Closed-form solutions for the piezoresistivity properties of short-fiber reinforced composites with percolation-type behavior

A new analytical formulation for the modeling of piezoresistive fiber-reinforced composites with percolation-type behavior is presented in this work. Firstly, we develop a closed-form solution of the electrical conductivity of oriented short-fiber reinforced composites by using generalized spherical harmonics series expansions of a Mori-Tanaka (MT) model. Piezoresistive effects are accounted for by means of three distinct mechanisms, namely filler reorientation, volume expansion, and breakage/formation of conductive paths. Then, this solution is used to derive simple analytical formulas to estimate the linear piezoresistivity coefficients. To illustrate the potentials of the proposed formulation, numerical results and discussion are presented on its application to the modeling of the piezoresistive composites doped with carbon nanotubes (CNTs). The presented formulation is also inlaid in a standard 3D finite element code to simulate the electromechanical response of full-scale CNT-based structural elements. The reported results demonstrate the capabilities of the proposed formulation to link the microstructural properties of short-fiber composites with the macroscopic response of structural systems with extraordinarily fast computation times and accuracy.Feder (UE) 2014e2020Ministerio de Economía y Competitividad (España) DPI2017-89162-RConsejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía (España) P18-RT-312

Modelado y Diseño Computacional de Materiales Micro-porosos con Distribución Aleatoria Utilizando una Formulación de Elementos de Contorno

Este trabalho apresenta uma Formulação de Elementos de Contorno para a modelagem em duas dimensões de microestruturas multi-fase que contêm furos e inclusões cilíndricas de raio variável, baseada nos trabalhos de Henry & Banerjee, 1991 e Banerjee & Henry, 1992 para sólidos tridimensionais. Na presente formulação, a inomogeneidade não é discretizada da forma convencional, com vistas a um método computacionalmente mais eficiente. Na implementação numérica são considerados materiais micro-porosos com distribuição aleatória. Cada micro-poro é modelado como um furo cilíndrico utilizando um único elemento de furo o qual interpola as variáveis físicas com funções de forma de base trigonométrica. Neste trabalho são propostas funções de forma para elementos de furo com graus de liberdade adicionais ao elemento proposto originalmente por Henry & Banerjee, 1991, obtendo-se elementos de 4, 5 e 6 nós. A integração dos núcleos singulares no elemento de furo é realizada pelo método direto, resultando um elemento regularizado. Essa integração é analisada em detalhes através de diversos experimentos numéricos, e sua eficiência é comprovada. Vários exemplos são estudados para ilustrar o desempenho do método. A formulação é utilizada para resolver o problema elástico de um Elemento de Volume Representativo (EVR) e aplicar a Teoria de Campos Médios para encontrar propriedades efetivas de materiais micro-porosos com matriz homogênea e isotrópica. Expressões para avaliar propriedades efetivas sob hipótese de Estado Plano de Tensões (EPT) e Estado Plano de Deformações (EPD) para materiais isotrópicos e transversalmente isotrópicos são desenvolvidas. Como exemplo de aplicação é modelado um material com propriedades e distribuição estatística de furos tomadas da microestrutura de um ferro fundido nodular ferrítico e considerando a hipótese de EPT. Neste exemplo é obtida a quantidade de inomogeneidades que é preciso modelar para considerar um EVR. Mostra-se que a formulação é eficiente e especialmente adequada para estimar propriedades efetivas em materiais micro-porosos. Adicionalmente, desenvolve-se também um procedimento geral para o projeto computacional de materiais compostos micro-heterogêneos com propriedades elásticas de acordo com os requerimentos pré-especificados. Esse procedimento combina Algoritmos Genéticos (AG) e o Método de Busca Direta (MBD) com a formulação de Elementos de Contorno desenvolvida. Visando acelerar o AG uma estratégia para aproximar o valor da função objetivo é proposta e implementada. A eficiência e capacidade da ferramenta desenvolvida são ilustradas mediante a solução de um problema inverso. Os resultados obtidos mostram importantes melhorias no tempo computacional, em comparação com a implementação do AG convencional, sem grandes perdas na precisão.In this work a Boundary Element Formulation for modeling two-dimensional multi-phase microstructure containing cylindrical holes and inclusions of variable radii is presented. The formulation is based on the works of Henry & Banerjee, 1991 and Banerjee & Henry, 1992 for three-dimensional solids. In the proposed approach it is not necessary to discretize the inhomogeneity like in conventional boundary methods, leading to a computationally more efficient method. In the numerical implementation, random micro-porous materials are considered. Each micro-pore is modeled as a cylindrical hole using a single hole element which use trigonometric shape function to interpolate physical unknowns. In addition, the original elements proposed by Henry & Banerjee, 1991 are further developed, and higher order elements of 4, 5 and 6 nodes are introduced. The integration of singular kernels of the hole element is accomplished by the direct method, resulting a regularized element. The integration results are analyzed in detail through various numerical experiments, and its efficiency is proved. Several examples are studied in order to illustrate the performance of the method. The formulation is used to solve the elastic problem in a Representative Volume Element (RVE) and the Theory of Mean Fields is applied in order to obtain effective properties of random micro-porous materials with homogeneous and isotropic matrix. Expressions for the evaluation of effective properties over Plane Stress and Strain Plane hypothesis for isotropic and transversally isotropic materials are developed. As an application example, a material with the same properties and microstructure distribution of the austempered ductile iron is modeled considering the Plane Stress hypothesis. In this example the quantity of inhomogeneities necessary for the model to deliver a RVE is obtained. It is demonstrated that the formulation is efficient and specially adapted for effective property estimation in micro-porous materials. Additionally, a general procedure to perform the computational design of micro-heterogeneous composite materials with pre-specified elastic properties requirements is developed. It combines the use of Genetic Algorithms (GA) and the Direct Search Method along with the proposed boundary element formulation. To accelerate the GA a strategy for approximation of the objective function is proposed and implemented. The performance and capabilities of the devised tool are illustrated by solving an inverse problem. The results obtained show important savings in computing time in comparison to the standard GA, with only minor accuracy degradation.En este trabajo se presenta una Formulación de Elementos de Contorno para el modelado en dos dimensiones de microestructuras multifase que contienen agujeros e inclusiones cilíndricas de radio variable, basándose en los trabajos de Henry & Banerjee, 1991 y Banerjee & Henry, 1992 para sólidos tridimensionales. En esta formulación, la inhomogeneidad no es discretizada en la forma convencional, buscando un método más eficiente computacionalmente. En la implementación numérica son considerados materiales micro-porosos con distribución aleatoria. Cada micro-poro es modelado como un agujero cilíndrico utilizando un único elemento de agujero, el cual interpola las variables físicas con funciones de forma de base trigonométrica. En este trabajo son propuestas funciones de forma para elementos de agujero con grados de libertad adicionales al elemento propuesto originalmente por Henry & Banerjee, 1991, obteniéndose elementos de 4, 5 e 6 nodos. La integración de los núcleos singulares en el elemento de agujero se realiza por el método directo, resultando un elemento regularizado. Esta integración es analizada en detalle a través de diversos experimentos numéricos, y su eficiencia es comprobada. Varios ejemplos son estudiados para ilustrar el desempeño del método. La formulación es utilizada para resolver el problema elástico de un Elemento de Volumen Representativo (EVR) y aplicar la Teoría de Campos Promedios para encontrar propiedades efectivas de materiales micro-porosos con matriz homogénea e isotrópica. Expresiones para evaluar propiedades efectivas sobre la hipótesis de Estado Plano de Tensiones (EPT) y Estado Plano de Deformaciones (EPD) para materiales isotrópicos e isotrópicos transversales son desarrolladas. Como ejemplo de aplicación se modela un material con propiedades y distribución estadística de agujeros tomadas de la microestructura de hierro fundido nodular ferrítico, considerando a hipótesis de EPT. En este ejemplo se obtiene la cantidad de inhomogeneidades que es preciso modelar para considerar un EVR. Se muestra que la formulación es eficiente y especialmente adecuada para estimar propiedades efectivas de materiales micro-porosos. Adicionalmente, también se desarrolla un procedimiento general para el diseño de materiales compuestos micro-heterogéneos con propiedades elásticas de acuerdo con requerimientos pre-especificados. Este procedimiento combina Algoritmos Genéticos (AG) y el Método de Búsqueda Directa con la Formulación de Elementos de Contorno desarrollada. Buscando acelerar el AG una estrategia para aproximar el valor de la función objetivo es propuesta e implementada. La eficiencia y capacidad de la herramienta desarrollada es ilustrada mediante la solución de un problema inverso. Los resultados obtenidos muestran importantes mejoras en el tiempo computacional en comparación con la implementación del AG convencional, sin grandes pérdidas en la precisión

Modelado y Diseño Computacional de Materiales Micro-porosos con Distribución Aleatoria Utilizando una Formulación de Elementos de Contorno

Este trabalho apresenta uma Formulação de Elementos de Contorno para a modelagem em duas dimensões de microestruturas multi-fase que contêm furos e inclusões cilíndricas de raio variável, baseada nos trabalhos de Henry & Banerjee, 1991 e Banerjee & Henry, 1992 para sólidos tridimensionais. Na presente formulação, a inomogeneidade não é discretizada da forma convencional, com vistas a um método computacionalmente mais eficiente. Na implementação numérica são considerados materiais micro-porosos com distribuição aleatória. Cada micro-poro é modelado como um furo cilíndrico utilizando um único elemento de furo o qual interpola as variáveis físicas com funções de forma de base trigonométrica. Neste trabalho são propostas funções de forma para elementos de furo com graus de liberdade adicionais ao elemento proposto originalmente por Henry & Banerjee, 1991, obtendo-se elementos de 4, 5 e 6 nós. A integração dos núcleos singulares no elemento de furo é realizada pelo método direto, resultando um elemento regularizado. Essa integração é analisada em detalhes através de diversos experimentos numéricos, e sua eficiência é comprovada. Vários exemplos são estudados para ilustrar o desempenho do método. A formulação é utilizada para resolver o problema elástico de um Elemento de Volume Representativo (EVR) e aplicar a Teoria de Campos Médios para encontrar propriedades efetivas de materiais micro-porosos com matriz homogênea e isotrópica. Expressões para avaliar propriedades efetivas sob hipótese de Estado Plano de Tensões (EPT) e Estado Plano de Deformações (EPD) para materiais isotrópicos e transversalmente isotrópicos são desenvolvidas. Como exemplo de aplicação é modelado um material com propriedades e distribuição estatística de furos tomadas da microestrutura de um ferro fundido nodular ferrítico e considerando a hipótese de EPT. Neste exemplo é obtida a quantidade de inomogeneidades que é preciso modelar para considerar um EVR. Mostra-se que a formulação é eficiente e especialmente adequada para estimar propriedades efetivas em materiais micro-porosos. Adicionalmente, desenvolve-se também um procedimento geral para o projeto computacional de materiais compostos micro-heterogêneos com propriedades elásticas de acordo com os requerimentos pré-especificados. Esse procedimento combina Algoritmos Genéticos (AG) e o Método de Busca Direta (MBD) com a formulação de Elementos de Contorno desenvolvida. Visando acelerar o AG uma estratégia para aproximar o valor da função objetivo é proposta e implementada. A eficiência e capacidade da ferramenta desenvolvida são ilustradas mediante a solução de um problema inverso. Os resultados obtidos mostram importantes melhorias no tempo computacional, em comparação com a implementação do AG convencional, sem grandes perdas na precisão.In this work a Boundary Element Formulation for modeling two-dimensional multi-phase microstructure containing cylindrical holes and inclusions of variable radii is presented. The formulation is based on the works of Henry & Banerjee, 1991 and Banerjee & Henry, 1992 for three-dimensional solids. In the proposed approach it is not necessary to discretize the inhomogeneity like in conventional boundary methods, leading to a computationally more efficient method. In the numerical implementation, random micro-porous materials are considered. Each micro-pore is modeled as a cylindrical hole using a single hole element which use trigonometric shape function to interpolate physical unknowns. In addition, the original elements proposed by Henry & Banerjee, 1991 are further developed, and higher order elements of 4, 5 and 6 nodes are introduced. The integration of singular kernels of the hole element is accomplished by the direct method, resulting a regularized element. The integration results are analyzed in detail through various numerical experiments, and its efficiency is proved. Several examples are studied in order to illustrate the performance of the method. The formulation is used to solve the elastic problem in a Representative Volume Element (RVE) and the Theory of Mean Fields is applied in order to obtain effective properties of random micro-porous materials with homogeneous and isotropic matrix. Expressions for the evaluation of effective properties over Plane Stress and Strain Plane hypothesis for isotropic and transversally isotropic materials are developed. As an application example, a material with the same properties and microstructure distribution of the austempered ductile iron is modeled considering the Plane Stress hypothesis. In this example the quantity of inhomogeneities necessary for the model to deliver a RVE is obtained. It is demonstrated that the formulation is efficient and specially adapted for effective property estimation in micro-porous materials. Additionally, a general procedure to perform the computational design of micro-heterogeneous composite materials with pre-specified elastic properties requirements is developed. It combines the use of Genetic Algorithms (GA) and the Direct Search Method along with the proposed boundary element formulation. To accelerate the GA a strategy for approximation of the objective function is proposed and implemented. The performance and capabilities of the devised tool are illustrated by solving an inverse problem. The results obtained show important savings in computing time in comparison to the standard GA, with only minor accuracy degradation.En este trabajo se presenta una Formulación de Elementos de Contorno para el modelado en dos dimensiones de microestructuras multifase que contienen agujeros e inclusiones cilíndricas de radio variable, basándose en los trabajos de Henry & Banerjee, 1991 y Banerjee & Henry, 1992 para sólidos tridimensionales. En esta formulación, la inhomogeneidad no es discretizada en la forma convencional, buscando un método más eficiente computacionalmente. En la implementación numérica son considerados materiales micro-porosos con distribución aleatoria. Cada micro-poro es modelado como un agujero cilíndrico utilizando un único elemento de agujero, el cual interpola las variables físicas con funciones de forma de base trigonométrica. En este trabajo son propuestas funciones de forma para elementos de agujero con grados de libertad adicionales al elemento propuesto originalmente por Henry & Banerjee, 1991, obteniéndose elementos de 4, 5 e 6 nodos. La integración de los núcleos singulares en el elemento de agujero se realiza por el método directo, resultando un elemento regularizado. Esta integración es analizada en detalle a través de diversos experimentos numéricos, y su eficiencia es comprobada. Varios ejemplos son estudiados para ilustrar el desempeño del método. La formulación es utilizada para resolver el problema elástico de un Elemento de Volumen Representativo (EVR) y aplicar la Teoría de Campos Promedios para encontrar propiedades efectivas de materiales micro-porosos con matriz homogénea e isotrópica. Expresiones para evaluar propiedades efectivas sobre la hipótesis de Estado Plano de Tensiones (EPT) y Estado Plano de Deformaciones (EPD) para materiales isotrópicos e isotrópicos transversales son desarrolladas. Como ejemplo de aplicación se modela un material con propiedades y distribución estadística de agujeros tomadas de la microestructura de hierro fundido nodular ferrítico, considerando a hipótesis de EPT. En este ejemplo se obtiene la cantidad de inhomogeneidades que es preciso modelar para considerar un EVR. Se muestra que la formulación es eficiente y especialmente adecuada para estimar propiedades efectivas de materiales micro-porosos. Adicionalmente, también se desarrolla un procedimiento general para el diseño de materiales compuestos micro-heterogéneos con propiedades elásticas de acuerdo con requerimientos pre-especificados. Este procedimiento combina Algoritmos Genéticos (AG) y el Método de Búsqueda Directa con la Formulación de Elementos de Contorno desarrollada. Buscando acelerar el AG una estrategia para aproximar el valor de la función objetivo es propuesta e implementada. La eficiencia y capacidad de la herramienta desarrollada es ilustrada mediante la solución de un problema inverso. Los resultados obtenidos muestran importantes mejoras en el tiempo computacional en comparación con la implementación del AG convencional, sin grandes pérdidas en la precisión

Piezoresistive modeling of orthotropic 3D-printing composite line extrusions with non-affine reorientation of fibers

© 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/bync/4.0/).This work introduces a micromechanical semi-analytical model to estimate piezoresistive constants in 3D-printable short-fiber composites, extending the Mori–Tanaka scheme with two innovative components. Firstly, the model incorporates non-affine reorientation of fibers, controlled by a parameter , which significantly impacts texture coefficients. This parameter introduces an additional layer of complexity, modifying texture coefficients beyond traditional affine fiber reorientations. Observational data illustrate a linear relationship between piezoresistivities and , leading to the development of a statistical model that calculates based on the fiber aspect ratio. Secondly, the model accounts for preferential fiber orientation in the undeformed state, resulting in an orthotropic material model. This anisotropy is quantified using an Orientation Distribution Function (ODF) derived from X-ray scans, which is parameterized to maintain symmetry across three orthogonal planes. Initial investigations indicate that the inherent anisotropy introduced by the printing process substantially affects the initial resistivity of 3D-printed composites. Utilizing a Gaussian transversely isotropic model, we find that neglecting this anisotropy could lead to significant errors in understanding and predicting piezoresistive behavior. The model’s incorporation into advanced nanocomposite frameworks offers preliminary reconciliation with experimental data. Additionally, our results emphasize the necessity to refine wavy fiber models by incorporating non-affine rotations, particularly crucial for fibers with small aspect ratios, thereby enhancing predictive accuracy. The findings lay a robust foundation for future research, emphasizing the need for comprehensive experimental validation and setting the stage for nuanced applications in piezoresistive composite material

Averaging material tensors of any rank in textured polycrystalline materials: Extending the scope beyond crystallographic proper point groups

In many modern micromechanical applications, there is usually a need to perform the orientational averaging of certain material tensors weighted by an orientation distribution function (ODF). The computation of these averages is seen to be very simplified by means of the so-called generalized spherical harmonic method (GSHM), which is based on the classical assumption that the ODFs are defined in the rotation group . A priori, this makes the averaging strictly applicable only to polycrystals with crystallite symmetry defined by one of the 11 proper point groups. Despite the interest in the study of materials belonging to any of the other 21 point groups, few studies have properly considered such cases. This is crucial for physical properties represented by odd-order tensors such as the third-order linear piezoelectric tensor. The goal of this work is to extend the applicability of the GSHM to crystallites with symmetry that belongs to the orthogonal group . Thus, a simple formula is provided for the averaging of material tensors of any rank in textured polycrystals containing crystallites with symmetry defined by any of the 32 crystallographic point groups. Our work presents a simple yet rigorous method for indirect averaging on the orthogonal group, using averaging on . We demonstrate the full equivalence between our proposed method and averaging properly on through a detailed proof provided in this paper. This proof represents a significant contribution to the field, providing a practical and reliable approach for researchers working with crystalline materials. The results reported here confirm the validity of closed-form expressions previously derived by the authors for piezoelectric materials.Consejería de Economía, Conocimiento, Empresas y Universidad, Junta de Andalucía P18-RT-312

Closed-form expressions for computing flexoelectric coefficients in textured polycrystalline dielectrics

The volume average of the constituting properties of a textured polycrystalline aggregate (the Voigt average) provides a simple way to estimate the effective properties of such an aggregate in terms of its crystallographic symmetry, texture symmetry, and the properties of the constituting crystallites. Despite the interest in estimate or characterizing flexoelectric properties, few studies have addressed the problem of finding effective flexoelectric properties from the continuum phases of the constituents. Instead, most of them focus on single-crystals at the atomic level. This study provides, for the first time, closed-form expressions that allow the direct flexoelectric coefficients to be computed for textured polycrystalline dielectrics. Crystallites can belong to any class of cubic symmetry or to the tetragonal TI Laue group, while no restriction is imposed on texture symmetry. These expressions are reduced to very simple ones when cubic crystallites with fiber texture are considered. The proposed formulation provides information on the matrix structure of the homogenized material related to the crystallographic symmetry and texture. We close with some illustrative examples

Construction of computational models for the stress analysis of the bones using CT imaging: application in the gleno-humeral joint

Se presenta en este trabajo una metodología para el procesamiento de imágenes de estudios de TC para la construcción de modelos computacionales de piezas óseas. Los modelos computacionales son utilizados para el análisis de esfuerzos utilizando el Método de los Elementos Finitos. Las constantes elásticas del tejido óseo son calculadas a partir de los datos de densidad de las TC. La metodología propuesta es aplicada en la construcción de un modelo para el análisis de la articulación gleno-humeral.A methodology for the construction of computational models from CT images is presented in this work. Computational models serve for the stress analysis of the bones using the Finite Element Method. The elastic constants of the bone tissue are calculated using the density data obtained in from the CTs. The proposed methodology is demonstrated in the construction of a model for the gleno-humeral joint.Fil: Cisilino, Adrian Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones en Ciencia y Tecnología de Materiales. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones en Ciencia y Tecnología de Materiales; ArgentinaFil: D'amico, Diego. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones en Ciencia y Tecnología de Materiales. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones en Ciencia y Tecnología de Materiales; ArgentinaFil: Buroni, Federico. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones en Ciencia y Tecnología de Materiales. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones en Ciencia y Tecnología de Materiales; ArgentinaFil: Commisso, Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Mar del Plata. Instituto de Investigaciones en Ciencia y Tecnología de Materiales. Universidad Nacional de Mar del Plata. Facultad de Ingeniería. Instituto de Investigaciones en Ciencia y Tecnología de Materiales; ArgentinaFil: Sammartino, Mario. Clínica de Fracturas y Ortopedia; ArgentinaFil: Capiel, Carlos. Instituto Radiológico; Argentin

Connectivity patterns in lead-free piezocomposites: A critical analysis for 0-3 and 1-3 configurations

This is an open access article under the CC BY-NC license.In the quest for eco-friendly alternatives within materials science, the development of sustainable and non-toxic piezoelectric composites is of utmost importance. This study undertakes a computational exploration to elucidate the influence of phase connectivity on the engineering performance of lead-free piezocomposites. Employing a combination of analytical and numerical methodologies, we critically evaluated various figures of merit across different microstructural configurations, juxtaposing these findings with traditional lead zirconate titanate (PZT)-based materials. Our analysis considers 0-3 and 1-3 connectivity patterns, incorporating active phases in the form of spherical particles and cylindrical fibers. We also examine the impact of carbon nanotubes (CNTs) in enhancing the polymeric matrix, which introduces the potential for network percolation and further mechanical and electrical property optimization. The study yields pivotal insights into the phase connectivity of lead-free piezocomposites, with direct implications for their application in sensing, actuating, and energy harvesting domains. We ascertain that the electromechanical performance of these composites is contingent upon the connectivity pattern and the proportion of active phase. Notably, the KNNS-BNZH & Polyethylene composite demonstrates exceptional potential in 1-3 configurations, while the BTO & PVDF composite distinguishes itself with superior dielectric and piezoelectric responses across varying volume fractions. The strategic infusion of CNTs into the PDMS matrix emerges as a significant enhancer of electromechanical attributes, albeit with performance improvements that are specific to the type of coefficient and CNT concentration. This investigation underscores the nuanced interplay between composite design and microstructural attributes, reinforcing the critical role these factors play in the advancement of effective and eco-conscious piezoelectric materials

XFEM crack growth virtual monitoring in self-sensing CNT reinforced polymer nanocomposite plates using ANSYS

This paper presents an eXtended Finite Element Method (XFEM)-based numerical scheme to compute electrical resistivity changes caused by the presence of cracks and the crack growth. Using the commercial finite element package ANSYS, the virtual continuous monitoring of the structure is solved in two steps. First, the strain response of the cracked composite domain is computed by means of the XFEM. In the second step, the electrical conductivity of the piezorresistive elements located in the domain are updated according to the strain state and the electric resistance between two electrodes of the damaged plate is computed. The comparison with the electric resistance measured for the undamaged plate allows us to detect the presence of a crack and its severity. Moreover, the crack growth process can be also monitored via the electric resistance increments. Several numerical studies are provided to show the capabilities of this computational framework.Ministerio de Economía y Competitividad DPI2017-89162-RaJunta de Andalucía P18-RT-312