Rashid K. Abu Al-Rub

On the coupling of anisotropic damage and plasticity models for ductile materials

In this contribution various aspects of an anisotropic damage model coupled to plasticity are considered. The model is formulated within the thermodynamic framework and implements a strong coupling between plasticity and damage. 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. The damaged material is modeled using the constitutive laws of the effective undamaged material in which the nominal stresses are replaced by the effective stresses. The model considers different interaction mechanisms between damage and plasticity defects in such a way that two-isotropic and two-kinematic hardening evolution equations are derived, one of each for the plasticity and the other for the damage. An additive decomposition of the total strain into elastic and inelastic parts is adopted in this work. The elastic part is further decomposed into two portions, one is due to the elastic distortion of the material grains and the other is due to the crack closure and void contraction. The inelastic part is also decomposed into two portions, one is due to nucleation and propagation of dislocations and the other is due to the lack of crack closure and void contraction. Uniaxial tension tests with unloadings have been used to investigate the damage growth in high strength steel. A good agreement between the experimental results and the model is obtained.

Carbon Nanotubes and Carbon Nanofibers for Enhancing the Mechanical Properties of Nanocomposite Cementitious Materials

Carbon nanotubes (CNTs) and carbon nanofibers (CNFs) are quickly becoming two of the most promising nanomaterials because of their unique mechanical properties. The size and aspect ratio of CNFs and CNTs mean that they can be distributed on a much finer scale than commonly used microreinforcing fibers. As a result, microcracks are interrupted much more quickly during propagation in a nano- reinforced matrix, producing much smaller crack widths at the point of first contact between the moving crack front and the reinforcement. In this study, untreated CNTs and CNFs are added to cement matrix composites in concentrations of 0.1 and 0.2% by weight of cement. The nanofilaments are dispersed by using an ultrasonic mixer and then cast into molds. Each specimen is tested in a custom-made three-point flexural test fixture to record its mechanical properties; namely, the Youngs modulus, flexural strength, ultimate strain capacity, and fracture toughness, at 7, 14, and 28 days. A scanning electron microscope (SEM) is used to discern the difference between crack bridging and fiber pullout. Test results show that the strength, ductility, and fracture toughness can be improved with the addition of low concentrations of either CNTs or CNFs. DOI: 10.1061/(ASCE)MT.1943-5533.0000266.

A Finite Strain Plastic-damage Model for High Velocity Impacts using Combined Viscosity and Gradient Localization Limiters: Part II - Numerical Aspects and Simulations

In this companion article, we present within the finite element context the numerical algorithms for the integration of the thermodynamically consistent formulation of geometrically nonlinear gradient-enhanced viscoinelasticity derived in the first part of the article. The proposed unified integration algorithms are extensions of the classical rate-independent return mapping algorithms to the rate-dependent problems. An operator split structure is used consisting of a trial state followed by the return map by imposing the generalized viscoplastic and visco-damage consistency conditions simultaneously. Furthermore, a trivially incrementally objective integration scheme is established for the rate constitutive relations. The proposed finite deformation scheme is based on hypoelastic stress-strain representations and the proposed elastic predictor and coupled viscoplastic-viscodamage corrector algorithm allows for the total uncoupling of geometrical and material nonlinearities. A simple and direct computational algorithm is also used for calculation of the higher-order gradients. This algorithm can be implemented in the existing finite element codes without numerous modifications as compared to the current numerical approaches for integrating gradient-dependent models. The nonlinear algebraic system of equations is solved by consistent linearization and the Newton-Raphson iteration. The proposed model is implemented in the explicit finite element code ABAQUS via the user subroutine VUMAT. Model capabilities are preliminarily illustrated for the dynamic localization of inelastic flow in adiabatic shear bands and the perforation of a 12 mm thick Weldox 460E steel plates by deformable blunt projectiles at various impact speeds. The simulated shear band results well illustrated the potential of the proposed model in dealing with the well-known mesh sensitivity problem. Consequently, the introduced implicit and explicit length-scale measures are able to predict size effects in localization failures. Moreover, good agreement is obtained between the numerical simulations and experimental results of the perforation problem.

Mechanical Properties of Nanocomposite Cement Incorporating Surface-Treated and Untreated Carbon Nanotubes and Carbon Nanofibers

To study the effects of functionalized carbon nanotubes (CNTs) and carbon nanofibers (CNFs) on the mechanical properties of cement composites, both untreated and treated CNFs and CNTs were added to cement paste in concentrations of 0.1% and 0.2% by weight of cement. The surface-treated nanofilaments were functionalized in a solution of sulfuric acid (H2SO4) and nitric acid (HNO3). The nano- filaments were dispersed by using an ultrasonic mixer and were then cast into molds. Each specimen was tested in a custom-made three-point flexural test fixture to record the mechanical properties (i.e., the Youngs modulus, flexural strength, ductility, and modulus of toughness) at the age of 7, 14, and 28 days. The microstructure was analyzed by using a scanning electron microscope. Untreated CNTs and CNFs were found to enhance the mechanical properties of cementitious materials, whereas the acid-treated CNTs and CNFs degraded the mechanical properties. DOI: 10.1061/(ASCE)NM.2153-5477.0000041.

Distribution of Carbon Nanofibers and Nanotubes in Cementitious Composites

Carbon nanofibers (CNFs) and nanotubes (CNTs) are known to be extremely strong and stiff, and their potential as reinforcement has been of interest to many investigators in the past decade. One of the most important keys for fully harnessing the properties of any type of fiber is to control the distribution in the material matrix. As far as CNFs–CNTs are concerned, the strong attraction among nanoscale fibers due to van der Waals forces makes this task difficult. This study focuses on some of the problems that prevent a uniform distribution of CNFs–CNTs in cement paste and the methods used in the past to enhance dispersion. The first phase of the experimental program investigates the effect of using superplasticizers accompanied by sonication on the dispersion of CNFs in water and paste. The second phase focuses on the problem of cement grain size and limitations that the use of fine grain cement causes. Finally, on the basis of results and past studies, suggestions are made for achieving enhanced dispersion of CNFs–CNTs in cement paste.

Determination of the Material Intrinsic Length Scale of Gradient Plasticity Theory

The enhanced strain-gradient plasticity theories formulate a constitutive framework on the continuum level that is used to bridge the gap between the micromechanical plasticity and the classical continuum plasticity. To assess the size effects it is indispensable to incorporate an intrinsic material length parameter into the constitutive equations. However, the full utility of gradient-type theories hinges on one’s ability to determine the constitutive length-scale parameter. The classical continuum plasticity is unable to predict properly the evolution of the material flow stress since the local deformation gradients at a given material point are not accounted for. The gradient-based flow stress is commonly assumed to rely on a mixed type of dislocations: statistically stored dislocations (SSDs) and geometrically necessary dislocations (GNDs). In this work a micromechanical model to assess the coupling between SSDs and GNDs, which is based on the Taylor’s hardening law, is used to identify the deformation-gradient-related intrinsic length-scale parameter in terms of measurable microstructural physical parameters. This work also presents a method for identifying the length-scale parameter from micro-indentation tests.

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.

Thermodynamic based model for the evolution equation of the backstress in cyclic plasticity

Abstract A nonlinear kinematic hardening rule is developed here within the framework of thermodynamic principles. The derived kinematic hardening evolution equation has three distinct terms: two strain hardening terms and a dynamic recovery term that operates at all times. The proposed hardening rule, which is referred in this paper as the FAPC (Fredrick and Armstrong–Phillips–Chaboche) kinematic hardening rule, shows a combined form of the Frederick and Armstrong backstress evolution equation, Phillips evolution equation, and Chaboche series rule. A new term is incorporated into the Frederick and Armstrong evolution equation that appears to have agreement with the experimental observations that show the motion of the center of the yield surface in the stress space is directed between the gradient to the surface at the stress point and the stress rate direction at that point. The model is further modified in order to simulate nonproportional cyclic hardening by proposing a measure representing the degree of nonproportionality of loading. This measure represents the topology of the incremental stress path. Numerically, it represents the angle between the current stress increment and the previous stress increment, which is interpreted through the material constants of the kinematic hardening evolution equation. This new kinematic hardening rule is incorporated in a material constitutive model based on the von Mises plasticity type and the Chaboche isotropic hardening type. Numerical integration of the incremental elasto-plastic constitutive equations is based on a simple semi-implicit return-mapping algorithm and the full Newton–Raphson iterative method is used to solve the resulting nonlinear equations. Experimental simulations are conducted for proportional and non-proportional cyclic loadings. The model shows good correlation with the experimental results.

Three-Dimensional Simulations of Asphalt Pavement Permanent Deformation Using a Nonlinear Viscoelastic and Viscoplastic Model

The writers recently developed a nonlinear viscoelastic-viscoplastic constitutive model, in order to represent the response of asphalt mixtures under different temperatures and rates of loading. This model has been implemented in the finite-element (FE) code Abaqus via the user material subroutine UMAT, and it was verified through comparisons with experimental data of asphalt mixtures at various stress levels and temperatures. This research develops a three-dimensional FE model using Abaqus to represent a three-layer pavement structure and to simulate the viscoelastic and viscoplastic responses under repeated loading at different temperatures. The results demonstrate the capability of the model in simulating the influence of temperature on permanent deformation and in predicting viscoelastic and viscoplastic strain distributions in the asphalt layer. The simulations show that tensile viscoplastic strain accumulates at the pavement surface, a phenomenon that could be associated with cracking of asphalt pavements. In addition, the results show that at high pavement temperature (40°C), tensile viscoplastic strain develops at the sides of the applied load due to asphalt mixture heave associated with permanent deformation and dilation.

Gradient-enhanced Coupled Plasticity-anisotropic Damage Model for Concrete Fracture: Computational Aspects and Applications

It is widely studied that classical continuum damage theory for concrete fracture exhibits an extreme sensitivity to the spatial discretization in the finite element simulations. This sensitivity is caused by the fact that the mathematical description becomes ill-posed at a certain level of accumulated damage. A well-posed problem can be recovered by using a gradient-enhanced damage model in which a material length scale is introduced as a localization limiter. In this work, a nonlocal gradient-enhanced fully coupled plastic-damage constitutive model for plain concrete is developed. Anisotropic damage with a plasticity yield criterion and a damage criterion are introduced to be able to adequately describe the plastic and damage behavior of concrete. In order to account for different effects under tensile and compressive loadings, nonlocal damage variables that account for the progressive degradation of mechanical properties under stress states of prevailing tension and compression and two internal length scales, one for tension and the other for compression, are introduced as localization limiters. Therefore, two nonlocal damage criteria are used: one for compression and a second for tension such that the total stress is decomposed into tensile and compressive components. In order to solve the time step problem, a decoupled elastic predictor and plastic corrector steps are performed first in the effective configuration where damage is absent, and then a nonlocal damage corrector step is applied in order to update the final stress state. The algorithmic treatment of both tension and compression is presented in a unified way. A simple procedure to calculate the gradient of the tensile/compressive damage variables is described which can be used directly without the need of intensive numerical modifications of an existing finite element code. The effectiveness of the proposed local model has been demonstrated in both uniaxial and biaxial tension and compression problems and compared with experimental data. Numerical results obtained with the proposed nonlocal model are compared with experimental results concerning bending of three-point notched and four-point notched concrete beams. As the mesh is refined, convergence of numerical results is observed both in terms of damage patterns and of the global response.