Continuum Damage Mechanics Model Research Articles

A thermodynamically consistent anisotropic continuum damage mechanics model with damage-dependent equivalent strain

Abstract This paper proposes a new thermodynamically consistent anisotropic continuum damage mechanics model that the degradation of material is described by a set of damage variables. We adopt a set of novel equivalent strains which are functions of both the strain and the damage state variables as well. A viscous stabilization method is considered to improve the convergence during the material softening. Thus, the evolution of damage state variables are updated through a Newton-Raphson iterative process. The expression of the consistent tangent stiffness matrix is also derived. A smoothed exponential damage evolution shape function with rescaling regularization method is proposed to describe the material softening process and the mesh dependency phenomenon is relieved. Several sets of numerical examples including three-point-bending of a notched-beam, uniaxial tension of unnotched and notched plate of composite material with multiple layups are presented to verify the anisotropic damage models.

A continuum damage mechanics model for fatigue degradation and failure of short fiber reinforced composites

The present study is concerned with the development and implementation of a continuum damage mechanics model describing the degradation and failure of short fiber reinforced composites subjected to static and fatigue loads. Towards a numerically efficient formulation, the model is defined entirely on the macroscopic scale, although being motivated by microstructural considerations. Based on experimental observations, an anisotropic linear elastic formulation is adopted as a base formulation. The elastic model is coupled with a damage formulation assuming that damage is driven by the approach of a state point in strain space to a Tsai-Hill type failure envelope. The model is implemented as a user defined subroutine into a finite element formulation. As a numerically highly efficient alternative, an enhanced Wöhler-Miner type post-processing procedure is proposed. Both numerical approaches are found in good agreement with experimental data obtained in coupon experiments as well as in experiments on specimens with more complex geometry and loading conditions.

Lifetime assessment of deep-sea manned cabin with PMMA pressure-resistant hull prior to micro-cracks

Fully-transparent cabin type, utilized in manned submersibles, is typically constructed using viscoelastic PMMA (polymethyl methacrylate) material. Accurate fatigue life assessment of manned cabin is of paramount importance to ensure structural safety. In this study, a calculation process for evaluating the fatigue life of PMMA manned cabins based on a continuum damage mechanics model combined with the viscoelastic constitutive relationship of the PMMA material in response to the absence of related specification is proposed. Fatigue tests of PMMA material are performed for determination of damage evolution model parameters and the calculation process is validated by the fatigue test data. Tailored subroutines are then developed by using ABAQUS software tool, respectively for structural analysis of a typical manned cabin with PMMA pressure-resistant hull, openings, hatches and penetration plates and simulating trapezoidal loading history acting on manned cabin. Subroutine coupling is accordingly employed to conduct parametric analysis on fatigue crack initiation life of the fully-transparent manned cabin with consideration of different outside pressure level and operation time and valuable results are obtained for identifying key areas of concern. This study intends to contribute to further cabins safety enhancement, as well as to design optimization and fatigue analysis of underwater structures involving PMMA materials.

Development of a novel continuum damage mechanics‐based machine learning approach for vibration fatigue assessment of fastener clip subjected to high‐frequency vibration

AbstractThis 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.

Creep‐fatigue deformation characteristics and life prediction model of Inconel 718 superalloy under hybrid stress–strain‐controlled mode

AbstractIn 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 parametric study on patch repaired quasi‐isotropic laminate in high‐velocity impact environment: Experimental and numerical investigation

AbstractIn this work, the post‐repair behavior of a quasi‐isotropic laminate was studied in a high‐velocity impact environment. The composite laminates were repaired using an external bonded patch repair procedure. Impact tests were performed to understand the effect of patch size and the result of filling the damaged region with a neat epoxy and hardener mixture. Two different size square patches, such as 30 and 50 mm, were used to analyze the effect of patch size. A continuum damage mechanics‐based model was developed and incorporated into a finite element model interface to estimate the optimum patch size and patch stacking sequence for the impact environment. In the finite element model, the intralaminar failure criteria were implemented using the ABAQUS‐VUMAT user subroutine, and the interlaminar failure criteria were implemented using the inbuilt cohesive contact properties. The 30 mm patch with resin filled in the damaged region performed better than the other combination patches. Also, placing 45° plies toward the outer layer of the patches showed lower damage compared to the patches in which 45° plies were placed at the inner surfaces.Highlights Post‐repair impact behavior of repaired laminates was examined. Impact of patch dimension and resin filling in the damaged area was analyzed. A 3‐D finite element model was developed to optimize patch parameters. Continuum damage mechanics model has been incorporated with VUMAT subroutine. Interface between lamina has been modeled using cohesive contact.

Mechanics of Pure Bending and Eccentric Buckling in High-Strain Composite Structures.

To maximize the capabilities of nano- and micro-class satellites, which are limited by their size, weight, and power, advancements in deployable mechanisms with a high deployable surface area to packaging volume ratio are necessary. Without progress in understanding the mechanics of high-strain materials and structures, the development of compact deployable mechanisms for this class of satellites would be difficult. This paper presents fabrication, experimental testing, and progressive failure modeling to study the deformation of an ultra-thin composite beam. The research study examines the deformation modes of a post-deployed boom under repetitive pure bending loads using a four-point bending setup and bending collapse failure under eccentric buckling. The material and fabrication challenges for ultra-thin, high-stiffness (UTHS) composite boom are discussed in detail. The continuum damage mechanics (CDM) model for the beam is calibrated using experimental coupon testing and was used for a finite element explicit analysis of the boom. It is shown that UTHS can sustain a bending radius of 14 mm without significant fiber and matrix damage. The finite element model accurately predicts the localized transverse fiber damage under eccentric buckling and buckling stiffness of 15.6 N/mm. The results of the bending simulation were found to closely match the experimental results, indicating that the simulation accurately shows deformation stages and predicts damage to the material. The findings of this research provide a better understanding of the structure characteristics with the progressive damage model of the UTHS boom, which can be used for designing a complex deployable payload for nano-micro-class satellites.

Microcapsule Triggering Mechanics in Cementitious Materials: A Modelling and Machine Learning Approach.

Self-healing cementitious materials containing microcapsules filled with healing agents can autonomously seal cracks and restore structural integrity. However, optimising the microcapsule mechanical properties to survive concrete mixing whilst still rupturing at the cracked interface to release the healing agent remains challenging. This study develops an integrated numerical modelling and machine learning approach for tailoring acrylate-based microcapsules for triggering within cementitious matrices. Microfluidics is first utilised to produce microcapsules with systematically varied shell thickness, strength, and cement compatibility. The capsules are characterised and simulated using a continuum damage mechanics model that is able to simulate cracking. A parametric study investigates the key microcapsule and interfacial properties governing shell rupture versus matrix failure. The simulation results are used to train an artificial neural network to rapidly predict the triggering behaviour based on capsule properties. The machine learning model produces design curves relating the microcapsule strength, toughness, and interfacial bond to its propensity for fracture. By combining advanced simulations and data science, the framework connects tailored microcapsule properties to their intended performance in complex cementitious environments for more robust self-healing concrete systems.

Rock fracture propagation in a damage zone by a constitutive model coupling volume and cohesive elements

AbstractThis paper presents a method to simulate fracture propagation in a damage zone in 2D, whereby Cohesive Zone (CZ) elements are assigned the same Continuum Damage Mechanics (CDM) model as their adjacent finite elements. Material hardening is a result of damage‐induced stiffness reduction in the finite and cohesive elements, while softening is modeled by cohesive debonding only. The damage components calculated at the numerical integration points of each finite element are averaged and that average is assigned to the nodes of the adjacent CZ elements. CZ element debonding triggers when a critical damage threshold is reached, beyond which, a linear softening law is used in both mode I and mode II. The proposed model is calibrated against published triaxial compression tests conducted on shale. Single‐CZ element simulations confirm that, below the critical damage threshold, elastic deformation energy is stored and energy is dissipated by damage propagation and irreversible deformation. Passed the critical damage threshold, stored elastic energy is released in the form of cohesive debonding energy. Borehole breakout simulations indicate that the proposed CDM‐based CZ model can predict the formation of spiral‐shaped fractures around a free circular cavity under biaxial stress. Simulations of fracture propagation in textured materials suggest that cohesive debonding within and between grains is triggered by the coalescence of localized zones of continuum damage, and that the grain size distribution affects both the fracture pattern and the softening behavior of the specimen considered.

Predicting damage behaviors of composite laminates under multiple low‐velocity impacts

AbstractIn present investigation, experimental and numerical approaches were combined to investigate the distance between impact positions (DBIP) effects upon impact response and damage accumulation regarding carbon/glass hybrid laminates with multiple low‐velocity impacts. Force/energy‐time curves as well as force‐displacement curves with respect to the laminates with three various impact energies at four different impact positions were obtained from experiment data. Then C‐scans were made to quantify the damage of all the impacted specimens. Additionally, an integrated multi‐impact model was created considering both interlaminar delamination and intralaminar damage. Continuum damage mechanics model (CDM) was developed to calculate the impact resistance regarding laminates under various impact loadings with high efficiency. The simulation and experimental results agreed well. Moreover, the qualitative effects of impact positions on the laminates' mechanical responses were summarized utilizing predictive model that validated.Highlights The effect of the DBIP on the impact response and damage accumulation of carbon/glass hybrid laminates subjected to multiple low‐velocity impacts was studied. An integrated multi‐impact model was developed to predict the impact resistance of laminates under various impact loadings with high efficiency. The qualitative effects of impact positions on the laminates' mechanical responses were summarized using a validated predictive model.

Impact of daily and seasonal temperature variation on rolling contact fatigue damage in the rail

Rolling contact fatigue is the most frequent degradation mode observed on the rail. Many mechanical studies have revealed the main physical parameters that generate the crack initiation, which are all linked with the track geometry, the dynamic of the wheel and contact conditions. Except for extreme climate, exogenous phenomena such as ambient temperature have not been analysed. The main reason is that temperature variations induce stress cycles in the rail that are under the endurance limit of the material. Therefore, most fatigue life estimation methods do not consider these cycles. In this paper, 6 years of daily temperature measurement over all the French railway network are used and integrated in a machine learning model developed for computing residual lifespan of each rail segment before crack initiation by rolling contact fatigue. Temperature data are aggregated using the continuum damage mechanics model developed by Lemaitre and Chaboche that gives the possibility to let low amplitude cycles to contribute to fatigue damage. Classification models are then implemented, and importance variables are finally quantifed. Results show that daily temperature variations have more impact than seasonal variations and that its contribution to fatigue is significant.

An enhanced constitutive model to predict plastic deformation and multiple failure mechanisms in fibre-reinforced polymer composite materials

Spurious damage modes in continuum damage mechanics models for fiber-reinforced polymer composite materials based on the effective stress tensor can be generated when large strains occur. A methodology to prevent this spurious phenomenon is developed in the present work. The longitudinal damage activation functions are based on the effective stress tensor, however, nominal stresses are used on the transverse damage activation functions. The proposed method can be straightforwardly implemented on previously-developed constitutive models which use the effective stress tensor, an explicit implementation of the proposed constitutive model is presented. The enhancement of the predicted failure mechanisms obtained from the present constitutive model, with respect to the models which use the effective stress tensor, is demonstrated. The proposed constitutive model presents a good agreement of the predicted failure pattern obtained from open-hole compressive experimental tests, as well as on the predicted failure strength.

Progressive Fatigue Modelling of Open-Hole Glass-Fibre Epoxy Laminates

The failure of composite laminates under cyclic fatigue loads is complex, as multiple failure mechanisms are in play at different scales and interact with each other. Predicting the remaining fatigue life as well as the residual capacities of a composite laminate or component is crucial, particularly for safety-critical applications. A progressive fatigue model is proposed to describe the catastrophic failure of open-hole laminates under tensile cyclic fatigue. To represent both intra-laminar and inter-laminar damage, a combination of a continuum damage mechanics model (CDM) and a discrete cohesive zone model (CZM) is implemented in the finite element (FE) software Abaqus. The CDM combines fibre- and matrix-dominated S-N curves with the Palmgren–Miner accumulation rule and Hashin’s residual strength to form a fatigue failure criterion differentiating between fibre failure (FF) and matrix failure (MF). The CZM implemented in this work is the CF20 model proposed by NASA. Fatigue cycling is simulated using an external cycle-jump scheme, where the stiffness degradation is conducted between the FE simulations outside of the implicit solver [90/0] s. Glass fibre reinforced polymer (GFRP) open-hole specimens were tested in tensile cyclic fatigue at a load ratio of 0.1. The experiments were reproduced numerically and the results compared. After calibration of a set of parameters based on one load level, the model was able to reproduce the experimental S-N curve very well, predicting a slope of −0.10, while the experimental value was −0.11. The failure sequence of the laminate was also successfully reproduced. The growth of the split from the hole, and its interaction with inter-laminar delamination, was successfully captured. The proposed approach was able to describe the fatigue failure of an open-hole laminate with a minimal set of material inputs using a simplified fatigue damage model while avoiding convergence issues.

An insight into the mechanical behavior and failure mechanism of CFRP/Al adhesively bonded structures subjected to low-velocity impact

Experimental and numerical methods are used to investigate the effects of different impact surfaces and energies on the mechanical properties and failure behavior of CFRP/Al adhesively bonded structures. To predict the damage initiation and propagation of CFRP laminates, interlaminar and adhesive layer, and aluminum alloy plates, the integrated finite element model containing both low-velocity impact (LVI) and axial tensile after impact (TAI) processes is developed by combining continuum damage mechanics model, cohesive zone model, and Johnson-Cook model. Additionally, LVI and TAI tests have been performed on the corresponding adhesively bonded structures with aluminum alloy 5182 (AA5182) and CFRP laminate as impact surfaces, which verify the effectiveness of the proposed integrated FEM. The results indicate that AA5182, as the impact surface, produces greater impact damage (maximum crater depth and opening width) in the center of the overlap region in comparison with the CFRP. The residual bearing capacity of the joints is the result of the competition between impact behavior and mechanical interlocking. The tensile strength, stiffness, and failure displacement of the joints decrease with the increase of impact energy. The failure modes of the joints include annular and circular cohesive failure of the adhesive layer in the impact region, interface failure, cohesive failure and peeling failure in the non-impact region, delamination damage, and fiber pullout and matrix cracking of the CFRP laminate. This investigation can provide good technical support for the structural design of heterogeneous material connections.

A dynamic constitutive model for carbon nanotubes (CNTs) modified unidirectional fibre reinforced plastic laminates

A constitutive model is developed herein for carbon nanotubes (CNTs) modified unidirectional fibre reinforced plastic (FRP) laminates subjected to impact loading. In the model, intralaminar damage is described by continuum damage mechanics (CDM) model which is modified by introducing two parameters (namely, matrix correction factor (S) and residual strength factor (RSF)); interlaminar damage (delamination) is depicted by cohesive element damage model. In order to verify the model, numerical simulations are carried out on the response and failure of unidirectional FRP laminates with or without CNTs modification under both ballistic and low-speed impact. Furthermore, parametric study is performed on the influences of impact velocity. It is demonstrated that the present model predictions are in good agreement with available experimental data in terms of residual velocity, energy absorption, failure patterns and delamination. It is also demonstrated that, with the same impact energy, larger mass impact causes greater intralaminar damage whilst lower mass higher velocity ballistic impact leads to more interlaminar damage and less intralaminar damage.

A progressive damage model for carbon fiber-reinforced polymer laminates subjected to fastener pull-through failure

Most studies on composite joints focus on in-plane bearing mode, while little attention to fastener pull-through failure mode from out-of-plane loads. This paper presents a 3D continuum damage mechanics (CDM) model to investigate and assess the mechanical behaviors of carbon fiber-reinforced polymer (CFRP) laminates subjected to pull-through failure. The proposed model based on the use of intralaminar and interlaminar damage models is established and implemented by means of a user-defined subroutine VUMAT for Abaqus/Explicit. The novelty lies in the material model integrating the thresholds preset by the semianalytical approximation approach, which largely eliminates unnecessary fracture angle enumerations in Puck’s IFF theory, coupled with the simple parabolic interpolation search (SPIS) technique for optimal fracture angle search, considerably reducing the computational burden. Moreover, the numerical results compare well with experiments, effectively replicating the load–displacement response, damage modes, failure behaviors and damaged areas. This study provides an accurate and efficient computational methodology to analyze pull-through failures, enhancing the damage modeling and predictive capabilities for composite joints under out-of-plane loads.

Numerical predictions of intralaminar and interlaminar damage in thin composite shells subjected to impact loads

This paper investigates the impact dynamics of thin semicylindrical woven composite laminate shells, with a particular focus on understanding the influence of thickness. Utilizing the finite element (FE) method, the study evaluates both intralaminar and interlaminar damage associated with the impact response. The findings show that the implementation of the explicit FE method, along with a continuum damage mechanics model, for the intralaminar damage, and a surface-based cohesive model, for the interlaminar damage, may be used to correctly predict the load histories, as well as the maximum impact force, maximum displacement, contact time and impact bending stiffness (IBS). The numerical predictions reproduce well the linear response of the maximum impact force and maximum displacement to the thickness variation, as well as the 2nd order polynomial curve of the IBS, with errors ranging from 2.7% to 15.9%. Moreover, the damage areas and the effect of thickness on the damage severity were accurately replicated. These results’ validation enables the prediction of the energy histories and the ensuing energy dissipation forms. According to the findings, the intralaminar damage is around 5 times more significant than the other energy dissipation forms. The accuracy of the simulation creates the possibility for more impact investigations using a similar numerical approach, reducing the expenditures of experimental testing.

A failure mode dependent continuum damage model for thick laminated composite plates

The fiber reinforced composite materials show stiffness degradation (termed as “Damage”) upon loading, which can be modeled mathematically using Continuum Damage Mechanics (CDM). However, the available CDM models for composites are not characterized using all the uniaxial/shear stress-strain curves, therefore, do not account for the non-linear stress-strain behavior from all uniaxial/shear material characterization tests. Therefore, the novelty of this article involves the development of a failure mode-dependent phenomenological continuum damage model involving coupling of damage in different directions for generalized plane stress conditions and the characterization of model parameters based on experimental uniaxial fiber tension-compression, matrix tension-compression, in-plane, and out-of-plane shear stress-strain curves. The finite element method with load, and displacement control is used to predict the evolution of damage in a moderately thick laminated composite plate subjected to quasi-static loading. The damage prediction capability of the developed model is compared with a failure mode independent damage model. The reported results depict the capability of the developed model to predict different damage in tension and compression in accordance with the strength values. The developed model is also tested using different coupling vectors (responsible for coupling of damage in different directions) to assess their accuracy in capturing the effect of coupling of damage in different directions.

Buckling assessment of GFRP and carbon fiber-reinforced plastic filament-wound tubes using an acoustic emission-based methodology

The aim of this research is to investigate the failure mechanisms of the filament-wound composite tubes under axial compressional loading by using an acoustic emission approach. First, the mechanical properties of ±45°C composite tubes were obtained experimentally. Then, failure due to the buckling phenomenon and crashworthiness characteristics were studied utilizing numerical simulation and experimental methods. Tubes were next simulated in ABAQUS software, and a continuum damage mechanics model was implemented in a progressive framework to assess the failure modes. From the macroscale view, results showed that the damage behavior of composite tubes turned out to be dominated by local buckling followed by a post-buckling field, which is generated by longitudinal cracks along the winding direction. On the micro-scale, the acoustic emission-based procedure based on the wavelet packet transform method was adopted. The hierarchical modeled assessment resulted in the identity of four clusters of AE signals. In GFRP tubes, the fiber breakage and fiber/matrix separation could mostly control the higher percentage of damage and cause to increase the energy absorption. Finally, by comparing the results obtained from micro and macro scales, the local buckling failure mode was attributed to the low content of fiber/matrix debonding in the structure.

Continuum Damage Mechanics Modeling of High-Temperature Flaw Propagation: Application to Creep Crack Growth in 316H Standardized Specimens and Nuclear Reactor Components

Abstract Predicting creep crack growth (CCG) of flaws found during operation in high-temperature alloy components is essential for assessing the remaining lifetime of those components. While defect assessment procedures are available for this purpose in design codes, these are limited in their range of applicability. This study assesses the application of a local damage-based finite element methodology as a more general technique for the prediction of CCG at high temperatures on a variety of structural configurations. Numerical results for stainless steel 316H, which are validated against experimental data, show the promise of this approach. This integration of continuum damage mechanics (CDM) based methodologies, together with adequate inelastic models; into assessment procedures can therefore inform the characterization of CCG under complex operating conditions while avoiding excessive conservatism. This article shows that such modeling frameworks can be calibrated to experimental data and used to demonstrate that the degree of triaxiality ahead of a growing creep crack affects its rate of growth. The framework is also successfully employed in characterizing CCG in realistic reactor pressure vessel geometry under an arbitrary loading condition. These results are particularly relevant to the nuclear power industry for defect assessment and inspections as part of codified practices of structural components with flaws in high-temperature reactors.