CONSTRUCTION OF ELASTO-PLASTIC BOUNDARIES using CONSERVATION LAWS

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Introduction. Solution of elasto-plastic problems is one of the most complicated and actual problems of solid mechanics. It is determined by the fact that elasto-plastic boundary is not known in advance and should be defined during the solution of a problem. The elasto-plastic problems were considered by many well-known mechanicians. One can find a good review in works of B. D. Annin and G. P. Cherepanov [1], L. A. Galin [2; 3]. For the moment, a common approach for solving such problems has not been worked out yet. There are only a few single solutions for different special cases. As classical results one should consider an exact solution for the problem of elasto-plastic torsion of a rod with oval cross-section constructed by V. V. Sokolovsky, as well as solution of L. A. Galin for the problem of straining of a plane with circular hole. An interesting theoretical result was obtained by B. D. Annin [1]. He proved the unique existence for the problem of elasto-plastic torsion of the rod with oval cross-section. For solving of the elasto-plastic problem the methods of complex analysis, calculus of variations or semi-inverse methods were applied. In this paper, for construction of the elasto-plastic boundary the conservation laws were used. The conservation laws were applied in works [4-6] for the solving of the problem of 2-dimensional ideal plasticity; they allowed to obtain analytical solutions of Cauchy and Riemann problems. In following works of one of the co-authors of the present article the conservation laws were used for solving of some elasto-plastic problems [7; 8]. The obtained method allowed to write an algorithm (computer programs) for construction of elasto-plastic boundaries of twisted straight rods. For these programs the certificates of State registration are got [9; 10]. In present paper the elasto-plastic boundary for the problem of torsion of a straight beam of arbitrary cross-section, which is limited by a piecewise smooth contour is constructed; and the elasto-plastic boundaries for the problems of a consol bending and anti-plane deformation are found. For convenience, the article is divided into sections. In the first section the basic equations of elasticity and boundary problems are considered; in the second section the basic equations of the theory of ideal plasticity of von Mises are given; in the third section the conditions on the boundaries of the elastic and plastic domains are formulated. The fourth section is devoted to torsion of elastic prismatic rods; the fifth one describes elastic bending of bars; in the sixth section the plane problem of theory of elasticity is given. The seventh section covers an anti-plane problem of elasticity theory; in the eighth section, conservation laws for the equations of elasticity are constructed; in the ninth one, conservation laws of two-dimensional equations of plasticity are discussed. In the tenth section an elasto-plastic boundary of a twisted straight rod is found; in the eleventh one an elasto-plastic boundary in the bended console is given; and finally, in the twelfth section a method for the construction of elasto-plastic boundaries for large areas is described. 1. The Basic Equations of Elasticity and Boundary Problems. Let’s consider steady-state equations of linear isotropic elasticity. The equilibrium equations look like: (1) here are components of a stress tensor, are components of an external force affected to a unit of volume. The components of a stress tensor related to components of a strain tensor by means of Hook’s law: (2) Here and are constants of Lamé, are components of a vector of deformations, are components of a strain tensor. Taking into account (2), equations of theory of elasticity can be written using displacements: (3) here ∆ is the Laplace operator. On account of (2) the components of a stress tensor are in accord with the compatibility equations along with the equilibrium equations (1). The compatibility equations here are written in case of absence of external forces: (4) , is a Poisson’s ratio. Problems for elasticity equations are usually pose either using displacements (in this case one have to solve equations (3)) or using stresses (in that case one solves equations (1)-(3)). If a problem is written using stresses, one should add boundary conditions to equations (1), (3): (5) here are direction cosines of an external normal line to the boundary surface at point under study, are components of a vector of superficial forces affected to a unit of area. If the problem is written in displacements then on a boundary S these displacements are specified: (6) here are certain functions on S. Remark. There are others problems in the theory of elasticity, but they are not adduced in this article. 2. The Basic Equations of the Theory of Plasticity of Mises. For steady-state equations of the theory of plasticity of Mises one should add the plasticity law of Mises to the equilibrium equations (1). This law looks like: (7) here k is a yield point under simple shear. In the case of plane deformation the plasticity law (7) can be reduce to form: (8) In plastic domain, the components of deviator of the strain tensor relate to the components of tensor of a strain rate with correlations (9) here is a nonnegative function obtained from (7): (10) here are components of the vector of strain rate. 3. Conditions on the Boundaries of the Elastic and Plastic domains. Determination of a boundary separating elastic and plastic domains is one of the most difficult problems of the solid mechanics. The boundary is not known in advance and is defined during the elasto-plastic problem solving. In some cases a shape of the boundary can be guessed by general considerations. Assume that an elastic state of a medium continuously changes over to a yield state. In this case close to elasto-plastic boundary and on each side of it the Hook’s law is applies. This fact leads to the continuity of all components of the stress tensor and strain tensor, on the elasto-plastic boundary. 4. Torsion of Elastic Prismatic Rods. Let’s consider an elastic prismatic rod with a cross-section of a variable form. It’s lateral surface is free from efforts, face planes have forces equivalent to rotational moment M. Let the coordinate origin is placed in an arbitrary point of the face plane and axis z is parallel to generatrix of the rod. The boundary conditions (3) will look like: (11) and on the face planes of the rod () (12) (13) here is the area of a cross-section. As is the convention in the theory of torsion: (14) and the remaining components of the stress tensor are in accord with equilibrium equations (1) which are the following form taking into account (14): (15) Compatibility equations (4) will look like (16) From the equations (16) one can get (17) here A, B, D, E, F, H are arbitrary constants. By substituting (17) into (13) on gets that in all alternate cross-sections of the rod. Therefore equations (15), (16) are reduced to the following: (18) (19) Let’s transform the equations (18), (19). For this purpose we’ll derive the equation (18) on x and subtract from it the first equation (19): (20) Now we’ll derive the equation (18) on y and subtract from it the second equation (19). (21) It follows from (19) and (20) (22) here C is an arbitrary constant. A system (18), (19) may be replaced be equations (18) and (22). As from (12) one can get (23) then (24) It is known that is the third component of the vector . We’ll obtain (25) here is the angle of a torsion per unit length of fiber of the rod. This angle is called twist and is denoted . From (25) and (22) one gets (26) We obtained finally that the problem of torsion of elastic prismatic rod comes to integration the following equations (27) with boundary conditions (28) As the equations (27) come to Poisson’s equation then it is the base of numerous examples of solving the problem of torsion of elastic prismatic rods. This fact leads to analogies which permit to reduce the problem (27), (28) to others mechanical problems which solution is described by the same equations. Here are some of them: membrane analogy, some fluid-flow analogies, electrodynamic analogy. One can introduce a term torsional hardness It is considered, the bigger a torsional hardness the better a rod resists to the torsion. It was shown that among all the prismatic rods with the same area of lateral face, the biggest torsional hardness belongs to a rod with a circular cross-section. Moreover, it is proved that among all the prismatic rods with multiply connected cross-section of the defined area and the defined total area of holes, the rod with ring-shaped cross-section which is bounded by two concentric circles has the highest torsional hardness. These and other problems of the theory of torsion of elastic bodies one can find in [11]. Saint-Venant noted an interesting fact: the maximum tangential stress as a rule is achieved upon the lateral face of a rod in the points the closest to a center of gravity of a cross-section. 5. Elastic Bending of Bars. Let’s consider a prismatic rod bending by two equal and opposite moments M in one of the principal plane (fig. 1). The coordinate origin is in the centre of gravity of a cross-section, the plane is in the main plane of bending. One gets the following elementary solution of the equations (1) in case of absence of body forces: (29) here R is a radius of curvature of the bended rod; E is Young’s modulus of stretch and compression. Let’s consider a common case of bending of a console with the constant cross-section, which is under the action of a force P applied to an end and which is parallel to one of a main axes of the cross-section (fig. 2). Let’s suppose that in console case, stresses allocate in a distance z from the fixed end in the same way as (29): (30) Let’s suppose now that in every point of the cross-sections tangential stresses and affected and the other components of the stress tensor are equal to zero. Fig. 1. Bending of a Bar Fig. 2. Bending of a Console By such suppositions, in case of absence of volume forces one gets from the equations (1) (31) It is follows from (31) that tangent stresses do not depend on z, and they are the same for every cross-section. The compatibility equations come to following: (32) One gets in the same way as in the previous paragraph: (33) It is obtained from the formulas (33) here C is a constant. It is possible to show that C = 0 [11]. Then equations of the bending of console look like: (34) One should add a boundary condition to these equations which is the following on the frontier of the contour . 6. A Plane Problem of Elasticity Theory. In this section the equations of a plane problem of elasticity theory in displacements are given and some boundary problems are posed. For a plane problem the following conditions are valid: Then from (3) one gets (35) The boundary conditions (5) look like (36) 7. An Anti-plane Problem of Elasticity Theory. Let’s consider equations (1), (2) when This case corresponds to so-called anti-plane elastic state. Equations (1) come to and compatibility conditions for deformations come to equation Let the elastic body be affected by only its dead weigh, then if the axis Oz is up-directed one receives , here is a constant density. Finally equations describing elastic state on condition that the deformation is anti-plane look as follow (37) 8. Conservation Laws for the Equations of Elasticity. Conservation laws are the fundamental laws of nature, they were determined in the beginning of the XXth century. A concept of the conservation laws appeared later after researches of E. Noether and her followings. The wide application of these laws to solving and investigations of some differential equations is relative to the last quarter of the XXth century. But significance and usefulness of this concept is not properly understood by majority of researchers even nowadays. In this work the simplest definition of the conservation laws is given. For more details see [11] and cited literature there. Let’s is a system of two differential equations for two sought functions . Definition. Conserved current for the system is a vector which is (38) here are some differentiation operators. It is assumed that both of them are not equal to zero simultaneously. Let’s find conservation laws for equations from the sections 4-7. 1. The equations describing the elastic torsion (27) in convenient denotation look as follows (39) here Let (40) where are arbitrary functions of . From (38) with respect to (40) and (39) one obtains Here and further, subscript signifies a corresponding variable derivative. One can get hence Or after simple conversion (41) Therefore, a conserved current look as follows (42) here is a solution of the Cauchy-Riemann system; , are determined from the equation (41). With respect to (42) the conservation law may be defined in the following form (43) where Г is an arbitrary piecewise smooth closed contour. Let a point inside the domain bounded by Г. One can choose (44) Let Г1 is a circle (fig. 3). Fig. 3. A Circle Г1 It’s not complicated to indicate that integral (44) (45) Let’s calculate the circulation integral on Г1 using polar coordinates In the last expression R tends to zero (), and using mean-value one gets Now from (45) (46) Let this time (47) Let’s calculate the circulation integral on Г1 in this case. In polar coordinates On conditions that one obtains from the last formula and (46): (48) Expressions (47) and (48) allow to calculate values at any internal point of the domain enclosed by Г if the values on the contour are known. But therefore these two values are not known on Г, it known only the expression Hence formulas (47), (48) don’t allow to calculate the values of the stress tensor inside the domain and then don’t allow to solve the problem of the rod torsion. But as one will see later these formulas allow to resolve the elasto-plastic problem which is more complicated. Remark. The similar formulas can be obtained easily for the equations of anti-plane theory of elasticity, which is given in the section 7. Equations describing the torsion of a bar have a form (34). Let us assume in these equations then (49) The conserved current for this system will be found in a form (50) here are fuctions of only. Analogously to the previous clause, one can get One can obtain from here After not complicated calculations (51) Hence, the conserved current looks like (52) here is an arbitrary solution of Cauchy-Riemann equations; , are determined from the equation (52). For the current (52) a conservation law can be written as follows Acting much as the previous clause one can obtain finally Remarks of the previous item are also correct for this problem. Conservation laws of the plane theory of elasticity. Let’s find some conservation laws for equations describing 2-dimensional resilience (35) as it is in the work [12]. The conserved current is looked for in a form here some functions of From the relation (53) From (53) one can obtain (54) (55) Making substitution of (54) into (55) one can get or It means that and are arbitrary solutions of the equations (35) which are coupled by correlations (55). This fact permits to construct an infinite system of conservation laws on a base of the exact solutions of equations of elasticity. 9. Conservation Laws of Two-dimensional Equations of Plasticity. Let us consider the following equations of two-dimensional plane theory of plasticity which can be obtained with ease from the equation of the section 2: (56) (57) Here σ is a hydrostatic pressure, is an angle between the main direction of the stress tensor and the axis Ox. Let’s find conservation laws of a system (86), (57) in the form for which equality or owing to Green’s formula (58) is correct on account of system, i. e. relation (59) has to be performed for all its solutions in a domain bounded by the smooth contour Г. Let’s determine a system of plasticity in a normal matrix form [5]: (60) Multiplying this system by the vector one can get the following equation (61) Comparing the equations (60) and (61) it is possible to obtain two exppressions for the functions C and D: (62) Let’s express the components of the linear system (62) in an explicit form: (63) It is possible to remark that by substitution the system (63) coincides with a linearized plasticity system (64) This fact permits to use all the proprieties of this system during the conservation laws construction. Thus, the linearization of the plasticity system is achieved without the requirement of the non being zero to Jacobian. Further using the substitution the system (63) comes to equations: (65) If insert new independent functions (66) it is possible to obtain the system (67) Finally, by setting one can come to equation: (68) And this is the well-known telegraph equation. Thereby the construction of conservation laws for the plasticity equations comes to solving of the linear systems for which a lot of methods of resolution of equations and boundary problems are developed. 10. Elasto-plastic Boundary of a Twisted Straight Rod. Let’s consider an elasto-plastic torsion of a straight rod which cross-section is bounded by a convex contour Г. If the twisting moment is rather significant, a plastic domain P forms in the rod. This domain arises on the external contour Г. Suppose that the plastic domain is covered completely by the contour. In this case in the cross-section two domains appear, a plastic one P and an elastic one F. L is a boundary of these domains (fig. 4). Fig. 4. Cross-section of the Twisted Rod There are a lot of works devoted to solution of the problem of the stressed state of an elasto-plastic rod, but most of them are based on some assumptions concerning the form of boundary L which is not known in advance. A novel method of the determination of unknown boundary is proposed by B. D. Annin [1]. This method is based on contact transformations and it permits to define the boundary between elastic and plastic domains in the rods with oval cross-section. This problem one can find in [1] and in the bibliography cited there. In the present work the stress state is defined in all internal points of the rod by means of conservation laws, and formulas for analytical calculations of these stresses are proposed in the case of a piecewise-smooth directed boundary of the cross-section. The conservation laws is used for a long time in a fruitful way for solving of various mathematical and mechanical problems. A summary of results and solved problems in different domains of mechanics can be found in [4; 5; 7]. Problem Definition Let are single non-zero components of the stress tensor. In the elastic domain they satisfy the equilibrium equation (69) and the equations (70) Here function determines a deplanation warpingof the cross-section, is a constant, G is a Young modulus by shear. Let’s introduce the stress function φ as following (71) then to determine of φ in the elastic domain one can get the equation (72) here is non-zero constant. In the plastic domain the components along with the equilibrium equation satisfy the plasticity condition (73) Here, to simplify the further calculations, the plasticity constant equals to one. By introducing the stress function in this equation one can get (74) Boundary Conditions. Let the lateral surface be free from stresses. It means that on the contour Г. Here is a tangent vector to the contour Г. It follows that among the contour. As Г is a simply connected contour then on it. Finally one gets the following problem. It is necessary to resolve the following equation in the domain bounded by the curve L: (75) In the domain bounded by curves L and Г, i. e. in the plastic domain, the function φ satisfies the equation (76) The following conditions apply on the contour Г for the function φ φ = 0, (77) or , (78) on the frontier L the function φ is continued. It is necessary to find φ in elastic and plastic domains and to determine a frontier L. Let’s introduce the denotation, . Then equations (75), (76) come to (79) (80) Owing to the denotation the following equality occurs: (81) Definition. A vector (A, B) is a conserved current for the system of the equations (79), (81) if there is the following correlation (82) Here are some linear differential operators. It means that for functions A and B the conservation is correct law for all solutions of the system (79), (81): (83) The conservation law (83) owing to the equations (79), (77) look like or taking into account and From the last expression follows that functions A and B satisfy the equations (84) (85) (84), (85) are Cauchy-Riemann equations. Let’s consider a domain D with the boundary Г on condition that plastic domain P comprises completely the elastic domain F. Let Г be a smooth directed contour, i. e. continuously differentiable without singular points. It follows from the conservation law (86) From (86), using Green’s formula one can obtain (87) Our objective is to find a domain F belonging with its boundary to the domain D where inequality applies. Let then (88) (89) According to the conservation law (83) one can get the equation (90) which contains conditions on functions α, β and γ. (91) Let’s consider two solutions of the system (90) The first one is (92) then (93) Respectively, the second one is (94) then Let’s note the equation (87) for the functions A and B (95) Let’s decompose the boundary Г into parts, i. e. Г = = Г1 + Г2 + Г3 + Г4; Г3 is a circle (x - x0)2 + (y - y0)2 = R2 (fig. 5). Fig. 5. Boundary Г In this case (96) Obviously, - (αv + + γ)dx = 0. Taking into account this condition the equation (93) looks like (97) Let’s calculate an integral Г1 is a circle of the radius R. Let (98) Introduce the polar coordinate system (99) then (100) As a result with one obtains (101) Analogically using (102) Finally one can get from (82) (103) (104) Let’s determine the curve in parametric form: (105) are derivatives of the functions and respectively. Hen functions from (98), (102) are found from the formulas (106) The solutions (102) and (103) were used respectively to obtain these relations. Let’s calculate now a value of the expression (107) in a point The points where (107) is greater than or equal to one belong to the plastic domain, the points where the expression (107) is less than one belong to the elastic domain. On the base of the formulas (103), (104) the programs were developed; they permit to construct plastic and elastic domains in a twisted rod with indicated accuracy. The solutions obtained using the programs coincident rather well with the known solutions. In this journal one can find some examples of calculation of elasto-plastic boundaries for some cross-section of the rolling section. These results belong to A. V. Kondrin and to the authors of the article. The article [13] gives examples of the calculation of elastic - plastic rods borders for rolling profile. 11. Elasto-plastic Boundary in the Bended Consol. Let’s consider a consol with the permanent cross-section bounded by the contour Г. The consol is under the concentrated force P on a free end in parallels to principal axes (fig. 6). Let the component of the stress tensor is distributed along the consol as in the case of pure bending Let components of the stress tensor are Then the residual components of the stress tensor satisfy the equations (108) Usually the equations (108) replace by two compatibility equations here is the Laplacian, is a Poisson’s ratio. This system is usually resolved by semi-inverse Saint-Venant method. Let’s rewrite the system (108) in terms of the vector of deformations (u; v; w). A boundary problem will be posed and resolved using conservation laws. Using the formulas (2) which connect components of the stress tensor and strain tensor one can get (109) here λ, μ are Lamé coeffisients, are sought functions of x, y. Fig. 6. Elasto-plastic Boundary in the Cross-section of the Consol From the first, the second and the third equation one can get (110) here constants can be evaluated with ease per From the equations (110) one can get here are sought functions. From the relation one gets Relations are substituted in the previous equation; the result is Suppose that the lateral surface of the beam if free from the stresses. It means that (111) here are an external normal line to the contour Г. Suppose also that the plastic flow begins from the external side of the lateral surface of the beam. In this case the plasticity condition of Von Mises looks like (112) here k is a constant of plasticity. Solving the system (111), (112) one can get Choosing the upper sign in these relations, one can pose the following problem. It is necessary to resolve the equation (113) under the following conditions on Г: (114) Remark. There are two domains in the cross-section of the beam, a plastic one and an elastic one. If the following condition satisfies in a point of the cross-section Then the point falls into the elastic domain. The other points including the boundary of the contour Г belong to the plastic domain. The relation of a form (115) is called a conservation law of (115) for the equation (113) if the equation (115) is correct for all solutions of the equation (113). Let the conserved current looks like One can obtain from (113) and (115) (116) The relation (116) is correct for all solutions of the equation (113) therefore it is follows from (116) (117) The conservation law (115) can be written using the Green’s formula: Let’s consider two solutions of the equations (117). The first one is the following The second one looks like Using the conservation law and applying it to the contour represented on the fig. 6 one can obtain on the analogy with the previous section Let’s calculate the second integral for the first and the second solutions of the equation (117). As a result the formulas for finding of are obtained. These formulas permit to find a stress state in any point . It means that it is possible to determine for every point of the domain its belonging to either elastic or plastic zone. The expounded method allow to construct a boundary between an elastic and a plastic domains with any prescribed accuracy using the computer. Preliminary calculations confirm this conclusion. 12. Elasto-plastic Boundaries for Large Areas. In this section only domains with smooth convex boundaries are regarded. Let’s consider a domain bounded by the contour Г (fig. 7). Fig. 7. Domain with the Contour Г Let boundary conditions (5) applies on Г. They will look like (118) Moreover, suppose that the loadings (118) bring the entire boundary and the nearby points to the plastic state. Then it is possible to introduce variables by the following way (119) In this case conditions (118) look like These conditions can be written as (120) Thus, one gets the Cauchy problem on the boundary Г in the plastic domain. Solving this problem using formulas of the section 9, one obtains two families of characteristic curves (fig. 8). Fig. 8. Characteristic Curves For these curves one can construct an envelope curve L. This line is the sought elasto-plastic boundary. It is enough to solve an elastic problem inside the domain to resolve completely elasto-plastic problem. It is possible to set Cauchy problem for the system of plasticity (56), (57). Let on the contour curve Г denominated as SP there are starting data: (121) Let describe a characteristic curve from the point P and a characteristic curve from the point S. Then a solution of Cauchy problem is determined in a curvilinear triangle (fig. 9). Fig. 9. Cauchy Problem Thus it is necessary to find coordinates of a cross point R of the characteristic curves. If the coordinates of the point and values of are known then it is possible to find values of functions σ, θ. One can calculate an integral over the closed contour SPR. Using the correlation (58) in the Stokes theorem for a plane [11], one can conclude that this integral is equal to zero: 1. For the coordinate one can get: Integrals and are integrated by parts: Assuming one can get the boundary conditions for the system (65) in the form: (122) Under such assumption the final expression for the coordinate looks like: (123) 2. Similarly fort the coordinate : Integrals and are integrated by parts: Assuming one can get the boundary conditions for the system (65) in the form: (124) Under such assumption the final expression for the coordinate looks like: (125) It remains to resolve the problems (65), (122) and (65), (124). These problems can be redused to the equation (66). Namely, taking into account that fuctions are related with function ρ in the following way (126) one can obtain the problems (127) and (128) A general solution of the first problem (128) (for the coordinate ) is the following function: moreover The solution of the second problem (128) (for the coordinate ) is the function: when In all solutions the function is the Bessel function of the first kind of an imaginary argument The functions can be found using formulas (126). From the relation (66) one can obtain the components of conservation laws: Substituting the obtained C and D in (123) and (124) one can get the coordinates of the point R. Thus Cauchy problem for the system of plasticity (56), (57) with starting data (121) is resolved completely. Conclusion. The small range of problems considered in this article, concerning the construction of elasto-plastic boundaries reveals good prospects of the application of conservation laws for the the boundary problems solving. By now, the authors have solved some other problems of solid mechanics and prepare them to publish. More results in the study of equations of elasticity and plasticity can be found in articles [14-17]. Acknowledgements. Research is supported by Ministry of Education and Science of Russian Federation, the project Б-180-14 and the FSP “Researches and Elaboration on the Priority Directions of Development of a Scientific and Technological Complex of Russia for 2014-2020”, the project No. 14.574.21.0082.

About the authors

S. I. Senashov

Reshetnev Siberian State Aerospace University

Email: sen@sibsau.ru
31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

E. V. Filyushina

Reshetnev Siberian State Aerospace University

31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

O. V. Gomonova

Reshetnev Siberian State Aerospace University

31, Krasnoyarsky Rabochy Av., Krasnoyarsk, 660037, Russian Federation

References

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