Continuum Damage Mechanics—Modelling and Simulation

Over the range of variables (temperature and stress) normally encountered in service applications, creep behaviors of 9Cr-1Mo steel were investigated with various models, such as Wilshire model, combination of exponential form and omega (CEO) model, and continuum damage mechanics (CDM) model. First, a series of short-term creep data was prepared to evaluate the material parameters occurring in these models. Then, creep curve, minimum creep rate, and long-term creep life of present 9Cr-1Mo steel were extrapolated with these estimated models. Based on the analysis of obtained results in detail, it suggested that both CDM model and CEO model can give reliable minimum creep rate predictions. However, the most reliable values of long-term creep life are obtained by the CDM model, followed by the Wilshire model and then the CEO model. In particular, the physically based CDM model can provide useful insights into the underlying creep mechanisms. Therefore, the CDM model has promising potential to study the long-term creep behaviors of 9Cr-1Mo steels.

ABSTRACT: The objective of this paper is to develop a two-dimensional Cohesive Element Model (CZM) coupled with a Continuum Damage Mechanics (CDM) model. The novel cohesive element model allows expressing the separation criterion in terms of the damage tensor used in the CDM model implemented in the neighboring volume elements. Above a critical damage value, debonding triggers. The residual tensile and shearing strength of the CZM follow a linear softening law. The proposed model was calibrated against published triaxial compression tests conducted on shale. Single-cohesive zone tests highlight that the proposed approach allows simulation of hardening during damage propagation followed by softening and failure. The model predicts different strengths in tension and compression and different stiffnesses and strengths for different confinements. A boundary-value problem was solved by using a mesh with embedded cohesive zones at all interfaces. It is shown that the 2D model can predict realistic stress/strain curves, stress and strain fields, and fracture patterns for biaxial compression tests. The advantage of the proposed approach is that it can be used to simulate diffuse damage followed by localized fracture with a smooth transfer of field variables between the finite and cohesive elements. 1 INTRODUCTION Simulating fracture propagation from the inception of microscopic cracks to the localization of macroscopic fractures remains a challenge. To date, finite element modeling approaches based on Continuum Damage Mechanics (CDM) and Cohesive Zone Models (CZM) emerge as the most widely used approaches. The goal of this study is to couple both, in order to take advantage of the benefits that each technique can bring. In CDM, sets micro-cracks are represented by a damage tensor, which is modeled as an internal variable within the framework of thermodynamic irreversible processes. Stiffness and strength decrease as damage accumulates. CDM models can be formulated based on phenomenological principles (Xu and Arson, 2014) or micromechanics principles (Jin and Arson, 2017). Exhaustive reviews of damage rock mechanics are available in (Arson and Gatmiri, 2008; Arson, 2020).

Modeling the high-temperature creep behaviors of 7075 and 2124 aluminum alloys by continuum damage mechanics model

In this paper, the creep‐fatigue responses of the nickel‐based superalloy Inconel 718 have been investigated by conducting hybrid stress–strain‐controlled creep‐fatigue interaction (HCFI) loading experiments at 650°C with different dwell stresses and dwell periods. Deformation responses demonstrate that stress dwell leads to complex cyclic responses, such as an increase in softening rate and changes in inelastic strain rate. Then, a novel creep equivalent stress determination method is proposed based on stress relaxed characteristics. After modifying the continuum damage mechanics (CDM) model with this method, accurate prediction results of the creep‐fatigue cyclic life under strain‐controlled creep‐fatigue loading and HCFI loading can be obtained. Finally, the damage evolutions of Inconel 718 under HCFI loading are analyzed by CDM model. The results of damage evolution indicate that the trend of damage accumulation under different test conditions is consistent with the trend of inelastic strain in the material.

A recently developed anisotropic model of continuum damage mechanics has been applied successfully to characterize ductile fracture of cracked plates under mode I and mixed mode failures. The damage model is further extended in this investigation to examine its applicability to include notch ductile fracture of thin plates containing a circular hole. Two hole sizes of 16 mm and 24 mm diameters are chosen and the specimen material is aluminum alloy 2024-T3. Fracture loads of the plates are predicted by the damage model and compared satisfactorily with those determined experimentally. This investigation provides an important confirmation that not only the anisotropic model of continuum damage mechanics but also the same failure criterion developed can be effectively employed to characterize both ductile fracture for plates containing an isolated macro-crack or circular hole which would otherwise not be possible using the conventional theory of fracture mechanics. The successful development of the unified approach to characterize ductile failure provides a vital impetus for design engineers in the general application of the theory of continuum damage mechanics to solve practical engineering problems.

Using ultrasonic investigations to develop anisotropic damage models for initially transverse isotropic materials undergoing damage to remain transverse isotropic

A temperature-dependent constitutive model based on the continuum damage mechanics (CDM) theory is developed to model the stress/strain behavior of C/SiC composites. A formulation of Helmholtz free energy with initial modulus and tangent modulus is proposed to describe the bilinear constitutive behavior of such materials. The Lagrange interpolation is used to describe the relations between material parameters and temperature. The effect of temperature is introduced into the theoretical framework of continuum damage mechanics. The constitutive model is applied to predict the stress/strain curves of two-dimensional woven C/SiC composite at room and elevated temperatures. Good agreement was found between the experimental data generated and the results predicted by the continuum damage mechanics model. This demonstrates that the methods developed in this paper are effective. The numerical analysis of two-dimensional woven C/SiC laminate with a hole shows that the temperature effect will reduce stress concentration near this hole.

To predict the damage variable of a 90° ply in an arbitrary lay-up laminate subjected to a certain loading, we formulated a damage progression law that represents the damage variable as a function of the damage-associated variable based on continuum damage mechanics (CDM). Utilizing the three-dimensional laminate theory and this CDM model, we predicted the damage progression of a 90° ply in cross-ply and quasi-isotropic laminates. In this analysis, the effective compliance of Lopes’ model was adopted. Additionally, the failure pressure of a filament-wound (FW) cylindrical vessel was predicted by coupling the analytical solution with the present CDM model, and the predicted failure pressure was compared with the results obtained using Christensen’s failure stress criteria. As a result, it was indicated that the prediction of the final fracture of FW vessel requires the CDM approach.

Fracture simulation of restored teeth using a continuum damage mechanics failure model

Bayesian calibration of continuum damage model parameters for an oxide-oxide ceramic matrix composite using inhomogeneous experimental data

Numerical simulation of strength test on graphite moderator bricks using a continuum damage mechanics model

Computational modelling of repetitive raindrop impact and predicting the ensuing damage in the coating and substrate due to fatigue is an essential step in determining the life of the wind turbine blade. The blade is multi-layered with a viscoelastic coating layer, a viscoelastic putty layer, and an elastic composite substrate with an interface between the layers. The damage usually takes the form of erosion of the coating and debonding at the various interfaces between the layers of the wind turbine blade. The modelling of impact and damage involves several steps beginning with generating a stochastic rain scenario which provides information on raindrop size distribution and velocity hitting a volume of the blade. A Coupled Eulerian-Lagrangian (CEL) analysis of a single raindrop and a representative region of the blade provides impact forces on and resulting stresses in the blade. The CEL analysis is repeated for drops of different diameters to generate a library of stresses. This library, combined with a stochastic rain scenario, generates stresses at any location of the representative blade due to the repetitive impact of different-sized raindrops hitting arbitrary locations. Continuum damage mechanics is used to predict the damage evolution in the coating, and when damage reaches a critical value in any element within the developed finite element model of the multilayered blade sample represents the erosion of the coating. A cohesive zone model for fatigue debonding is employed for interfaces between coating, putty, and composite substrate. Both the Continuum damage mechanics model and a Cohesive zone model for fatigue are implemented through a user-defined subroutine in ABAQUS/Explicit. Using the model, the interaction between erosion and debonding is studied. Using the properties of materials from the literature, the analysis shows that damage always initiates from the bottom portion of the coating layer close to the adhesive interface. This eventually creates displacement jumps at the interface of coating and putty, causing damage initiation at the interfaces. Repetitive raindrop impact of different sizes at arbitrary locations causes damage accumulation and propagation faster at the interface, finally damaging the coating layer.

This paper proposes a novel method based on continuum damage mechanics (CDM) and machine learning (ML) models to evaluate the vibration fatigue behavior of W1‐type railway fastener clips subjected to high‐frequency vibration. Firstly, static and fatigue tests are conducted on 60Si2Mn spring steel to acquire elastic modulus, tensile strength, and P‐S‐N curves. Subsequently, a CDM model is established, and numerical simulations are performed under various working conditions to obtain the fatigue characteristics of the clips. Finally, the ML model is trained using numerical simulation results, thereby establishing a mapping model between the working conditions and fatigue characteristics. The developed ML model demonstrates high accuracy in predicting the vibration fatigue life of the clips. Moreover, the Shapley Additive Explanations (SHAP) algorithm is employed to elucidate the ML model, revealing that the vibration frequency has a greater impact on the fatigue life of the clips compared to the vibration displacement.

Professor Sadeghi and his coworkers have been developing a method to estimate the rolling contact fatigue life of rolling bearings in recent years using a continuum damage mechanics (CDM) approach. Development of the method has involved progression from a defect-free and inclusion-free model to ones incorporating grain boundary cracks and inclusions.The most recent paper on this topic is authored by Walvekar et al. [1]. In Sec. 2.2, the authors stated “It was hypothesised that failure mechanism for torsional fatigue and rolling contact fatigue are equivalent because the mechanism for damage accumulation is similar for both types of fatigue phenomena.” It is necessary to verify whether such a hypothesis is correct or not but this has not been carried out within the paper.This hypothesis was postulated by Raje and Sadeghi in 2009 in a paper published in the IMechE Journal of Engineering Tribology entitled “Statistical Numerical Modelling of Sub-Surface Initiated Spalling in Bearing Contacts” [2]. In Sec. 3, the authors stated “It is hypothesised that since fatigue damage under rolling contact conditions is caused purely by the action of shear stresses, the mechanism of failure is similar to that prevalent in torsional fatigue. Thus, by knowing the stress-life relationship for a material in torsional fatigue, the fatigue damage parameters can be extracted.”Torsional fatigue data on SAE 52100 bearing steel from Styri were used to provide the shear stress to failure [3], but Styri stated that in about 90% of cases, failures start below the surface, presumably at inclusions or other defects that act as stress concentrating features. However, the model being validated by Raje and Sadeghi is for a material free of defects, voids, or inclusions, while the steel used by Styri would be of 1940 s pedigree but with no statement of its cleanliness.Would the authors comment upon this observation, particularly as, intuitively, the torsional fatigue life of a pristine steel would be expected to be much longer than the one containing defects or inclusions while failures would be surface-initiated?Furthermore, results from the CDM model of pristine steel are compared to L10 bearing lives measured by Lundberg and Palmgren for NU 209 cylindrical roller bearings. The authors present a method in Sec. 4.3 for the conversion of L10 life in shaft revolutions to inner race stress cycles. A comparison of the model results and the bearing lives of Lundberg–Palmgren is then presented. The authors stated that “Stress-life results for this bearing are shown in Fig. 7 and are found to be in good agreement with stress-life results from the current model.”Would the authors comment upon this conclusion, taking into account that the results from the CDM model of pristine steel are being compared to bearings of 1940 s pedigree with no statement of cleanliness?In Sec. 4.2, the authors claim that the model is found to fit within the range of experimental data for cylindrical roller bearings reported by Harris and Barnsby [4]. Reference to this data appears to suggest that shaft revolutions have been plotted as opposed to stress cycles in Fig. 7. Would the authors confirm whether this is the case or not?Intuitively, a defect-free and inclusion-free model should predict lives that are much longer than both the experimental data from Harris–Barnsby and Lundberg–Palmgren. Once defects such as inclusions are introduced into the model, comparisons may then be made with the experimental data. It is necessary to model the size, number, shape, and composition of the inclusions (and other defects, such as carbide stringers) within the steels tested by Harris–Barnsby (1980 s/1990 s pedigree) and Lundberg–Palmgren (1940 s pedigree) in order to carry out such a comparison. Only at this point the claims can be made for validation (or otherwise) of the model. Would the authors comment upon this observation?It also appears that the CDM model has not been validated at lower contact stresses. It may be difficult to validate the model at contact stresses below 2200 MPa (the main region of interest for industrial machines) for the following reasons:Would the authors comment upon validation of the model at lower contact stresses?

A general stochastic dynamic model of continuum damage mechanics