Isotropic Damage Model Research Articles

An isotropic unilateral damage model coupled with frictional sliding for quasi-brittle materials

In this paper, we present an original extension of an isotropic damage model for quasi-brittle materials and assess its predictive capabilities. The proposed model accounts not only for unilateral behavior related to the opening and closure of microcracks but also for inelastic strains reflecting the frictional sliding along closed microcracks. More importantly, owing to its careful mathematical formulation with a particular attention paid to the continuous differentiability of the underlined thermodynamic potential, the model ensures the continuity of the inelastic stress–strain response. First applications show that it is able to predict the asymmetric behavior and hysteretic response of microcracked materials such as concrete and some rocks

Static and dynamic physically non-linear analysis of concrete structures using a hybrid mixed finite element model

A new hybrid-mixed stress finite element model for the static and dynamic non-linear analysis of concrete structures is presented and discussed in this paper. The main feature of this model is the simultaneous and independent approximation of the stress, the strain and the displacement fields in the domain of each element. The displacements along the static boundary, which is considered to include inter-element boundaries, are also directly approximated. To define the approximation bases in the space domain, complete sets of orthonormal Legendre polynomials are used. The adoption of these functions enables the use of analytical closed form solutions for the computation of all linear structural operators and leads to the development of very effective p-refinement procedures. To represent the material quasi-brittle behaviour, a physically non-linear model is considered by using damage mechanics. A simple isotropic damage model is adopted and to control strain localisation problems a non-local integral formulation is considered. To solve the dynamic non-linear governing system, a time integration procedure based on the use of the @a-HHT method is used. For each time step, the solution of the non-linear governing system is achieved using an iterative algorithm based on a secant method. The model being discussed is applied to the solution of two-dimensional structures. To validate the model, to illustrate its potential and to assess its accuracy and numerical efficiency, several numerical examples are discussed and comparisons are made with solutions provided by experimental tests and with other numerical results obtained using conventional finite elements (CFE).

A new hybrid-mixed stress model for the analysis of concrete structures using damage mechanics

This paper presents a new hybrid-mixed stress finite element model for the static analysis of concrete structures. A non-linear behaviour at the material level is assumed and to model the material’s quasi-brittle behaviour damage mechanics is considered. In this model, the stress, the strain and the displacement fields are independently approximated in the domain of each element. The displacements along the static boundary, which is considered to include inter-element boundaries, are also directly approximated. Complete sets of orthonormal Legendre polynomials are used to define all approximation bases required by the finite element model. The adoption of these functions enables the use of analytical closed form solutions for the computation of all linear structural operators and leads to the development of very effective p-refinement procedures. A simple isotropic damage model is adopted and to control strain localization problems a non-local integral formulation is considered. The model being discussed is applied to the solution of two-dimensional structures. To validate the model, to illustrate its potential and to assess its accuracy and numerical efficiency, several numerical examples are discussed and comparisons are made with solutions provided by experimental tests and with other numerical results obtained using conventional finite elements.

A numerical model of delamination in composite laminated beams using the LRZ beam element based on the refined zigzag theory

A method based on the Refined Zigzag Theory (RZT) to model delamination in composite laminated beam structures is presented. The novelty of this method is the use of one-dimensional finite elements to discretize the geometry of the beam. The key property of this beam element, named LRZ [1], is the possibility to capture the relative displacement between consecutive layers which occurs during delamination. The fracture mode that the LRZ element is capable to predict is mode II. In order to capture the relative displacement using the LRZ element it is necessary to adapt the RZT theory as presented in this paper. The mechanical properties of the layers are modeled using a continuum isotropic damage model [2]. The modified Newton–Raphson method is used for solving the non-linear problem.The RZT theory, the LRZ finite element and the isotropic damage model are described in the paper. Also, the implicit integrations algorithm is presented. The performance of the LRZ element is analyzed by studying the delamination in a beam for two different laminates, using the plane stress solution as a reference.

A micromechanical analysis of time-dependent behavior based on subcritical damage in claystones

In this paper, a micromechanical model is proposed for time-dependent strain in claystones. The representative volume element of claystone is characterized by a three-phase composite. Quartz and calcite grains are embedded inside clay matrix. The time-dependent strain is related to subcritical propagation of microcracks inside the clay matrix. A two-step homogenization procedure is proposed. A micromechanics-based isotropic damage model is first formulated for the subcritical growth of microcracks inside the clay matrix. Different homogenization schemes are considered by taking into account opening and closure of microcracks. Two driving forces are used respectively for the subcritical evolution of damage under tensile and compressive stresses. Then the macroscopic properties of the claystone are determined by a second-level homogenization procedure using three different schemes. The numerical algorithm for numerical implementation of the proposed model in standard finite element code is proposed. Numerical evaluations of the proposed model are performed through simulations of creep tests under different loading paths and using different homogenization schemes. Finally, comparisons between numerical results and experimental data are presented.

Relevance of a mesoscopic modeling for the coupling between creep and damage in concrete

In its service-life concrete is loaded and delayed strains appear due to creep phenomenon. Some theories suggest that micro-cracks nucleate and grow when concrete is submitted to a high sustained loading, thereby contributing to the weakening of concrete. Thus, it is important to understand the interaction between the viscoelastic deformation and damage in order to design reliable civil engineering structures. Several creep-damage theoretical models have been proposed in the literature. However, most of these models are based on empirical relations applied at the macroscopic scale. Coupling between creep and damage is mostly realized by adding some parameters to take into account the microstructure effects. In the authors’ opinion, the microstructure effects can be modeled by taking into account the effective interactions between the concrete matrix and the inclusions. In this paper, a viscoelastic model is combined with an isotropic damage model. The material volume is modeled by a Digital Concrete Model which takes into account the “real” aggregate size distribution of concrete. The results show that stresses are induced by strain incompatibilities between the matrix and aggregates at mesoscale under creep and lead to cracking.

Surrounding Rock Stability Analysis and Rock Burst Prediction of Underground Cavity in High <i>In Situ</i> Stress

Based on Sidoroff isotropic elastic damage model, the damage deterioration constitutive equation was constructed for rigid and brittle rock mass. The excavation numerical simulation of a diversion tunnel with large section in Jinping hydro station was carried on. The displacement and stress evolution was analyzed considering stress transfer and dissipation of the tunnel surrounding rock mass. The stability evaluation parameters were put forward such as stress relaxation coefficient of surrounding rock mass, safety factor, the scale and distribution of plastic region of surrounding rock mass. The rock burst predication index was presented in viewpoint of energy such as sub-range statistic average energy release rate and sub-range statistic failure volume. The excavation methods for the diversion tunnel with large section were evaluated with multi criteria and the guideline for large section cavity in high in-situ stress was summarized.

Prediction of anisotropic material behavior based on multiresolution continuum mechanics in consideration of a characteristic length scale

New advanced materials have received more attention from many scientists and engineers because of their outstanding chemical, electrical, thermal, optical, and mechanical properties. Since the design of advanced material by experiments requires high cost and time, numerical approaches have always been of great interest. In this paper, finite element analysis of anisotropic material behavior has been carried out based on a multiresolution continuum theory. Gurson-Tvergaard-Needleman (GTN) damage model has been applied as a constitutive model at macroscale. Effects of plastic anisotropy on deformation behavior are assessed using Hill’s 48 yield function for anisotropic material and von Mises yield function for isotropic material, respectively. The material parameters for both isotropic and anisotropic damage models have systematically been determined from microstructure through unit cell modeling. The newly proposed linear approximation of local velocity gradient resolved the underdetermined problem of the previous homogenization process. Anisotropic material behaviors of a tensile specimen have been investigated by the proposed multiresolution continuum theory.

Two-phase damage modeling of concrete affected by alkali–silica reaction under variable temperature and humidity conditions

In the present work the concrete affected by alkali–silica reaction (ASR) is represented as a two-phase material made of a solid skeleton and a wet expanding gel, which exerts a pressure capable of severely damaging the concrete surrounding the reactive sites. Both the effects of temperature and humidity conditions on the kinetic of the chemical reaction and on the final value of the consequent expansion are included in the proposed model. The mechanical degradation induced by the ASR is described by a phenomenological isotropic damage model. The constitutive model, implemented in a finite element code, is used for the analyses of structures made of reactive concrete in the presence of temperature and moisture gradients. Firstly the temperature and humidity fields are obtained through uncoupled heat and moisture transport analyses and then the chemo-mechanical analysis is performed starting from the values of temperature and humidity preliminary calculated.

New Strain Energy–Based Coupled Elastoplastic Damage-Healing Formulations Accounting for Effect of Matric Suction during Earth-Moving Processes

AbstractInnovative initial elastic strain energy–based coupled elastoplastic hybrid isotropic damage and healing models for partially saturated soils have been developed and implemented for numerical simulation of two-dimensional earth-pushing processes. A class of elastoplastic constitutive damage-healing models, based on a continuum thermodynamic framework, is proposed within an initial elastic strain energy–based formulation. In particular, the change of effective stress caused by matric suction in the formulation is considered, and the governing incremental damage and healing evolutions are coupled and characterized through the effective stress concept in conjunction with the hypothesis of strain equivalence. Further, plastic flow is introduced by means of an additive split of the stress tensor. In this innovative formulation, two characteristic energy norms of the tensile and compressive strain tensors, respectively, are introduced for the corresponding damage and healing mechanisms. By incorporating...

A numerical model of the structural behavior of buckling-restrained braces

This work presents a numerical model of the cyclic structural behavior of dissipative buckling-restrained braces, commonly used as an alternative to classical concentric braces for seismic protection of building frames and other structures. Such devices are usually composed of a slender steel core embedded in a stockiest casing that is intended to prevent its buckling when it is under compression. The casing is made either of mortar or steel, and a sliding interface is interposed between the core and the casing to prevent excessive shear stress transfer. The behavior of the steel core is described by a damage and plasticity model; the behavior of the mortar casing is described by an isotropic damage model and the sliding behavior of the interface is described by a contact penalty model. These three models are implemented in the Abaqus software package following an explicit formulation. The ability of this algorithm to reproduce the cyclic behavior of buckling-restrained braces is verified in a number of representative yet simple situations. The accuracy of the proposed model is checked by comparison with experimental results; a satisfactory agreement is obtained. Preliminary conclusions are issued and further research needs are identified.

Damage analysis of concrete structures using polynomial wavelets

This paper presents and discusses a hybrid-mixed stress finite element model based on the use of polynomial wavelets for the physically non-linear analysis of concrete structures. The effective stress and the displacement fields in the domain of each element and the displacements on the static boundary are independently approximated. As none of the fundamental equations is locally enforced a priori, the hybrid-mixed stress formulation enables the use of a wide range of functions. In the numerical model reported here, all approximations are defined using complete sets of polynomial wavelets. These bases present some important features. In one hand, the functions are orthogonal, which is an important issue when implementing hybrid-mixed stress elements as it ensures high levels of sparsity. On the other hand, the polynomial wavelet basis is defined through linear combinations of Legendre polynomials. This fact enables the use of closed-form solutions for the computation of the integrations involved in the definition of all linear structural operators. A simple isotropic damage model is adopted and a non-local integral formulation where the strain energy release rate is taken as the non-local variable is considered. The numerical model is both incremental and iterative and is solved with a modified Newton–Raphson method that uses the secant matrix. Classical benchmark tests are chosen to illustrate the use of the model under discussion and to assess its numerical performance.

Characterization and Modeling of Ductile Damage in Structural Steel at Low and Intermediate Strain Rates

AbstractIn this research, the concept of continuum damage mechanics (CDM) was implemented to evaluate damage initiation and evolution in structural steel rebars at low and intermediate loading rates. The work was divided into experimental and theoretical phases to evaluate the ductile damage behavior. The experimental part involved uniaxial tensile tests for steel bars at the ambient temperature over a range of strain rates between 3.0×10−4 and 0.1 s−1. Scanning electron microscopy (SEM), coupled with image processing techniques, was utilized to characterize the damage at several loading stages. The theoretical part of this paper introduced a new energy-based isotropic damage model capable of capturing the evolution of damage throughout the deformation process. The proposed damage model was verified via comparisons with the present SEM results and with other data available in the literature. The comparisons showed good agreement, and results from both approaches indicated that cracks and voids intensified...

Simplified damage models applied in the numerical analysis of reinforced concrete structures

This work presents one and two-dimensional numerical analyses using isotropic and anisotropic damage models for the concrete in order to discuss the advantages of these modeling. Initially, it is shortly described the damage model proposed by Mazars. This constitutive model assumes the concrete as isotropic and elastic material, where locally the damage is due to extensions. On the other hand, the damage model proposed by Pituba, the material is assumed as initial elastic isotropic medium presenting anisotropy, plastic strains and bimodular response (distinct elastic responses whether tension or compression stress states prevail) induced by the damage. To take into account for bimodularity two damage tensors governing the rigidity in tension and compression regimes, respectively, are introduced. Damage activation is expressed by two criteria indicating the initial and further evolution of damage. Soon after, the models are used in numerical analyses of the mechanical behavior of reinforced concrete structures. Accordingly with comparison of the obtained responses, considerations about the application of the isotropic and anisotropic damage models are presented for 1D and 2D reinforced concrete structures modeling as well as the potentialities of the simplified versions of damage models applied in situations of structural engineering.

Determination of representative volume element in concrete under tensile deformation

The 2D representative volume element (RVE) for softening quasi-brittle materials like concrete is determined. Two alternative methods are presented to determine a size of RVE in concrete subjected to uniaxial tension by taking into account strain localization. Concrete is described as a heterogeneous three-phase material composed of aggregate, cement matrix and bond. The plane strain FE calculations of strain localization at meso-scale are carried out with an isotropic damage model with non-local softening.

Failure Predictions for DP Steel Cross-die Test using Anisotropic Damage

The Lemaitre's continuum damage model is well known in the field of damage mechanics. The anisotropic damage model given by Lemaitre is relatively simple, applicable to nonproportional loads and uses only four damage parameters. The hypothesis of strain equivalence is used to map the effective stress to the nominal stress. Both the isotropic and anisotropic damage models from Lemaitre are implemented in an in-house implicit finite element code. The damage model is coupled with an elasto-plastic material model using anisotropic plasticity (Hill-48 yield criterion) and strain-rate dependent isotropic hardening. The Lemaitre continuum damage model is based on the small strain assumption; therefore, the model is implemented in an incremental co-rotational framework to make it applicable for large strains. The damage dissipation potential was slightly adapted to incorporate a different damage evolution behavior under compression and tension. A tensile test and a low-cycle fatigue test were used to determine the damage parameters. The damage evolution was modified to incorporate strain rate sensitivity by making two of the damage parameters a function of strain rate. The model is applied to predict failure in a cross-die deep drawing process, which is well known for having a wide variety of strains and strain path changes. The failure predictions obtained from the anisotropic damage models are in good agreement with the experimental results, whereas the predictions obtained from the isotropic damage model are slightly conservative. The anisotropic damage model predicts the crack direction more accurately compared to the predictions based on principal stress directions using the isotropic damage model. The set of damage parameters, determined in a uniaxial condition, gives a good failure prediction under other triaxiality conditions.

Modelling and computation of curing and damage of thermosets

The curing of thermosets is a complex process involving the transition from a fluid into a (visco-) elastic solid. This phase transition comes along with an increase in stiffness and a volume shrinkage of the polymer. The latter may lead to severe residual strains and stresses, which in turn can cause damage in the final, usually quasi-brittle material. In this contribution a constitutive model is developed which takes into account the curing of a thermosetting material together with the process-induced damage as resulting from curing shrinkage. The curing of the material is governed by a phenomenological hypoelastic constitutive equation which includes temporal evolutions for stiffness and volume shrinkage. Thermal and viscous effects are neglected in the present study. An isotropic gradient-enhanced damage model is adapted to describe the damage evolution. The curing-damage model is implemented into a finite element code and numerical examples for thermosetting materials demonstrate that the proposed model is capable to predict cure-induced damage in thermosets.

Estimation of crack opening from a two‐dimensional continuum‐based finite element computation

SUMMARYDamage models are capable of representing crack initiation and mimicking crack propagation within a continuum framework. Thus, in principle, they do not describe crack openings. In durability analyses of concrete structures however, transfer properties are a key issue controlled by crack propagation and crack opening. We extend here a one‐dimensional approach for estimating a crack opening from a continuum‐based finite element calculation to two‐dimensional cases. The technique operates in the case of mode I cracking described in a continuum setting by a nonlocal isotropic damage model. We used the global tracking method to compute the idealized crack location as a post‐treatment procedure. The original one‐dimensional problem devised in Dufour et al. [4] is recovered as profiles of deformation orthogonal to the idealized crack direction are computed. An estimate of the crack opening and an error indicator are computed by comparing finite element deformation profiles and theoretical profiles corresponding to a displacement discontinuity. Two estimates have been considered: In the strong approach, the maxima of the profiles are assumed to be equal; in the weak approach, the integrals of each profile are set equal. Two‐dimensional numerical calculations show that the weak estimates perform better than do the strong ones. Error indicators, defined as the distance between the numerical and theoretical profiles, are less than a few percentages. In the case of a three‐point bending, test results are in good agreement with experimental data, with an error lower than 10% for widely opened crack (> 40µm). Copyright © 2011 John Wiley & Sons, Ltd.

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

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

A continuum description of the damage process in the arterial wall of abdominal aortic aneurysms.

In the present work, we develop a three-dimensional isotropic finite-strain damage model for abdominal aortic aneurysm (AAA) wall that considers both the characteristic softening of the material caused by damage and the spatial variation of the material properties. A strain energy function is formulated that accounts for a hyperelastic, slightly compressible, isotropic material behavior during the elastic phase, whereas the damage process only contributes to the material response when the elastic limit of the AAA wall is exceeded. Material and damage parameters are obtained by fitting the strain energy function to the experimental data obtained by uniaxial tensile tests of freshly harvested AAA wall samples. The damage model extends the validity of the material law to a strain range of up to 50%. Purely elastic material laws for AAA wall are only valid for a strain range of up to 17%. In a series of finite element simulations of patient-specific AAAs, serving as numerical examples, we investigate the applicability of the damage model. The use of the damage model does not yield a more distinct identification of rupture-prone AAAs than other computational-based risk indices. However, the benefit of the finite-strain damage model is the potential capability to trigger growth and remodeling processes in mechanobiological models.