Damage Formulation Research Articles

Constitutive description of snow at finite strains by the modified cam‐clay model and an implicit gradient damage formulation

AbstractSnow, characterized as a unique granular and low‐density material, exhibits intricate behavior influenced by the proximity to its melting point and its three‐phase composition. This composition entails a structured ice skeleton surrounded by voids filled with air and spread with liquid water. Mechanically, snow experiences dynamic transformations, including bonding/degradation between its grains, significant inelastic deformations, and a distinct rate sensitivity. Given snow's varied structures and mechanical strengths in natural settings, a comprehensive constitutive model is necessary. Our study introduces a pioneering formulation grounded on the modified Cam‐Clay model, extended to finite strains. This formulation is further enriched by an implicit gradient damage modeling, creating a synergistic blend that offers a detailed representation of snow behavior. The versatility of the framework is emphasized through the careful calibration of damage parameters. Such calibration allows the model to adeptly capture the effects of diverse strain rates, particularly at high magnitudes, highlighting its adaptability in replicating snow's unique mechanical responses across various conditions. Upon calibration against established experimental benchmarks, the model demonstrates a suitable alignment with observed behavior, underscoring its potential as a comprehensive tool for understanding and modeling snow behavior with precision and depth.

A finite element framework for thermo‐mechanically coupled gradient‐enhanced damage formulations

AbstractThe solution of coupled multi‐field problems by means of commercially available finite element codes such as Abaqus constitutes an important part in the study of industrial processes. Against this background, a comprehensive implementation framework for a gradient‐enhanced continuum damage formulation in a thermo‐mechanically coupled finite deformation setting is proposed in this contribution. To assess the applicability of the implementation framework to industrial processes, a thermo‐mechanically coupled, gradient‐enhanced ductile damage model is exemplarily employed. The resulting three‐field problem, consisting of the balance of linear momentum, the heat equation and the balance of micromorphic momentum, is implemented based on an Abaqus user material by making use of a novel two‐instance formulation. Representative simulation results are discussed, with the specific focus lying on the application to manufacturing processes involving complex contact interactions. The Abaqus framework is made available as an open‐source code on GitHub, see Sobisch et al., Finite Elements in Analysis and Design 232 (2024) 104105.

Modeling of textile composite using analytical network-averaging and gradient damage approach

In this contribution, we present a gradient damage model for anisotropic textile reinforcements including fiber inextensibility and fiber sliding. In contrast to previous works, the gradient damage formulation stems not from a numerical regularization basis but from the thermodynamics of internal variables. It results in a nonlocal term as the internal energy of fiber bending with measurable nonlocal parameter. Furthermore, to guarantee a priori that rotations and reflections determined by orthogonal tensors among the symmetry group do not affect the response function of the anisotropic constitutive law, a novel mesoscopic kinematic measure for the representative volume element of the fabric is defined on the basis of the analytical network-averaging concept. Such kinematic measure is of crucial importance for material modeling of damage-elastoplasticity in anisotropic textile reinforcements, and allows for analytical descriptions of inter- and intra-ply sliding of fibers. A mixed finite element formulation is then presented for textile reinforcements taking into account fiber inextensibility. The predictive capability of the computational model is demonstrated by comparing with multiple experimental datasets of dry textile fabrics.

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.

Impact of Tissue Damage and Hemodynamics on Restenosis Following Percutaneous Transluminal Angioplasty: A Patient-Specific Multiscale Model.

Multiscale agent-based modeling frameworks have recently emerged as promising mechanobiological models to capture the interplay between biomechanical forces, cellular behavior, and molecular pathways underlying restenosis following percutaneous transluminal angioplasty (PTA). However, their applications are mainly limited to idealized scenarios. Herein, a multiscale agent-based modeling framework for investigating restenosis following PTA in a patient-specific superficial femoral artery (SFA) is proposed. The framework replicates the 2-month arterial wall remodeling in response to the PTA-induced injury and altered hemodynamics, by combining three modules: (i) the PTA module, consisting in a finite element structural mechanics simulation of PTA, featuring anisotropic hyperelastic material models coupled with a damage formulation for fibrous soft tissue and the element deletion strategy, providing the arterial wall damage and post-intervention configuration, (ii) the hemodynamics module, quantifying the post-intervention hemodynamics through computational fluid dynamics simulations, and (iii) the tissue remodeling module, based on an agent-based model of cellular dynamics. Two scenarios were explored, considering balloon expansion diameters of 5.2 and 6.2mm. The framework captured PTA-induced arterial tissue lacerations and the post-PTA arterial wall remodeling. This remodeling process involved rapid cellular migration to the PTA-damaged regions, exacerbated cell proliferation and extracellular matrix production, resulting in lumen area reduction up to 1-month follow-up. After this initial reduction, the growth stabilized, due to the resolution of the inflammatory state and changes in hemodynamics. The similarity of the obtained results to clinical observations in treated SFAs suggests the potential of the framework for capturing patient-specific mechanobiological events occurring after PTA intervention.

Multi-stress damage and healing mechanics in quasi-brittle materials: Theoretical overview

This work introduces a theoretical framework for continuum damage and healing mechanics by extending stress decomposition to account for tensile, compressive, and shear stresses. In addition to the spectral stress decomposition into tensile and compressive components, we extend the existing stress decomposition method to address shear stresses. The extraction of shear stresses employs two hypotheses, considering both same-signed and opposite-signed principal stresses. This stress decomposition approach yields three damage variables [Formula: see text] and three healing variables [Formula: see text]. The damage formulation is discussed in terms of equivalent strain and conjugate force, while the healing formulation is based on the initial damage state and healing time. We explore the influence of material parameters and healing time on damage and healing evolution. Furthermore, we analyze the relationship between nominal stress-to-effective stress ratio, damage variables, and healing time. Lastly, we present a thermodynamically consistent formulation for damage-healing processes, acknowledging that this work establishes a theoretical formulation. The proposed method is validated by analyzing the performance of an L-shaped concrete specimen using three damage variables and one healing variable. These results illustrate the model's ability to effectively capture the damage and healing phenomena. The practical implementation of the proposed formulation will be pursued numerically using innovative healing techniques and a pseudo-damage healing approach, which will be detailed in future work.

Block preconditioning strategies for generalized continuum models with micropolar and nonlocal damage formulations

AbstractIn this work, preconditioning strategies are developed in the context of generalized continuum formulations used to regularize multifield models for simulating localized failure of quasi‐brittle materials. Specifically, a micropolar continuum extended by a nonlocal damage formulation is considered for regularizing both, shear dominated failure and tensile cracking. For such models, additional microrotation and nonlocal damage fields, and their interactions, increase the complexity and size of the arising linear systems. This increases the demand for specialized preconditioning strategies when iterative solvers are adopted. Herein, a block preconditioning strategy, employing algebraic multigrid methods (AMG) for approximating the application of sub‐block inverses, is developed and tested in three steps. First, a block preconditioner is introduced for linear systems resulting from micropolar models. For this case, a simple sparse Schur complement approximation, which is practical to compute, is proposed and analyzed. It is tested for three different discretizations. Second, the developed preconditioner is extended to reflect the additional nonlocal damage field. This extended three‐field preconditioner is tested on the simulation of a compression test on a sandstone sample. All numerical tests show an improved performance of the block preconditioning approach in comparison to a black‐box monolithic AMG approach. Finally, a problem‐adapted preconditioner setup strategy is proposed, which involves a reuse of the multigrid hierarchy during nonlinear iterations, and additionally accounts for the different stages occurring in the simulation of localized failure. The problem‐adapted preconditioning strategy has the potential to further reduce the total computation time.

Nonlocal interfaces accounting for progressive damage within continuum-kinematics-inspired peridynamics

In this work, we present a modeling approach to nonlocal material interfaces in the framework of continuum-kinematics-inspired peridynamics. The nonlocal model accounts for progressive damage within a finite-thickness interface, as opposed to the more common practice of abrupt bond breakage across a zero-thickness interface. Our approach is based on an overlap of the constituents within the interface. Interfacial bonds between initially overlapping partner points are governed by a constitutive law reminiscent of a traction-separation-law. The governing equations for continuum-kinematics-inspired peridynamics in the presence of an interface are derived using a rate-variational principle. The damage formulation is established using the classical concept of internal variables. Following the notion of a standard dissipative material, thermodynamic consistency of the constitutive laws and the evolution of the internal variables is ensured. The latter results in a straightforward evaluation of a damage function. We give details about the computational implementation comprising a peridynamic discretization and a Newton–Raphson scheme. A sound approach to approximate the interface normal during deformation is presented, which allows to penalize material penetration across the interface. The proposed model is explored in a series of numerical examples, i.e., classical peeling and shearing tests, for a variety of damage functions. A key feature of our interface model are the nonlocal characteristics that are assumed to play a role especially at small scales. We, first, observe that an increasing thickness of the nonlocal interface leads to stronger interfacial bonding and less damage. Second, an increase in horizon size results in stiffer material behavior. When studying the wrinkling and delamination behavior of a compressed bilayer, it is found that an increase in interface stiffness leads to a smaller wrinkling wavelength. Moreover, delamination due to progressive damage of interfacial bonds in the post-wrinkling regime is observed, which, to the best of our knowledge, has not been studied in a nonlocal model before.

A consistent multi-phase-field formulation for anisotropic brittle fracture

Anisotropic fracture modelling constitutes an important problem in fracture mechanics as many natural, composite and architected materials exhibit fracture anisotropy. The multi-phase-field technique offers a seemingly straightforward and easy-to-implement approach for handling a range of fracture anisotropies, some of which would otherwise necessitate the use of higher-order damage formulation. However, a closer examination reveals that the formulation contains a few inconsistencies, which hinder the establishment of precise definitions for crack through phase-fields and fracture toughness. In this article, we present modifications to the multi-phase-field approach aimed at rectifying these issues. Our model introduces a new degradation function, enabling an unambiguous definition of crack and leading to an analytical expression of the modelled fracture toughness. We conduct an in-depth analysis of several key components of the present formulation and discuss the nature of anisotropy it can handle in its current form, and also outline a roadmap for expanding its capability. A limited set of numerical simulations is also provided to illustrate the effectiveness of the current proposition.

Simulation of fracture in vascular tissue: coupling a continuum damage formulation with an embedded representation of fracture

The fracture of vascular tissue, and load-bearing soft tissue in general, is relevant to various biomechanical and clinical applications, from the study of traumatic injury and disease to the design of medical devices and the optimisation of patient treatment outcomes. The fundamental mechanisms associated with the inception and development of damage, leading to tissue failure, have yet to be wholly understood. We present the novel coupling of a microstructurally motivated continuum damage model that incorporates the time-dependent interfibrillar failure of the collagenous matrix with an embedded phenomenological representation of the fracture surface. Tissue separation is therefore accounted for through the integration of the cohesive crack concept within the partition of unity finite element method. A transversely isotropic cohesive potential per unit undeformed area is introduced that comprises a rate-dependent evolution of damage and accounts for mixed-mode failure. Importantly, a novel crack initialisation procedure is detailed that identifies the occurrence of localised deformation in the continuum material and the orientation of the inserted discontinuity. Proof of principle is demonstrated by the application of the computational framework to two representative numerical simulations, illustrating the robustness and versatility of the formulation.

Low cycle fatigue of components manufactured by rod extrusion: Experiments and modeling

The performance of formed components is significantly influenced by the initiation of ductile damage. Preceding forming operations, for instance, affect the service life determined in fatigue tests. In the current investigation, the effect of ductile damage in forming is isolated by changing the shoulder opening angle in forward rod extrusion. Forming-induced ductile damage is then related to measurements of void area fraction, density and Young’s modulus. Subsequent fatigue tests in the low cycle range indicate that the service life of the extruded components can be improved through a reduction of the forming-induced damage. A novel constitutive model considering forming-induced damage and fatigue damage is proposed to account for the observed behavior in axial fatigue tests of extruded components. The non-local ductile damage formulation is formulated in the framework of Generalized Standard Materials. Kinematic and isotropic hardening are considered. Based on earlier work of Lemaitre and Desmorat, the fatigue damage initiation criterion is extended to take the observed mechanical behavior in low cycle axial fatigue tests of formed components into account. The extended model is able to capture the effect of forming-induced damage on the service life.

Modelling of damage and failure within polymeric adhesives

AbstractWithin this contribution, we propose a fully thermo‐mechanical coupled isotropic damage model for polymeric adhesives under finite strains. This model is based on the multiplicative decomposition of the deformation gradient into mechanical and thermal parts. We consider rate‐dependent damage behaviour (e.g. creep damage) of the material by applying a Perzyna‐type ansatz for the damage evolution equations. To overcome the pronounced mesh dependencies which would result from the usage of a local damage model, we make use of a gradient‐extended damage formulation. Besides the main aspects of model, we show the thermodynamically consistent derivation of constitutive quantities as well as the numerical treatment of the governing equations. Finally, we show selected numerical examples to demonstrate the capabilities of the model.

Probabilistic capacity models and fragility estimate for NRC and UHSC panels subjected to contact blast

A Performance-based Design (PBD) framework for protective structural walls/slabs susceptible to contact blast has been developed. Performance-based capacity models are predicted for three performance levels namely, operation with minimum damage (P1), medium damage (P2) and collapse prevention (P3) allied to four damage states. The current study examines multi-level PBD instead of single-level conventional collapse-based design. All these probabilistic models are developed by combining mechanical models with dimensionless explanatory functions. Bayesian inference based on numerical data is used to evaluate the unknown model parameters. The constructed models take into account all inherent aleatoric and epistemic uncertainties of material and geometry. For Normal Reinforced Concrete (NRC) and Ultra-high Strength Concrete (UHSC) panels subjected to contact blast, finite element (FE) modelling (LS-DYNA) is used to obtain the necessary data. The validated models have a high level of agreement with chosen experimentation. In addition, local damage formulations, such as crater diameter, crater depth, spall diameter, spall depth, and perforation diameter, are also developed. The developed capacity models are used to estimate fragility, and the obtained plots demonstrate the consistency of the evaluated equations. When subjected to a contact blast, these probabilistic models can be used to construct protective structures based on the expected demand.

From ductile damage to unilateral contact via a point-wise implicit discontinuity

Ductile damage models and cohesive laws incorporate the material plasticity entailing the growth of irrecoverable deformations even after complete failure. This unrealistic growth remains concealed until the unilateral effects arising from the crack closure emerge. We address this issue by proposing a new strategy to cope with the entire process of failure, from the very inception in the form of diffuse damage to the final stage, i.e. the emergence of sharp cracks. To this end, we introduce a new strain field, termed discontinuity strain, to the conventional additive strain decomposition to account for discontinuities in a continuous sense so that the standard principle of virtual work applies. We treat this strain field similar to a strong discontinuity, yet without introducing new kinematic variables and nonlinear boundary conditions. In this paper, we demonstrate the effectiveness of this new strategy at a simple ductile damage constitutive model. The model uses a scalar damage index to control the degradation process. The discontinuity strain field is injected into the strain decomposition if this damage index exceeds a certain threshold. The threshold corresponds to the limit at which the induced imperfections merge and form a discrete crack. With three-point bending tests under pure mode I and mixed-mode conditions, we demonstrate that this augmentation does not show the early crack closure artifact which is wrongly predicted by plastic damage formulations at load reversal. We also use the concrete damaged plasticity model provided in Abaqus commercial finite element program for our comparison. Lastly, a high-intensity low-cycle fatigue test demonstrates the unilateral effects resulting from the complete closure of the induced crack.

TP53 and the Ultimate Biological Optimization Steps of Curative Radiation Oncology.

The new biological interaction cross-section-based repairable-homologically repairable (RHR) damage formulation for radiation-induced cellular inactivation, repair, misrepair, and apoptosis was applied to optimize radiation therapy. This new formulation implies renewed thinking about biologically optimized radiation therapy, suggesting that most TP53 intact normal tissues are low-dose hypersensitive (LDHS) and low-dose apoptotic (LDA). This generates a fractionation window in LDHS normal tissues, indicating that the maximum dose to organs at risk should be ≤2.3 Gy/Fr, preferably of low LET. This calls for biologically optimized treatments using a few high tumor dose-intensity-modulated light ion beams, thereby avoiding secondary cancer risks and generating a real tumor cure without a caspase-3-induced accelerated tumor cell repopulation. Light ions with the lowest possible LET in normal tissues and high LET only in the tumor imply the use of the lightest ions, from lithium to boron. The high microscopic heterogeneity in the tumor will cause local microscopic cold spots; thus, in the last week of curative ion therapy, when there are few remaining viable tumor clonogens randomly spread in the target volume, the patient should preferably receive the last 10 GyE via low LET, ensuring perfect tumor coverage, a high cure probability, and a reduced risk for adverse normal tissue reactions. Interestingly, such an approach would also ensure a steeper rise in tumor cure probability and a higher complication-free cure, as the few remaining clonogens are often fairly well oxygenated, eliminating a shallower tumor response due to inherent ion beam heterogeneity. With the improved fractionation proposal, these approaches may improve the complication-free cure probability by about 10-25% or even more.

Concrete failure based on a localizing gradient‐enhanced damage formulation coupled with plasticity

AbstractInvestigations of the mechanical behavior of concrete, including plastic deformability and failure mechanisms, have always been of vital importance. In this work, a modified yield criterion with three surfaces is proposed to capture the plastic yielding behavior under different loading conditions. The failure, that is, the gradual degradation and the final loss of material integrity, can be described through a damage concept. For the purpose of robust finite element (FE) implementations as well as the failure description at an appropriate scale, a localizing gradient‐enhanced damage model has been adopted. As such, one obtains mesh‐insensitive material responses, while ruling out the unphysical damage broadening observed in conventional gradient‐enhanced damage models with a constant regularization length. Following a consistent derivation, the plasticity‐damage coupled model is implemented into an in‐house FE framework, based on which an experiment‐inspired example is simulated to illustrate the capability of the proposed description in concrete modeling, especially the mixed‐mode failure. Lastly, the conclusion provides some final insights.

Characterisation of damage by means of electrical measurements: Numerical predictions

AbstractUnderstanding damage mechanisms and quantifying damage is important in order to optimise structures and to increase their reliability. To achieve this goal, experimental‐ and simulation‐based techniques are to be combined. Different methods exist for the analysis of damage phenomena such as fracture mechanics, phase field models, cohesive zone formulations and continuum damage modelling. Assuming a typical ‐type damage formulation, the governing equations of continua that account for gradient‐enhanced ductile damage under mechanical and electrical loads are derived. The mechanical and electrical sub‐problems give rise to the local form of the balance equation of linear momentum, the micromorphic balance relation and the continuity equation for the electric charge, respectively. Experimental investigations indicate that changes in electrical conductivity arise due to the evolution of the underlying microstructure, for example, of cracks and dislocations. Therefore, motivated by deformation‐induced property changes, the effective electrical conductivity is assumed to be a function of the damage variable. This eventually allows the prediction of experimentally recorded changes in the electrical resistance due to mechanically‐induced damage processes. Interpreting the resistivity as a fingerprint of the material microstructure, the simulation approach proposed in the present work contributes to the development of non‐destructive electrical‐resistance‐based characterisation methods. To demonstrate the applicability of the proposed framework, different representative simulations are studied.

A new implicit gradient damage model based on energy limiter for brittle fracture: Theory and numerical investigation

We present a general form of the gradient-enhanced damage theory and its numerical implementation using finite element method (FEM) in modeling quasi-static brittle crack growth in one- (1D), two- (2D) and three-dimensional (3D) bodies. Coupled equations of the equilibrium and a new implicit gradient damage formulation are introduced to govern the deformation of the solid and evolution of the damage. The resulting nonlocal damage evolution equation featuring the growth of diffusive crack is integrated with a characteristic length scale to eliminate the common mesh-bias issue in FEM implementation. In contrast to the traditional gradient-enhanced damage approaches, the nonlocal damage field here is defined as the primary variable of the damage evolution equation without interpolation mismatch between the displacement and nonlocal damage fields. For derivation of the material constitutive law and local damage parameter, a novel strain energy density (SED) function based on the energy limiter theory for brittle crack growth problems under small strain regime is introduced. To further improve the performance of the developed model, an initial SED threshold, which is used for determining the critical point when damage starts to initiate in the material, is integrated into the novel energy limiter theory. For preventing nonphysical failure in compression domains, the spectral decomposition technique for the strain tensor is adopted to split the reference SED. With integrating the energy limiter into the developed theory, unlike the conventional nonlocal damage theories where the interpretation of the length scale is still ambiguous, the developed nonlocal damage model defines the length scale parameter as the problem-dependent factor. The performance and ability of the proposed model are demonstrated via a set of representative numerical examples in 1D, 2D and 3D fracture problems.

Creep-fatigue assessment of martensitic welds based on numerically determined local deformation

Abstract The low cycle fatigue and creep-fatigue behavior of base material and similar welded joints of martensitic steel Grade 92 were examined in this work. Several low cycle fatigue tests with different loading scenarios: fatigue tests with strain-controlled hold time, fatigue tests with strain-controlled amplitudes, and load-controlled hold times, were performed at 873 K. The tests were evaluated with regard to fatigue lifetime, peak stress, the evolution of relaxation stress or creep strain, and failure location. The effect of amplitude level, hold time modus and hold time length on the lifetime of the specimens was investigated using lifetime and damage assessment methods based on strain energy approaches. The numerical simulations for the determination of the evolution of local strains and stresses were done using a viscoplastic material model of Chaboche type, including a strain energy based damage formulation. For the parameter identification, tests were carried out on the specimens made of base metal, welded joint, weld metal, and with microstructure simulated specimens aiming to obtain representative microstructures of the fine grain, intercritical, and coarse grain heat affected zone. The numerical assessment of the local stress, strain, and damage distribution explains the smaller lifetime of the welded joint specimens.

Relaxed Incremental Formulations for Damage at Finite Strains Including Strain Softening

AbstractRelaxation is a promosing technique to overcome mesh‐dependency in computational damage mechanics originating from the non‐convexity of an underlying incremental variational formulation. This technique does not require an internal length scale parameter. However, in case of damage formulations, for many years the decrease of stresses with an increase of strains, referred to as strain‐softening, could not be modeled in the relaxed regime. This contribution discusses several possibilities of relaxation that lead to suitable models for stress‐ and strain‐softening.