Pub Date : 2024-09-04DOI: 10.1177/10812865241270732
Idan Z Friedberg, Gal deBotton
An equibiaxially stretched thin neo-Hookean circular membrane with a hole at its center under plane-stress condition is analyzed within the framework of finite deformation elasticity. Initially, we introduce a novel form for the differential governing equation to the problem. This enables the introduction of a closed-form solution in the limit of infinite stretch. Comparison of this solution to corresponding finite element simulations reveals a neat agreement for stretch ratios larger than 2.5. In the practically important case of a small hole, at the circumference of the hole, the stress concentration factor is 4 and the tangential stretch ratio is twice the applied far-field stretch ratio. These values are double the corresponding ratios in the well-known limit of infinitesimal deformation.
{"title":"Plane-stress analysis of a holed membrane at finite equibiaxial stretch","authors":"Idan Z Friedberg, Gal deBotton","doi":"10.1177/10812865241270732","DOIUrl":"https://doi.org/10.1177/10812865241270732","url":null,"abstract":"An equibiaxially stretched thin neo-Hookean circular membrane with a hole at its center under plane-stress condition is analyzed within the framework of finite deformation elasticity. Initially, we introduce a novel form for the differential governing equation to the problem. This enables the introduction of a closed-form solution in the limit of infinite stretch. Comparison of this solution to corresponding finite element simulations reveals a neat agreement for stretch ratios larger than 2.5. In the practically important case of a small hole, at the circumference of the hole, the stress concentration factor is 4 and the tangential stretch ratio is twice the applied far-field stretch ratio. These values are double the corresponding ratios in the well-known limit of infinitesimal deformation.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"41 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201730","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 : 2024-08-22DOI: 10.1177/10812865241276440
Milan Batista
This comment discusses the derivation procedure of stress distribution formulas for an infinite elastic plate with an elliptic hole under uniform tension, as presented by M. Oore and S. Oore. While the authors use a heuristic three-step procedure, it is shown that these derivations can be simplified using Maple 2023 or manually. This confirms the exactness of the authors’ formulas, asserting their role as definitive closed-form solutions.
本评论讨论了 M. Oore 和 S. Oore 提出的均匀拉伸下带椭圆孔的无限弹性板应力分布公式的推导过程。虽然作者使用了启发式的三步程序，但研究表明，这些推导可以使用 Maple 2023 或手动进行简化。这证实了作者公式的精确性，使其成为明确的闭式解。
{"title":"Comment on “Explicit solutions in Cartesian coordinates for an elliptic hole in an infinite elastic plate” by M. Oore and S. Oore","authors":"Milan Batista","doi":"10.1177/10812865241276440","DOIUrl":"https://doi.org/10.1177/10812865241276440","url":null,"abstract":"This comment discusses the derivation procedure of stress distribution formulas for an infinite elastic plate with an elliptic hole under uniform tension, as presented by M. Oore and S. Oore. While the authors use a heuristic three-step procedure, it is shown that these derivations can be simplified using Maple 2023 or manually. This confirms the exactness of the authors’ formulas, asserting their role as definitive closed-form solutions.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"13 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201731","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 : 2024-08-19DOI: 10.1177/10812865241259129
Hadi Asghari, Heiko Topol, Jesús Lacalle, José Merodio
In this article, we apply sensitivity analysis (SA) to study the pressure–inflation relation and axial force in a pressurized and extended cylindrical tube. The material consists of an isotropic ground substance that is reinforced in the azimuthal direction with one family of fibers which are taken to be dispersed about that (mean) direction. The natural configuration of the fibers may differ from that of the ground substance, either because the fibers are pre-stretched or because the bonding between the fibers and the ground substance is considered to be imperfect. The axial stretch of the cylindrical membrane is given by a constant value. The input parameters data of the mechanical system, namely, the azimuthal stretch of the cylinder, the fiber dispersion, and the fiber natural configurations, are assumed to be distributed according to three probability distribution functions. In the sensitivity analysis, we apply the Sobol method as well as the Fourier amplitude sensitivity test (FAST) method to determine the way in which variations of the input parameters affect the required inflation pressure and corresponding axial force (output variables). The implementation of the Sobol and FAST methods allows us to account for the interplay of different parameters as well as to identify the most influential parameters in both the pressure–inflation relation and the axial force. The analysis singles out all these aspects showing a rich variety of results.
在本文中，我们运用灵敏度分析法（SA）研究了加压伸长圆柱管中的压力-膨胀关系和轴向力。材料由各向同性的基体物质组成，基体物质在方位角方向上由一系纤维增强，这些纤维被认为围绕该（平均）方向分散。纤维的自然构造可能不同于研磨材料的自然构造，这可能是因为纤维是预先拉伸的，也可能是因为纤维与研磨材料之间的粘合被认为是不完美的。圆柱形薄膜的轴向拉伸由一个恒定值给出。机械系统的输入参数数据，即圆柱体的方位拉伸、纤维离散度和纤维自然配置，被假定为按照三个概率分布函数分布。在灵敏度分析中，我们采用了 Sobol 方法和傅立叶振幅灵敏度测试（FAST）方法，以确定输入参数的变化如何影响所需的充气压力和相应的轴向力（输出变量）。通过使用索博尔和 FAST 方法，我们可以考虑不同参数之间的相互作用，并确定对压力-充气关系和轴向力影响最大的参数。所有这些方面的分析都显示出丰富多样的结果。
{"title":"Sensitivity analysis of an inflated and extended fiber-reinforced membrane with different natural configurations of its constituents","authors":"Hadi Asghari, Heiko Topol, Jesús Lacalle, José Merodio","doi":"10.1177/10812865241259129","DOIUrl":"https://doi.org/10.1177/10812865241259129","url":null,"abstract":"In this article, we apply sensitivity analysis (SA) to study the pressure–inflation relation and axial force in a pressurized and extended cylindrical tube. The material consists of an isotropic ground substance that is reinforced in the azimuthal direction with one family of fibers which are taken to be dispersed about that (mean) direction. The natural configuration of the fibers may differ from that of the ground substance, either because the fibers are pre-stretched or because the bonding between the fibers and the ground substance is considered to be imperfect. The axial stretch of the cylindrical membrane is given by a constant value. The input parameters data of the mechanical system, namely, the azimuthal stretch of the cylinder, the fiber dispersion, and the fiber natural configurations, are assumed to be distributed according to three probability distribution functions. In the sensitivity analysis, we apply the Sobol method as well as the Fourier amplitude sensitivity test (FAST) method to determine the way in which variations of the input parameters affect the required inflation pressure and corresponding axial force (output variables). The implementation of the Sobol and FAST methods allows us to account for the interplay of different parameters as well as to identify the most influential parameters in both the pressure–inflation relation and the axial force. The analysis singles out all these aspects showing a rich variety of results.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"39 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201732","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 : 2024-08-19DOI: 10.1177/10812865241263788
Andrea Chiesa, Martin Kružìk, Ulisse Stefanelli
We analyze the finite-strain Poynting–Thomson viscoelastic model. In its linearized small-deformation limit, this corresponds to the serial connection of an elastic spring and a Kelvin–Voigt viscoelastic element. In the finite-strain case, the total deformation of the body results from the composition of two maps, describing the deformation of the viscoelastic element and the elastic one, respectively. We prove the existence of suitably weak solutions by a time-discretization approach based on incremental minimization. Moreover, we prove a rigorous linx earization result, showing that the corresponding small-strain model is indeed recovered in the small-loading limit.
{"title":"Finite-strain Poynting–Thomson model: Existence and linearization","authors":"Andrea Chiesa, Martin Kružìk, Ulisse Stefanelli","doi":"10.1177/10812865241263788","DOIUrl":"https://doi.org/10.1177/10812865241263788","url":null,"abstract":"We analyze the finite-strain Poynting–Thomson viscoelastic model. In its linearized small-deformation limit, this corresponds to the serial connection of an elastic spring and a Kelvin–Voigt viscoelastic element. In the finite-strain case, the total deformation of the body results from the composition of two maps, describing the deformation of the viscoelastic element and the elastic one, respectively. We prove the existence of suitably weak solutions by a time-discretization approach based on incremental minimization. Moreover, we prove a rigorous linx earization result, showing that the corresponding small-strain model is indeed recovered in the small-loading limit.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"14 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201733","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 : 2024-08-16DOI: 10.1177/10812865241269732
Kosar Samadi-Aghdam, Chongqing Ru, Peter Schiavone
We use an effective medium model to study the problem of reflection of plane waves from the free surface of a half-space occupied by an elastic particulate metacomposite. This problem has received little attention in the recent literature despite its significance from both practical and theoretical points of view. Classical formulas for the reflection angles and amplitudes of the reflected waves for a homogeneous elastic half-space with no wave attenuation are extended to a particulate metacomposite half-space with wave attenuation. We also include a detailed discussion concerning the reflected plane shear wave and surface compressional wave in the case of an incident shear wave propagating at an incident angle smaller than the critical angle. The efficiency and accuracy of the model are demonstrated via detailed comparisons between the predicted phase velocity and attenuation coefficient of plane waves in an (infinite) entire space and the corresponding results available in the literature. The implications of our results on the reflection of plane waves from the free surface of a hard sphere-filled elastic metacomposite are discussed. We mention that a quantitative validation of our results cannot be made here as a result of the lack of availability of established data in the existing literature.
{"title":"Reflection of plane waves from the free surface of a hard sphere-filled elastic metacomposite","authors":"Kosar Samadi-Aghdam, Chongqing Ru, Peter Schiavone","doi":"10.1177/10812865241269732","DOIUrl":"https://doi.org/10.1177/10812865241269732","url":null,"abstract":"We use an effective medium model to study the problem of reflection of plane waves from the free surface of a half-space occupied by an elastic particulate metacomposite. This problem has received little attention in the recent literature despite its significance from both practical and theoretical points of view. Classical formulas for the reflection angles and amplitudes of the reflected waves for a homogeneous elastic half-space with no wave attenuation are extended to a particulate metacomposite half-space with wave attenuation. We also include a detailed discussion concerning the reflected plane shear wave and surface compressional wave in the case of an incident shear wave propagating at an incident angle smaller than the critical angle. The efficiency and accuracy of the model are demonstrated via detailed comparisons between the predicted phase velocity and attenuation coefficient of plane waves in an (infinite) entire space and the corresponding results available in the literature. The implications of our results on the reflection of plane waves from the free surface of a hard sphere-filled elastic metacomposite are discussed. We mention that a quantitative validation of our results cannot be made here as a result of the lack of availability of established data in the existing literature.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"1 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201734","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 : 2024-08-14DOI: 10.1177/10812865241263039
Wenqiang Xiao, Min Ling
In this paper, we use the virtual element method to solve a history-dependent hemivariational inequality arising in contact problems. The contact problem concerns the deformation of a viscoelastic body with long memory, subjected to a contact condition with non-monotone normal compliance and unilateral constraints. A fully discrete scheme based on the trapezoidal rule for the discretization of the time integration and the virtual element method for the spatial discretization are analyzed. We provide a unified priori error analysis for both internal and external approximations. For the linear virtual element method, we obtain the optimal order error estimate. Finally, three numerical examples are reported, providing numerical evidence of the theoretically predicted optimal convergence orders.
{"title":"Virtual element method for solving a viscoelastic contact problem with long memory","authors":"Wenqiang Xiao, Min Ling","doi":"10.1177/10812865241263039","DOIUrl":"https://doi.org/10.1177/10812865241263039","url":null,"abstract":"In this paper, we use the virtual element method to solve a history-dependent hemivariational inequality arising in contact problems. The contact problem concerns the deformation of a viscoelastic body with long memory, subjected to a contact condition with non-monotone normal compliance and unilateral constraints. A fully discrete scheme based on the trapezoidal rule for the discretization of the time integration and the virtual element method for the spatial discretization are analyzed. We provide a unified priori error analysis for both internal and external approximations. For the linear virtual element method, we obtain the optimal order error estimate. Finally, three numerical examples are reported, providing numerical evidence of the theoretically predicted optimal convergence orders.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"14 4 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201736","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 : 2024-08-14DOI: 10.1177/10812865241266809
Pham Thi Ha Giang, Pham Chi Vinh
The existence of Rayleigh waves (propagating in isotropic elastic half-spaces) with the tangential and normal impedance boundary conditions was investigated. It has been shown that for the tangential impedance boundary condition (TIBC), there always exists a unique Rayleigh wave, while for the normal impedance boundary condition (NIBC), there exists a domain (of impedance and material parameters) in which exactly one Rayleigh wave is possible and outside this domain a Rayleigh wave is impossible. In this paper, we consider the existence of Rayleigh waves with the full impedance boundary condition (FIBC) that contains both TIBC and NIBC. It is shown that the existence picture of Rayleigh waves for this general case is more complicated. It contains domain for which exactly one Rayleigh wave exists, domain where a Rayleigh wave is impossible, and domain for which all three possibilities may occur: two Rayleigh waves exist, one Rayleigh wave exists, and no Rayleigh wave exists at all. The obtained existence results recover the existence results established previously for the cases of TIBC and NIBC. The formulas for the Rayleigh wave velocity are derived. As these formulas are totally explicit, they are very useful in various practical applications, especially in the non-destructive evaluation of the mechanical properties of structures. In order to establish the existence results and derive formulas for the Rayleigh wave velocity, the complex function method, which is based on the Cauchy-type integrals, is employed.
{"title":"On the existence of Rayleigh waves with full impedance boundary condition","authors":"Pham Thi Ha Giang, Pham Chi Vinh","doi":"10.1177/10812865241266809","DOIUrl":"https://doi.org/10.1177/10812865241266809","url":null,"abstract":"The existence of Rayleigh waves (propagating in isotropic elastic half-spaces) with the tangential and normal impedance boundary conditions was investigated. It has been shown that for the tangential impedance boundary condition (TIBC), there always exists a unique Rayleigh wave, while for the normal impedance boundary condition (NIBC), there exists a domain (of impedance and material parameters) in which exactly one Rayleigh wave is possible and outside this domain a Rayleigh wave is impossible. In this paper, we consider the existence of Rayleigh waves with the full impedance boundary condition (FIBC) that contains both TIBC and NIBC. It is shown that the existence picture of Rayleigh waves for this general case is more complicated. It contains domain for which exactly one Rayleigh wave exists, domain where a Rayleigh wave is impossible, and domain for which all three possibilities may occur: two Rayleigh waves exist, one Rayleigh wave exists, and no Rayleigh wave exists at all. The obtained existence results recover the existence results established previously for the cases of TIBC and NIBC. The formulas for the Rayleigh wave velocity are derived. As these formulas are totally explicit, they are very useful in various practical applications, especially in the non-destructive evaluation of the mechanical properties of structures. In order to establish the existence results and derive formulas for the Rayleigh wave velocity, the complex function method, which is based on the Cauchy-type integrals, is employed.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"10 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201735","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 : 2024-08-14DOI: 10.1177/10812865241266992
Noelia Bazarra, José R Fernández, Ramón Quintanilla
In this work, we study, from both analytical and numerical points of view, a heat conduction model which is based on the Moore-Gibson-Thompson equation. The second gradient effects are also included. First, the existence of a unique solution is proved by using the theory of linear semigroups, and the exponential energy decay is also shown when the constitutive tensors are homogeneous. The analyticity of the semigroup is also discussed in the isotropic case, and its spatial behavior is studied. The spatial exponential decay is also proved. Then, we provide the numerical analysis of a fully discrete approximation obtained by using the finite element method and an implicit Euler scheme. A discrete stability property is shown, and some a priori error estimates are derived, from which the linear convergence is concluded under suitable regularity conditions. Finally, some one-dimensional numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy.
{"title":"A Moore-Gibson-Thompson heat conduction problem with second gradient","authors":"Noelia Bazarra, José R Fernández, Ramón Quintanilla","doi":"10.1177/10812865241266992","DOIUrl":"https://doi.org/10.1177/10812865241266992","url":null,"abstract":"In this work, we study, from both analytical and numerical points of view, a heat conduction model which is based on the Moore-Gibson-Thompson equation. The second gradient effects are also included. First, the existence of a unique solution is proved by using the theory of linear semigroups, and the exponential energy decay is also shown when the constitutive tensors are homogeneous. The analyticity of the semigroup is also discussed in the isotropic case, and its spatial behavior is studied. The spatial exponential decay is also proved. Then, we provide the numerical analysis of a fully discrete approximation obtained by using the finite element method and an implicit Euler scheme. A discrete stability property is shown, and some a priori error estimates are derived, from which the linear convergence is concluded under suitable regularity conditions. Finally, some one-dimensional numerical simulations are presented to demonstrate the accuracy of the approximations and the behavior of the discrete energy.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"41 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201737","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 : 2024-08-13DOI: 10.1177/10812865241265049
Mujan N Seif, Jake Puppo, Metodi Zlatinov, Denver Schaffarzick, Alexandre Martin, Matthew J Beck
Investigating the mechanical properties of complex, porous microstructures by assessing model representative volumes is an established method of determining materials properties across a range of length scales. An understanding of how behavior evolves with length scale is essential for evaluating the material’s suitability for certain applications where the interaction volume is so small that the mechanical response originates from individual features rather than a set of features. Here, we apply the Kentucky Random Structure Toolkit (KRaSTk) to metallic foams, which are crucial to many emerging applications, among them shielding against hypervelocity impacts caused by micrometeoroids and orbital debris (MMOD). The variability of properties at feature-scale and mesoscale lengths originating from the inherently random microstructure makes developing predictive models challenging. It also hinders the optimization of components fabricated with such foams, an especially serious problem for spacecraft design where the benefit–cost–mass optimization is overshadowed by the catastrophic results of component failure. To address this problem, we compute the critical transition between the feature-scale, where mechanical properties are determined by individual features, and the mesoscale, where behavior is determined by ensembles of features. At the mesoscale, we compute distributions of properties—with respect to both expectation value and standard variability—that are consistent and predictable. A universal transition is found to occur when the side length of a cubic sample volume is ~10× greater than the characteristic length. Comparing KRaSTk-computed converged stiffness distributions with experimental measurements of a commercial metallic foam found an excellent agreement for both expectation value and standard variability at all reduced densities. Lastly, we observe that the diameter of a representative MMOD strike is ~30× shorter than the feature-scale to mesoscale transition for the foam at any reduced density, strongly implying that individual features will determine response to hypervelocity impacts, rather than bulk, or even mesoscale, structure.
{"title":"Stochastic mesoscale mechanical modeling of metallic foams","authors":"Mujan N Seif, Jake Puppo, Metodi Zlatinov, Denver Schaffarzick, Alexandre Martin, Matthew J Beck","doi":"10.1177/10812865241265049","DOIUrl":"https://doi.org/10.1177/10812865241265049","url":null,"abstract":"Investigating the mechanical properties of complex, porous microstructures by assessing model representative volumes is an established method of determining materials properties across a range of length scales. An understanding of how behavior evolves with length scale is essential for evaluating the material’s suitability for certain applications where the interaction volume is so small that the mechanical response originates from individual features rather than a set of features. Here, we apply the Kentucky Random Structure Toolkit (KRaSTk) to metallic foams, which are crucial to many emerging applications, among them shielding against hypervelocity impacts caused by micrometeoroids and orbital debris (MMOD). The variability of properties at feature-scale and mesoscale lengths originating from the inherently random microstructure makes developing predictive models challenging. It also hinders the optimization of components fabricated with such foams, an especially serious problem for spacecraft design where the benefit–cost–mass optimization is overshadowed by the catastrophic results of component failure. To address this problem, we compute the critical transition between the feature-scale, where mechanical properties are determined by individual features, and the mesoscale, where behavior is determined by ensembles of features. At the mesoscale, we compute distributions of properties—with respect to both expectation value and standard variability—that are consistent and predictable. A universal transition is found to occur when the side length of a cubic sample volume is ~10× greater than the characteristic length. Comparing KRaSTk-computed converged stiffness distributions with experimental measurements of a commercial metallic foam found an excellent agreement for both expectation value and standard variability at all reduced densities. Lastly, we observe that the diameter of a representative MMOD strike is ~30× shorter than the feature-scale to mesoscale transition for the foam at any reduced density, strongly implying that individual features will determine response to hypervelocity impacts, rather than bulk, or even mesoscale, structure.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"5 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201756","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 : 2024-08-13DOI: 10.1177/10812865241261619
Xilu Wang, Xiaoliang Cheng, Hailing Xuan
In this paper, we consider a new parabolic bilateral obstacle model. Both upper and lower obstacles are elastic-rigid and assign a non-monotone reactive normal pressure with respect to the interpenetration. The weak form of the model is a parabolic variational–hemivariational inequality with non-monotone multivalued relations in the domain. We show the existence and uniqueness of the solution. Then, a fully discrete numerical method is introduced, with the approximations can be internal or external. We bound the error estimates and obtain the Céa type inequality. Using the linear finite elements, the optimal-order error estimates are derived. Finally, we report the numerical simulation results.
{"title":"Analysis of a parabolic bilateral obstacle problem with non-monotone relations in the domain","authors":"Xilu Wang, Xiaoliang Cheng, Hailing Xuan","doi":"10.1177/10812865241261619","DOIUrl":"https://doi.org/10.1177/10812865241261619","url":null,"abstract":"In this paper, we consider a new parabolic bilateral obstacle model. Both upper and lower obstacles are elastic-rigid and assign a non-monotone reactive normal pressure with respect to the interpenetration. The weak form of the model is a parabolic variational–hemivariational inequality with non-monotone multivalued relations in the domain. We show the existence and uniqueness of the solution. Then, a fully discrete numerical method is introduced, with the approximations can be internal or external. We bound the error estimates and obtain the Céa type inequality. Using the linear finite elements, the optimal-order error estimates are derived. Finally, we report the numerical simulation results.","PeriodicalId":49854,"journal":{"name":"Mathematics and Mechanics of Solids","volume":"19 1","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142201755","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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