Research

15 pages, 1699 KiB

Open AccessArticle

An Accurate Limit Load Solution for an Anisotropic Highly Undermatched Tension Specimen with a Crack

by Sergei Alexandrov, Yun-Che Wang and Lihui Lang

Symmetry 2021, 13(10), 1941; https://doi.org/10.3390/sym13101941 - 15 Oct 2021

Viewed by 952

Plastic anisotropy significantly influences the behavior of structures subjected to various loading conditions. The extremum principles in the theory of rigid plastic solids are convenient and reliable tools for plastic design. The present paper combines the upper bound theorem and Hill’s quadratic yield [...] Read more.

Plastic anisotropy significantly influences the behavior of structures subjected to various loading conditions. The extremum principles in the theory of rigid plastic solids are convenient and reliable tools for plastic design. The present paper combines the upper bound theorem and Hill’s quadratic yield criterion for orthotropic materials to evaluate the plastic collapse load of a highly undermatched welded tensile panel with a crack in the weld. The base material is supposed to be rigid. The shape of the crack is quite arbitrary. The orientation of the principal axes of anisotropy varies through the thickness of the weld. The upper bound solution is based on an exact solution for a layer of an anisotropic material. This feature of the upper bound solution is advantageous for increasing its accuracy. A numerical treatment is only necessary to find the solution for the uncracked specimen. This specimen has two axes of symmetry, which simplifies the solution. Simple analytic formulae transform this solution into a solution for the cracked specimens with one axis of symmetry and no symmetry. It is shown that the through-thickness distribution of anisotropic properties significantly affects the limit load. Full article

(This article belongs to the Special Issue Analysis and Design of Structures Made of Plastically Anisotropic Materials 2020)

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12 pages, 1150 KiB

Open AccessArticle

Effect of Plastic Anisotropy on the Collapse of a Hollow Disk under Thermal and Mechanical Loading

by Elena Lyamina

Symmetry 2021, 13(5), 909; https://doi.org/10.3390/sym13050909 - 20 May 2021

Cited by 3 | Viewed by 1398

Abstract

Plastic anisotropy significantly affects the behavior of structures and machine parts. Given the many parameters that classify a structure made of anisotropic material, analytic and semi-analytic solutions are very useful for parametric analysis and preliminary design of such structures. The present paper is [...] Read more.

Plastic anisotropy significantly affects the behavior of structures and machine parts. Given the many parameters that classify a structure made of anisotropic material, analytic and semi-analytic solutions are very useful for parametric analysis and preliminary design of such structures. The present paper is devoted to describing the plastic collapse of a thin orthotropic hollow disk inserted into a rigid container. The disk is subject to a uniform temperature field and a uniform pressure is applied over its inner radius. The condition of axial symmetry in conjunction with the assumption of plane stress, permits an exact analytic solution. Two plastic collapse mechanisms exist. One of these mechanisms requires that the entire disk is plastic. According to the other mechanism, plastic deformation localizes at the inner radius of the disk. Additionally, two special solutions are possible. One of these solutions predicts that the entire disk becomes plastic at the initiation of plastic yielding (i.e., plastic yielding simultaneously initiates in the entire disk). The other special solution predicts that the plastic localization occurs at the inner radius of the disk with no plastic region of finite size. An essential difference between the orthotropic and isotropic disks is that plastic yielding might initiate at the outer radius of the orthotropic disk. Full article

(This article belongs to the Special Issue Analysis and Design of Structures Made of Plastically Anisotropic Materials 2020)

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15 pages, 2196 KiB

Open AccessArticle

An Exact Axisymmetric Solution in Anisotropic Plasticity

by Yaroslav Erisov, Sergei Surudin, Fedor Grechnikov and Elena Lyamina

Symmetry 2021, 13(5), 825; https://doi.org/10.3390/sym13050825 - 08 May 2021

Cited by 1 | Viewed by 1407

Abstract

A hollow cylinder of incompressible material obeying Hill’s orthotropic quadratic yield criterion and its associated flow rule is contracted on a rigid cylinder inserted in its hole. Friction occurs at the contact surface between the hollow and solid cylinders. An axisymmetric boundary value [...] Read more.

A hollow cylinder of incompressible material obeying Hill’s orthotropic quadratic yield criterion and its associated flow rule is contracted on a rigid cylinder inserted in its hole. Friction occurs at the contact surface between the hollow and solid cylinders. An axisymmetric boundary value problem for the flow of the material is formulated and solved, and the solution is in closed form. A numerical technique is only necessary for evaluating ordinary integrals. The solution may exhibit singular behavior in the vicinity of the friction surface. The exact asymptotic representation of the solution shows that some strain rate components and the plastic work rate approach infinity in the friction surface’s vicinity. The effect of plastic anisotropy on the solution’s behavior is discussed. Full article

(This article belongs to the Special Issue Analysis and Design of Structures Made of Plastically Anisotropic Materials 2020)

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12 pages, 422 KiB

Open AccessArticle

Solution Behavior Near Very Rough Walls under Axial Symmetry: An Exact Solution for Anisotropic Rigid/Plastic Material

by Sergei Alexandrov, Elena Lyamina and Pierre-Yves Manach

Symmetry 2021, 13(2), 184; https://doi.org/10.3390/sym13020184 - 24 Jan 2021

Cited by 2 | Viewed by 1296

Abstract

Rigid plastic material models are suitable for modeling metal forming processes at large strains where elastic effects are negligible. A distinguished feature of many models of this class is that the velocity field is describable by non-differentiable functions in the vicinity of certain [...] Read more.

Rigid plastic material models are suitable for modeling metal forming processes at large strains where elastic effects are negligible. A distinguished feature of many models of this class is that the velocity field is describable by non-differentiable functions in the vicinity of certain friction surfaces. Such solution behavior causes difficulty with numerical solutions. On the other hand, it is useful for describing some material behavior near the friction surfaces. The exact asymptotic representation of singular solution behavior near the friction surface depends on constitutive equations and certain conditions at the friction surface. The present paper focuses on a particular boundary value problem for anisotropic material obeying Hill’s quadratic yield criterion under axial symmetry. This boundary value problem represents the deformation mode that appears in the vicinity of frictional interfaces in a class of problems. In this respect, the applied aspect of the boundary value problem is not essential, but the exact mathematical analysis can occur without relaxing the original system of equations and boundary conditions. We show that some strain rate and spin components follow an inverse square rule near the friction surface. An essential difference from the available analysis under plane strain conditions is that the system of equations is not hyperbolic. Full article

(This article belongs to the Special Issue Analysis and Design of Structures Made of Plastically Anisotropic Materials 2020)

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16 pages, 1678 KiB

Open AccessArticle

Finite Pure Plane Strain Bending of Inhomogeneous Anisotropic Sheets

by Sergei Alexandrov, Elena Lyamina and Yeong-Maw Hwang

Symmetry 2021, 13(1), 145; https://doi.org/10.3390/sym13010145 - 16 Jan 2021

Cited by 5 | Viewed by 2035

Abstract

The present paper concerns the general solution for finite plane strain pure bending of incompressible, orthotropic sheets. In contrast to available solutions, the new solution is valid for inhomogeneous distributions of plastic properties. The solution is semi-analytic. A numerical treatment is only necessary [...] Read more.

The present paper concerns the general solution for finite plane strain pure bending of incompressible, orthotropic sheets. In contrast to available solutions, the new solution is valid for inhomogeneous distributions of plastic properties. The solution is semi-analytic. A numerical treatment is only necessary for solving transcendent equations and evaluating ordinary integrals. The solution’s starting point is a transformation between Eulerian and Lagrangian coordinates that is valid for a wide class of constitutive equations. The symmetric distribution relative to the center line of the sheet is separately treated where it is advantageous. It is shown that this type of symmetry simplifies the solution. Hill’s quadratic yield criterion is adopted. Both elastic/plastic and rigid/plastic solutions are derived. Elastic unloading is also considered, and it is shown that reverse plastic yielding occurs at a relatively large inside radius. An illustrative example uses real experimental data. The distribution of plastic properties is symmetric in this example. It is shown that the difference between the elastic/plastic and rigid/plastic solutions is negligible, except at the very beginning of the process. However, the rigid/plastic solution is much simpler and, therefore, can be recommended for practical use at large strains, including calculating the residual stresses. Full article

(This article belongs to the Special Issue Analysis and Design of Structures Made of Plastically Anisotropic Materials 2020)

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16 pages, 1136 KiB

Open AccessArticle

Influence of the Replacement of the Actual Plastic Orthotropy with Various Approximations of Normal Anisotropy on Residual Stresses and Strains in a Thin Disk Subjected to External Pressure

by Yaroslav Erisov, Sergei Surudin, Sergei Alexandrov and Lihui Lang

Symmetry 2020, 12(11), 1834; https://doi.org/10.3390/sym12111834 - 05 Nov 2020

Cited by 1 | Viewed by 1428

Abstract

Plastic anisotropy is very common to metallic materials. This property may significantly affect the performance of structures. However, the actual orthotropic yield criterion is often replaced with a criterion based on the assumption of normal anisotropy. The present paper aims to reveal the [...] Read more.

Plastic anisotropy is very common to metallic materials. This property may significantly affect the performance of structures. However, the actual orthotropic yield criterion is often replaced with a criterion based on the assumption of normal anisotropy. The present paper aims to reveal the influence of this replacement on the distribution of strains and residual strains in a thin hollow disk under plane stress conditions. The boundary-value problem is intentionally formulated such that it is possible to obtain an exact semi-analytical solution without relaxing the boundary conditions. It is assumed that the disk is loaded by external pressure, followed by elastic unloading. The comparative analysis of the distributions of residual strains shows a significant deviation of the distribution resulting from the solutions based on the assumption of normal anisotropy from the distribution found using the actual orthotropic yield criterion. This finding shows that replacing the actual orthotropic yield criterion with the assumption of normal anisotropy may result in very inaccurate predictions. The type of anisotropy accepted is of practical importance because it usually results from such processes as drawing end extrusion with an axis of symmetry. Full article

(This article belongs to the Special Issue Analysis and Design of Structures Made of Plastically Anisotropic Materials 2020)

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12 pages, 2166 KiB

Open AccessArticle

A Limit Load Solution for Anisotropic Welded Cracked Plates in Pure Bending

by Sergei Alexandrov, Elena Lyamina, Alexander Pirumov and Dinh Kien Nguyen

Symmetry 2020, 12(11), 1764; https://doi.org/10.3390/sym12111764 - 24 Oct 2020

Cited by 3 | Viewed by 1501

Abstract

The present paper’s main objective is to derive a simple upper bound solution for a welded plate in pure bending. The plate contains a crack located in the weld. Both the weld and base materials are orthotropic. Hill’s quadratic yield criterion is adopted. [...] Read more.

The present paper’s main objective is to derive a simple upper bound solution for a welded plate in pure bending. The plate contains a crack located in the weld. Both the weld and base materials are orthotropic. Hill’s quadratic yield criterion is adopted. The solution is semi-analytic. A numerical method is only required for minimizing a function of two independent variables. Six independent dimensionless parameters classify the structure. Therefore, the complete parametric analysis of the solution is not feasible. However, for a given set of parameters, the numerical solution is straightforward, and the numerical method is fast. A numerical example emphasizes the effect of plastic anisotropy and the crack’s location on the bending moment at plastic collapse. In particular, the bending moment for the specimen having a vertical axis of symmetry is compared with that of the asymmetric specimen. It is shown that the latter is smaller for all considered cases. The solution found can be used in conjunction with flaw assessment procedures. Full article

(This article belongs to the Special Issue Analysis and Design of Structures Made of Plastically Anisotropic Materials 2020)

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12 pages, 9668 KiB

Open AccessArticle

Influence of Plastic Anisotropy on the Limit Load of an Overmatched Cracked Tension Specimen

by Elena Lyamina, Nataliya Kalenova and Dinh Kien Nguyen

Symmetry 2020, 12(7), 1079; https://doi.org/10.3390/sym12071079 - 01 Jul 2020

Cited by 2 | Viewed by 1506

Abstract

Plastic anisotropy is a common property of many metallic materials. This property affects many aspects of structural analysis and design. In contrast to the isotropic case, there is a great variety of yield criteria proposed for anisotropic materials. Moreover, even if one specific [...] Read more.

Plastic anisotropy is a common property of many metallic materials. This property affects many aspects of structural analysis and design. In contrast to the isotropic case, there is a great variety of yield criteria proposed for anisotropic materials. Moreover, even if one specific yield criterion is selected, several constitutive parameters are involved in it. Therefore, parametric analysis of structures made of anisotropic materials is quite cumbersome. The present paper demonstrates the effect of the constitutive parameters involved in Hill’s quadratic yield criterion on the upper bound limit load for weld stretched overmatched tension specimens containing a crack of arbitrary shape, assuming that the crack is located inside the weld. Different sets of the constitutive parameters are involved in the yield criteria for weld and base materials. Since the limit load is an input parameter of many flaw assessment procedures, the final result of the present paper shows that it is necessary to take into account plastic anisotropy in these procedures. It is worthy of note that the limit load is involved in the flaw assessment procedures in combination with the stress and strain fields near the tip of a crack. In anisotropic materials, these fields may become non-symmetric even under symmetric loading. This behavior affects the propagation of cracks. Full article

(This article belongs to the Special Issue Analysis and Design of Structures Made of Plastically Anisotropic Materials 2020)

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