Homogenized Gurson-type behavior equations for strain rate sensitive materials

In this paper, the classical Gurson model for ductile porous media is extended for strain-rate-dependent materials. Based on micromechanical considerations, approximate closed-form macroscopic behavior equations are derived to describe the viscous response of a ductile metallic material. To this end, the analysis of the expansion of a long cylindrical void in an ideally plastic solid introduced by McClintock (J Appl Mech 35:363, 1968) is revisited. The classical Gurson yield locus has been modified to explicitly take into account the strain rate sensitivity parameter for strain rate power-law solids. Two macroscopic approaches are proposed in this work. Both models use the first term of a Taylor series expansion to approximate integrals to polynomial functions. The first proposed closed-form approach is analytically more tractable than the second one. The second approach is more accurate. In order to compare the proposed approximate Gurson-type macroscopic functions with the behavior of the original Gurson yield locus, numerical finite element analyses for cylindrical cells have been conducted for a wide range of porosities, triaxialities, and strain rate sensitivity parameters. The results presented evidence that, for large values of the rate sensitivity parameters, the proposed extended Gurson-type models have the important quality to better predict the behavior of rate sensitive materials than the classical one. They also provide simpler and accurate alternatives to more traditional viscoplastic models.The authors are indebted to the Spanish Ministry of Economy and Competitiveness (Projects EUIN2015-62556 and DPI2014-57989-P) for the financial support received which allowed conducting part of this work. The research leading to these results has received funding from the European Union’s Horizon2020 Programme (Excellent Science, Marie-Sklodowska-Curie Actions) under REA Grant Agreement 675602 (Project OUTCOME)

An analysis of Gurson model with parameters dependent on triaxiality based on unitary cells

The Gurson model is widely used in the continuum-mechanics framework to analyse the ductile fracture process promoted by the nucleation, growth, and coalescence of voids. Further works improved the original Gurson model by introducing two parameters, q1 and q2, to adjust model predictions to the numerical results of a periodic array of cylindrical and spherical voids in hardening materials. This modified model is known as the Gurson–Tvergaard (GT) model. Commonly, these parameters are considered constants or dependent only on the material-hardening properties. However, there is evidence that these parameters also depend on the triaxiality of the stress field, as well as on initial porosity. In this work, a consistent fully implicit integration of the constitutive equations of the GT model, considering the q-parameters dependent on the triaxiality and the initial porosity of the stress field, is presented, and the corresponding consistent tangent operator is proposed. The model is validated by comparing the stress–strain behaviour, as well as the evolution of void volume fraction, of a voided cell and the equivalent cell of GT material with dependent parameters. The cases considered correspond to variable triaxiality stress fields, present in non-proportional loading conditions.This research was performed with the financial support of the Spanish Ministry of Education under Project reference DPI2005- 06769, and of the Region of Madrid, under Project reference CCG06-UC3M/DPI-0796.Publicad

Modelos de fractura dúctil en condiciones estáticas y dinámicas

En esta tesis se han abordado algunos problemas de simulación numérica del comportamiento de materiales cuyos micromecanismos de fractura están regidos por la nucleación, crecimiento y coalescencia de microvacíos. Este tipo de fenómenos aparecen en muchas aplicaciones de ingeniería como son los procesos de fabricación por conformado y corte, el análisis del comportamiento frente a choque de estructuras de vehículos ligeros (automóviles, helicópteros), la predicción de la propagación de fisuras en paneles de pequeñoo espesor, típicos en la industria aeronáutica, el diseño de protecciones contra impacto balístico, entre otros. En particular, se ha propuesto un algoritmo consistente para integrar las ecuaciones constitutivas de materiales de Gurson aplicable a problemas termoviscoplásticos, teniendo en cuenta la influencia de la velocidad de deformación y la temperatura en el comportamiento. También se ha formulado una variante del modelo de Gurson considerando que algunos parámetros del mismo no son constantes, sino dependientes de la triaxialidad del campo tensional. Se ha desarrollado el correspondiente algoritmo de integración de las ecuaciones de este modelo modificado y, en este caso, se ha aplicado a problemas estáticos. Finalmente, se ha aplicado el modelo de Gurson para analizar la influencia de la porosidad inicial en las inestabilidades por cavitación en metales dúctiles. Los algoritmos desarrollados se han implementado en códigos comerciales de elementos finitos, se ha comprobado su funcionamiento y se han validado con resultados experimentales ___________________________________________In this thesis, some problems about numerical simulation of the mechanical behaviour of materials whose fracture mechanisms are related to the nucleation, growth and coalescence of voids have been analized. This kind of analysis must be performed in many engineering applications like metal forming and cutting, light vehicles structures (automobiles, helicopters) under crashing, crack growth in thin panels, typical in the aircraft industry, high-speed impact on metallic armours and others. In particular, a consistent integration algorithm of Gurson's constitutive equations considering strain rate and thermal effects has been developed. Also, a modified GTN model that considers that some parameters of the model are no constant but dependent of stress state has been formulated. Finally, the influence of the porosity on cavitation instabilities in metallic materials has been analyzed. In previous work, this kind of instabilities has been predicted considering only one void contained in an unbounded solid. The developed algorithms have been implemented in a Finite Element commercial code and they have been validated with experimental result

A closed-form yield criterion for porous materials with Mises-Schleicher-Burzynski matrix containing cylindrical voids

This work develops a closed-form yield criterion applicable to porous materials with pressure-dependent matrix presenting tension-compression asymmetry (Mises-Schleicher-Burzyski material) containing parallel cylindrical voids. To develop the strength criterion, the stress-based variational homogenization approach due to Cheng et al. (Int J Plast 55:133-151, 2014) is extended to the case of a hollow cylinder under generalized plane strain conditions subjected to axisymmetric loading. Adopting a strictly statically admissible trial stress field, the homogenization procedure results in an approximate yield locus depending on the current material porosity, tension-compression material asymmetry, the mean lateral stress, and an equivalent shear stress. The analytical criterion provides exact solutions for purely hydrostatic loading. Theoretical results are compared with finite element (FE) simulations considering cylindrical unit-cells with distinct porosity levels, different values of the tension-compression asymmetry, and a wide range of stress triaxialities. Based on comparisons, the theoretical results are found to be in good agreement with FE simulations for most of the loading conditions and material features considered in this study. More accurate theoretical predictions are provided when higher material porosities and/or lower tension-compression asymmetries are considered. Overall, the main outcome of this work is a closed-form yield function proving fairly accurate predictions to engineering applications, in which pressure-dependent and tension-compression asymmetric porous materials with cylindrical voids are dealt with. This can be the case of honeycomb structures or additively manufactured materials, in which metal matrix composites are employed

Some applications of Burzynski yield condition in metal plasticity

The classical J2 plasticity theory is widely used to describe the plastic response of metallic materials. However, this theory does not provide satisfactory predictions for materials which exhibit pressure sen sitive yielding or plastic dilatancy. Another difficulty is the difference between the values of yield stresses in tension and compression for isotropic materials, the so called strength differential effect (SD), leading to the asymmetry of the elastic range. The Burzyn´ ski yield condition, proposed in 1928, can be used to overcome some of these problems. In this paper an implicit integration of the elasto plastic constitutive equations for the paraboloid case of Burzyn´ ski’s yield condition is formulated. Also, the tangent operator consistent with the integration algorithm was developed and is presented. The proposed model was implemented in a commercial Finite Element code and different kinds of tests reported in the literature were simulated. The comparison between the numerical and experimental results shows that the plastic ity theory with the paraboloid case of Burzyn´ ski’s yield condition describes adequately the strength dif ferential effect, which is present in many kinds of materials significant for recent applications.The authors gratefully acknowledge the financial support given by the Spanish Ministerio de Educación y Ciencia, Project Reference DPI/2008 06408.Publicad

Consistent integration of the constitutive equations of Gurson materials under adiabatic conditions

16 pages, 12 figures.-- MSC2000 codes: 74S20, 74C10, 74A15.-- Zbl#: Zbl 1159.74447In this paper, a consistent integration procedure for the thermoviscoplastic version of the complete Gurson model is proposed. With adiabatic conditions considered and with the use of the backward Euler integration scheme, a numerical algorithm implicit in all variables as well as the corresponding algorithmic operator have been developed. The proposed algorithm was implemented in a finite element code and its performance is demonstrated with the numerical simulation of different examples.This research was done with the financial support of the Spanish Ministry of Education under Project Reference DPI2005-06769, and of the Region of Madrid, under Project Reference UC3M-IME-05-054

Influence of crystallographic orientation on the void growth at the grain boundaries in bi-crystals

Void growth and morphology evolution in fcc bi-crystals are investigated using crystal plasticity finite element method. For that purpose, representative volume element of bi-crystals with a void at the grain boundary are considered in the analysis. Grain boundary is assumed initially perpendicular/coaxial with the straight sides of the cell. Fully periodic boundary conditions are prescribed in the representative volume element and macroscopic stress triaxiality and Lode parameter are kept constant during the whole deformation process. Three different pairs of crystal orientations characterized as hard-hard, soft-soft and soft-hard have been employed for modelling the mechanical response of the bi-crystal. Simulations are performed to study the implications of triaxiality, Lode parameter and crystallographic orientation on slip mechanism, hardening and hence void evolution. The impact of void presence and its growth on the heterogeneity of lattice rotation and resulting grain fragmentation in neighbouring areas is also analysed and discussed.Manjunath Dakshinamurthy and Guadalupe Vadillo kindly acknoledge the funding obtained from the European Union’s Horizon 2020 Program (Excellent Science, Marie-Sklodowska-Curie Actions) under REA grant agreement 675602 (Project OUTCOME). The research of Katarzyna Kowalczyk-Gajewska was partially supported by the project No. 2016/23/B/ST8/03418 of the National Science Centre, Poland. We are thankful to Dr. Jose A. Rodríguez-Martínez of University Carlos III, Madrid, for numerous fruitful discussions

A modified Gurson model to account for the influence of the Lode parameter at high triaxialities

The influence of the Lode parameter on ductile failure has been pointed out by different authors even at high triaxiality stress states. However, one of the most widely used model for ductile damage, like the Gurson-Tvergaard (GT) model, systematically disregard the role played by the third stress invariant. In this paper, an improvement of the classical Gurson-Tvergaard model is proposed. The new relation takes into account the effect of triaxiality and Lode parameter through the q(1) and q(2) GT parameters. The convexity of the proposed yield surface has been examined and ensured. The integration of the new constitutive equations as well as the consistent tangent modulus have been formulated and implemented in a Finite Element code. A computational 3D cell has been used to prescribe both macroscopic triaxiality and Lode parameter during loading. Numerical simulations are presented for Weldox 960 steel with different initial porosities and for different prescribed macroscopic triaxialities and Lode parameters using a computational 3D cell methodology. The results are compared with those obtained with a J(2) voided cell. These comparisons show that the improved model captures adequately the Lode effect on the stress-strain curves and on the void growth.The authors are indebted to the Ministerio de Economía y Competitividad (Projects DPI2011-23191 and DPI/2011-24068) for the financial support received which allowed condicting part of this work.Publicad

On the interplay between strain rate and strain rate sensitivity on flow localization in the dynamic expansion of ductile rings

In this work a stability analysis on flow localization in the dynamic expansion of ductile rings is con ducted. Within a 1 D theoretical framework, the boundary value problem of a radially expanding thin ring is posed. Based on a previous work, the equations governing the stretching process of the expanding ring are derived and solved using a linear perturbation method. Then, three different perfectly plastic material constitutive behaviours are analysed: the rate independent material, the rate dependent mate rial showing constant logarithmic rate sensitivity and the rate dependent material showing non constant and non monotonic logarithmic rate sensitivity. The latter allows to investigate the interaction between inertia and strain rate sensitivity on necking formation. The main feature of this work is rationally dem onstrate that under certain loading conditions and material behaviours: (1) decreasing rate sensitivity may not lead to more unstable material, (2) increasing loading rate may not lead to more stable material. This finding reveals that the relation between rate sensitivity and loading rate controls the unstable flow growth. Additionally a finite element model of the ring expansion problem is built in ABAQUS/Explicit. The stability analysis properly reflects the results obtained from the numerical simulations. Both proce dures, perturbation analysis and numerical simulations, allow for emphasizing the interplay between rate sensitivity and inertia on strain localization.The financial support of the Comunidad Autónoma de Madrid (Project CCG10 UC3M/DPI 5596) and of the Ministerio de Ciencia e Innovación de España (Project DPI/2008 06408) is kindly acknowledged.Publicad

First-Order Solutions for the Buckling Loads of Euler-Bernoulli Weakened Columns

In this work, closed-form expressions for the buckling loads of a weakened column with different boundary conditions are presented. The cracked-column model is based on the well-known method consisting of dividing the column into two segments connected by a rotational linear spring whose flexibility is related to the crack size and the geometry of the cross section. For the formulation of closed-form expressions, the perturbation method is used and the results are compared with those found by directly solving the eigenvalue problem.Publicad