Rate-dependent Damage Research Articles

Blast resistance of unidirectional fiber reinforced composites

Three-dimensional transient deformations of a laminate comprised of several unidirectional fiber reinforced layers perfectly bonded to each other and subjected to a blast load are analyzed by the finite element method with an in-house developed, verified and fully validated code with rate-dependent damage evolution equations for anisotropic bodies. The continuum damage mechanics approach employing three internal variables is used to characterize damage due to fiber/matrix debonding, fiber breakage, and matrix cracking. The delamination between two adjoining layers is assumed to ensue when the stress state there satisfies a failure criterion, and may initiate simultaneously at several points. The relative sliding between adjoining layers is simulated by the nodal release technique. The interaction among fiber/matrix debonding, fiber breakage, matrix cracking and delamination, and the possibility of their initiating concurrently at one or more points in the composite are considered. The effect of different material, geometric and loading parameters on the damage development and propagation, and the energy absorbed in each one of the four failure modes have been examined. These results give preliminary information on composite structure’s design for maximizing the energy absorption and hence increasing structure’s resistance to blast loads. The paper is a sequel to Hassan and Batra’s paper [Hassan NM, Batra RC. Modeling damage development in polymeric composites. Composites B, doi:10.1016/j.compositesb.2007.02.001] wherein details of the damage evolution equations, verification of the code, and the validation of the mathematical model are given.

A coupled viscoplastic rate-dependent damage model for the simulation of fatigue failure of cement–bone interfaces

Clinical loosening of cemented hip replacements is often associated with mechanical failure of the cement–bone interface. Although different computational models trying to simulate the interface behavior can be found in the literature, plastic deformations have not been considered yet, despite the experimental results available demonstrating their importance. The main purpose of this work is to develop a computational model for bone–cement interfaces able to capture the behavior of the interface under static and cyclic mixed-mode loading, incorporating stiffness degradation (damage) and permanent deformation (plasticity). With this aim, a cohesive zone model that incorporates a coupled damage-viscoplastic constitutive law has been developed. A detailed description of the corresponding mathematical and computational framework is here included. A parameter fitting has been also performed, obtaining a correlation for every parameter with the quantity of interdigitated bone that can be easily obtained from computer tomographies. 3D finite element simulations of several experimental tests for cement–bone specimens have been carried out in order to show the ability of the model to reproduce the initiation and progression of failure as well as fatigue life of the interface. The predicted results are in close agreement with such experimental observations.

The Improved Johnson-Cook’s Strength Model Taking Account of the Rate-Dependent Micro-Damage Evolution for C30 Concrete

The dynamic mechanical behavior of C30 concrete under a wide range of strain rates from 10-4s-1 up to 102s-1 is studied. According to Johnson-Cook’s strength model, the strain rate coefficients and related material constants of C30 concrete subjected to large strains, high strain rates and high pressures are determined experimentally: C=0.34*10-1, A=1.05, B=1.65, N=0.76, TC =3.162MPa, fc’=39.2MPa. The damage evolution for C30 concrete is a rate-dependent process, which can be formulated to a rate-dependent damage evolution law in a simple form for engineering application. When ε > ε th, ( ) 1 D th D = K ε&α − ε −ε . The corresponding dynamic coefficients of C30 concrete are also obtained from impact experimental results: KD=530.2, a=0.83. Due to a<1, the damage evolution corresponds to an impact toughening process that coincide well with the dynamic experimental results for C30 concrete.

Response of fiber reinforced composites to underwater explosive loads

An in-house developed, verified and fully validated three-dimensional finite element code, with rate dependent damage evolution equations for anisotropic bodies, is used to numerically ascertain the damage developed in a fiber-reinforced composite due to shock loads representative of those produced by an underwater explosion. Three internal variables characterize damage due to fiber/matrix debonding, fiber breakage, and matrix cracking. The delamination and relative sliding of adjoining layers has been simulated by the nodal release technique. The interaction among the four failure modes, and the possibility of their initiating concurrently at one or more points in the composite is considered. The effect of different parameters on the damage development and propagation, and energy absorbed in each one of the four failure modes has been examined. These results give preliminary information on composite structure’s design for maximizing the energy absorption and hence increasing structure’s resistance to blast loads. The paper is a sequel to Hassan and Batra’s paper [Composites B, 2007] wherein details of the damage model, verification of the code, and the validation of the mathematical model are given.

The Impact Damage Evolution of Steel Fiber Reinforced and Polymer Modified Concrete

Impact tests were carried out by use of a 74-mm-diameter split Hopkinson pressure bar to investigate the impact damage behaviors of plain concrete (PC), steel fiber reinforced concrete (SFRC) and steel fiber reinforced and polymer modified concrete (SFRPMC). The results show that all three kinds of materials appear strain rate strengthening effects, and SFRPMC appears a better impact resistance and energy absorbing ability than PC and SFRC. Based on the analysis of experimental results, a rate-dependent damage model is suggested to depict the dynamic behaviors of SFRC and SFRPMC, which derives the impact damage evolution of the materials. It shows that the damages in SFRPMC develop more slowly than that in SFRC.

A Finite Strain Plastic-damage Model for High Velocity Impact using Combined Viscosity and Gradient Localization Limiters: Part I - Theoretical Formulation

During dynamic loading processes, large inelastic deformation associated with high strain rates leads, for a broad class of ductile metals, to degradation and failure by strain localization. However, as soon as material failure dominates a deformation process, the material increasingly displays strain softening and the finite element computations are considerably affected by the mesh size and alignment. This gives rise to a non-physical description of the localized regions. This article presents a theoretical framework to solve this problem with the aid of nonlocal gradient-enhanced theory coupled to viscoinelasticity. Constitutive equations for anisotropic thermoviscodamage (rate-dependent damage) mechanism coupled with thermo-hypoelasto-viscoplastic deformation are developed in this work within the framework of thermodynamic laws, nonlinear continuum mechanics, and nonlocal continua. Explicit and implicit microstructural length-scale measures, which preserve the well-posedness of the differential equations, are introduced through the use of the viscosity and gradient localization limiters. The gradient-enhanced theory that incorporates macroscale interstate variables and their high-order gradients is developed here to describe the change in the internal structure and to investigate the size effect of statistical inhomogeneity of the evolution related plasticity and damage. The gradients are introduced in the hardening internal state variables and are considered dependent on their local counterparts. The derived microdamage constitutive model is destined to be applied in the context of high velocity impact and penetration damage mechanics. The theoretical framework presented in this article can be considered as a feasible thermodynamic approach that enables to derive various gradient (visco) plasticity/(visco) damage theories by introducing simplifying assumptions. Besides the clear physical significance of the proposed framework, it also defines a very convenient context for the efficient numerical integration of the resulting constitutive equations. This aspect is explored in Part II of this work and the application of the framework proposed herein to the numerical simulation of high velocity impacts on metal plates.

A micromechanical finite element model for linear and damage-coupled viscoelastic behaviour of asphalt mixture

This study presents a finite element (FE) micromechanical modelling approach for the simulation of linear and damage-coupled viscoelastic behaviour of asphalt mixture. Asphalt mixture is a composite material of graded aggregates bound with mastic (asphalt and fine aggregates). The microstructural model of asphalt mixture incorporates an equivalent lattice network structure whereby intergranular load transfer is simulated through an effective asphalt mastic zone. The finite element model integrates the ABAQUS user material subroutine with continuum elements for the effective asphalt mastic and rigid body elements for each aggregate. A unified approach is proposed using Schapery non-linear viscoelastic model for the rate-independent and rate-dependent damage behaviour. A finite element incremental algorithm with a recursive relationship for three-dimensional (3D) linear and damage-coupled viscoelastic behaviour is developed. This algorithm is used in a 3D user-defined material model for the asphalt mastic to predict global linear and damage-coupled viscoelastic behaviour of asphalt mixture. For linear viscoelastic study, the creep stiffnesses of mastic and asphalt mixture at different temperatures are measured in laboratory. A regression-fitting method is employed to calibrate generalized Maxwell models with Prony series and generate master stiffness curves for mastic and asphalt mixture. A computational model is developed with image analysis of sectioned surface of a test specimen. The viscoelastic prediction of mixture creep stiffness with the calibrated mastic material parameters is compared with mixture master stiffness curve over a reduced time period. In regard to damage-coupled viscoelastic behaviour, cyclic loading responses of linear and rate-independent damage-coupled viscoelastic materials are compared. Effects of particular microstructure parameters on the rate-independent damage-coupled viscoelastic behaviour are also investigated with finite element simulations of asphalt numerical samples. Further study describes loading rate effects on the asphalt viscoelastic properties and rate-dependent damage behaviour. Copyright © 2006 John Wiley & Sons, Ltd.

A rate-dependent damage model for brittle materials based on the dominant crack

A rate-dependent, continuum damage model is developed for brittle materials under dynamic loading. This model improves on the approach (ISOSCM) of [Addessio, F.L., Johnson, J.N., 1990. A constitutive model for the dynamic response of brittle materials. Journal of Applied Physics 67, 3275–3286] in several respects. (1) A new damage surface is found by applying the generalized Griffith instability criterion to the dominant crack (having the most unstable orientation), rather than by averaging the instability condition over all crack orientations as done previously. The new surface removes a discontinuity in the damage surface in ISOSCM when the pressure changes sign. (2) The strain due to crack opening is more consistent with crack mechanics, with only the tensile principal stresses contributing to the crack opening strain. This is achieved by incorporating a projection operator in the equation for the crack opening strain. One consequence of incorporating the projection operator is a prediction of shear dilatancy, which is not accounted for in ISOSCM. (3) The evolution of damage, which is based on the energy-release rate for the dominant crack, has a physical basis, whereas in the previous approach the damage growth rate was assumed to be an exponential function of the distance from the stress state to the damage surface without specific physical justification. An implicit algorithm has been developed so that a larger time step can be used than with the explicit algorithm used in ISOSCM. The numerical results of a silicon carbide (SiC) ceramic under several loading paths (hydrostatic tension/compression, uniaxial strain, uniaxial stress, and shear) and strain rates are presented to illustrate the main features of the model.

On numerical implementation of a coupled rate dependent damage‐plasticity constitutive model for concrete in application to high‐rate dynamics

PurposeTo provide an efficient and robust constitutive equations for concrete ion application to high rate dynamics.Design/methodology/approachDevelops an explicit‐implicit integration scheme for a concrete model. This robust integration scheme ensures computational efficiency. Comparison between simulations of impact of equivalent aircraft engine projectiles and the tests carried out in Sandia laboratory also demonstrate its efficiency.FindingsShows that modeling transient high rate dynamic behavior of concrete is very important to take into account for design concrete structures in the cases of dynamic loading conditions, such as an impact on the structure.Originality/valueProposes an original integration scheme for a coupled rate dependent damage plasticity model. Also provides a detailed consideration of the numerical stability of this kind of scheme for rate‐dependent damage model.

Thermodynamic framework for coupling of non-local viscoplasticity and non-local anisotropic viscodamage for dynamic localization problems using gradient theory

This study develops a general consistent and systematic framework for the analysis of heterogeneous media that assesses a strong coupling between rate-dependent plasticity and anisotropic rate-dependent damage for dynamic problems within the framework of thermodynamic laws and gradient theories. The proposed formulation includes thermo-elasto-viscoplasticity (rate-dependent plasticity) with anisotropic thermo-viscodamage (rate-dependent damage); a dynamic yield criterion of a von Mises type and a dynamic damage growth criterion; the associated flow rules; thermal softening; non-linear strain hardening; strain-rate hardening; strain hardening gradients; and strain-rate hardening gradients. Since the material macroscopic thermomechanical response under dynamic loading is governed by different physical mechanisms on the meso- and macroscale levels, the proposed three-dimensional kinematical model is introduced with manifold structure accounting for discontinuous fields of dislocation interactions (plastic flow) and crack and void interactions (damage growth). The gradient theory of rate-independent plasticity and rate-independent damage that incorporates macroscale interstate variables and their higher-order gradients is generalized here for rate-dependent plasticity and rate-dependent damage to properly describe the change in the internal structure and in order to investigate the size effect of statistical inhomogeneity of the evolution-related rate- and temperature dependent materials. The idea of bridging length-scales is made more general and complete by introducing spatial higher-order gradients in the temporal evolution equations of the internal state variables that describe hardening in coupled viscoplasticity and viscodamage models, which are considered here physically and mathematically related to their local counterparts. Furthermore, the constitutive equations for the damaged material are written according to the principle of strain energy equivalence between the virgin material and the damaged material; that is, the damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses and strains are replaced by their effective ones. In addition, computational issues concerned with the current gradient-dependent formulation of initial-boundary value problems are introduced in a finite element context. A weak (virtual work) formulation of the non-local dynamic viscoplastic and viscodamage conditions is derived, which can serve as a basis for the numerical solution of initial boundary value problems in the sense of the finite element method. Explicit expressions for the generalized tangent stiffness matrix and the generalized nodal forces are given. The model presented in this paper can be considered as a feasible thermodynamic approach that enables one to derive various coupled gradient viscoplasticity and viscodamage theories by introducing simplifying assumptions.

A Unified Viscoplastic Fatigue Damage Model for 63Sn-37Pb Solder Alloy Under Cyclic Stress Control

A damage coupling viscoplastic model is developed to predict fatigue life of solder alloy 63Sn-37Pb under stress control. The viscoplastic flow rule chosen employs a hyperbolic sine function. A damage evolution equation is formulated based on three distinct material deformation behaviors: (i) stress rate independent damage evolution; (ii) stress rate dependent cyclic damage evolution; and (iii) stress rate dependent ductile damage evolution. The cyclic stress testing with different stress waveforms was first conducted to investigate their progressive viscoplastic deformations of the solder alloy. The investigation reveals that the material constants used in the model can be adequately determined from the results of standard creep tests. The constitutive model is validated by comparing the predicted and measured ratchetting results of the solder alloy under different forms of stress cycling. The proposed model is found to be capable of satisfactorily describing the viscoplastic deformation and ratchetting failure behaviors of the solder alloy under the conditions of the cyclic stress loading.

Bifurcation Analyses of Steel and Concrete with Rate-Dependent Properties Part Two: Bifurcation Analyses and Demonstration

Based on the continuum tangent stiffness tensors derived in part one for steel and concrete, bifurcation analyses of rate-dependent elasto-plasticity and elasto-damage are performed in (-(and (-(spaces of Mohr, respectively. Localization ellipse osculating Mohr's circle enables us to determine the critical state and failure angle. It is shown that strain rate has no effect on the failure angle, and that the parameters in rate-dependent damage model are restricted by the results of bifurcation analysis.

The adequacy of GWPs as indicators of damage costsincurred by global warming

This article looks at the ability of Global Warming Potentials (GWPs) towork as indicators of equivalence for temperature development and damagecosts. We look at two abatement scenarios that are equivalent when using100-year GWPs: one scenario reduces short-lived gases, mainly methane(CH4); the other scenario reduces carbon dioxide (CO2).Despite their equivalence in terms of CO2 equivalents, the scenariosdo not result in equal rates or levels of temperature change. The disparitiescontinue as we move further down the chain of causality toward damagecosts, measured either in terms of rate of climate change or level of climatechange. Compared to the CH4 mitigation scenario, the CO2mitigation scenario gives present value costs 1.3 and 1.5 times higher forlevel- and rate-dependent damage costs, respectively, assuming a discountrate of 3%. We also test the GWPs for other time horizons and theconclusions remain the same; using GWP as an index to reflect equivalentclimate effects and damage costs from emissions is questionable.

Coupled damage and plasticity modelling in transient dynamic analysis of concrete

AbstractIn a concrete structure subjected to an explosion, for example a concrete slab, the material is subjected to various states of stress which lead to many modes of rupture. Closer to the explosive, a state of strong hydrostatic compression is observed. This state of stress produces an irreversible compaction of the material. Away from the zone of explosion, confinement decreases and the material undergoes compression with a state of stress, which is slightly triaxial. Finally, the compression wave can be reflected on a free surface and becomes a tensile wave, which by interaction with the compression wave, produces scabbing. We present, in this paper, a model aimed at describing these three failure modes. It is based on visco‐plasticity and rate dependent damage in which a homogenization method is used in order to include the variation of the material porosity due to compaction. The model predictions are compared with several experiments performed on the same concrete. Computations of split Hopkinson tests on confined concrete, a tensile test with scabbing, and an explosion on a concrete slab are presented. Copyright © 2001 John Wiley & Sons, Ltd.

Rate-dependent transition from tensile damage to discrete fracture in dynamic brittle failure

Based on the work for a combined damage/plasticity model of geologic materials and the bifurcation analysis of material failure, an analytical framework is established to study the rate-dependent transition from continuum damage to discrete fracture in dynamic brittle failure. Because of the simple formulation, a vectorized constitutive model solver can be designed for large-scale computer simulation. A continuum tangent stiffness tensor is invoked for the tensile damage evolution such that the bifurcation analysis can be performed to identify the initiation and orientation of tensile failure. It is shown that the orientation of tensile failure is rate-independent although the limit state is rate-dependent for the rate-dependent tensile damage model. Sample problems are considered to demonstrate the features of the proposed approach.

Simulating the evolution of localization based on the diffusion of damage

Based on the failure mechanisms of brittle solids under dynamic loading conditions, a strain-based damage diffusion model is proposed to simulate the evolution of localization due to microcracking. A three-dimensional diffusion equation is formulated with local rate-independent and rate-dependent damage evolution laws, respectively. The diffusion equation governing the evolution of damage is incorporated into the hyperbolic equation governing the wave propagation in a parallel setting. A parametric study is performed to investigate the effects of model parameters on the diffusion of damage. One- and two-dimensional sample problems are considered to illustrate how the dynamic evolution of localization can be simulated without using nonlocal models in the strain–stress space. The relationship between the proposed approach and the existing ones for localization problems is also discussed based on the preliminary results obtained in this paper. It appears that the proposed approach might be an effective numerical procedure to simulate the evolution of localization, with parallel computing, in a single computational domain involving different lower-order governing differential equations.

The calculation of dynamic JR-curves from the finite element analysis of a Charpy test using a rate-dependent damage model

A two-dimensional analysis of the Charpy V-notch specimen subjected to impact loading, according to the standard DIN EN 10045-1, is carried out, using a transient explicit dynamic finite element program. An elastic-viscoplastic, temperature dependent, constitutive relation for a porous plastic solid based on the Gurson damage model is developed. Ductile fracture of the matrix material will be described by the nucleation and subsequent growth of voids to coalescence. An updated Lagrange–Jaumann formulation is employed accounting for large strain and rotation. The discretization is based on four-node plane strain solid elements with one Gauss point. The equations of motion are integrated numerically by an explicit integration algorithm utilising a lumped mass matrix. The predictions of the numerical analysis in terms of force deflection response, crack resistance behaviour and deformation energy absorbtion are compared with results from Charpy tests which were carried out according to the low-blow technique.

Viscoelastic Continuum Damage Model of Asphalt Concrete with Healing

A viscoelastic constitutive model of asphalt mixtures that accounts for the rate-dependent damage growth and microdamage healing is presented in this paper. An elastic continuum damage model, which is based on thermodynamics of irreversible processes with internal state variables, is first reviewed and extended to a corresponding viscoelastic model using an elastic-viscoelastic correspondence principle. A rate-type internal state evolution law is employed to describe the damage growth and microdamage healing in asphalt concrete. An analytical representation of the model is established for the uniaxial loading condition. Tensile uniaxial cyclic tests were performed under the controlled-strain mode with different strain amplitudes to determine model parameters. The resulting constitutive model successfully predicts the damage growth and recovery in asphalt concrete under multilevels of loading, varying rates of loading, different modes of loading (controlled-strain and controlled-stress), and random rest periods.

Closure to “Rate Dependent Damage Model for Concrete in Dynamics” by Jean‐François Dubé, Gilles Pijaudier‐Cabot, and Christian La Borderie

Closure to “Rate Dependent Damage Model for Concrete in Dynamics” by Jean‐François Dubé, Gilles Pijaudier‐Cabot, and Christian La Borderie

Discussion: Rate Dependent Damage Model for Concrete in Dynamics

Discussion: Rate Dependent Damage Model for Concrete in Dynamics