Pub Date : 2024-07-26DOI: 10.1177/10812865241259134
Péter Máté, András Szekrényes
Cylindrical shells curved in only one direction show an interesting behaviour when bent, especially if they remain completely in the elastic region and do not undergo plastic forming. This can be observed in their most common application: the measuring tape. They can be coiled easily because of the loss of stability of their cross-sections, which makes transportation of long shells efficient. This property could be very useful if one could use such a one-way curved shell as a beam, which could be transported and deployed easily. The aim of this study is to observe the behaviour of such a shell, under pure bending load, with special emphasis on the stability loss of the cross-section. In this paper, analytical, semi-analytical, and finite-element methods are used for the description of the shell. The solution derived here uses a variable cross-section Euler–Bernoulli beam model combined with elements of the Kirchhoff plate theory without the shallow shell assumption. It is assumed that the cross-section remains circular and does not change its length. For a universal description, dimensionless parameters and variables are introduced. The semi-analytical investigation revealed that the snap-through ability of the shell may not exist for certain cross-sections which can be presented on a stability map. The derived model reveals the existence of a limiting point between the cross-section deformation modes for larger cross-section angles. In the article, ready-to-use analytical and semi-analytical solutions are given for the critical load and stability map of these shells, which are compared to similar shallow shell models from the literature and the finite-element solution of the problem. The finite-element method also revealed that for a dimensionless description, a length-cross-section radius parameter should be introduced to describe the three-point bending scenario.
{"title":"Stability analysis of thin cylindrical shells under pure and three-point bending","authors":"Péter Máté, András Szekrényes","doi":"10.1177/10812865241259134","DOIUrl":"https://doi.org/10.1177/10812865241259134","url":null,"abstract":"Cylindrical shells curved in only one direction show an interesting behaviour when bent, especially if they remain completely in the elastic region and do not undergo plastic forming. This can be observed in their most common application: the measuring tape. They can be coiled easily because of the loss of stability of their cross-sections, which makes transportation of long shells efficient. This property could be very useful if one could use such a one-way curved shell as a beam, which could be transported and deployed easily. The aim of this study is to observe the behaviour of such a shell, under pure bending load, with special emphasis on the stability loss of the cross-section. In this paper, analytical, semi-analytical, and finite-element methods are used for the description of the shell. The solution derived here uses a variable cross-section Euler–Bernoulli beam model combined with elements of the Kirchhoff plate theory without the shallow shell assumption. It is assumed that the cross-section remains circular and does not change its length. For a universal description, dimensionless parameters and variables are introduced. The semi-analytical investigation revealed that the snap-through ability of the shell may not exist for certain cross-sections which can be presented on a stability map. The derived model reveals the existence of a limiting point between the cross-section deformation modes for larger cross-section angles. In the article, ready-to-use analytical and semi-analytical solutions are given for the critical load and stability map of these shells, which are compared to similar shallow shell models from the literature and the finite-element solution of the problem. The finite-element method also revealed that for a dimensionless description, a length-cross-section radius parameter should be introduced to describe the three-point bending scenario.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"19 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776120","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-26DOI: 10.1177/10812865241257850
Jacobo Ayensa-Jiménez, Javier Orera-Echeverría, Manuel Doblare
Nonlinear materials are often difficult to model with classical state model theory because they have a complex and sometimes inaccurate physical and mathematical description, or we simply do not know how to describe such materials in terms of relations between external and internal variables. In many disciplines, neural network methods have emerged as powerful tools to identify very complex and nonlinear correlations. In this work, we use the very recently developed concept of physically guided neural networks with internal variables (PGNNIVs) to discover constitutive laws using a model-free approach and training solely with measured force–displacement data. PGNNIVs make a particular use of the physics of the problem to enforce constraints on specific hidden layers and are able to make predictions without internal variable data. We demonstrate that PGNNIVs are capable of predicting both internal and external variables under unseen loading scenarios, regardless of the nature of the material considered (linear, with hardening or softening behavior and hyperelastic), unravelling the constitutive law of the material hence explaining its nature altogether, endowing the method with some explanatory character that distances it from the traditional black box approach.
{"title":"Predicting and explaining nonlinear material response using deep physically guided neural networks with internal variables","authors":"Jacobo Ayensa-Jiménez, Javier Orera-Echeverría, Manuel Doblare","doi":"10.1177/10812865241257850","DOIUrl":"https://doi.org/10.1177/10812865241257850","url":null,"abstract":"Nonlinear materials are often difficult to model with classical state model theory because they have a complex and sometimes inaccurate physical and mathematical description, or we simply do not know how to describe such materials in terms of relations between external and internal variables. In many disciplines, neural network methods have emerged as powerful tools to identify very complex and nonlinear correlations. In this work, we use the very recently developed concept of physically guided neural networks with internal variables (PGNNIVs) to discover constitutive laws using a model-free approach and training solely with measured force–displacement data. PGNNIVs make a particular use of the physics of the problem to enforce constraints on specific hidden layers and are able to make predictions without internal variable data. We demonstrate that PGNNIVs are capable of predicting both internal and external variables under unseen loading scenarios, regardless of the nature of the material considered (linear, with hardening or softening behavior and hyperelastic), unravelling the constitutive law of the material hence explaining its nature altogether, endowing the method with some explanatory character that distances it from the traditional black box approach.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"69 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776124","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-07-25DOI: 10.1177/10812865241257268
Luis Dorfmann, José Merodio, Raimondo Penta, Prashant Saxena
{"title":"In recognition of the 80th birthday of Ray Ogden","authors":"Luis Dorfmann, José Merodio, Raimondo Penta, Prashant Saxena","doi":"10.1177/10812865241257268","DOIUrl":"https://doi.org/10.1177/10812865241257268","url":null,"abstract":"","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"53 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141776127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-22DOI: 10.1177/10812865241257534
Xu Wang, Peter Schiavone
We derive a closed-form solution to the plane strain problem of a partially debonded rigid elliptical inclusion in which the debonded portion is filled with a liquid slit inclusion when the infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. The original boundary value problem is reduced to a Riemann–Hilbert problem with discontinuous coefficients, and its analytical solution is derived. By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, we obtain a set of two coupled linear algebraic equations for the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. As a result, these two unknowns can be uniquely determined revealing the elastic field in the matrix.
{"title":"A partially debonded rigid elliptical inclusion with a liquid slit inclusion occupying the debonded portion","authors":"Xu Wang, Peter Schiavone","doi":"10.1177/10812865241257534","DOIUrl":"https://doi.org/10.1177/10812865241257534","url":null,"abstract":"We derive a closed-form solution to the plane strain problem of a partially debonded rigid elliptical inclusion in which the debonded portion is filled with a liquid slit inclusion when the infinite isotropic elastic matrix is subjected to uniform remote in-plane stresses. The original boundary value problem is reduced to a Riemann–Hilbert problem with discontinuous coefficients, and its analytical solution is derived. By imposing the incompressibility condition of the liquid slit inclusion and balance of moment on a circular disk of infinite radius, we obtain a set of two coupled linear algebraic equations for the two unknowns characterizing the internal uniform hydrostatic tension within the liquid slit inclusion and the rigid body rotation of the rigid elliptical inclusion. As a result, these two unknowns can be uniquely determined revealing the elastic field in the matrix.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"2 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141505237","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-21DOI: 10.1177/10812865241250015
Xin-Lin Gao
Critical velocities of a three-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are obtained in closed-form expressions. A Love–Kirchhoff thin shell model including the rotary inertia and material anisotropy effects is used in the formulation. The composite tube is made of three perfectly bonded cylindrical layers of dissimilar materials, each of which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form formulas for the critical velocities are first derived for the general case by incorporating the effects of material anisotropy, rotary inertia and radial stress. Specific formulas are then obtained for composite tubes without the rotary inertia effect and/or the radial stress effect and with various types of material symmetry for each layer as special cases. It is also shown that the current model for three-layer tubes can be reduced to those for single- and two-layer tubes. To illustrate the newly derived formulas, an example is provided for a composite tube consisting of an isotropic inner layer, an orthotropic core, and an isotropic outer layer. All four critical velocities of the composite tube are computed using the new closed-form formulas. Three values of the lowest critical velocity of the three-layer composite tube are analytically obtained from three sets of the new formulas, which agree well with the value computationally determined by others.
{"title":"Critical velocities of a three-layer composite tube incorporating the rotary inertia and material anisotropy","authors":"Xin-Lin Gao","doi":"10.1177/10812865241250015","DOIUrl":"https://doi.org/10.1177/10812865241250015","url":null,"abstract":"Critical velocities of a three-layer composite tube subjected to a uniform internal pressure moving at a constant velocity are obtained in closed-form expressions. A Love–Kirchhoff thin shell model including the rotary inertia and material anisotropy effects is used in the formulation. The composite tube is made of three perfectly bonded cylindrical layers of dissimilar materials, each of which can be orthotropic, transversely isotropic, cubic or isotropic. Closed-form formulas for the critical velocities are first derived for the general case by incorporating the effects of material anisotropy, rotary inertia and radial stress. Specific formulas are then obtained for composite tubes without the rotary inertia effect and/or the radial stress effect and with various types of material symmetry for each layer as special cases. It is also shown that the current model for three-layer tubes can be reduced to those for single- and two-layer tubes. To illustrate the newly derived formulas, an example is provided for a composite tube consisting of an isotropic inner layer, an orthotropic core, and an isotropic outer layer. All four critical velocities of the composite tube are computed using the new closed-form formulas. Three values of the lowest critical velocity of the three-layer composite tube are analytically obtained from three sets of the new formulas, which agree well with the value computationally determined by others.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"24 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529206","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-02DOI: 10.1177/10812865241253208
Rashmi Prasad, Roushan Kumar
This study aims to establish the convolutional-type variational and reciprocity theorems within the framework of the hyperbolic two-temperature generalized thermoelasticity theory for an isotropic thermoelastic material, with the help of alternate formulation of the mixed boundary initial value problem, in which initial conditions are combined with field equations (using the Laplace transform). The convolutional-type variational principle adapts readily to numerical solutions based on the Ritz method and is useful in the finite element method. The reciprocity theorem is helpful in the theoretical development of boundary and finite element methods. The current effort can be valuable for the problem of coupling effects of thermal and mechanical fields, especially in geophysics and mining.
{"title":"Some qualitative results in hyperbolic two-temperature generalized thermoelasticity","authors":"Rashmi Prasad, Roushan Kumar","doi":"10.1177/10812865241253208","DOIUrl":"https://doi.org/10.1177/10812865241253208","url":null,"abstract":"This study aims to establish the convolutional-type variational and reciprocity theorems within the framework of the hyperbolic two-temperature generalized thermoelasticity theory for an isotropic thermoelastic material, with the help of alternate formulation of the mixed boundary initial value problem, in which initial conditions are combined with field equations (using the Laplace transform). The convolutional-type variational principle adapts readily to numerical solutions based on the Ritz method and is useful in the finite element method. The reciprocity theorem is helpful in the theoretical development of boundary and finite element methods. The current effort can be valuable for the problem of coupling effects of thermal and mechanical fields, especially in geophysics and mining.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"35 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141253819","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-01DOI: 10.1177/10812865241253520
Xinze Guo, Kemin Zhou
In the design optimization of fiber-reinforced composites, spatial material discontinuity is considered intractable within the manufacturing reality imposed by advanced technologies. This paper presents a topological optimization framework based on truss-like material to design composite structures with continuous fiber. Specifically, the fiber morphology at the scattered design points, which controls the orientation and volume fraction, is taken as design variables. Using the Shepard interpolant scheme, the fiber morphology at any given computational point is interpolated by scattered design variables within a certain circular influence domain. The employed interpolation inherently ensures the spatial continuity and range-restricted of the physical field in an element-independent manner. Since separating the design variable field and analysis mesh on two independent sets of points, this method is well suited for using a sparse design variable field. The computational savings are compelling due to the reduced number of design variables without significantly restricting the design freedom. Numerical instability such as checkerboard and mesh dependencies vanished as no intermediate densities are suppressed in optimization. The continuous fiber-reinforced composites (CFRCs) in the form of truss-like continua are ready to be manufactured with the aid of a simple post-processing. Several numerical examples are investigated to demonstrate the feasibility and effectiveness of the proposed formulation and numerical techniques.
{"title":"Topology optimization of continuous fiber-reinforced composites using Shepard interpolation and its design variable reduction","authors":"Xinze Guo, Kemin Zhou","doi":"10.1177/10812865241253520","DOIUrl":"https://doi.org/10.1177/10812865241253520","url":null,"abstract":"In the design optimization of fiber-reinforced composites, spatial material discontinuity is considered intractable within the manufacturing reality imposed by advanced technologies. This paper presents a topological optimization framework based on truss-like material to design composite structures with continuous fiber. Specifically, the fiber morphology at the scattered design points, which controls the orientation and volume fraction, is taken as design variables. Using the Shepard interpolant scheme, the fiber morphology at any given computational point is interpolated by scattered design variables within a certain circular influence domain. The employed interpolation inherently ensures the spatial continuity and range-restricted of the physical field in an element-independent manner. Since separating the design variable field and analysis mesh on two independent sets of points, this method is well suited for using a sparse design variable field. The computational savings are compelling due to the reduced number of design variables without significantly restricting the design freedom. Numerical instability such as checkerboard and mesh dependencies vanished as no intermediate densities are suppressed in optimization. The continuous fiber-reinforced composites (CFRCs) in the form of truss-like continua are ready to be manufactured with the aid of a simple post-processing. Several numerical examples are investigated to demonstrate the feasibility and effectiveness of the proposed formulation and numerical techniques.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"42 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141193874","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-30DOI: 10.1177/10812865241256800
Roberto Fedele, Gabriele Milani, Francesco dell’Isola
{"title":"Special issue: “Integrated approaches for the mechanics of masonry structures: Novel strategies for experimental characterization, mathematical modelling and computer simulations”","authors":"Roberto Fedele, Gabriele Milani, Francesco dell’Isola","doi":"10.1177/10812865241256800","DOIUrl":"https://doi.org/10.1177/10812865241256800","url":null,"abstract":"","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"18 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141194030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-27DOI: 10.1177/10812865241251470
Xiang Zhou, Guoshuang Shui
Considering the importance of understanding the propagation of transient waves in the piezomagnetic solids, the thermal effect on the transient behavior of a piezomagnetic half-space subjected to dynamic anti-plane load is investigated analytically in this paper. Using one-sided, two-sided Laplace transformation and Cagniard–de Hoop (CH) technique, an efficient and accurate analytical derivation for the solution of the anti-plane displacement, shear stress, magnetic potential, and induction in Laplace domain is presented. The study shows that the thermal stresses developed in x-axis and y-axis directions have significant influence on the transient response of the half-space. The magnetic induction [Formula: see text] increases obviously when the thermal stress is applied in x-axis direction, while it decreases when the thermal stress is applied in y-axis direction. Approaching time of magnetic induction [Formula: see text] and [Formula: see text] will become longer with higher thermal stress in x-axis direction. With the growth of the thermal stress in x-direction, contribution from the electromagnetic–elastic head (EH) wave increases, while the contribution from the shear elastic (SE) wave and the static value of shear stress decrease.
考虑到理解瞬态波在压磁固体中传播的重要性，本文分析研究了热效应对承受动态反面载荷的压磁半空间瞬态行为的影响。利用单面、双面拉普拉斯变换和 Cagniard-de Hoop（CH）技术，给出了拉普拉斯域中反平面位移、剪应力、磁势和感应的高效、精确的分析推导。研究表明，在 x 轴和 y 轴方向产生的热应力对半空间的瞬态响应有显著影响。当在 x 轴方向施加热应力时，磁感应强度[计算公式：见正文]明显增加，而当在 y 轴方向施加热应力时，磁感应强度降低。随着 x 轴方向热应力的增加，磁感应强度[计算公式：见正文]和[计算公式：见正文]的接近时间会变长。随着 x 轴方向热应力的增加，电磁弹性头（EH）波的贡献增大，而剪切弹性（SE）波的贡献和剪应力的静态值减小。
{"title":"Thermal effect on the transient behavior of a piezomagnetic half-space subjected to dynamic anti-plane load","authors":"Xiang Zhou, Guoshuang Shui","doi":"10.1177/10812865241251470","DOIUrl":"https://doi.org/10.1177/10812865241251470","url":null,"abstract":"Considering the importance of understanding the propagation of transient waves in the piezomagnetic solids, the thermal effect on the transient behavior of a piezomagnetic half-space subjected to dynamic anti-plane load is investigated analytically in this paper. Using one-sided, two-sided Laplace transformation and Cagniard–de Hoop (CH) technique, an efficient and accurate analytical derivation for the solution of the anti-plane displacement, shear stress, magnetic potential, and induction in Laplace domain is presented. The study shows that the thermal stresses developed in x-axis and y-axis directions have significant influence on the transient response of the half-space. The magnetic induction [Formula: see text] increases obviously when the thermal stress is applied in x-axis direction, while it decreases when the thermal stress is applied in y-axis direction. Approaching time of magnetic induction [Formula: see text] and [Formula: see text] will become longer with higher thermal stress in x-axis direction. With the growth of the thermal stress in x-direction, contribution from the electromagnetic–elastic head (EH) wave increases, while the contribution from the shear elastic (SE) wave and the static value of shear stress decrease.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"48 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141168571","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-05-13DOI: 10.1177/10812865241243086
Erick Pruchnicki
In this work, we present a new two-scale finite-strain plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. Two scales exist, the macroscopic scale is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. This work aims to propose such a theory for thick plates in a nonlinear setting when the thickness and the size of heterogeneities are of the same order of magnitude. The homogenization theory for large deformation with growth is suitable for the modelization of nearly incompressible plant tissue. This model is suitable for wavy leaves. For thick plates, the transverse normal stress and transverse shearing are modelized at both microscopic and macroscopic levels. At the macroscopic level, we consider a nonlinear Cosserat plate model. At the microscopic level, we impose that the average of contribution of the microscopic displacement to rotation angles is equal to zero. We also deal with the problem of boundary layer problem near the lateral boundary. The model recently proposed by Pruchnicki is valid for thin heterogeneous plates; we present an extension for thick plates that takes into account both transverse normal stress and shearing. This model is equivalent to the first model presented but it involves a second-order derivative of the macroscopic displacement field.
{"title":"Homogenization of nonlinear Cosserat plate including growth theory when the thickness of the plate and the size of the in-plane heterogeneities are of the same order of magnitude","authors":"Erick Pruchnicki","doi":"10.1177/10812865241243086","DOIUrl":"https://doi.org/10.1177/10812865241243086","url":null,"abstract":"In this work, we present a new two-scale finite-strain plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. Two scales exist, the macroscopic scale is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. This work aims to propose such a theory for thick plates in a nonlinear setting when the thickness and the size of heterogeneities are of the same order of magnitude. The homogenization theory for large deformation with growth is suitable for the modelization of nearly incompressible plant tissue. This model is suitable for wavy leaves. For thick plates, the transverse normal stress and transverse shearing are modelized at both microscopic and macroscopic levels. At the macroscopic level, we consider a nonlinear Cosserat plate model. At the microscopic level, we impose that the average of contribution of the microscopic displacement to rotation angles is equal to zero. We also deal with the problem of boundary layer problem near the lateral boundary. The model recently proposed by Pruchnicki is valid for thin heterogeneous plates; we present an extension for thick plates that takes into account both transverse normal stress and shearing. This model is equivalent to the first model presented but it involves a second-order derivative of the macroscopic displacement field.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"68 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140940646","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"工程技术","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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