Una ecuación de dureza unificada para ensayos de indentación esférica

El presente proyecto tiene su marco teórico en las técnicas de indentación instrumentada. Mediante este tipo de ensayos pueden relacionarse la dureza y las propiedades mecánicas fundamentales de un material, tales como el módulo de Young, el límite de fluencia o el coeficiente de endurecimiento por deformación. En el caso de indentaciones puntiagudas en las que se utilizan pirámides Vickers o Berkovich esta relación se puede establecer mediante el conocimiento de las curvas de carga aplicada (P)-profundidad de penetración (h). En estos casos no existe variación de dureza conforme la profundidad de penetración aumenta. Por otro lado, en el caso de indentación esférica o indentación Brinell la dureza sí varía con la profundidad de penetración. El propósito de este proyecto es llegar a establecer una ecuación general de dureza, medida con penetradores esféricos según ensayo Brinell, que permita extraer las propiedades mecánicas de un material a partir de tres indentaciones es decir, mediante tres valores de dureza y sus correspondientes radios de contacto será posible determinar propiedades tales como el módulo de Young, la tensión de fluencia y el coeficiente de endurecimiento por deformación. Para alcanzar este objetivo se utiliza el análisis dimensional a fin de establecer las dependencias del problema y, sistemáticamente, se realizan simulaciones por elementos finitos del ensayo de indentación en sólidos que poseen un amplio abanico de propiedades. El resultado es una función en R4 que relaciona la dureza y el tamaño de la correspondiente huella inducida por la indentación con las propiedades mecánicas anteriormente citadas. La función predice correctamente la evolución del modo de contacto elasto-plástico a profundidades pequeñas, al completamente plástico a medida que se incrementa la profundidad de penetración del indentador en el material. En consecuencia, la solución obtenida suministra un tratamiento unificado al problema del contacto esférico entre sólidos que era inexistente en la literatura especializada

The contact deformation regimes in brinell indentation and the finding of a general hardness equation

This work concerns analysis of spherical indentation experiments through extensive finite element simulations involving the J2 flow and the J2 deformation plasticity theories both under finite and infinite deformations to gain a fundamental comprehension into the mechanics of the transition between elasto-plastic and fully-plastic contacts. A decrease in hardness with increasing penetration is found to be a manifestation of the differences in material pileup responses between the two plasticity theories, so that in contrast to prior investigations, a peak in hardness cannot be taken to mark onset of a so-called finite deformation fully-plastic regime. The accuracy of Tabor’s hardness relation is examined in detail in light of the simulations and a general relation is proposed through dimensional analysis to correlate hardness with the uniaxial mechanical properties for any arbitrary elasto-plastic or fully-plastic contact. Experiments are also performed in different groups of metallic materials and a methodology is proposed to extract yield strength ys and power-law strain hardening parameter n from a minimum of two hardness measurements performed at different penetration depths. Influence of pressure sensitivity in the extracted properties is then examined through the experimental results. The issue of the uniqueness in the extracted properties and of frictional effects between indenter and material are briefly covered. The investigation ends with a discussion on the robustness of mechanical property extractions through single crystal spherical indentation experiments. Along these lines, consistency is found between simulations with the flow theory of plasticity and the crystal plasticity model for fcc metals. Finally, the potential of spherical indentation to distinguish between single crystal elastic and plastic anisotropy is considered

Una ecuación de dureza unificada para ensayos de indentación esférica

El presente proyecto tiene su marco teórico en las técnicas de indentación instrumentada. Mediante este tipo de ensayos pueden relacionarse la dureza y las propiedades mecánicas fundamentales de un material, tales como el módulo de Young, el límite de fluencia o el coeficiente de endurecimiento por deformación. En el caso de indentaciones puntiagudas en las que se utilizan pirámides Vickers o Berkovich esta relación se puede establecer mediante el conocimiento de las curvas de carga aplicada (P)-profundidad de penetración (h). En estos casos no existe variación de dureza conforme la profundidad de penetración aumenta. Por otro lado, en el caso de indentación esférica o indentación Brinell la dureza sí varía con la profundidad de penetración. El propósito de este proyecto es llegar a establecer una ecuación general de dureza, medida con penetradores esféricos según ensayo Brinell, que permita extraer las propiedades mecánicas de un material a partir de tres indentaciones es decir, mediante tres valores de dureza y sus correspondientes radios de contacto será posible determinar propiedades tales como el módulo de Young, la tensión de fluencia y el coeficiente de endurecimiento por deformación. Para alcanzar este objetivo se utiliza el análisis dimensional a fin de establecer las dependencias del problema y, sistemáticamente, se realizan simulaciones por elementos finitos del ensayo de indentación en sólidos que poseen un amplio abanico de propiedades. El resultado es una función en R4 que relaciona la dureza y el tamaño de la correspondiente huella inducida por la indentación con las propiedades mecánicas anteriormente citadas. La función predice correctamente la evolución del modo de contacto elasto-plástico a profundidades pequeñas, al completamente plástico a medida que se incrementa la profundidad de penetración del indentador en el material. En consecuencia, la solución obtenida suministra un tratamiento unificado al problema del contacto esférico entre sólidos que era inexistente en la literatura especializada

Avaliação e redesenho de uma unidade sobre proporcionalidade, usando a ferramenta de Idoneidade Didática

The objective of this paper is to describe and analyze the reflection made by a teacher while using the Didactical Suitability Criteria (DS) to assess and redesign a didactical unit of proportionality. The qualitative analysis points out that the assessment made by the teacher on the aforementioned unit is rather balanced, as it contemplates all the DS components. However, while weighting the different DS items used to redesign the unit, the teacher puts more emphasis on epistemic suitability, particularly, on redesigning different tasks typologies in order to explore a broader significance of proportionality, in particular, the geometric and arithmetic ones. It is concluded that this type of result can be observed in other research projects dealing with the reflection that the teacher makes when using the didactical suitability tool.El objetivo de este trabajo es describir y analizar la reflexión que hace un profesor de su práctica, cuando utiliza los Criterios de Idoneidad Didáctica (CI) para valorar y rediseñar una unidad didáctica sobre la proporcionalidad. El análisis cualitativo del caso apunta que la valoración que hace el profesor de la unidad didáctica es bastante equilibrada, pues se pauta en todos los componentes de los CI. Sin embrago, al atribuir el peso al uso de los criterios en el rediseño de la unidad, el profesor pone más énfasis en la idoneidad epistémica, en particular, en el rediseño de diferentes tipologías de tareas con la finalidad de trabajar diferentes significados parciales de la proporcionalidad, en particular, el geométrico y el aritmético. Se concluye que este tipo de resultado puede ser observado en otras investigaciones que tratan de la reflexión que hace el profesor al utilizar la herramienta idoneidad didáctica.O objetivo deste trabalho é descrever e analisar a reflexão que um professor faz de sua prática quando utiliza os Critérios de Idoneidade Didática (CI) para avaliar e redesenhar uma unidade didática sobre a proporcionalidade. A análise qualitativa do caso indica que a avaliação do professor sobre a unidade didática é bastante equilibrada, pois se baseia em todos os componentes do CI. No entanto, ao atribuir o peso ao uso dos critérios no redesenho da unidade, o professor enfatiza mais a adequação epistêmica, especialmente, o redesenho de diferentes tipologias de tarefas, a fim de trabalhar diferentes significados parciais da proporcionalidade, em particular, geométrico e aritmético. Conclui-se que esse tipo de resultado pode ser observado em outras pesquisas que tratam da reflexão que o professor faz ao utilizar a ferramenta de idoneidade didática

The influence of pore size on the indentation behavior of metallic nanoporous materials : a molecular dynamics study

In general, the influence of pore size is not considered when determining the Young's modulus of nanoporous materials. Here, we demonstrate that the pore size needs to be taken into account to properly assess the mechanical properties of these materials. Molecular Dynamics simulations of spherical indentation experiments on single crystalline nanoporous Cu have been undertaken in systems with: (i) a constant degree of porosity and variable pore diameter; and (ii) a constant pore diameter and variable porosity degree. The classical Gibson and Ashby expression relating Young's modulus with the relative density of the nanoporous metal is modified to include the influence of the pore size. The simulations reveal that, for a fixed porosity degree, the mechanical behavior of materials with smaller pores differs more significantly from the behavior of the bulk, fully dense counterpart. This effect is ascribed to the increase of the overall surface area as the pore size is reduced, together with the reduced coordination number of the atoms located at the pores edges

Nanomechanical behaviour of open-cell nanoporous metals: homogeneous versus thickness-dependent porosity

Two dierent nanoporous materials, porous copper prepared by dealloying and porous nickel prepared by electrodeposition, have been studied by means of nanoindentation experiments at dierent maximum applied loads. While nanoporous Cu is homogeneous along its cross-section, the electrodeposited Ni lms show a graded porosity, with smaller pores and thicker pore walls close to the lm's surface. The mechanical properties of the two materials have been extracted using a methodology based on scaling laws and subsequent interpretation has been performed using nite element simulations. Two dierent deformation mechanisms are observed for nanoporous Cu and nanoporous Ni, respectively. Dealloyed porous copper behaves as an homogeneous material without evident eect of densication and with mechanical properties that are independent of the applied load. Given this homogeneity, it is possible to t the entire loading - unloading curve for dierent maximum applied loads with a single set of mechanical properties. Conversely, electrodeposited porous nickel shows a decrease in the reduced Young's modulus, an increase in yield stress and a constant hardness when the maximum applied load during nanoindentation is increased. While the decrease in the reduced Young's modulus can be explained in the context of thickness inhomogeneity of the electrodeposited porous nickel (i.e., increase of porosity with depth), this cannot explain, and actually would go against, the observed increase in the yield stress, which is instead associated to the decrease in the ligament size

Understanding the mechanical behaviour of fiber/matrix interfaces during push-in tests by means of finite element simulations and a cohesive zone model

The present work represents a progress towards the understanding of the mechanical behavior of the fiber/matrix interface during push-in tests of fiber-reinforced polymer-matrix composites. Finite element simulations incorporating a cohesive zone model are used for this purpose. Different values of interface strength, interface fracture toughness, fiber diameter and friction coefficient are considered to study how they affect the load-displacement curves. A critical value of the displacement exists, being independent of the fiber diameter for given values of interface strength and fracture toughness, marking the separation between two regimes: (i) a cohesive-dominated zone interaction and (ii) a frictional contact between debonded fiber and matrix. Maps showing the different regimes are constructed, proving their helpfulness to tune the mechanical properties of the interface in order to favor a certain mechanical response. Finally, we study the debonding velocity and how this is affected by the mechanical properties of the interface providing an empirical relation

The contact deformation regimes in brinell indentation and the finding of a general hardness equation

This work concerns analysis of spherical indentation experiments through extensive finite element simulations involving the J2 flow and the J2 deformation plasticity theories both under finite and infinite deformations to gain a fundamental comprehension into the mechanics of the transition between elasto-plastic and fully-plastic contacts. A decrease in hardness with increasing penetration is found to be a manifestation of the differences in material pileup responses between the two plasticity theories, so that in contrast to prior investigations, a peak in hardness cannot be taken to mark onset of a so-called finite deformation fully-plastic regime. The accuracy of Tabor’s hardness relation is examined in detail in light of the simulations and a general relation is proposed through dimensional analysis to correlate hardness with the uniaxial mechanical properties for any arbitrary elasto-plastic or fully-plastic contact. Experiments are also performed in different groups of metallic materials and a methodology is proposed to extract yield strength ys and power-law strain hardening parameter n from a minimum of two hardness measurements performed at different penetration depths. Influence of pressure sensitivity in the extracted properties is then examined through the experimental results. The issue of the uniqueness in the extracted properties and of frictional effects between indenter and material are briefly covered. The investigation ends with a discussion on the robustness of mechanical property extractions through single crystal spherical indentation experiments. Along these lines, consistency is found between simulations with the flow theory of plasticity and the crystal plasticity model for fcc metals. Finally, the potential of spherical indentation to distinguish between single crystal elastic and plastic anisotropy is considered

The contact deformation regimes in brinell indentation and the finding of a general hardness equation

This work concerns analysis of spherical indentation experiments through extensive finite element simulations involving the J2 flow and the J2 deformation plasticity theories both under finite and infinite deformations to gain a fundamental comprehension into the mechanics of the transition between elasto-plastic and fully-plastic contacts. A decrease in hardness with increasing penetration is found to be a manifestation of the differences in material pileup responses between the two plasticity theories, so that in contrast to prior investigations, a peak in hardness cannot be taken to mark onset of a so-called finite deformation fully-plastic regime. The accuracy of Tabor’s hardness relation is examined in detail in light of the simulations and a general relation is proposed through dimensional analysis to correlate hardness with the uniaxial mechanical properties for any arbitrary elasto-plastic or fully-plastic contact. Experiments are also performed in different groups of metallic materials and a methodology is proposed to extract yield strength ys and power-law strain hardening parameter n from a minimum of two hardness measurements performed at different penetration depths. Influence of pressure sensitivity in the extracted properties is then examined through the experimental results. The issue of the uniqueness in the extracted properties and of frictional effects between indenter and material are briefly covered. The investigation ends with a discussion on the robustness of mechanical property extractions through single crystal spherical indentation experiments. Along these lines, consistency is found between simulations with the flow theory of plasticity and the crystal plasticity model for fcc metals. Finally, the potential of spherical indentation to distinguish between single crystal elastic and plastic anisotropy is considered