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http://hdl.handle.net/10985/10376
Détermination des diagrammes de perte d’ellipticité par une approche micromécanique
FRANZ, Gérald; ABED-MERAIM, Farid; BEN ZINEB, Tarak; BERVEILLER, Marcel; LEMOINE, Xavier
La striction et la rupture au cours de l’opération d’emboutissage figurent parmi les principaux phénomènes limitant les déformations maximales admises par les métaux. Ces phénomènes sont liés à la microstructure des matériaux ainsi qu’aux conditions de sollicitation. Afin de caractériser l’aptitude au formage d’un matériau, et ce pour différents modes de déformations, Keeler (1965) et Goodwin (1968) ont introduit la notion de Courbe Limite de Formage (CLF). L'inconvénient de cette représentation est sa forte dépendance au chemin de déformation, ce qui suppose qu’elle doit être déterminée pour chaque type de trajet de déformation. L’idée d’Arrieux (1982) fut de rechercher une représentation indépendante du trajet de chargement, ce qui donna naissance aux courbes limites de formage en contraintes. Les diagrammes de perte d'ellipticité (PDE) représentés dans l’espace des déformations principales dans celui des contraintes principales à partir d’une approche micromécanique sont présentés dans ce poster. Ces diagrammes sont qualitativement similaires aux CLF mais beaucoup plus restrictifs. L’influence de certains paramètres sur le tracé de ces courbes est étudiée.
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/10985/103762006-01-01T00:00:00ZFRANZ, GéraldABED-MERAIM, FaridBEN ZINEB, TarakBERVEILLER, MarcelLEMOINE, XavierLa striction et la rupture au cours de l’opération d’emboutissage figurent parmi les principaux phénomènes limitant les déformations maximales admises par les métaux. Ces phénomènes sont liés à la microstructure des matériaux ainsi qu’aux conditions de sollicitation. Afin de caractériser l’aptitude au formage d’un matériau, et ce pour différents modes de déformations, Keeler (1965) et Goodwin (1968) ont introduit la notion de Courbe Limite de Formage (CLF). L'inconvénient de cette représentation est sa forte dépendance au chemin de déformation, ce qui suppose qu’elle doit être déterminée pour chaque type de trajet de déformation. L’idée d’Arrieux (1982) fut de rechercher une représentation indépendante du trajet de chargement, ce qui donna naissance aux courbes limites de formage en contraintes. Les diagrammes de perte d'ellipticité (PDE) représentés dans l’espace des déformations principales dans celui des contraintes principales à partir d’une approche micromécanique sont présentés dans ce poster. Ces diagrammes sont qualitativement similaires aux CLF mais beaucoup plus restrictifs. L’influence de certains paramètres sur le tracé de ces courbes est étudiée.Numerical integration of rate-independent BCC single crystal plasticity models: comparative study of two classes of numerical algorithms
http://hdl.handle.net/10985/13535
Numerical integration of rate-independent BCC single crystal plasticity models: comparative study of two classes of numerical algorithms
AKPAMA, Holanyo K.; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
In an incremental formulation suitable to numerical implementation, the use of rate-independent theory of crystal plasticity essentially leads to four fundamental problems. The first is to determine the set of potentially active slip systems over a time increment. The second is to select the active slip systems among the potentially active ones. The third is to compute the slip rates (or the slip increments) for the active slip systems. And the last problem is the possible non-uniqueness of slip rates. The purpose of this paper is to propose satisfactory responses to the above-mentioned first three issues by presenting and comparing two novel numerical algorithms. The first algorithm is based on the usual return-mapping integration scheme, while the second follows the so-called ultimate scheme. The latter is shown to be more relevant and efficient than the former. These comparative performances are illustrated through various numerical simulations of the mechanical behavior of single crystals and polycrystalline aggregates subjected to monotonic and complex loadings. Although these algorithms are applied in this paper to Body-Centered-Cubic (BCC) crystal structures, they are quite general and suitable for integrating the constitutive equations for other crystal structures (e.g., FCC and HCP).
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/135352016-01-01T00:00:00ZAKPAMA, Holanyo K.BEN BETTAIEB, MohamedABED-MERAIM, FaridIn an incremental formulation suitable to numerical implementation, the use of rate-independent theory of crystal plasticity essentially leads to four fundamental problems. The first is to determine the set of potentially active slip systems over a time increment. The second is to select the active slip systems among the potentially active ones. The third is to compute the slip rates (or the slip increments) for the active slip systems. And the last problem is the possible non-uniqueness of slip rates. The purpose of this paper is to propose satisfactory responses to the above-mentioned first three issues by presenting and comparing two novel numerical algorithms. The first algorithm is based on the usual return-mapping integration scheme, while the second follows the so-called ultimate scheme. The latter is shown to be more relevant and efficient than the former. These comparative performances are illustrated through various numerical simulations of the mechanical behavior of single crystals and polycrystalline aggregates subjected to monotonic and complex loadings. Although these algorithms are applied in this paper to Body-Centered-Cubic (BCC) crystal structures, they are quite general and suitable for integrating the constitutive equations for other crystal structures (e.g., FCC and HCP).Taylor Meshless Method for bending and buckling of thin plates
http://hdl.handle.net/10985/20359
Taylor Meshless Method for bending and buckling of thin plates
TIAN, Haitao; POTIER-FERRY, Michel; ABED-MERAIM, Farid
This paper introduces a new meshless method named Taylor Meshless Method (TMM) using Taylor series to deduce the shape functions. Next the problem is discretized by point-collocation only on the boundary and without integration. The discrete boundary problem is solved by least-squares method. In this talk, this method is applied to bending and buckling of isotropic and anisotropic plates. The shape functions are polynomials that coincide with harmonic polynomials in the case of Laplace equation. These polynomials are computed numerically by solving the PDE approximately in the sense of Taylor series. Of course, this leads to an error that decreases asymptotically with the degree. TMM can be considered as a Trefftz Method, but we search approximated solutions in the sense of Taylor series while Trefftz Method is generally based on exact solutions of the PDE. As a counterpart, one is able to build this complete family of approximated solutions, whatever be the studied equation. A strong reduction of the number of degrees of freedom is the main advantage of this class of discretization techniques. The main drawback is the ill-conditioning of the final matrix, what can limit the size of the solved problem, but it was established that TMM is able to solve large scale problems. In the case of nonlinear PDEs as Föppl-Von Karman plate models, one applies first a linearization technique as Newton iterative technique. Here we apply Asymptotic Numerical Method. Next the resulting linear equations have variable coefficients are they are solved by TMM. Several linear and nonlinear numerical results will be presented for isotropic and anisotropic laminated plates.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/203592017-01-01T00:00:00ZTIAN, HaitaoPOTIER-FERRY, MichelABED-MERAIM, FaridThis paper introduces a new meshless method named Taylor Meshless Method (TMM) using Taylor series to deduce the shape functions. Next the problem is discretized by point-collocation only on the boundary and without integration. The discrete boundary problem is solved by least-squares method. In this talk, this method is applied to bending and buckling of isotropic and anisotropic plates. The shape functions are polynomials that coincide with harmonic polynomials in the case of Laplace equation. These polynomials are computed numerically by solving the PDE approximately in the sense of Taylor series. Of course, this leads to an error that decreases asymptotically with the degree. TMM can be considered as a Trefftz Method, but we search approximated solutions in the sense of Taylor series while Trefftz Method is generally based on exact solutions of the PDE. As a counterpart, one is able to build this complete family of approximated solutions, whatever be the studied equation. A strong reduction of the number of degrees of freedom is the main advantage of this class of discretization techniques. The main drawback is the ill-conditioning of the final matrix, what can limit the size of the solved problem, but it was established that TMM is able to solve large scale problems. In the case of nonlinear PDEs as Föppl-Von Karman plate models, one applies first a linearization technique as Newton iterative technique. Here we apply Asymptotic Numerical Method. Next the resulting linear equations have variable coefficients are they are solved by TMM. Several linear and nonlinear numerical results will be presented for isotropic and anisotropic laminated plates.Buckling and wrinkling of thin membranes by using a numerical solver based on multivariate Taylor series
http://hdl.handle.net/10985/20655
Buckling and wrinkling of thin membranes by using a numerical solver based on multivariate Taylor series
TIAN, Haitao; POTIER-FERRY, Michel; ABED-MERAIM, Farid
Buckling and wrinkling of thin structures often lead to very complex response curves that are hard to follow by standard path-following techniques, especially for very thin membranes in a slack or nearly slack state. Many recent papers mention numerical difficulties encountered in the treatment of wrinkling problems, especially with path-following procedures and often these authors switch to pseudo-dynamic algorithms. Moreover, the numerical modeling of many wrinkles leads to very large size problems. In this paper, a new numerical procedure based on a double Taylor series is presented, that combines path-following techniques and discretization by a Trefftz method: Taylor series with respect to a load parameter (Asymptotic Numerical Method) and with respect to space variables (Taylor Meshless Method). The procedure is assessed on buckling benchmarks and on the difficult problem of a sheared rectangular membrane.
Fri, 01 Jan 2021 00:00:00 GMThttp://hdl.handle.net/10985/206552021-01-01T00:00:00ZTIAN, HaitaoPOTIER-FERRY, MichelABED-MERAIM, FaridBuckling and wrinkling of thin structures often lead to very complex response curves that are hard to follow by standard path-following techniques, especially for very thin membranes in a slack or nearly slack state. Many recent papers mention numerical difficulties encountered in the treatment of wrinkling problems, especially with path-following procedures and often these authors switch to pseudo-dynamic algorithms. Moreover, the numerical modeling of many wrinkles leads to very large size problems. In this paper, a new numerical procedure based on a double Taylor series is presented, that combines path-following techniques and discretization by a Trefftz method: Taylor series with respect to a load parameter (Asymptotic Numerical Method) and with respect to space variables (Taylor Meshless Method). The procedure is assessed on buckling benchmarks and on the difficult problem of a sheared rectangular membrane.Combined effect of damage and plastic anisotropy on the ductility limit of thin metal sheets
http://hdl.handle.net/10985/20264
Combined effect of damage and plastic anisotropy on the ductility limit of thin metal sheets
MSOLLI, Sabeur; BEN BETTAIEB, Mohamed; ABED-MERAIM, Farid
It is well known that both damage and plastic anisotropy strongly affect the ductility limit of thin metal sheets. Due to the manufacturing processes, initial defects, such as inclusions and voids, are commonly present in the produced sheet metals. Plastic anisotropy is a direct outcome of the rolling process, where the resulting metal sheets exhibit preferred crystallographic orientations or strong texture. In the present study, the combined effect of plastic anisotropy and damage on localized necking is numerically investigated and analyzed. To this aim, an improved version of the Gurson—Tvergaard—Needleman (GTN) constitutive framework is used to model the mechanical behavior of the studied sheet. This version, which is an extension of the original GTN model, incorporates Hill’s anisotropic yield function to take into account the plastic anisotropy of the matrix material. Particular attention is devoted to the derivation of the analytical tangent modulus associated with this constitutive model. This extended GTN model is successfully coupled with bifurcation theory to predict sheet metal ductility limits, which are represented in terms of forming limit diagrams (FLDs). The effect of some material parameters (e.g., anisotropy parameters of the metallic matrix) on the shape and the location of the predicted FLDs is then investigated and discussed through numerical simulations.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/202642016-01-01T00:00:00ZMSOLLI, SabeurBEN BETTAIEB, MohamedABED-MERAIM, FaridIt is well known that both damage and plastic anisotropy strongly affect the ductility limit of thin metal sheets. Due to the manufacturing processes, initial defects, such as inclusions and voids, are commonly present in the produced sheet metals. Plastic anisotropy is a direct outcome of the rolling process, where the resulting metal sheets exhibit preferred crystallographic orientations or strong texture. In the present study, the combined effect of plastic anisotropy and damage on localized necking is numerically investigated and analyzed. To this aim, an improved version of the Gurson—Tvergaard—Needleman (GTN) constitutive framework is used to model the mechanical behavior of the studied sheet. This version, which is an extension of the original GTN model, incorporates Hill’s anisotropic yield function to take into account the plastic anisotropy of the matrix material. Particular attention is devoted to the derivation of the analytical tangent modulus associated with this constitutive model. This extended GTN model is successfully coupled with bifurcation theory to predict sheet metal ductility limits, which are represented in terms of forming limit diagrams (FLDs). The effect of some material parameters (e.g., anisotropy parameters of the metallic matrix) on the shape and the location of the predicted FLDs is then investigated and discussed through numerical simulations.Investigation of ductility limits based on bifurcation theory coupled with continuum damage mechanics
http://hdl.handle.net/10985/10440
Investigation of ductility limits based on bifurcation theory coupled with continuum damage mechanics
BOUKTIR, Yasser; CHALAL, Hocine; HADDAD, Moussa; ABED-MERAIM, Farid
The ductility limits of an St14 steel are investigated using an elastic‒plastic‒damage model and bifurcation theory. An associative J2-flow theory of plasticity is coupled with damage within the framework of continuum damage mechanics. For strain localization prediction, the bifurcation analysis is adopted. Both the constitutive equations and the localization bifurcation criterion are implemented into the finite element code ABAQUS, within the framework of large strains and a fully three-dimensional formulation. The material parameters associated with the fully coupled elastic‒plastic‒damage model are calibrated based on experimental tensile tests together with an inverse identification procedure. The above-described approach allows the forming limit diagrams of the studied material to be determined, which are then compared with experimental measurements. A main conclusion of the current study is that the proposed approach is able to provide predictions that are in good agreement with experiments under the condition of accurate material parameter calibration. The latter requires a careful identification strategy based on both calibrated finite element simulations of tensile tests at large strains and appropriately selected necking measurements. The resulting approach represents a useful basis for setting up reliable ductility limit prediction tools as well as effective parameter identification strategies.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10985/104402016-01-01T00:00:00ZBOUKTIR, YasserCHALAL, HocineHADDAD, MoussaABED-MERAIM, FaridThe ductility limits of an St14 steel are investigated using an elastic‒plastic‒damage model and bifurcation theory. An associative J2-flow theory of plasticity is coupled with damage within the framework of continuum damage mechanics. For strain localization prediction, the bifurcation analysis is adopted. Both the constitutive equations and the localization bifurcation criterion are implemented into the finite element code ABAQUS, within the framework of large strains and a fully three-dimensional formulation. The material parameters associated with the fully coupled elastic‒plastic‒damage model are calibrated based on experimental tensile tests together with an inverse identification procedure. The above-described approach allows the forming limit diagrams of the studied material to be determined, which are then compared with experimental measurements. A main conclusion of the current study is that the proposed approach is able to provide predictions that are in good agreement with experiments under the condition of accurate material parameter calibration. The latter requires a careful identification strategy based on both calibrated finite element simulations of tensile tests at large strains and appropriately selected necking measurements. The resulting approach represents a useful basis for setting up reliable ductility limit prediction tools as well as effective parameter identification strategies.Evaluation of a new solid-shell finite element on the simulation of sheet metal forming processes
http://hdl.handle.net/10985/10071
Evaluation of a new solid-shell finite element on the simulation of sheet metal forming processes
CHALAL, Hocine; SALAHOUELHADJ, Abdellah; ABED-MERAIM, Farid
In this paper, the performance of the solid-shell finite element SHB8PS is assessed in the context of sheet metal forming simulation using anisotropic elastic-plastic behavior models. This finite element technology has been implemented into the commercial implicit finite element code Abaqus/Standard via the UEL subroutine. It consists of an eight-node three-dimensional hexahedron with reduced integration, provided with an arbitrary number of integration points along the thickness direction. The use of an in-plane reduced integration scheme prevents some locking phenomena, resulting in a computationally efficient formulation when compared to conventional 3D solid elements. Another interesting feature lies in the possibility of increasing the number of through-thickness integration points within a single element layer, which enables an accurate description of various phenomena in sheet forming simulations. A general elastic-plastic model has been adopted in the constitutive modeling for sheet forming applications with plastic anisotropy. As an illustrative example, the performance of the element is shown in the earing prediction of a cylindrical cup drawing process.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/10985/100712012-01-01T00:00:00ZCHALAL, HocineSALAHOUELHADJ, AbdellahABED-MERAIM, FaridIn this paper, the performance of the solid-shell finite element SHB8PS is assessed in the context of sheet metal forming simulation using anisotropic elastic-plastic behavior models. This finite element technology has been implemented into the commercial implicit finite element code Abaqus/Standard via the UEL subroutine. It consists of an eight-node three-dimensional hexahedron with reduced integration, provided with an arbitrary number of integration points along the thickness direction. The use of an in-plane reduced integration scheme prevents some locking phenomena, resulting in a computationally efficient formulation when compared to conventional 3D solid elements. Another interesting feature lies in the possibility of increasing the number of through-thickness integration points within a single element layer, which enables an accurate description of various phenomena in sheet forming simulations. A general elastic-plastic model has been adopted in the constitutive modeling for sheet forming applications with plastic anisotropy. As an illustrative example, the performance of the element is shown in the earing prediction of a cylindrical cup drawing process.Strain localization analysis using a large deformation anisotropic elastic-plastic model coupled with damage
http://hdl.handle.net/10985/10208
Strain localization analysis using a large deformation anisotropic elastic-plastic model coupled with damage
HADDAG, Badis; ABED-MERAIM, Farid; BALAN, Tudor
Sheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic-plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice's localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic-plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results - e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.
Thu, 01 Jan 2009 00:00:00 GMThttp://hdl.handle.net/10985/102082009-01-01T00:00:00ZHADDAG, BadisABED-MERAIM, FaridBALAN, TudorSheet metal forming processes generally involve large deformations together with complex loading sequences. In order to improve numerical simulation predictions of sheet part forming, physically-based constitutive models are often required. The main objective of this paper is to analyze the strain localization phenomenon during the plastic deformation of sheet metals in the context of such advanced constitutive models. Most often, an accurate prediction of localization requires damage to be considered in the finite element simulation. For this purpose, an advanced, anisotropic elastic-plastic model, formulated within the large strain framework and taking strain-path changes into account, has been coupled with an isotropic damage model. This coupling is carried out within the framework of continuum damage mechanics. In order to detect the strain localization during sheet metal forming, Rice's localization criterion has been considered, thus predicting the limit strains at the occurrence of shear bands as well as their orientation. The coupled elastic-plastic-damage model has been implemented in Abaqus/implicit. The application of the model to the prediction of Forming Limit Diagrams (FLDs) provided results that are consistent with the literature and emphasized the impact of the hardening model on the strain-path dependency of the FLD. The fully three-dimensional formulation adopted in the numerical development allowed for some new results - e.g. the out-of-plane orientation of the normal to the localization band, as well as more realistic values for its in-plane orientation.Développement d’un nouvel élément fini prismatique « SHB6 » de type solide–coque : formulation et évaluation à travers des cas tests
http://hdl.handle.net/10985/10441
Développement d’un nouvel élément fini prismatique « SHB6 » de type solide–coque : formulation et évaluation à travers des cas tests
TRINH, Vuong-Dieu; ABED-MERAIM, Farid; COMBESCURE, Alain
Cet article décrit le développement d’un nouvel élément fini prismatique SHB6 de type solide-coque, obtenu à partir d’une formulation purement tridimensionnelle. Cet élément possède six nœuds et cinq points d’intégration répartis selon la direction de l’épaisseur. L’objectif étant d’avoir des éléments à géométrie volumique capables de modéliser des structures minces, tout en prenant correctement en compte les différents phénomènes à travers l’épaisseur. Afin d’améliorer ses performances de calcul et d’éviter certains blocages, l’intégration réduite a été employée. On montre d’abord que cette sous-intégration ne génère pas de modes de hourglass. Ensuite, on met en évidence que l’élément SHB6, sans aucune modification ou projection de son opérateur gradient discrétisé, peut souffrir de certains verrouillages de type cisaillement transverse ou membrane.; This paper presents the development of a new solid-shell finite element “SHB6” derived from a purely three-dimensional formulation. It has six nodes as well as five integration points, all distributed along the “thickness” direction. The main goal of this research is to develop low-order solid elements that are able to model thin structures while correctly taking into account the various through-thickness phenomena. In order to improve its calculation performances and to prevent some locking phenomena, reduced integration was used. We demonstrate first that there are no hourglass modes generated by the reduced integration. On the other hand, we show that, without any modification or projection of its discrete gradient operator, the SHB6 element could suffer from some membrane and shear locking phenomena.
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/10985/104412007-01-01T00:00:00ZTRINH, Vuong-DieuABED-MERAIM, FaridCOMBESCURE, AlainCet article décrit le développement d’un nouvel élément fini prismatique SHB6 de type solide-coque, obtenu à partir d’une formulation purement tridimensionnelle. Cet élément possède six nœuds et cinq points d’intégration répartis selon la direction de l’épaisseur. L’objectif étant d’avoir des éléments à géométrie volumique capables de modéliser des structures minces, tout en prenant correctement en compte les différents phénomènes à travers l’épaisseur. Afin d’améliorer ses performances de calcul et d’éviter certains blocages, l’intégration réduite a été employée. On montre d’abord que cette sous-intégration ne génère pas de modes de hourglass. Ensuite, on met en évidence que l’élément SHB6, sans aucune modification ou projection de son opérateur gradient discrétisé, peut souffrir de certains verrouillages de type cisaillement transverse ou membrane.
This paper presents the development of a new solid-shell finite element “SHB6” derived from a purely three-dimensional formulation. It has six nodes as well as five integration points, all distributed along the “thickness” direction. The main goal of this research is to develop low-order solid elements that are able to model thin structures while correctly taking into account the various through-thickness phenomena. In order to improve its calculation performances and to prevent some locking phenomena, reduced integration was used. We demonstrate first that there are no hourglass modes generated by the reduced integration. On the other hand, we show that, without any modification or projection of its discrete gradient operator, the SHB6 element could suffer from some membrane and shear locking phenomena.Analysis of primal and dual variables in structural shape control by piezoelectric patches using solid-shell finite elements
http://hdl.handle.net/10985/13034
Analysis of primal and dual variables in structural shape control by piezoelectric patches using solid-shell finite elements
KPEKY, Fessal; ABED-MERAIM, Farid; DAYA, El Mostafa
This paper presents an assessment of the performances of new piezoelectric solid−shell finite elements. Compared to conventional solid and shell elements, the solid–shell concept reveals to be very attractive, due to a number of well-established advantages and computational capabilities. This paper focuses on two element formulations, denoted SHB15E and SHB20E, which represent a quadratic prismatic solid−shell element and its hexahedral counterpart, respectively. The current analysis consists in an evaluation of primal and dual variables during the process of shape control of structures. The interest in this solid–shell approach is shown through a set of selective and representative plate and shell benchmark problems. The results obtained by the proposed formulations are compared with those given by state-of-the-art piezoelectric elements available in ABAQUS.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10985/130342017-01-01T00:00:00ZKPEKY, FessalABED-MERAIM, FaridDAYA, El MostafaThis paper presents an assessment of the performances of new piezoelectric solid−shell finite elements. Compared to conventional solid and shell elements, the solid–shell concept reveals to be very attractive, due to a number of well-established advantages and computational capabilities. This paper focuses on two element formulations, denoted SHB15E and SHB20E, which represent a quadratic prismatic solid−shell element and its hexahedral counterpart, respectively. The current analysis consists in an evaluation of primal and dual variables during the process of shape control of structures. The interest in this solid–shell approach is shown through a set of selective and representative plate and shell benchmark problems. The results obtained by the proposed formulations are compared with those given by state-of-the-art piezoelectric elements available in ABAQUS.