Pub Date : 2024-03-08DOI: 10.1177/10812865241236861
Kostas P Soldatos
This short communication rectifies an issue that may cause controversy in a recent publication and thus removes some doubt recorded in the same regarding its ability to determine the spherical part of the couple-stress in the case of large elastic deformations of isotropic polar materials.
{"title":"On the role of the invariant “I4” in conventional couple-stress hyperelasticity","authors":"Kostas P Soldatos","doi":"10.1177/10812865241236861","DOIUrl":"https://doi.org/10.1177/10812865241236861","url":null,"abstract":"This short communication rectifies an issue that may cause controversy in a recent publication and thus removes some doubt recorded in the same regarding its ability to determine the spherical part of the couple-stress in the case of large elastic deformations of isotropic polar materials.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"19 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140071178","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 : 2024-03-08DOI: 10.1177/10812865241233013
Jicheng Li, Hongling Ye, Nan Wei, Xingyu Zhang
Topology optimization is one of the most common methods for design of material distribution in mechanical metamaterials, but resulting in expensive computational cost due to iterative simulation of finite element method. In this work, a novel deep learning-based topology optimization method is proposed to design mechanical microstructure efficiently for metamaterials with extreme material properties, such as maximum bulk modulus, maximum shear modulus, or negative Poisson’s ratio. Large numbers of microstructures with various configurations are first simulated by modified solid isotropic material with penalization (SIMP), to construct the microstructure data set. Subsequently, the ResUNet involved generative and adversarial network (ResUNet-GAN) is developed for high-dimensional mapping between optimization parameters and corresponding microstructures to improve the design accuracy of ResUNet. By given optimization parameters, the well-trained ResUNet-GAN is successfully applied to the microstructure design of metamaterials with different optimization objectives under proper configurations. According to the simulation results, the proposed ResUNet-GAN-based topology optimization not only significantly reduces the computational duration for the optimization process, but also improves the structure precise and mechanical performance.
{"title":"ResUNet involved generative adversarial network-based topology optimization for design of 2D microstructure with extreme material properties","authors":"Jicheng Li, Hongling Ye, Nan Wei, Xingyu Zhang","doi":"10.1177/10812865241233013","DOIUrl":"https://doi.org/10.1177/10812865241233013","url":null,"abstract":"Topology optimization is one of the most common methods for design of material distribution in mechanical metamaterials, but resulting in expensive computational cost due to iterative simulation of finite element method. In this work, a novel deep learning-based topology optimization method is proposed to design mechanical microstructure efficiently for metamaterials with extreme material properties, such as maximum bulk modulus, maximum shear modulus, or negative Poisson’s ratio. Large numbers of microstructures with various configurations are first simulated by modified solid isotropic material with penalization (SIMP), to construct the microstructure data set. Subsequently, the ResUNet involved generative and adversarial network (ResUNet-GAN) is developed for high-dimensional mapping between optimization parameters and corresponding microstructures to improve the design accuracy of ResUNet. By given optimization parameters, the well-trained ResUNet-GAN is successfully applied to the microstructure design of metamaterials with different optimization objectives under proper configurations. According to the simulation results, the proposed ResUNet-GAN-based topology optimization not only significantly reduces the computational duration for the optimization process, but also improves the structure precise and mechanical performance.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"12 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140070912","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 : 2024-03-05DOI: 10.1177/10812865231225769
Gongye Zhang, Xin-Lin Gao, Ziwen Guo
A new non-classical model for spatial rods incorporating surface energy effects is developed using a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equations and complete boundary conditions. The newly developed spatial rod model contains three surface elasticity constants to account for surface energy effects. The new model recovers the classical elasticity-based Kirchhoff rod model as a special case when the surface energy effects are not considered. To illustrate the new spatial rod model, two sample problems are analytically solved by directly applying the general formulas derived. The first one is the buckling of an elastic rod of circular cross-section with fixed-pinned supports, and the other is the equilibrium analysis of a helical rod deformed from a straight rod. An analytical formula is derived for the critical buckling load required to perturb the axially compressed straight rod, and two closed-form expressions are obtained for the force and couple needed in deforming the helical rod. These formulas reduce to those based on classical elasticity when the surface energy effects are suppressed. For the buckling problem, it is found that the critical buckling load predicted by the current new model is always higher than that given by the classical elasticity-based model, and the difference between the two sets of predicted values is significantly large when the radius of the rod is sufficiently small but diminishes as the rod radius increases. For the helical rod problem, the numerical results reveal that the force and couple predicted by the current model are, respectively, significantly larger and smaller than those predicted by the classical model when the rod radius is very small, but the difference is diminishing with the increase of the rod radius.
{"title":"A new model for spatial rods incorporating surface energy effects","authors":"Gongye Zhang, Xin-Lin Gao, Ziwen Guo","doi":"10.1177/10812865231225769","DOIUrl":"https://doi.org/10.1177/10812865231225769","url":null,"abstract":"A new non-classical model for spatial rods incorporating surface energy effects is developed using a surface elasticity theory. A variational formulation based on the principle of minimum total potential energy is employed, which leads to the simultaneous determination of the equilibrium equations and complete boundary conditions. The newly developed spatial rod model contains three surface elasticity constants to account for surface energy effects. The new model recovers the classical elasticity-based Kirchhoff rod model as a special case when the surface energy effects are not considered. To illustrate the new spatial rod model, two sample problems are analytically solved by directly applying the general formulas derived. The first one is the buckling of an elastic rod of circular cross-section with fixed-pinned supports, and the other is the equilibrium analysis of a helical rod deformed from a straight rod. An analytical formula is derived for the critical buckling load required to perturb the axially compressed straight rod, and two closed-form expressions are obtained for the force and couple needed in deforming the helical rod. These formulas reduce to those based on classical elasticity when the surface energy effects are suppressed. For the buckling problem, it is found that the critical buckling load predicted by the current new model is always higher than that given by the classical elasticity-based model, and the difference between the two sets of predicted values is significantly large when the radius of the rod is sufficiently small but diminishes as the rod radius increases. For the helical rod problem, the numerical results reveal that the force and couple predicted by the current model are, respectively, significantly larger and smaller than those predicted by the classical model when the rod radius is very small, but the difference is diminishing with the increase of the rod radius.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"270 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-03-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140045047","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 : 2024-02-28DOI: 10.1177/10812865241230269
Mariam Mubarak Almudarra, Ariel Ramírez-Torres
This study investigates the growth of an avascular tumour described through the interchange of mass among its constituents and the production of inelastic distortions induced by growth itself. A key contribution of this research examines the role of non-local diffusion arising from the complex and heterogeneous tumour micro-environment. In our context, the non-local diffusion is enhanced by a variable-order fractional operator that incorporates crucial information about regions of limited nutrient availability within the tissue. Our research also focuses on the identification of an evolution law for the growth-induced inelastic distortions recast through the identification of non-conventional forces dual to suitable kinematic descriptors associated with the growth tensor. The establishment of such evolution law stems from examining the dissipation inequality and subsequently determining a posteriori connections between the inelastic distortions and the source/sink terms in the mass balance laws. To gain insights into the dynamics of tumour growth and its response to the proposed modelling framework, we first study how the variables governing the tissue evolution are affected by the introduction of the new growth law. Second, we investigate how regions of limited diffusion within the tumour, encoded into a fractional operator of variable-order, influence its growth.
{"title":"Examining avascular tumour growth dynamics: A variable-order non-local modelling perspective","authors":"Mariam Mubarak Almudarra, Ariel Ramírez-Torres","doi":"10.1177/10812865241230269","DOIUrl":"https://doi.org/10.1177/10812865241230269","url":null,"abstract":"This study investigates the growth of an avascular tumour described through the interchange of mass among its constituents and the production of inelastic distortions induced by growth itself. A key contribution of this research examines the role of non-local diffusion arising from the complex and heterogeneous tumour micro-environment. In our context, the non-local diffusion is enhanced by a variable-order fractional operator that incorporates crucial information about regions of limited nutrient availability within the tissue. Our research also focuses on the identification of an evolution law for the growth-induced inelastic distortions recast through the identification of non-conventional forces dual to suitable kinematic descriptors associated with the growth tensor. The establishment of such evolution law stems from examining the dissipation inequality and subsequently determining a posteriori connections between the inelastic distortions and the source/sink terms in the mass balance laws. To gain insights into the dynamics of tumour growth and its response to the proposed modelling framework, we first study how the variables governing the tissue evolution are affected by the introduction of the new growth law. Second, we investigate how regions of limited diffusion within the tumour, encoded into a fractional operator of variable-order, influence its growth.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"135 1 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140006869","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 : 2024-02-26DOI: 10.1177/10812865241231208
Pietro Liguori, Massimiliano Gei
We study the nonlinear deformation of a soft dielectric tube subjected to an external electric field induced by two outer fixed electrodes. The tube follows an electro-elastic, ideal dielectric, neo-Hookean free energy and is longitudinally either constrained or free; in general, it deforms by shrinking and finding an equilibrium configuration closer to the inner electrode. The non-homogeneous system of governing equations is solved numerically with details given for each type of boundary conditions. The notion of contraction limit is introduced and emergence of electromechanical instability for both problems is noted with the relevant modes studied at some relevant points of the stretch–voltage actuation curve.
{"title":"Large deformation of soft dielectric cylindrical tubes under external radial electric field","authors":"Pietro Liguori, Massimiliano Gei","doi":"10.1177/10812865241231208","DOIUrl":"https://doi.org/10.1177/10812865241231208","url":null,"abstract":"We study the nonlinear deformation of a soft dielectric tube subjected to an external electric field induced by two outer fixed electrodes. The tube follows an electro-elastic, ideal dielectric, neo-Hookean free energy and is longitudinally either constrained or free; in general, it deforms by shrinking and finding an equilibrium configuration closer to the inner electrode. The non-homogeneous system of governing equations is solved numerically with details given for each type of boundary conditions. The notion of contraction limit is introduced and emergence of electromechanical instability for both problems is noted with the relevant modes studied at some relevant points of the stretch–voltage actuation curve.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"21 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139977256","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 : 2024-02-19DOI: 10.1177/10812865241228825
Karim Ehab Moustafa Kamel, Thierry J Massart
The models developed for masonry and historical structures in the literature are usually classified into macromodels considering masonry as an equivalent continuum; and micromodels in which brick, blocks, or stones and mortar joints are represented explicitly. In this second category, many contributions dealt with regular bond masonry for which the geometrical description and the discretization are rather straightforward. For irregular masonry structures however, even though both finite element method (FEM) and discrete element method (DEM) approaches have been developed, obtaining a versatile geometry representation and its discretization remains much less straightforward. This becomes an important issue nowadays with the availability of image acquisition techniques, based on which computational models could be derived. The present contribution proposes an automated methodology to produce a line description of the mortar joints of an irregular masonry blocks/stones and mortar assembly, which can subsequently be used in modeling approaches. The starting point of the development is a generation technique based on inclusions packings which uses distance fields to the nearest neighboring inclusions to describe an assembly of blocks or stones geometrically. It is shown that such an assembly and the corresponding distance fields can be used to extract efficiently and in an automated way a line or lumped description of the corresponding mortar joint based on the concept of a medial axis. This line description can subsequently be used to define computational models. This is illustrated by the automated generation of FEM models that represent mortar joints by interface elements equipped with a cohesive law. Simulations on representative volume elements (RVEs) and on a wall are shown to illustrate the methodology that paves the way towards the image-based modeling of irregular masonry structures through the automated generation of cohesive zone-based models.
{"title":"Towards automated image-based cohesive zone modeling of cracking in irregular masonry","authors":"Karim Ehab Moustafa Kamel, Thierry J Massart","doi":"10.1177/10812865241228825","DOIUrl":"https://doi.org/10.1177/10812865241228825","url":null,"abstract":"The models developed for masonry and historical structures in the literature are usually classified into macromodels considering masonry as an equivalent continuum; and micromodels in which brick, blocks, or stones and mortar joints are represented explicitly. In this second category, many contributions dealt with regular bond masonry for which the geometrical description and the discretization are rather straightforward. For irregular masonry structures however, even though both finite element method (FEM) and discrete element method (DEM) approaches have been developed, obtaining a versatile geometry representation and its discretization remains much less straightforward. This becomes an important issue nowadays with the availability of image acquisition techniques, based on which computational models could be derived. The present contribution proposes an automated methodology to produce a line description of the mortar joints of an irregular masonry blocks/stones and mortar assembly, which can subsequently be used in modeling approaches. The starting point of the development is a generation technique based on inclusions packings which uses distance fields to the nearest neighboring inclusions to describe an assembly of blocks or stones geometrically. It is shown that such an assembly and the corresponding distance fields can be used to extract efficiently and in an automated way a line or lumped description of the corresponding mortar joint based on the concept of a medial axis. This line description can subsequently be used to define computational models. This is illustrated by the automated generation of FEM models that represent mortar joints by interface elements equipped with a cohesive law. Simulations on representative volume elements (RVEs) and on a wall are shown to illustrate the methodology that paves the way towards the image-based modeling of irregular masonry structures through the automated generation of cohesive zone-based models.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"115 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139948421","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 : 2024-02-19DOI: 10.1177/10812865241228230
Meir Shillor, Ian Pahuja
This work introduces and studies a novel normal compliance contact condition with damping, the Damped Normal Compliance (DNC), which is more realistic than the usual normal compliance condition that is often used in modeling contact between solids. The condition is applied in a model for the contact of a rigid mass with a reactive obstacle and allows for energy dissipation during contact. We analyze the model and show that the condition is of mathematical, as well as applied, interest. Then, we establish its relationship with the so-called “coefficient of restitution,”[Formula: see text], and show that the concept is not well defined, since [Formula: see text] depends on the system parameters, applied force, and initial data. The various theoretical results are depicted using computer simulations.
{"title":"Damped Normal Compliance (DNC) and the restitution coefficient","authors":"Meir Shillor, Ian Pahuja","doi":"10.1177/10812865241228230","DOIUrl":"https://doi.org/10.1177/10812865241228230","url":null,"abstract":"This work introduces and studies a novel normal compliance contact condition with damping, the Damped Normal Compliance (DNC), which is more realistic than the usual normal compliance condition that is often used in modeling contact between solids. The condition is applied in a model for the contact of a rigid mass with a reactive obstacle and allows for energy dissipation during contact. We analyze the model and show that the condition is of mathematical, as well as applied, interest. Then, we establish its relationship with the so-called “coefficient of restitution,”[Formula: see text], and show that the concept is not well defined, since [Formula: see text] depends on the system parameters, applied force, and initial data. The various theoretical results are depicted using computer simulations.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"48 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139948521","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 : 2024-02-16DOI: 10.1177/10812865231226183
Xiang Yu, Yibin Fu
We derive the incremental equations for a hyperelastic solid that incorporate surface tension effect by assuming that the surface energy is a general function of the surface deformation gradient. The incremental equations take the same simple form as their purely mechanical counterparts and are valid for any geometry. In particular, for isotropic materials, the extra surface elastic moduli are expressed in terms of the surface energy function and the two surface principal stretches. The effectiveness of the resulting incremental theory is illustrated by applying it to study the Plateau–Rayleigh and Wilkes instabilities in a solid cylinder.
{"title":"On the incremental equations in surface elasticity","authors":"Xiang Yu, Yibin Fu","doi":"10.1177/10812865231226183","DOIUrl":"https://doi.org/10.1177/10812865231226183","url":null,"abstract":"We derive the incremental equations for a hyperelastic solid that incorporate surface tension effect by assuming that the surface energy is a general function of the surface deformation gradient. The incremental equations take the same simple form as their purely mechanical counterparts and are valid for any geometry. In particular, for isotropic materials, the extra surface elastic moduli are expressed in terms of the surface energy function and the two surface principal stretches. The effectiveness of the resulting incremental theory is illustrated by applying it to study the Plateau–Rayleigh and Wilkes instabilities in a solid cylinder.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"42 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139948424","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 : 2024-02-16DOI: 10.1177/10812865231226311
Chiara Lonati, Alfredo Marzocchi
In this review, we discuss some conditions for achieving non-interpenetration and self-contact of solids, in particular for regular, inextensible, and closed elastic rods. We establish some equivalences between conditions that were stated sometimes independently, underlying their local or global character. We then examine three conditions related to virtual displacements and to topological characters of knots, that can be generalized to filaments, considering the midline of the loop as an inextensible regular knot.
{"title":"On self-contact and non-interpenetration of elastic rods","authors":"Chiara Lonati, Alfredo Marzocchi","doi":"10.1177/10812865231226311","DOIUrl":"https://doi.org/10.1177/10812865231226311","url":null,"abstract":"In this review, we discuss some conditions for achieving non-interpenetration and self-contact of solids, in particular for regular, inextensible, and closed elastic rods. We establish some equivalences between conditions that were stated sometimes independently, underlying their local or global character. We then examine three conditions related to virtual displacements and to topological characters of knots, that can be generalized to filaments, considering the midline of the loop as an inextensible regular knot.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"48 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139948422","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 : 2024-02-14DOI: 10.1177/10812865231221994
Sankalp Tiwari, Anindya Chatterjee
The use of global displacement basis functions to solve boundary-value problems in linear elasticity is well established. No prior work uses a global stress tensor basis for such solutions. We present two such methods for solving stress problems in linear elasticity. In both methods, we split the sought stress σ into two parts, where neither part is required to satisfy strain compatibility. The first part, σ p, is any stress in equilibrium with the loading. The second part, σ h is a self-equilibrated stress field on the unloaded body. In both methods, σ h is expanded using tensor-valued global stress basis functions developed elsewhere. In the first method, the coefficients in the expansion are found by minimizing the strain energy based on the well-known complementary energy principle. For the second method, which is restricted to planar homogeneous isotropic bodies, we show that we merely need to minimize the squared L2 norm of the trace of stress. For demonstration, we solve nine stress problems involving sharp corners, multiple-connectedness, non-zero net force and/or moment on an internal hole, body force, discontinuous surface traction, material inhomogeneity, and anisotropy. The first method presents a new application of a known principle. The second method presents a hitherto unreported principle, to the best of our knowledge.
使用全局位移基函数来求解线性弹性中的边界值问题已得到广泛认可。之前还没有工作使用全局应力张量基础来求解此类问题。我们提出了两种解决线性弹性中应力问题的方法。在这两种方法中，我们将寻求的应力 σ 分成两部分，其中任何一部分都不需要满足应变兼容性。第一部分，σ p，是与载荷平衡的任何应力。第二部分，σ h 是未加载体上的自平衡应力场。在这两种方法中，σ h 都是使用其他地方开发的张量值全局应力基函数展开的。在第一种方法中，根据著名的互补能原理，通过最小化应变能找到展开中的系数。第二种方法仅限于平面均质各向同性体，我们证明只需最小化应力迹的平方 L2 准则即可。为了演示，我们解决了九个应力问题，涉及尖角、多连通性、内孔上的非零净力和/或力矩、体力、不连续表面牵引、材料不均匀性和各向异性。第一种方法是对已知原理的新应用。据我们所知，第二种方法提出了一种迄今为止尚未报道过的原理。
{"title":"Solution of planar elastic stress problems using stress basis functions","authors":"Sankalp Tiwari, Anindya Chatterjee","doi":"10.1177/10812865231221994","DOIUrl":"https://doi.org/10.1177/10812865231221994","url":null,"abstract":"The use of global displacement basis functions to solve boundary-value problems in linear elasticity is well established. No prior work uses a global stress tensor basis for such solutions. We present two such methods for solving stress problems in linear elasticity. In both methods, we split the sought stress σ into two parts, where neither part is required to satisfy strain compatibility. The first part, σ<jats:sub> p</jats:sub>, is any stress in equilibrium with the loading. The second part, σ<jats:sub> h</jats:sub> is a self-equilibrated stress field on the unloaded body. In both methods, σ<jats:sub> h</jats:sub> is expanded using tensor-valued global stress basis functions developed elsewhere. In the first method, the coefficients in the expansion are found by minimizing the strain energy based on the well-known complementary energy principle. For the second method, which is restricted to planar homogeneous isotropic bodies, we show that we merely need to minimize the squared L<jats:sup>2</jats:sup> norm of the trace of stress. For demonstration, we solve nine stress problems involving sharp corners, multiple-connectedness, non-zero net force and/or moment on an internal hole, body force, discontinuous surface traction, material inhomogeneity, and anisotropy. The first method presents a new application of a known principle. The second method presents a hitherto unreported principle, to the best of our knowledge.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"42 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139948423","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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