Pub Date : 2023-01-01DOI: 10.21638/11701/spbu10.2023.207
N. Egorov, A. G. Karpov, Sergey P. Sobolev
One of the most informative methods for studying dielectric materials is the analysis of their polarization and depolarization processes. The most suitable representation is the dielectric diagram in the complex plane (Argand diagram). Often, the dielectric diagram is represented as a collection of intersecting parts of circles, at least for preprocessing. The paper proposes means of modeling the dielectric diagram by means of a set of parts of circles.
{"title":"Mathematical methods for analyzing the results of investigation of dielectric","authors":"N. Egorov, A. G. Karpov, Sergey P. Sobolev","doi":"10.21638/11701/spbu10.2023.207","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2023.207","url":null,"abstract":"One of the most informative methods for studying dielectric materials is the analysis of their polarization and depolarization processes. The most suitable representation is the dielectric diagram in the complex plane (Argand diagram). Often, the dielectric diagram is represented as a collection of intersecting parts of circles, at least for preprocessing. The paper proposes means of modeling the dielectric diagram by means of a set of parts of circles.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81905984","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.21638/11701/spbu10.2023.110
A. Fominyh, V. Karelin, L. Polyakova
The article considers the problem of optimal control of an object described by a system of ordinary differential equations with a continuously differentiable right-hand side and with a nonsmooth (but only a quasidifferentiable) quality functional. The problem is in the Mayer form with either free or partially fixed right end. Piecewise-continuous and bounded controls are supposed to be admissible if they lie in some parallelepiped at any moment of time. The phase coordinates and controls are also subject to mixed pointwise constraints. Phase constraints are taken into account by introducing new variables with known boundary conditions into the system. The standard discretization of the original system and the parametrization of the control are carried out, theorems are given on the convergence of the solution of the discrete system obtained to the desired solution of the continuous problem. Further, in order to study the resulting discrete system, the apparatus of quasidifferential calculus is used and the method of the quasidifferential descent is applied. Examples illustrating the operation of the algorithm are given.
{"title":"Method for solving an optimal control problem in the Mayer form with a quasidifferentiable functional in the presence of phase constraints","authors":"A. Fominyh, V. Karelin, L. Polyakova","doi":"10.21638/11701/spbu10.2023.110","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2023.110","url":null,"abstract":"The article considers the problem of optimal control of an object described by a system of ordinary differential equations with a continuously differentiable right-hand side and with a nonsmooth (but only a quasidifferentiable) quality functional. The problem is in the Mayer form with either free or partially fixed right end. Piecewise-continuous and bounded controls are supposed to be admissible if they lie in some parallelepiped at any moment of time. The phase coordinates and controls are also subject to mixed pointwise constraints. Phase constraints are taken into account by introducing new variables with known boundary conditions into the system. The standard discretization of the original system and the parametrization of the control are carried out, theorems are given on the convergence of the solution of the discrete system obtained to the desired solution of the continuous problem. Further, in order to study the resulting discrete system, the apparatus of quasidifferential calculus is used and the method of the quasidifferential descent is applied. Examples illustrating the operation of the algorithm are given.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76169199","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.21638/11701/spbu10.2023.204
A. V. Kvasnov, Anatoliy A. Baranenko, Evgeniy Y. Butyrsky, Uliana P. Zaranik
The principle of maximum entropy (ME) has a number of advantages that allow it to be used in machine learning. The density distribution of maximum entropy (WEO) requires solving the problem of calculus of variations on the a priori distribution, where the central tendency can be used as a parameter. In Lebesgue space, the central tendency is described by the generalized Gelder average. The paper shows the evolution of the density of the ME distribution depending on the given norm of the average. The minimum Kulback — Leibler divergence between the WEO and the a prior density is achieved at the harmonic mean, which is effective in reducing the dimensionality of the training sample. At the same time, this leads to a deterioration in the function of loss in the conditions of machine learning by precedents.
{"title":"On the influence of the cental trend on the nature of the density distribution of maximum entropy in machine learning","authors":"A. V. Kvasnov, Anatoliy A. Baranenko, Evgeniy Y. Butyrsky, Uliana P. Zaranik","doi":"10.21638/11701/spbu10.2023.204","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2023.204","url":null,"abstract":"The principle of maximum entropy (ME) has a number of advantages that allow it to be used in machine learning. The density distribution of maximum entropy (WEO) requires solving the problem of calculus of variations on the a priori distribution, where the central tendency can be used as a parameter. In Lebesgue space, the central tendency is described by the generalized Gelder average. The paper shows the evolution of the density of the ME distribution depending on the given norm of the average. The minimum Kulback — Leibler divergence between the WEO and the a prior density is achieved at the harmonic mean, which is effective in reducing the dimensionality of the training sample. At the same time, this leads to a deterioration in the function of loss in the conditions of machine learning by precedents.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81902209","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-06-22DOI: 10.21638/11701/spbu10.2023.205
Almaz Mustafin, Aliya Kantarbayeva
Clearing functions (CFs), which express a mathematical relationship between the expected throughput of a production facility in a planning period and its workload (or work-inprogress, WIP) in that period have shown considerable promise for modeling WIP-dependent cycle times in production planning. While steady-state queueing models are commonly used to derive analytic expressions for CFs, the finite length of planning periods calls their validity into question. We apply a different approach to propose a mechanistic model for one-resource, one-product factory shop based on the analogy between the operation of machine and enzyme molecule. The model is reduced to a singularly perturbed system of two differential equations for slow (WIP) and fast (busy machines) variables, respectively. The analysis of this slow-fast system finds that CF is nothing but a result of the asymptotic expansion of the slow invariant manifold. The validity of CF is ultimately determined by how small is the parameter multiplying the derivative of the fast variable. It is shown that sufficiently small characteristic ratio ’working machines: WIP’ guarantees the applicability of CF approximation in unsteady-state operation.
{"title":"Clearing function in the context of the invariant manifold method","authors":"Almaz Mustafin, Aliya Kantarbayeva","doi":"10.21638/11701/spbu10.2023.205","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2023.205","url":null,"abstract":"Clearing functions (CFs), which express a mathematical relationship between the expected throughput of a production facility in a planning period and its workload (or work-inprogress, WIP) in that period have shown considerable promise for modeling WIP-dependent cycle times in production planning. While steady-state queueing models are commonly used to derive analytic expressions for CFs, the finite length of planning periods calls their validity into question. We apply a different approach to propose a mechanistic model for one-resource, one-product factory shop based on the analogy between the operation of machine and enzyme molecule. The model is reduced to a singularly perturbed system of two differential equations for slow (WIP) and fast (busy machines) variables, respectively. The analysis of this slow-fast system finds that CF is nothing but a result of the asymptotic expansion of the slow invariant manifold. The validity of CF is ultimately determined by how small is the parameter multiplying the derivative of the fast variable. It is shown that sufficiently small characteristic ratio ’working machines: WIP’ guarantees the applicability of CF approximation in unsteady-state operation.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87497215","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.21638/11701/spbu10.2022.304
A. Tur, O. Petrosian
In the paper we consider a class of the differential games with continuous updating with random information horizon. It is assumed that at each time instant, players have information about the game (motion equations and payoff functions) for a time interval with the length theta and as the time evolves information about the game updates. We first considered this type of games in 2019. Here we additionally assume that theta is a random variable. The subject of the current paper is definition of Nash equilibrium based solution concept and solution technique based on Hamilton—Jacobi—Bellman equations.
{"title":"Random information horizon for a class of differential games with continuous updating","authors":"A. Tur, O. Petrosian","doi":"10.21638/11701/spbu10.2022.304","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2022.304","url":null,"abstract":"In the paper we consider a class of the differential games with continuous updating with random information horizon. It is assumed that at each time instant, players have information about the game (motion equations and payoff functions) for a time interval with the length theta and as the time evolves information about the game updates. We first considered this type of games in 2019. Here we additionally assume that theta is a random variable. The subject of the current paper is definition of Nash equilibrium based solution concept and solution technique based on Hamilton—Jacobi—Bellman equations.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84940372","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.21638/11701/spbu10.2022.206
O. Mitrofanova, Eugenii P. Mitrofanov, Nataliya A. Bure
Thanks to the development of information technologies and computing resources, it became possible to obtain and process big data, including geospatial data. Most research in the field of precision farming is interdisciplinary in nature, with experimental field data used by disparate scientific groups. In this connection, it became necessary to develop a unified web-based system for storing, organizing, and exchanging experimental information between researchers. The first step in achieving this goal was to create a geospatial database. Since the system being developed in the future may require extensions, modifications, adjustments, integration into other projects, it seems appropriate to use the ontology to form the database structure. The most popular tools were used as the main tools: the ontology language OWL (Ontology Web Language), the Protege 5.5 development environment. The main initial information obtained in the course of experimental studies carried out at the biopolygon: weather data, agrochemical indicators (sampling of soil and plants with georeferencing), agrophysical parameters (humidity, electrical conductivity), remote sensing data. Based on the results of the analysis of the current state of research in the field of storage and systematization of experimental information in crop production, as well as a survey of ARI employees, a prototype of the database structure was formed based on the ontological approach. Nine parent classes were defined as the foundation: Field, Crop rotation - experience, Agrotechnology, Yield, Meteo, Ground samples, Orthophoto, Calendar, and Dictionary - units of measurement.
由于信息技术和计算资源的发展，获取和处理包括地理空间数据在内的大数据成为可能。精准农业领域的大多数研究本质上是跨学科的，不同的科学团体使用了实验现场的数据。在这方面，有必要开发一个统一的基于web的系统来存储、组织和交换研究人员之间的实验信息。实现这一目标的第一步是创建地理空间数据库。由于将来开发的系统可能需要扩展、修改、调整、集成到其他项目中，因此使用本体来形成数据库结构似乎是合适的。使用最流行的工具作为主要工具:本体语言OWL (ontology Web language)、Protege 5.5开发环境。在生物多边形进行的实验研究过程中获得的主要初步信息:天气数据、农业化学指标(土壤和植物的地理参照取样)、农业物理参数(湿度、电导率)、遥感数据。在分析作物生产实验信息存储与系统化研究现状的基础上，通过对ARI员工的调查，建立了基于本体论方法的数据库结构原型。9个父类被定义为基础:田地，作物轮作-经验，农业技术，产量，气象，地面样本，正射影像，日历和字典-测量单位。
{"title":"ONTOLOGICAL APPROACH APPLICATION TO THE DESIGN OF A GEOSPATIAL EXPERIMENTAL DATABASE FOR INFORMATION SUPPORT OF RESEARCH IN PRECISION AGRICULTURE","authors":"O. Mitrofanova, Eugenii P. Mitrofanov, Nataliya A. Bure","doi":"10.21638/11701/spbu10.2022.206","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2022.206","url":null,"abstract":"Thanks to the development of information technologies and computing resources, it became possible to obtain and process big data, including geospatial data. Most research in the field of precision farming is interdisciplinary in nature, with experimental field data used by disparate scientific groups. In this connection, it became necessary to develop a unified web-based system for storing, organizing, and exchanging experimental information between researchers. The first step in achieving this goal was to create a geospatial database. Since the system being developed in the future may require extensions, modifications, adjustments, integration into other projects, it seems appropriate to use the ontology to form the database structure. The most popular tools were used as the main tools: the ontology language OWL (Ontology Web Language), the Protege 5.5 development environment. The main initial information obtained in the course of experimental studies carried out at the biopolygon: weather data, agrochemical indicators (sampling of soil and plants with georeferencing), agrophysical parameters (humidity, electrical conductivity), remote sensing data. Based on the results of the analysis of the current state of research in the field of storage and systematization of experimental information in crop production, as well as a survey of ARI employees, a prototype of the database structure was formed based on the ontological approach. Nine parent classes were defined as the foundation: Field, Crop rotation - experience, Agrotechnology, Yield, Meteo, Ground samples, Orthophoto, Calendar, and Dictionary - units of measurement.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88302607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.21638/11701/spbu10.2022.408
L. Polyakova
For an arbitrary convex function, using the infimal convolution operation, a family of continuously differentiable convex functions approximating it is constructed. The constructed approximating family of smooth convex functions Kuratowski converges to the function under consideration. If the domain of the considered function is compact, then such smooth convex approximations are uniform in the Chebyshev metric. The approximation of a convex set by a family of smooth convex sets is also considered.
{"title":"Smooth approximations of nonsmooth convex functions","authors":"L. Polyakova","doi":"10.21638/11701/spbu10.2022.408","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2022.408","url":null,"abstract":"For an arbitrary convex function, using the infimal convolution operation, a family of continuously differentiable convex functions approximating it is constructed. The constructed approximating family of smooth convex functions Kuratowski converges to the function under consideration. If the domain of the considered function is compact, then such smooth convex approximations are uniform in the Chebyshev metric. The approximation of a convex set by a family of smooth convex sets is also considered.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89517350","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.21638/11701/spbu10.2022.414
E. Lezhnina, Gennady M. Khitrov, Elizaveta A. Kalinina, Ivan A. Kononov
In modern management, one of the most popular and effective approaches to organizing activities is the creation of temporary working groups. An important indicator is the effectiveness of such groups, which strongly depends on the internal structure of the working group and the relationship between team members. For the effective functioning of the group, it is necessary that the group consists of a leader and subordinates. The artificial appointment of a team leader often proves ineffective. The article solves the problem of dividing the team into working groups, taking into account the relationship between individual personalities. Two algorithms are proposed: the first one divides the team into separate groups of possible leaders and subordinates, the second one makes effective working groups. The effectiveness of the group is calculated based on the ee efficiency function, which takes into account interpersonal relationships. Practical and numerical experiments were carried out to verify the adequacy of the model.
{"title":"Working groups: Creation and management","authors":"E. Lezhnina, Gennady M. Khitrov, Elizaveta A. Kalinina, Ivan A. Kononov","doi":"10.21638/11701/spbu10.2022.414","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2022.414","url":null,"abstract":"In modern management, one of the most popular and effective approaches to organizing activities is the creation of temporary working groups. An important indicator is the effectiveness of such groups, which strongly depends on the internal structure of the working group and the relationship between team members. For the effective functioning of the group, it is necessary that the group consists of a leader and subordinates. The artificial appointment of a team leader often proves ineffective. The article solves the problem of dividing the team into working groups, taking into account the relationship between individual personalities. Two algorithms are proposed: the first one divides the team into separate groups of possible leaders and subordinates, the second one makes effective working groups. The effectiveness of the group is calculated based on the ee efficiency function, which takes into account interpersonal relationships. Practical and numerical experiments were carried out to verify the adequacy of the model.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89693249","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.21638/11701/spbu10.2022.412
Yu. A. Salamatov, E. S. Nikova, D. I. Devyaterikov, Evgeny A. Kravtsov
This work analyzes two mathematical algorithms for processing experimental curves in polarized neutron reflectometry, one of which makes it possible to determine the complex reflection coefficient. The approbation was carried out for a Fe/Cr type superlattice with an irregular non-collinear ordering of the magnetic moments of the Fe layers. The processing of the experiment was carried out both by direct refinement of the structure parameters and by the method of calculating the module and phase of the reflectometry signal using the Gd reference layer. The results obtained with different methods are compared with each other. To clarify the structural and magnetic characteristics, the Levenberg-Marquardt algorithm was applied in both cases. The obtained data on the magnetic structure is in agreement with the theoretical model of magnetization of a layered antiferromagnet of finite dimensions in a weak field. The presented modification of the reference layer method can be considered as the phase problem solution in polarized neutron reflectometry.
{"title":"Recovering complex reflection coefficient using the reference layer algorithm for multilayer systems with non-collinear magnetic ordering","authors":"Yu. A. Salamatov, E. S. Nikova, D. I. Devyaterikov, Evgeny A. Kravtsov","doi":"10.21638/11701/spbu10.2022.412","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2022.412","url":null,"abstract":"This work analyzes two mathematical algorithms for processing experimental curves in polarized neutron reflectometry, one of which makes it possible to determine the complex reflection coefficient. The approbation was carried out for a Fe/Cr type superlattice with an irregular non-collinear ordering of the magnetic moments of the Fe layers. The processing of the experiment was carried out both by direct refinement of the structure parameters and by the method of calculating the module and phase of the reflectometry signal using the Gd reference layer. The results obtained with different methods are compared with each other. To clarify the structural and magnetic characteristics, the Levenberg-Marquardt algorithm was applied in both cases. The obtained data on the magnetic structure is in agreement with the theoretical model of magnetization of a layered antiferromagnet of finite dimensions in a weak field. The presented modification of the reference layer method can be considered as the phase problem solution in polarized neutron reflectometry.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76741217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-01DOI: 10.21638/11701/spbu10.2022.109
Abdulla M. Abakarov, Y. Pronina
The effective properties of cracked solids are often expressed in terms of the crack density parameter or its tensor generalization, using the approximation of noninteracting cracks. This approximation remains accurate at sufficiently high crack densities, provided the location of cracks are random. The presented analysis confirms that the effective elastic moduli of a material with ordered fracture structures strongly depend on the linear dimensions of cracks and their mutual arrangement even at a constant crack density. A change in these parameters can cause a noticeable anisotropy of the effective properties of the material even when the eigenvalues of the crack density tensor are equal to each other. The effective elastic characteristics of a material with one doubly periodic system of parallel cracks are compared with those for a material with two mutually perpendicular systems of such cracks in a two-dimensional formulation. The calculations are carried out using the approximate method of M. Kachanov for determining the mean stresses at the cracks edges, applicable for large systems of interacting cracks. Analysis of the obtained results showed that the effective compliance of the material in a certain direction is largely determined by the effects of interaction (shielding and amplification) within a system of parallel cracks perpendicular to this direction. The interaction of this system of cracks with the perpendicular system has a weak effect on the indicated properties in the case of rectangular symmetry of the system. In this case, the interaction of mutually perpendicular systems of cracks leads to a violation of the symmetry of the tensor of effective elastic constants.
{"title":"On the effective elastic properties of a material with mutually perpendicular systems of parallel cracks","authors":"Abdulla M. Abakarov, Y. Pronina","doi":"10.21638/11701/spbu10.2022.109","DOIUrl":"https://doi.org/10.21638/11701/spbu10.2022.109","url":null,"abstract":"The effective properties of cracked solids are often expressed in terms of the crack density parameter or its tensor generalization, using the approximation of noninteracting cracks. This approximation remains accurate at sufficiently high crack densities, provided the location of cracks are random. The presented analysis confirms that the effective elastic moduli of a material with ordered fracture structures strongly depend on the linear dimensions of cracks and their mutual arrangement even at a constant crack density. A change in these parameters can cause a noticeable anisotropy of the effective properties of the material even when the eigenvalues of the crack density tensor are equal to each other. The effective elastic characteristics of a material with one doubly periodic system of parallel cracks are compared with those for a material with two mutually perpendicular systems of such cracks in a two-dimensional formulation. The calculations are carried out using the approximate method of M. Kachanov for determining the mean stresses at the cracks edges, applicable for large systems of interacting cracks. Analysis of the obtained results showed that the effective compliance of the material in a certain direction is largely determined by the effects of interaction (shielding and amplification) within a system of parallel cracks perpendicular to this direction. The interaction of this system of cracks with the perpendicular system has a weak effect on the indicated properties in the case of rectangular symmetry of the system. In this case, the interaction of mutually perpendicular systems of cracks leads to a violation of the symmetry of the tensor of effective elastic constants.","PeriodicalId":43738,"journal":{"name":"Vestnik Sankt-Peterburgskogo Universiteta Seriya 10 Prikladnaya Matematika Informatika Protsessy Upravleniya","volume":null,"pages":null},"PeriodicalIF":0.6,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78011232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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