectiveness and objectivity of the proposed material model are demonstrated by a series of test cases.

Comparison of the continuum damage and fracture mechanics in fatigue assessment of components containing residual stresses

In the present study, the fatigue crack growth life of standard and smooth notched specimens containing residual stresses are evaluated based on the continuum damage and fracture mechanics approaches. It is indicated that fatigue crack growth lives assessed by the continuum damage model are closer to the experimental results, with a difference of <10% in the specimens containing residual stresses. The total stress intensity factor range is used to consider the effects of residual stresses in the estimation of fatigue crack growth life based on the fracture mechanics. It is found that in existing residual stresses, the stress intensity factor is not a proper parameter to evaluate crack tip zone. Fatigue crack initiation life is experimentally obtained based on assessing variations of the slope of unloading curve in the force-displacement diagram of fatigue experiments. The cycle including a change in the slope is considered as fatigue crack initiation event. Isotropic and kinematic hardening parameters of ASTM A516 pressure vessel steel are considered in finite element analyses. Tensile residual stresses are introduced into the notched specimens by employing four-point-bending with a magnitude of around 90% of the yield stress of the material. The results indicate that at least 65% of the fatigue life of the specimens is decreased due to the presence of tensile residual stresses. Tensile residual stresses are measured using the hole-drilling technique.

Creep damage parameter extraction from ex-service 12% Cr steel using digital image correlation computed strain data

Several continuum damage mechanics (CDM) modelling approaches for predicting creep deformation of tempered ferritic steels have been developed in the literature, which have evolved from efforts to extend the operability of power plant components. Few of these models, however, focus on damage assessment of ex-service states of power plant steels through the extraction of damage parameters. Furthermore, few CDM approaches leverage the high density of creep curve data available through full-field strain measurement techniques such as digital image correlation (DIC). This work uses multiple creep curves obtained from DIC computed strain data at several stresses and temperatures from individual specimens of X20CrMoV12-1 (X20) piping steel. These curves serve as input data to a modified Oruganti continuum damage mechanics (CDM) model whereby microstructural-specific damage parameters can be extracted. Good agreement is noted between CDM-extracted parameters and microstructural, creep cavity density and hardness damage indicators. Damage parameters based on subgrain growth are particularly sensitive to the ex-service state of the X20 steel. The proposed CDM approach using DIC computed creep curves is shown to be a material efficient alternative to traditional damage assessment methods of ex-service material.

Interaction of compression failure mechanisms in composites with ply waviness defects

This article presents results from experimental and computational investigations of composite laminates with embedded ply waviness defects of different geometries under compression. The effect of the ply waviness defect on the initiation, interaction, and evolution of failure mechanisms of surrounding plies are studied. Using high-resolution imaging and digital correlation analysis, the strain localization phenomenon is monitored, and the sequence of failure mechanisms is established. The experimental observations are supported with computational finite element progressive failure analyses of the same composites with ply waviness defects using an energy based continuum damage mechanics approach. For the case of laminates with ply waviness defects of high angles, a delamination crack between the wavy plies and adjacent transverse plies is the dominant mode of failure, which interacts strongly with the fiber compression and matrix tension modes. For the case of laminates with ply waviness defects of low angles, failure does not initiate at the waviness defect but at locations where the laminate strength was locally reduced due to other microstructural defects. The combined experimental and computational analyses presented here advanced the understanding of the interaction of compression failure mechanisms in composites with ply waviness defects, which is highly challenging to experimentally measure.

Cumulative Fatigue Damage of Composite Laminates: Engineering Rule and Life Prediction Aspect.

The analysis of cumulative fatigue damage is an important factor in predicting the life of composite elements and structures that are exposed to field load histories. A method for predicting the fatigue life of composite laminates under varying loads is suggested in this paper. A new theory of cumulative fatigue damage is introduced grounded on the Continuum Damage Mechanics approach that links the damage rate to cyclic loading through the damage function. A new damage function is examined with respect to hyperbolic isodamage curves and remaining life characteristics. The nonlinear damage accumulation rule that is presented in this study utilizes only one material property and overcomes the limitations of other rules while maintaining implementation simplicity. The benefits of the proposed model and its correlation with other relevant techniques are demonstrated, and a broad range of independent fatigue data from the literature is used for comparison to investigate its performance and validate its reliability.

Numerical modelling of microstructure inhomogeneity to reproduce experimentally observed scatter in fretting fatigue lifetimes

Experimental fretting fatigue lives exhibit large degree of scatter. One of the aspects behind this is the microstructural inhomogeneity which affects the stresses in the material. Variance in stress values can greatly impact crack initiation speed, leading to changes in fretting fatigue lifetimes, as well as the crack propagation behaviour. This research aims to study this issue in more detail and recreate the experimentally observed scatter using numerical modelling. In this work, Voronoi tessellation is used to simulate microstructural topologies of low carbon steel, where each grain is assigned elasto-plastic properties at random, from a predetermined range. In total, 60 finite element models are created and the effect of microstructural inhomogeneity on fretting fatigue lifetime dispersion is evaluated using a Continuum Damage Mechanics approach. Additionally, crack propagation is studied with XFEM combined with Smith-Watson-Topper fatigue parameter to determine crack path under multiaxial, non-proportional loading conditions. For both crack initiation and propagation investigation, the Theory of Critical Distances is applied to overcome the effect of high stress gradients. Simulation results show that microstructural inhomogeneity has significant impact on the contact stress distributions and subsurface stress and strain fields. Scatter observed in the predicted lifetimes matches well with that observed in practical experiments and is used to propose a design curve for the studied material. The applied crack path prediction approach captures the variance in crack paths originating from microstructural inhomogeneity and shows good correlation with reference experimental results at low bulk stress levels.

Two-Scale analysis of fretting fatigue crack initiation in heterogeneous materials using Continuum Damage Mechanics

The phenomenon of fretting fatigue damage is related to tribological behaviour and commonly occurs at the contact interface between two bodies that sustain oscillation loads. Based on the complicated mechanical situation, several methods have been developed to predict the crack initiation location and life. The Continuum Damage Mechanics (CDM) approach in conjunction with Finite Element Method (FEM) is used in this paper. The fretting fatigue lifetime includes two parts, crack initiation lifetime and crack propagation lifetime, and this paper focuses on the crack initiation phase. Homogeneous materials are usually studied in fretting fatigue problems, and the influence of defects is necessary to be considered. In this study, the authors proposed several models to consider the effect of heterogeneity on crack initiation location and lifetime. The numerical model is a 2D one with plane strain assumption. To get more accurate numerical results with low computational cost, a Two-Scale Analysis approach is developed in this paper. Two macroscopic models and three microscopic models are proposed, including a Macroscopic homogeneous model (Macro-HOMO), a Multiscale Homogenization model with Direct Numerical Simulation (MH-DNS), a Microscopic homogeneous model (Micro-HOMO), a Microscopic Model with equivalent material properties by using Mean Field Homogenization method (Micro-MFH), and a microscopic model with Direct Numerical Simulation (Micro-DNS). The results obtained by using the CDM approach with different microscopic models are compared, and show that the prediction accuracy of the Micro-DNS is the best. In addition, in terms of computational cost, the Micro-DNS model shows great advantages as well comparing to the Macro-DNS model.

Continuum damage mechanics behavior of a carbon-fiber-reinforced epoxy composite fabricated by filament winding with different material and manufacturing conditions

The elastic, plastic, and in-plane damage evolutions of a carbon-fiber-reinforced epoxy composite fabricated via filament winding were studied based on continuum damage mechanics. The specific voids and resin-rich region distributions depended on the material and manufacturing conditions. The damage evolutions were expressed by a unified curve, indicating that the continuum damage mechanics approach gave a robust prediction of the damage evolution, although the critical damage at the ultimate failure depended on the material and manufacturing conditions. The damage coupling parameters were also affected by these conditions, which changed the damage evolution behavior under simultaneous shear and transverse damage development situations.

Prediction of Fatigue Crack Initiation under Variable Amplitude Loading: Literature Review

Metallic materials are widely employed in engineering constructions, and one of the most common failure mechanisms in metals is fatigue failure. Even though metal fatigue has been studied for almost 160 years, many problems remain unsolved. Fatigue in metal occurs when the metallic material is subjected to varying loads, resulting in failure due to damage accumulation. The fatigue process consists of a buildup of damage that leads to crack initiation, followed by a period of crack growth until the critical flaw size is reached. The sum of a start phase and a propagation phase represents the total life. To better understand the fatigue phenomenon at its different stages and predict the fatigue life, various types of prediction models have been developed and reported in the literature. This paper reviewed the different models that include microstructure scale parameters that can be used to predict how fatigue cracks start under variable amplitude loading, including the Modified Tanaka-Mura Model, Acoustic Second Harmonic Generation, and the Probability of Crack Initiation on Defects. For perfect life prediction under variable amplitude loading, a stress-based approach, a strain-based approach, and a continuum damage mechanics approach are reviewed. The purpose of this paper is to get overview of the current state of approaches to the life prediction of fatigue crack initiation with variable amplitude. Finally, gaps in knowledge about the prediction of fatigue crack initiation under variable loading at high temperatures are pointed out.

New Damage Accumulation Model for Spall Propagation Mechanism in Bearing Raceways

The aim of this study was to investigate the spall propagation mechanism in ball bearing raceways using physics-based models. Spalling is one of the most common types of bearing failures that can lead to catastrophic failure. This research takes a step forward toward developing a prognostic tool for ball bearings. It is first necessary to understand the spall progression process in order to formulate a constitutive law of spall deterioration and to estimate the amount of remaining useful life. Fragment formation in the vicinity of the spall edge was found to consist of surface and sub-surface cracks that eventually coalesce, and a fragment is released from the raceway, based on naturally-developed spalls. Here, we describe a physics-based model, integrating a dynamic model with a finite element one to simulate this process. A continuum damage mechanics (CDM) approach and fracture mechanics tools were embedded into the finite element model to simulate the damage propagation. The formation of cracks in the vicinity of the spall (surface and sub-surface cracks) were studied using this effective stress CDM model, and the propagation of the cracks was examined using two approaches: a fracture mechanics approach and an accumulated inelastic hysteresis energy CDM approach. The latter also predicts the overall process of a single fragment release. The simulation results of the spall propagation models are supported by experimental results of spalls from both laboratory experimental bearings and an in-service Sikorsky CH-53 helicopter swashplate bearing. The results obtained show that the impact of the ball on the spall edge affects the crack propagation and the appearance of the surface and sub-surface cracks. Both release the residual stresses and cause crack propagation until a fragment is released.

Continuum damage mechanics to predict rubber fatigue life under multiaxial loadings

The continuum damage mechanics approach allows quantifying the degradation of a material via a damage variable D. By developing a new equivalent stress, a novel way is proposed to analyze the multiaxial fatigue of rubbers. Fatigue experiments under multiaxial loading paths were carried out on a Natural Rubber. Moreover, data on a Styrene Butadiene Rubber available in the literature were also used to check the validity of this approach. The comparison between the experimental data and the model estimates in terms of fatigue life leads to a nice agreement making this approach a robust tool for designing rubber parts.

Determination of multiaxial stress rupture criteria for creeping materials: A critical analysis of different approaches

• Factors affecting different methods for the determination of MSRC are analyzed. • Principles in the determination of MSRC are recommended. • The factor, α , is strongly dependent on the operative creep failure mechanism. • A relation exists between multiaxial factor, α , and damage tolerance parameter, λ . Materials in engineering applications are rarely uniaxially-loaded. In reality, failures under multiaxial loading has been widely observed in engineering structures. The life prediction of a component under multiaxial stresses has long been a challenging issue, particularly for high temperature applications. To distinguish the mode of failure ranging from a maximum principal stress intergranular damage to von Mises effective stress rupture mode a multiaxial stress rupture criterion (MSRC) was originally proposed by Sdobyrev and then Hayhurst and Leckie (SHL MSRC). A multiaxial-factor, α , was developed as a result which was intended to be a material constant and differentiates the bias of the MSRC between maximum principal stress and effective stress. The success of the SHL MSRC relies on accurately calibrating the value of α to quantify the multiaxial response of the material/geometry combination. To find a more suitable approach for determining MSRC, the applicability of different methods are evaluated. Given that the resulting analysis of the various approaches can be affected by the creep failure mechanism, principles in the determination of MSRC with and without using continuum damage mechanics approaches are recommended. The viability of uniaxial material parameters in correlating with α through the analysis of available data in literature is also presented. It is found that the increase of the uniaxial creep damage tolerance parameter λ is accompanied by the decrease of the α –value, which implies that the creep ductility plays an important role in affecting the multiaxial rupture behavior of materials.

Microstructure-Based Computational Analysis of Deformation Stages of Rock-like Sandy-Cement Samples in Uniaxial Compression

This work presents a new finite-difference continuum damage mechanics approach for assessment of threshold stresses based on the mechanical response of a representative volume element of a sandy-cement rock-like material. An original experimental study allows validating the mathematical model. A new modification of the damage accumulation kinetic equation is proposed. Several approaches based on acoustic emission, instantaneous Poisson's ratio and reversal point method are employed to determine the threshold stresses. Relying on the numerical modeling of deformation and failure of model samples, the threshold stresses and the deformation stages are determined. The model predicts the crack initiation stress threshold with less than 10% error. The model prediction of the crack damage stress threshold corresponds to the upper boundary of the experimental range. The model predicts the peak stress threshold with less than 0.2% error in comparison with the average experimental peak stress. The results of numerical modeling are shown to correlate well with the available experimental and literature data and sufficiently complement them.

Creep life assessment of aero-engine recuperator based on continuum damage mechanics approach

Creep life assessment of aero-engine recuperator based on continuum damage mechanics approach

Modelling the creep curves of RAFM steel employing a dislocation density reliant model

The microstructure based creep modelling is one of the important tools to elucidate the creep deformation behavior with ongoing substructure evolution. Present research work embodied the development of hybrid creep model that combines a microstructure based creep model and continuum damage mechanics (CDM) approach for addressing the creep behavior of reduced activation ferritic-martensitic (RAFM) steel. By addition of these two methods, the model is capable to address the creep curves up to a large portion of tertiary creep regime. The model is capable of demonstrating the evolution of the microstructure based variables that are important to understand the microstructural degradation mechanisms during creep. The evolution of dislocation densities (mobile, dipole and boundary), glide velocity, dislocation mobility, mobile dislocation spacing and subgrain boundary energy/unit volume, with ongoing creep is discussed and the magnitude lies in the range of 6.37 × 1012-–1.19 × 1014 m−2, 4.66 × 109 – 4.38 × 1011 m−2, 5.91 × 1013 –2.86 × 1014 m−2, 3.34 × 10-12 − 3.99 × 10-12 m/s, 5.69 × 10-21-7.148 × 10-21 mPa-1s−1, 91.2–395 nm and 2.07 × 1015 - 2.42 × 1017 Jm−5, respectively, at the end of simulation. The simulated creep curves have shown a reasonable match with the experimental ones.

Micromechanical modeling of damage evolution and fracture behavior in particle reinforced metal matrix composites based on the conventional theory of mechanism-based strain gradient plasticity

The particle/matrix interface of metal matrix composites (MMCs) can give rise to extra strength and then affect the local deformation behavior. This strengthening effect originates from the plastic strain gradients due to the incompatibility of plastic deformation between the particle and matrix. However, only limited researches utilized the strain gradient plasticity to study the damage evolution and fracture behavior, in which only one or two of the damage mechanisms (i.e., matrix damage, interface debonding, and particle fracture) was considered. In this work, all of these damage mechanisms were coupled into the finite element model under conventional theory of mechanism-based strain-gradient (CMSG) plasticity. Besides, a new numerical algorithm of geometrically necessary dislocations (GNDs) was proposed for a multi-scale model, and then this model was used to analyze the damage evolution and failure behavior of SiCp/AZ91 composite. The results show that the strengthening effects of plastic strain gradients can describe interface debonding, break the monotonicity of the effective plastic strain with the effective stress at the local area close to the particle/matrix interface, and give a more reasonable distribution of stress and plastic strain compared to the classical J2 flow theory. If the probability distribution of the interfacial strength is considered, the CMSG model has the potential capacity to capture the crack initiation in the matrix. When using the continuum damage mechanics approach to describe the fracture process based on the multi-scale model, the weakening exponent value should be considered.

Numerical Analysis of Pillar Stability in Longwall Mining of Two Adjacent Panels of an Inclined Coal Seam

Longwall mining is one of the most widespread methods globally. During the preliminary development of the working, the coal seam is sectioned into panels divided by protective pillars. The pillars are necessary for maintaining the service life of underground mines, a highly productive stope, and personnel safety. In this work, we apply the finite-difference continuum damage mechanics approach to modeling the stress–strain evolution of the rock mass during the extraction of two adjacent longwall panels of an inclined seam. A new modification of the damage accumulation kinetic equation is proposed. The numerical-modeling approach accounts for an explicit number of numerous factors affecting the rock mass behavior. These factors are gravity forces, lithology, tectonic stresses, natural discontinuities, geotechnical, and mining parameters. When the model parameters are calibrated against the in situ observations, the results of the numerical-modeling approach provide a reliable basis for a pillar stability assessment. We build a structural model of a rock mass containing an underground working based on a simplified stratigraphy of the Kondomsky deposit, Kuznetsk coal basin, Russia. Based on the results of the numerical modeling, the stability of a pillar is analyzed. A new numerical technique extending the classical approach to the stability analysis is proposed and verified against the field data.