A thermodynamically consistent plastic-damage framework for localized failure in quasi-brittle solids: material model and strain localization analysis

Aiming for the modeling of localized failure in quasi-brittle solids, this paper addresses a thermodynamically consistent plastic-damage framework and the corresponding strain localization analysis. A unified elastoplastic damage model is first presented based on two alternative kinematic decompositions assuming infinitesimal deformations, with the evolution laws of involved internal variables characterized by a dissipative flow tensor. For the strong (or regularized) discontinuity to form in such inelastic quasi-brittle solids and to evolve eventually into a fully softened one, a novel strain localization analysis is then suggested. A kinematic constraint more demanding than the classical discontinuous bifurcation condition is derived by accounting for the traction continuity and the loading/unloading states consistent with the kinematics of a strong (or regularized) discontinuity. More specifically, the strain jumps characterized by Maxwell’s kinematic condition have to be completely inelastic (energy dissipative). Reproduction of this kinematics implies vanishing of the aforesaid dissipative flow tensorial components in the directions orthogonal to the discontinuity orientation. This property allows naturally developing a localized plastic-damage model for the discontinuity (band), with its orientation and the traction-based failure criterion consistently determined a posteriori from the given stress-based counterpart. The general results are then particularized to the 2D conditions of plane stress and plane strain. It is found that in the case of plane stress, strain localization into a strong (or regularized) discontinuity can occur at the onset of strain softening. Contrariwise, owing to an extra kinematic constraint, in the condition of plane strain some continuous inelastic deformations and substantial re-orientation of principal strain directions in general have to take place in the softening regime prior to strain localization. The classical Rankine, Mohr–Coulomb, von Mises (J2) and Drucker–Prager criteria are analyzed as illustrative examples. In particular, both the closed-form solutions for the discontinuity angles validated by numerical simulations and the corresponding traction-based failure criteria are obtained.Peer ReviewedPostprint (author's final draft

Effect of the tool tilt angle on the heat generation and the material flow in friction stir welding

This work studies the effect of the tool tilt angle on the generated heat and the material flow in the work pieces joint by Friction Stir Welding (FSW). An apropos kinematic framework together with a two-stage speed-up strategy is adopted to simulate the FSW problem. The effect of tilt angle on the FSWelds is modeled through the contact condition by modifying an enhanced friction model. A rotated friction shear stress is proposed, the angle of rotation depending on the process parameters and the tilt angle. The proposed rotation angle is calibrated by the experimental data provided for a tilt angle 2.5°. The differences of generated heat and material flow for the cases of tool with tilt angle of 0° and 2.5° are discussed. It is concluded that due to the higher temperature, softer material and greater frictional force in the trailing side of the tool, the material flow in the rear side of the FSW tool with the title angle is considerably enhanced, which assists to prevent the generation of defect.Peer ReviewedPostprint (published version

3D numerical modelling of twisting cracks under bending and torsion of skew notched beams

The testing of mode III and mixed mode failure is every so often encountered in the dedicated literature of mechanical characterization of brittle and quasi-brittle materials. In this work, the application of the mixed strain displacement e-ue-u finite element formulation to three examples involving skew notched beams is presented. The use of this FE technology is effective in problems involving localization of strains in softening materials. The objectives of the paper are: (i) to test the mixed formulation in mode III and mixed mode failure and (ii) to present an enhancement in terms of computational time given by the kinematic compatibility between irreducible displacement-based and the mixed strain-displacement elements. Three tests of skew-notched beams are presented: firstly, a three point bending test of a PolyMethyl MethaAcrylate beam; secondly, a torsion test of a plain concrete prismatic beam with square base; finally, a torsion test of a cylindrical beam made of plain concrete as well. To describe the mechanical behavior of the material in the inelastic range, Rankine and Drucker-Prager failure criteria are used in both plasticity and isotropic continuum damage formats. The proposed mixed formulation is capable of yielding results close to the experimental ones in terms of fracture surface, peak load and global loss of carrying capability. In addition, the symmetric secant formulation and the compatibility condition between the standard irreducible method and the strain-displacement one is exploited, resulting in a significant speedup of the computational procedure.Peer ReviewedPostprint (author's final draft

Explicit mixed strain–displacement finite elements for compressible and quasi-incompressible elasticity and plasticity

The final publication is available at Springer via http://dx.doi.org/ 10.1007/s00466-016-1305-zThis paper presents an explicit mixed finite element formulation to address compressible and quasi-incompressible problems in elasticity and plasticity. This implies that the numerical solution only involves diagonal systems of equations. The formulation uses independent and equal interpolation of displacements and strains, stabilized by variational subscales. A displacement sub-scale is introduced in order to stabilize the mean-stress field. Compared to the standard irreducible formulation, the proposed mixed formulation yields improved strain and stress fields. The paper investigates the effect of this enhancement on the accuracy in problems involving strain softening and localization leading to failure, using low order finite elements with linear continuous strain and displacement fields (P1P1 triangles in 2D and tetrahedra in 3D) in conjunction with associative frictional Mohr–Coulomb and Drucker–Prager plastic models. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to analytical solutions for plane stress and plane strain situations. Benchmark numerical examples show the capacity of the mixed formulation to predict correctly failure mechanisms with localized patterns of strain, virtually free from any dependence of the mesh directional bias. No auxiliary crack tracking technique is necessary.Peer ReviewedPostprint (author's final draft

Stabilized mixed explicit finite element formulation for compressible and nearly-incompressible solids

El presente estudio presenta una formulación mixta de elementos finitos capaz de abordar problemas quasiincompresibles en forma explícita. Esta formulación se aplica a elementos con interpolaciones independientes e iguales de desplazamientos y deformaciones, estabilizadas mediante subescalas variacionales (VMS). Como continuación del estudio presentado en la referencia [23] , en la que se introdujo la subescala de las deformaciones, en este trabajo se incluyen los efectos de la sub-escala de los desplazamientos, con el fin de estabilizar el campo de las presiones. La formulación evita la condición de Ladyzhenskaya-Babuska-Brezzi (LBB) y sólo requiere la resolución de un sistema diagonal de ecuaciones. En este artículo se tratan también los principales aspectos de implementación. Finalmente, ejemplos de validación numérica muestran el comportamiento de estos elementos en comparación con la formulación irreducible.This study presents a mixed finite element formulation able to address nearly-incompressible problems explicitly. This formulation is applied to elements with independent and equal interpolation of displacements and strains, stabilized by variational subscales (VMS). As a continuation of the study presented in reference [23], in which the strains sub-scale was introduced, in this work the effects of sub-scale displacements are included, in order to stabilize the pressure field. The formulation avoids the Ladyzhenskaya-Babuska-Brezzi (LBB) condition and only requires the solution of a diagonal system of equations. The main aspects of implementation are also discussed. Finally, numerical examples validate the behaviour of these elements compared with the irreductible formulation.Peer ReviewedPostprint (published version

Strain localization of elastic-damaging frictional-cohesive materials: analytical results and numerical verification

Damage-induced strain softening is of vital importance for the modeling of localized failure in frictional-cohesive materials. This paper addresses strain localization of damaging solids and the resulting consistent frictional-cohesive crack models. As a supplement to the framework recently established for stress-based continuum material models in rate form (Wu and Cervera 2015, 2016), several classical strain-based damage models, expressed usually in total and secant format, are considered. Upon strain localization of such damaging solids, Maxwell's kinematics of a strong (or regularized) discontinuity has to be reproduced by the inelastic damage strains, which are defined by a bounded characteristic tensor and an unbounded scalar related to the damage variable. This kinematic constraint yields a set of nonlinear equations from which the discontinuity orientation and damage-type localized cohesive relations can be derived. It is found that for thePeer ReviewedPostprint (published version

Simulación numérica del comportamiento termo-mecánico de presas de HCR Parte II: Aplicación a la Presa de Rialb

Se presenta en este trabajo un procedimiento numérico para la simulación del proceso de construcción de presas de Hormigón Compactado con Rodillo (HCR). El método se aplica a la presa de Rialb, recientemente contruida en España utilizando HCR con fuerte adición de cenizas volantes. Se obtienen como resultado la distribución y evolución de la temperatura en el interior de la presa durante y después de su construcción, así como la evolución de las resistencias a compresión y tracción y del módulo elástico. La metodología permite determinar el campo de tensiones en la presa en cualquier momento durante y después de su construcción. Para ello, se tienen en cuenta tanto los efectos mecánicos a largo plazo, incorporando un modelo de fluencia que tiene en cuenta el envejecimiento del hormigón, como el riesgo de fisuración a tracción, incorporando un modelo de degradación adecuado.Peer Reviewe

A novel positive/negative projection in energy norm for the damage modeling of quasi-brittle solids

The asymmetric tensile/compressive material behavior and microcracks closure-reopening (MCR) effects exhibited by quasi-brittle solids are of significant importance to the nonlinear responses of engineering structures under cyclic loading, e.g., earthquake excitations. Based on our previous work (Cervera et al., 1995; Faria et al., 1998; Wu et al., 2006) this work addresses a novel thermodynamically consistent unilateral damage model for concrete. In particular, the positive/negative projection (PNP) of the effective stress tensor and the additive bi-scalar damage constitutive relation are maintained owing to the conceptual simplicity and computational efficiency. It is found that the classical PNP widely adopted in the literature is not optimal for this damage model, since the resulting stiffness is not always of major symmetry. Consequently, a well-defined free energy potential does not exist in general cases and the model cannot be cast into the framework of thermodynamics with internal variables. Furthermore, the damage induced anisotropy cannot be captured, exhibiting excessive lateral deformations under uniaxial tension. To overcome the above issues, a novel PNP, variationally interpreted as the closest point projection of the effective stress in energy norm, is proposed with closed-form solution. With the novel PNP, the secant stiffness tensor of the proposed unilateral damage model always possesses major symmetry and exhibits orthotropic behavior under uniaxial tension and mixed tension/compression. The corresponding thermodynamics framework is then given, resulting in an energy release rate based rounded-Rankine type damage criterion appropriate for tensile failure in quasi-brittle solids. Several numerical examples of single-point verifications and benchmark tests are presented. It is demonstrated that the proposed model is capable of characterizing localized failure of concrete under proportional and non-proportional static loading, as well as the MCR effects under seismic cyclic loading.Peer ReviewedPostprint (author's final draft

Numerical modelling of microstructure evolution in friction stir welding (FSW)

This work studies the metallurgical and microstructural aspects of Friction Stir Welding (FSW) in terms of grain size and microhardness. The modelling is based on the combination of an apropos kinematic framework for the local simulation of FSW processes and a material particle tracing technique for tracking the material flow during the weld. The resulting grain size and microhardness values are validated with experimental observations from an identical processed sample. A Sheppard-Wright constitutive relation is adopted to describe the mechanical behavior of AZ31 Mg alloy considered in this work. The strain rate and temperature histories obtained from the numerical model are stored on the tracers. The relationship among the grain size, microhardness, strain rate, and temperature is obtained using Zener-Hollomon parameter and Hall-Petch relationship. A linear description relates the logarithm of average grain size to the logarithm of Zener-Hollomon parameter. The relationship between microhardness and average grain size stands away from the linear trend.Peer ReviewedPostprint (published version

High-fidelity prediction of crack formation in 2D and 3D pullout tests

This paper presents the 2D and 3D numerical analysis of pullout tests on steel anchorages in concrete blocks using standard and mixed finite elements. A novel (stabilized) mixed formulation in the variables of total strain 8 and displacements u is introduced to overcome the intrinsic deficiencies of the standard displacement-based one in the context of localization of strains, such as mesh dependency. The quasi brittle behavior of concrete is described through an elastoplastic constitutive law with a local Rankine yielding criterion. The proposed formulation is shown to be a reliable and accurate tool, sensitive to the physical parameters of the pullout tests, but objective with respect to the adopted FE mesh. Furthermore, the mixed epsilon/u finite element is able to capture the correct failure mechanism with relatively coarse discretizations. At the same time, the spurious behavior of the standard formulation is not alleviated by mesh-refinement.Peer ReviewedPostprint (author's final draft