Nonlocal damage plasticity

Based on the theory of Damage Mechanics, the damage of material is considered as an internal state variable. It describes the deterioration of material upon loading. If it is locally defined, then, the damage of a material point is completely determined by the state of its own. It is also known as local material damage model, a popular way in current modeling of material damage. On the other hand, the nonlocal damage model provides an alternative way to define the material damage of a point through certain domain on where it lays. This approach yields a more accurate stress description than that of the local model in describing high stress/damage gradient structures, such as an observed crack in a structure.

To describe the damage characteristics in high stress/damage gradient structures, a nonlocal damage model is developed in this work. The nonlocal damage is treated as weighting average over a prescribed domain in terms of explicitly expressed damage gradient. The proposed damage model together with the constitutive model of aluminum alloy are implemented in a commercial finite element code ABAQUS (Version 6.5) via its UMAT subroutine. Full implicit integration scheme is used to solve for internal variables of integration point (IP) at each time interval. Then the weighting average is carried out over a particular domain characterized by Characteristic Length for the processing IP. Finally, consistent tangent modulus (Jacobian) of this IP is provided to ABAQUS to assemble global stiffness matrix. A center- cracked specimen made of AL2024-T3 is chosen as a typical example to illustrate the advantage of introducing nonlocal damage and also to provide a validation of the proposed constitutive model. In addition, the effects of Characteristic Length and strain-hardening coefficient on the proposed nonlocal model are investigated and discussed.

One major consideration on the development of the nonlocal damage model is the stress/damage gradient observed in a loaded structure. As illustrated in chapter 2, there is no difference in the computed stress/damage distribution based on the nonlocal damage model and corresponding local ones in a homogeneous structure where no gradient exists when loaded. To examine the effect of stress/damage gradient on structures, notches of varying radii are designed to examine notch sensitivity on the nonlocal model.

Unlike a typical strain hardening material of AL2024-T3, 63Sn37Pb exhibits strain-softening behavior. The proposed nonlocal damage model is coupled into the constitutive model of 63Sn37Pb to show its applicability on strain softening materials. Again, the nonlocal damage model together with the constitutive model of 63Sn37Pb is incorporated into ABAQUS via its UMAT subroutine. Similar to the strain hardening material AL2024-T3, the mesh-sensitivity is remarkably reduced, demonstrating that the proposed nonlocal damage model is capable of being applied to both strain hardening materials and strain softening ones.

This article presents development of a damage-coupled viscoplastic constitutive model with temperature consideration. The model is subjected to an explicit nonlocal treatment within the characteristic length that is not limited to one local element as conventional ones. Temperature dependence of material behavior is incorporated into the model to account for the material property degradation at elevated temperature. A test program to determine the correlation between material parameters and temperature is also presented. The nonlocal damage-coupled viscoplastic material model is implemented in a commercial finite element program ABAQUS through its user-defined material subroutine UMAT using a semi-implicit time integration scheme. The model is applied to predict the behavior of 63Sn37Pb soldered structure. The mesh sensitivity of the model is discussed, as well as the efficiency of deduced consistent tangent modulus.

Dual-phase (DP) steel sheets have high potential for utilization as automotive structures due to their good combination of strength and ductility. As sheet metal forming processes induce complicated stress-strain states, determination of forming limit is vital, particularly using numerical approaches. This current study aims to examine the fracture behavior of DP600 steel sheets through several ductile fracture criteria in a wide range of stress states. For a better and more accurate understanding of the experimental tests, parallel numerical simulations were performed. First, the models were calibrated using the results of Nakazima tests, and then the fracture loci in principal strains, and equivalent strain-stress triaxiality spaces were predicted by each model. The capability of the criteria was verified through cross-die and bulge tests. Also, errors were quantified for the calculated results using correlation coefficient and relative error methods. The results reveal that Maximum Shear Stress, Modified Mohr Coulomb, and Lou fracture models were able to predict the onset of fracture with acceptable accuracy. However, Maximum Shear Stress required only one experimental test to be calibrated.

In this work, nonlocal nonlinear finite element analysis of laminated composite plates using Reddy's third-order shear deformation theory (TSDT) [1] and Eringen's nonlocality [2] is presented. The governing equations of third order shear deformation theory with the von Kármán strains are derived employing the Eringen's [2] stress-gradient constitu-tive model. The principle of virtual displacement is used to derive the weak forms, and the displacement finite element models are developed using the weak forms. Four-noded rectangular conforming element with 8 degrees of freedom per node has been used. The coefficients of stiffness matrix and tangent stiffness matrix are presented along with non-local force vector. The developed finite element model can be employed to capture the small scale deviations from local continuum models caused by material inhomogeneity and the inter atomic and inter molecular forces. Numerical examples are presented to illustrate the effects of nonlocality, anisotropy, and the von Kármán type nonlinearity on the bending behaviour of laminated composite plates.

The development of a coupled damage-plasticity constitutive model for concrete is presented. Emphasis is put on thermodynamic admissibility, rigour and consistency both in the formulation of the model, and in the identification of model parameters based on experimental tests. The key feature of the thermodynamic framework used in this study is that all behaviour of the model can be derived from two specified energy potentials, following procedures established beforehand. Based on this framework, a constitutive model featuring full coupling between damage and plasticity in both tension and compression is developed. Tensile and compressive responses of the material are captured using two separate damage criteria, and a yield criterion with a multiple hardening rule. A crucial part of this study is the identification of model parameters, with these all being shown to be identifiable and computable based on standard tests on concrete. Behaviour of the model is assessed against experimental data on concrete.

We develop a fully implicit stress-return algorithm for a wide range of nonlocal inelastic models (of integral type) with coupling among integration points. Two examples of nonlocal models are presented, which combine plasticity with either damage or breakage. Using these models, we emphasize the need for a versatile integration scheme for nonlocal incremental constitutive equations. Furthermore, these models are used to demonstrate the importance of adequately selecting the way constitutive equations for softening materials are regularized. The stress update process requires the construction of a nonlocal constitutive matrix: we highlight and take advantage of its sparsity. Numerical solutions to relevant structural and geomechanical boundary value problems are used to demonstrate the performance of the proposed approach.

Non-local regularization is applied to a new coupled damage–plasticity model (Int. J. Numer. Anal. Meth. Geomech. 2007; DOI: 10.1002/nag.627), turning it into a non-local model. This procedure resolves softening-related problems encountered in local constitutive models when dealing with softening materials. The parameter identification of the new non-local coupled damage–plasticity model is addressed, with all parameters being shown to be obtainable from the experimental data on concrete. Because of the appearance of non-local spatial integrals in the constitutive equations, a new implementation scheme is developed for the integration of the non-local incremental constitutive equations in nonlinear finite element analysis. The performance of the non-local model is assessed against a range of two-dimensional structural tests on concrete, illustrating the stability of the stress update procedure and the lack of mesh dependency of the model.

This paper presents a procedure for the determination of parameters of non-local damage models. This is to assure a consistent response of a non-local damage model, as choice of the internal length and other parameters of the model are varied. Correlations between the internal length and other parameters governing the local constitutive behaviour of the model are addressed and exploited. Focus is put on the relationship between the internal length of the non-local model and the width of the fracture process zone. Numerical examples are used to demonstrate the rigour of the proposed method.

This paper presents a new 1-D non-local damage-plasticity deformation model for ductile materials. It uses the thermodynamic framework described in and holds, nevertheless, some similarities with approach. A 1D finite element (FE) model of a bar fixed at one end and loaded in tension at the other end is introduced. This simple model demonstrates how the approach can be implemented within the finite element framework, and that it is capable of capturing both the pre-peak hardening and post-peak softening (generally responsible for models instability) due to damage-induced stiffness and strength reduction characteristic of ductile materials. It is also shown that the approach has further advantages of achieving some degree of mesh independence, and of being able to capture deformation size effects. Finally, it is illustrated how the model permits the calculation of essential work of rupture (EWR), i.e. the specific energy per unit cross-sectional area that is needed to cause tensile failure of a specimen.

This paper presents a new 1-D non-local damage-plasticity deformation model for ductile materials. It uses the thermodynamic framework described in Houlsby and Puzrin (2000) and holds, nevertheless, some similarities with Lemaitre’s (1971) approach. A 1D finite element (FE) model of a bar fixed at one end and loaded in tension at the other end is introduced. This simple model demonstrates how the approach can be implemented within the finite element framework, and that it is capable of capturing both the pre-peak hardening and post-peak softening (generally responsible for models instability) due to damage-induced stiffness and strength reduction characteristic of ductile materials. It is also shown that the approach has further advantages of achieving some degree of mesh independence, and of being able to capture deformation size effects. Finally, it is illustrated how the model permits the calculation of essential work of rupture (EWR), i.e. the specific energy per unit cross-sectional area that is needed to cause tensile failure of a specimen.

This paper focuses on the development of a thermodynamic approach to constitutive modelling of concrete materials, with emphasis on the use of non-local damage models. Effort is put on the construction of a consistent and rigorous thermodynamic framework, which readily allows the incorporation of non-local features into the constitutive modelling. This is an important feature in developing non-local constitutive models based on thermodynamics. Examples of non-local constitutive models derived from this framework and numerical examples are given to demonstrate the promising features of the proposed approach

In this work, nonlocal nonlinear finite element analysis of laminated composite plates using Reddy's third-order shear deformation theory (TSDT) [1] and Eringen's nonlocality [2] is presented. The governing equations of third order shear deformation theory with the von Kármán strains are derived employing the Eringen's [2] stress-gradient constitutive model. The principle of virtual displacement is used to derive the weak forms, and the displacement finite element models are developed using the weak forms. Four-noded rectangular conforming element with 8 degrees of freedom per node has been used. The coefficients of stiffness matrix and tangent stiffness matrix are presented along with nonlocal force vector. The developed finite element model can be employed to capture the small scale deviations from local continuum models caused by material inhomogeneity and the inter atomic and inter molecular forces. Numerical examples are presented to illustrate the effects of nonlocality, anisotropy, and the von Kármán type nonlinearity on the bending behaviour of laminated composite plates.

Most constitutive models for quasi-brittle materials ignore the fact that the measurable fracture energy represents more than one dissipative mechanism, including those from the creation of new surface areas and frictions between surfaces. Specifically, these models “see” the measurable fracture energy as one that comes only from the creation of new surface area. However, experimental deduction by Bažant indicates that the frictional dissipation is not at all negligible, even in mode I failure, where it contributes an energy loss of around 50–75% to the total dissipation budget. Here we study the consequences of this simplification by adopting simple local and nonlocal damage models, which include a new powerful parameter that represents the measurable partition of the dissipation. It is found that the oversight of frictional dissipation can lead to a critical over-prediction of both the magnitude and the spreading of damage.

The focus is on isotropic elastodamaging (softening) materials, where the damage parameter is expressed as a function of the total strain. By integrating the mechanical work in the strain space along a stepwise holonomic loading history, an incremental strain energy is obtained. A coaxiality condition for the incremental strain energy to be potential is identified, and its implications on the associativity of the damage evolution are discussed. Under some hypotheses, the increment of the mechanical work is shown to be minimum along strain radial paths. These results are used to construct a multifield variational framework supporting finite element (nonlocal) formulations.

Most constitutive models for quasi-brittle materials ignore the fact that the measurable fracture energy represents more than one dissipative mechanism, including those from the creation of new surface areas and frictions between surfaces. Specifically, these models ''see'' the measurable fracture energy as one that comes only from the creation of new surface area. However, experimental deduction by Bažant indicates that the frictional dissipation is not at all negligible, even in mode I failure, where it contributes an energy loss of around 50-75% to the total dissipation budget. Here we study the consequences of this simplification by adopting simple local and nonlocal damage models, which include a new powerful parameter that represents the measurable partition of the dissipation. It is found that the oversight of frictional dissipation can lead to a critical over-prediction of both the magnitude and the spreading of damage.

We develop a fully implicit stress-return algorithm for a wide range of nonlocal inelastic models (of integral type) with coupling among integration points. Two examples of nonlocal models are presented, which combine plasticity with either damage or breakage. Using these models, we emphasize the need for a versatile integration scheme for nonlocal incremental constitutive equations. Furthermore, these models are used to demonstrate the importance of adequately selecting the way constitutive equations for softening materials are regularized. The stress update process requires the construction of a nonlocal constitutive matrix: we highlight and take advantage of its sparsity. Numerical solutions to relevant structural and geomechanical boundary value problems are used to demonstrate the performance of the proposed approach.

Size dependence of the load-displacement and stress-strain curves is a phenomenon widely reported in the literature and readily observed in experiments. The present paper introduces two developments in the analysis of ductile rupture, one in the area of experimental technique and interpretation, and one in the area of numerical modelling. Firstly, the new experimental technique is described of multiple gauge length optical extensometry (laser scanning) that provides a uniquely powerful means of collecting the data needed for the study of size effects, from a single tensile test on a ductile specimen. The interpretation procedures are presented based on the considerations about the nature of deformation and energy expenditure within different material volumes in a tensile specimen. Consequently, the property of the specimen can be defined and identified that relates to the energy consumption per unit cross-sectional area required to rupture the specimen entirely (the essential work of rupture). Another energetic measure is also obtained from the test that relates to the plastic work dissipation per unit volume in pre-peak deformed material. Secondly, the observed size effects and specific energy determination procedures are discussed in the context of non-local coupled plasticity-damage modelling. Conventional local numerical models of inelastic deformation fail to predict the sequence of deformation responses observed in the tensile test, namely, uniform deformation followed by strain localisation, necking and rupture. It is shown here that coupled non-local damage-plasticity modelling is capable of capturing the hardening-softening and global-local plasticity transitions, and hence the trends observed in ductile fracture. The predictions of the coupled non-local damage-plasticity modelling for different deformation regimes are shown to agree qualitatively with the experimentally observed behaviour. The two aspects of the proposed approach, experimental and modelling, are presented here together since they are intimately linked at the fundamental level. The implications of the proposed approach are discussed for the deformation analysis of plastically deforming and damageable materials and structures at various scales, from macroscopic to miniature.

This article presents the development of a generalized nonlocaldamage-coupled material model. The model introduces the concept of cumulativedamage gradient through a set of damage evolution equations within the irreversiblethermodynamics framework. The proposed material model is implemented in a commercial finite ele-ment code ABAQUS (Version 6.5) via its UMAT subroutine. The implementation of this model on ABAQUS is described with a focus on the nonlocal treatment together with the derivation of the consistent tangent modulus (Jacobian). As a numerical example, the nonlocal damage model is applied to center-cracked specimen made of aluminum alloy 2024-T3. Comparison is made between the computed results andexperimental ones. The validity of the proposed model is examined, and its effectiveness for engineering application is elucidated.