Discussion on stress assumption and yield criterion determination rule in teaching of the PSM

Discussion on stress assumption and yield criterion determination rule in teaching of the PSM | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Discussion on stress assumption and yield criterion determination rule in teaching of the PSM Ruibin Mei, Li Bao, Tanqiu Chen, Lihao Chen, Xianghua Liu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4489595/v1 This work is licensed under a CC BY 4.0 License Status: Published Journal Publication published 08 Jan, 2025 Read the published version in Journal of the Brazilian Society of Mechanical Sciences and Engineering → Version 1 posted 4 You are reading this latest preprint version Abstract As one of the main methods to solve deformation mechanics, the principal stress method (PSM) is widely used in stress analysis and technology optimization of metal forming processes. The stress assumption, yield criterion and boundary conditions have been investigated according to the frequently asked questions in teaching and the application of the PSM, and then the determination rule was proposed and described. The yield criterion, boundary conditions and calculated results should keep the corresponding relationship with the initial stress assumption of the micro-element. The stress distribution and load in the plane deformation of the rectangular workpiece was successfully solved using the PSM based on the rule proposed and FEM. The average relative error is 8.3% between the predicted loads by FEM and PSM, and the stress and load predicted by PSM are in good agreement with the FEM. Furthermore, the same calculated results of stress were obtained under different conditions, indicating that the rule proposed is reliable and practical significance for the teaching and application of the plastic mechanics. PSM yield criterion plane deformation boundary condition stress assumption Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 1 Introduction The principal stress method (PSM) is the first approximate mathematic method used in the solution of engineering problems to obtain the stress distribution and deformation load. Based on a series of simplified assumptions, the mechanical characteristics and deformation behaviour are analyzed by establishing the equilibrium equations, simultaneous yield criteria, boundary conditions, and so on [1–2] . As one of the main methods to solve deformation mechanics, the PSM is widely used in stress analysis [3–4] and technology optimization [5–6] in metal forging, rolling, drawing, extrusion, deep drawing and other plastic processing. Therefore, it is not only an important content of the mechanics textbooks including the classical elastoplastic mechanics [1] , plastic mechanics of materials forming [7–8] , material processing principles [9–10] and so on, but also is the key contents in the teaching and analysis of materials forming technology [11–12] . With the development of computer technology, the finite element method (FEM) has more accurate and detailed calculated results of plastic deformation, which are applied in the field of plastic processing [13–14] . However, the FEM consumes much longer time and may not converge during iterations [15–16] , so the PSM has the significant advantage of the rapid prediction of plastic deformation and force parameters [17–18] . Bao [19] used the FE software including ANSYS, MARC, ABAQUS, DEFORM and PSM to solve the mechanical characteristics and plastic deformation of Fe-6.5%Si steel in cylinder compressing. The load solved by PSM is larger than about 10% compared with the FE software, but there is an advantage in computing time. Based on the PSM, the influence of different diameters of upper and lower rolls, different linear speeds and contact friction on the rolling force and torque during serpentine rolling was studied by Salimi [20] . The influences of rolling reduction and initial aluminium thickness on the thickness ratio of aluminium-steel in the deformation zone were investigated, and then the interfacial shear stress distribution in the deformation zone was calculated and explained under different conditions of the thickness variation of aluminium and steel layer [21] . Many researchers studied the theories and application of PSM, but the lack of rule description of stress setting and yield criterion in these papers [1,7–12] leads to it being difficult for students and researchers to understand them accurately and their application. The frequently asked questions such as unclear parsing rules in teaching and application of PSM are investigated, and then the stress assumption and yield criterion determination rule are proposed and described. Finally, the contact stress and load of the rectangular workpiece in plane strain compressing were calculated under different conditions, and the unique solution was obtained significantly. 2 Multivalued questions and determination rule 2.1 Solution steps and questions in teaching According to the simultaneous solution of the equilibrium differential equations and the yield criterion based on the simplified model, the key points of the PSM to solve engineering problems are described: (1) The engineering problem should be simplified into a plane or an axisymmetric problem according to the actual deformation. The complex deformation needs to be divided into several zones that are regarded as the plane or axisymmetric problem. (2) According to the metal flow tendency and the determined coordinate system, the normal stress on the contact surface is assumed to be the principal stress with a uniform distribution. A differential equation is established based on the static equilibrium condition and transformed to the ordinary differential equation subsequently. (3) When the yield criterion is used, the normal stress is considered as the principal stress without considering the influence of shear stress. Integrating the ordinary differential equation, and then the integral constant can be obtained according to the boundary conditions, to obtain the stress distribution and deformation load. If a rectangular workpiece with a larger size in one direction compared with two others in the deformation zone the deformation along the direction can be ignored. Then the problem can be simplified to plane deformation with zero strain in some direction. The rectangular workpiece compression and plane deformation simplification are shown in Fig. 1 . Taking the solution of rectangular compression processes as an example of a plane deformation problem to describe the solution steps and frequently asked questions. As an indispensable example in the PSM teaching of plastic mechanics [1,7–10] , the principle stress solution of rectangular workpieces is significant for the application of stress and load analysis in the forging processes [22–23] and plane deformation experiments [24–26] . The engineering problems should be simplified firstly to plane stress, plane strain or axial symmetry model, and the simplified plane strain model is established for the rectangular workpiece compression because the deformation in length direction can be ignored. The solution steps and main questions in the solution of rectangular workpieces by PSM can be described as shown in Fig. 2 . (1) Setting the micro-element, and assuming the stress direction and increment. The setting methods of compressive stress, tensile stress and the direction of stress increment are the main questions in stress assumption because the actual stress directions are usually unknown. (2) An equilibrium differential equations are established according to the force balance and stress assumption. (3) The equation is transformed to the ordinary differential equation based on the yield criterion. The Mises simplified yield criterion can be described as equation σ max - σ min = βσ s , but how to determine the first and third principal stress in the yield criterion under the condition of unknown stress value and direction. (4) Integrating the differential equation, and then the integral constant can be obtained according to the known boundary conditions. How to substitute the boundary stress into the integrated equation, especially if the boundary stress direction is inconsistent with the assumed stress. (5) Substituting the integral constant into the integrated equation, and then the contact stress can be described. (6) Finally, the deformation load can be obtained based on the definite integration of contact stress equations along the contact surface. Therefore, due to the unknown stress direction in actual deformation, conventional methods are easy to cause uncertainty in stress assumptions and incorrect use of yield criteria and boundary conditions, which leads to ambiguity errors in the calculation results in the solution of engineering problems by PSM. 2.2 Determination rule description 2.2.1 Stress assumption The equilibrium differential equation describes the relationship among the stress vector components, coordinates and volume forces (if it is considered). The general equilibrium differential equation in a three-dimensional rectangular coordinate system can be described as $$\left\{ {\begin{array}{*{20}{c}} {\frac{{\partial {\sigma _x}}}{{\partial x}}+\frac{{\partial {\tau _{yx}}}}{{\partial y}}+\frac{{\partial {\tau _{zx}}}}{{\partial z}}+{K_x}=0} \\ {\frac{{\partial {\tau _{xy}}}}{{\partial x}}+\frac{{\partial {\sigma _y}}}{{\partial y}}+\frac{{\partial {\tau _{zy}}}}{{\partial z}}+{K_y}=0} \\ {\frac{{\partial {\tau _{xz}}}}{{\partial x}}+\frac{{\partial {\tau _{yz}}}}{{\partial y}}+\frac{{\partial {\sigma _z}}}{{\partial z}}+{K_z}=0} \end{array}} \right.$$ 1 Where the K x , K y and K z are components of the volume forces in the x , y and z direction, respectively. The volume forces are gravity, magnetic force and so on, and the value usually can be ignored in the solution of plastic deformation. Equation ( 1 ) can be used directly or reconstructed according to the deformation model in the solution of engineering problems, but the stress direction and increment of micro-element should be described obviously to use the equilibrium differential equations correctly. For the first question in the solution step one (Fig. 2 ), it is well known that the stress increment shows only the change of stress in a certain direction and the value can be positive or negative so that the position exchange of the σ x and σ x + d σ x does not affect the solution result. However, the Eq. ( 1 ) is obtained under the conditions of stress assumption is positive, so that the stress increment should be consistent with the positive direction of the coordinate system (Fig. 3 ). For the second question in the solution steps one (Fig. 2 ), the compressive or tensile stress should be allowed to set because of the actual existence of tensile and compressive stress states. According to the physics theory, the positive calculated value means that the actual stress direction is consistent with the assumption, and the negative calculated result means the actual value is opposite to the assumed direction. 2.2.2 Yield criterion The typical yield criteria in plastic mechanics are the Tresca yield criterion and Mises yield criterion, and the Mises yield criterion can be described as the same form as the Tresca through introducing the Lode parameter. $${\sigma _{\hbox{max} }} - {\sigma _{\hbox{min} }}=\beta {\sigma _s}$$ 2 Where σ s is yield strength, σ max and σ min are the maximum and minimum of principal stress, respectively. When the first and third principal stress is known in the deformation zone the σ max and σ min equal to σ 1 and σ 3 , respectively. The β is coefficient related to Lode parameter µ d , \(\beta =\frac{2}{{\sqrt {3+\mu _{d}^{2}} }}\) , and the value equals 1 and 2/√3 under the axisymmetric and plane deformation condition, respectively. For solving the question in step 3 it is not necessary to set the stress direction of the micro-element, but it has an important influence on the application of yield criterion. The inaccurate determination of maximum and minimum value and direction of stress perhaps led to the multivalued solution and incorrect results. In general, the normal stress in the main deformation direction is either the first principal stress σ 1 or the third principal stress σ 3 . The positive calculated value means that the actual stress direction is consistent with the assumption, and the negative calculated result means the actual value is opposite to the assumed direction so that a minus should be added to the assumed compressive stress (Fig. 3 ) in the application of the yield criterion to correctly describe the physical meaning. As an example, the main compressive deformation is generated in the y and r direction (Fig. 3 ), and then the yield criterion can be described as $$\left\{ {\begin{array}{*{20}{c}} {{\sigma _x} - \left( { - {\sigma _y}} \right)=\beta {\sigma _s} \Rightarrow {\sigma _x}+{\sigma _y}=\beta {\sigma _s}} \\ {\left( { - {\sigma _\theta }} \right) - \left( { - {\sigma _r}} \right)=\beta {\sigma _s} \Rightarrow {\sigma _r} - {\sigma _\theta }=\beta {\sigma _s}} \end{array}} \right.$$ 3 2.2.3 Boundary condition For the question in solution step three, the free surface or the boundary with known stress is often used to obtain the integral constants. The stress is zero on the free surface, and then it is easy to solve the integral constant. If the boundary value is known and not equal to zero, the stress on the boundary surface should be consistent with the direction of assumed stress, and there is no need to consider the physical significance of known boundary values. As an example, the known compressive stress on the boundary surface, and then the use of boundary conditions can be described as shown in Fig. 4 . In summary, the rule of the stress assumption, yield criterion determination and boundary conditions through the above analysis can be described as follows: (Ⅰ) The stress increment should be in the positive direction of the coordinate system. (Ⅱ) The minus should be added to the assumed compressive stress of the micro-element before substituting it into the yield criterion. (Ⅲ) When the known boundary stress is inconsistent with the assumed stress direction, a minus should be added to it as the boundary value. (Ⅳ) The negative calculated results mean that the actual stress is opposite to the assumed value direction, and the minus is only added to the assumed compressive stress to demonstrate the actual physical meaning of the final calculated results. 3 Plane deformation solution of a rectangular workpiece The plane deformation problems of rectangular workpiece compression are shown in Fig. 5 . Where the stress direction of the micro-element is assumed to be compressive (Fig. 5 .b) and tensile (Fig. 5 .c), and the stress value on the boundary surface equals σ k (Fig. 5 .d). According to the rule (Ⅰ), the stress increment is set in the positive direction of the coordinate system. The contact friction is assumed to be the form of sliding friction met with Colulomb’s law τ = fσ y . 3.1 Assumed compressive stress in micro-element with free surface The micro-element and the assumed stress direction are shown in Fig. 5 .b, and the equilibrium differential equation is established according to the balance of force \(\sum {{F_x}} =0\) $${\sigma _x} \cdot h \cdot l=\left( {{\sigma _x}+{\text{d}}{\sigma _x}} \right) \cdot h \cdot l+2\tau \cdot {\text{d}}x \cdot l$$ 4 Organizing the Eq. ( 4 ), and then the friction condition \({\tau _{\text{f}}}=f{\sigma _y}\) is substituted into the equilibrium differential equation $$\frac{{{\text{d}}{\sigma _x}}}{{{\text{d}}x}}+\frac{{2f{\sigma _y}}}{h}=0$$ 5 The main compressive deformation is generated in the height direction, and the component of stress σ y is the minimum value of stress. Subsequently, according to the determination rule (Ⅱ), the minus is added to the assumed stress in the application of the yield criterion. $${\sigma _{\hbox{max} }} - {\sigma _{\hbox{min} }}=\beta {\sigma _{\text{s}}} \Rightarrow \left( { - {\sigma _x}} \right) - \left( { - {\sigma _y}} \right)=\beta {\sigma _{\text{s}}} \Rightarrow {\sigma _y} - {\sigma _x}=\beta {\sigma _{\text{s}}}$$ 6 According to the above Eq. ( 6 ), the d σ x = d σ y is obtained, and then the ordinary differential equation of the component of stress σ y can be described as Eq. ( 7 ). $$\frac{{{\text{d}}{\sigma _y}}}{{{\text{d}}x}}+\frac{{2f{\sigma _y}}}{h}=0$$ 7 The stress component in the y direction can be obtained subsequently by integrating the differential equation. $${\sigma _y}=C\exp \left( { - \frac{{2f}}{h}x} \right)$$ 8 Subsequently, the integral constant C is obtained according to the boundary conditions. The σ x = 0 with the x = w /2 so that the σ y = βσ s according to the Eq. ( 6 ), and then integral constant C is obtained from the above Eq. ( 8 ). $$C=\beta {\sigma _{\text{s}}}\exp \left( {\frac{{fw}}{h}} \right)$$ 9 Substituting the constant C into the Eq. ( 8 ), and then the contact stress in the y direction can be described as $${\sigma _y}=\beta {\sigma _{\text{s}}}\exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right]$$ 10 Substituting Eq. ( 10 ) into Eq. ( 6 ), and then the distribution of the x component stress σ x in the deformation body is described as $${\sigma _x}=\beta {\sigma _{\text{s}}}\left\{ {\exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right] - 1} \right\}$$ 11 The values of the Eqs. ( 10 ) and ( 11 ) are greater than 0 which demonstrates the assumed stress direction is consistent with that of the actual stress, and the stress state is compressive. According to the rule (Ⅳ), considering the physical meaning the distribution of actual stress can be described as $$\left\{ {\begin{array}{*{20}{c}} {{\sigma _x}= - \beta {\sigma _{\text{s}}}\left\{ {\exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right] - 1} \right\}} \\ {{\sigma _y}= - \beta {\sigma _{\text{s}}}\exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right]} \end{array}} \right.$$ 12 According to the plane deformation theory, the stress in the direction without strain can be obtained $${\sigma _z}=\frac{1}{2}\left( {{\sigma _x}+{\sigma _y}} \right)= - \beta {\sigma _{\text{s}}}\left\{ {\exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right] - \frac{1}{2}} \right\}$$ 13 The plane deformation of rectangular workpiece compression is three-dimensional compressive stress states according to the Eq. ( 11 ) to ( 13 ). 3.2 Assumed tensile stress in micro-element with free surface When the assumed stress of the micro-element is tensile, for the micro-element and the assumed stress direction shown in Fig. 5 .c, the equilibrium differential equation is established $${\sigma _x} \cdot h \cdot l+2\tau \cdot {\text{d}}x \cdot l=\left( {{\sigma _x}+{\text{d}}{\sigma _x}} \right) \cdot h \cdot l$$ 14 Organizing the Eq. ( 14 ), and then the friction condition \({\tau _{\text{f}}}=f{\sigma _y}\) is also substituted into the equilibrium differential equation $$\frac{{{\text{d}}{\sigma _x}}}{{{\text{d}}x}} - \frac{{2f{\sigma _y}}}{h}=0$$ 15 According to the determination rule (Ⅱ), the minus is not needed to add to the σ x in the application of the yield criterion. $${\sigma _{\hbox{max} }} - {\sigma _{\hbox{min} }}=\beta {\sigma _{\text{s}}} \Rightarrow {\sigma _x} - \left( { - {\sigma _y}} \right)=\beta {\sigma _{\text{s}}} \Rightarrow {\sigma _x}+{\sigma _y}=\beta {\sigma _{\text{s}}}$$ 16 It can be seen from equations ( 6 ) and ( 16 ) that the yield criterion expressions are different with the different assumed stress directions. According to the above Eq. ( 16 ), the d σ x =- d σ y is obtained, and then the ordinary differential equation is also obtained by substituting it to the Eq. ( 15 ). $$\frac{{{\text{d}}{\sigma _y}}}{{{\text{d}}x}}+\frac{{2f{\sigma _y}}}{h}=0$$ 17 The stress σ y can be obtained subsequently by integrating the differential Eq. ( 17 ). $${\sigma _y}=C\exp \left( { - \frac{{2f}}{h}x} \right)$$ 18 The integral constant C is obtained according to the boundary conditions. The σ x = 0 with the x = w /2 so that the σ y = βσ s according to the Eq. ( 16 ), and then integral constant C is obtained from the above Eq. ( 18 ). $$C=\beta {\sigma _{\text{s}}}\exp \left( {\frac{{fw}}{h}} \right)$$ 19 Substituting the C into the Eq. ( 18 ), and then the σ y can be described as the same as Eq. ( 10 ) $${\sigma _y}=\beta {\sigma _{\text{s}}}\exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right]$$ 20 Substituting the Eq. ( 20 ) into the Eq. ( 16 ), and then the σ x in the deformation body is described as $${\sigma _x}=\beta {\sigma _{\text{s}}}\left\{ {1 - \exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right]} \right\}$$ 21 The value of Eq. ( 21 ) is less than zero that demonstrates the assumed stress direction is inconsistent with that of the actual stress, and the actual stress state is compressive. According to the rule (Ⅳ), considering the physical meaning the distribution of actual stress can be described as $$\left\{ {\begin{array}{*{20}{c}} {{\sigma _x}=\beta {\sigma _{\text{s}}}\left\{ {1 - \exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right]} \right\}} \\ {{\sigma _y}= - \beta {\sigma _{\text{s}}}\exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right]} \end{array}} \right.$$ 22 It can be seen from equations ( 12 ) and ( 22 ) that no multivalued solution of stress results is generated according to the determination rules proposed and described. 3.3 Assumed compressive stress with the known stress boundary According to the rule (Ⅲ), the σ x = σ k with the x = w /2 so that the σ y = βσ s + σ k according to the Eq. ( 6 ), and then the integral constant C can be obtained from the above Eq. ( 8 ). $${\sigma _y}\left| {_{{x=w/2}}} \right.=C\exp \left( { - \frac{{2f}}{h}x} \right)\left| {_{{x=w/2}}} \right.=\beta {\sigma _s}+{\sigma _k} \Rightarrow C=\left( {\beta {\sigma _s}+{\sigma _k}} \right)\exp \left( {\frac{{fw}}{h}} \right)$$ 23 Substituting the C into the Eq. ( 8 ), and then the contact stress in the y direction can be described as $${\sigma _y}=\left( {\beta {\sigma _{\text{s}}}+{\sigma _k}} \right)\exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right]$$ 24 Substituting Eq. ( 24 ) into the Eq. ( 6 ), and then the distribution of the stress σ x in the deformation body is described as $${\sigma _x}=\beta {\sigma _{\text{s}}}\left\{ {\exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right] - 1} \right\}+{\sigma _k} \cdot \exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right]$$ 25 The values of the Eqs. ( 24 ) and ( 25 ) are greater than 0 which demonstrates the assumed stress direction is consistent with that of the actual stress, and the stress state is compressive. Furthermore, it can be seen from equations ( 10 ), ( 11 ), (24) and (25) that the application of boundary compressive stress enhances the stress states and increases the deformation load obviously, which is in agreement with the plastic deformation theory. 3.4 Deformation load To solve the deformation load, the Eq. ( 10 ) is used to integrate, and the β equals 2/√3 under plane deformation condition is also substituted. $$P=2\mathop \smallint \nolimits_{0}^{{\frac{w}{2}}} {\sigma _y}{\text{d}}x=2\mathop \smallint \nolimits_{0}^{{\frac{w}{2}}} \beta {\sigma _{\text{s}}}\exp \left[ {\frac{{2f}}{h}\left( {\frac{w}{2} - x} \right)} \right]{\text{d}}x{\text{=}}\frac{{2h{\sigma _{\text{s}}}}}{{\sqrt 3 f}}\left[ {\exp \left( {\frac{{fw}}{h}} \right) - 1} \right]$$ 26 Furthermore, the deformation load at different times is also solved according to the constant volume of plastic deformation. Assuming the initial height and width are h 0 and w 0 , respectively, the volume size meets the relationship w 0 . h 0 = w . h during the compressing processes. And then the relationship between the deformation load and the height of the rectangular workpiece can be described as $$P=\frac{{2h{\sigma _{\text{s}}}}}{{\sqrt 3 f}}\left[ {\exp \left( {\frac{{fw}}{h}} \right) - 1} \right]=\frac{{2h{\sigma _{\text{s}}}}}{{\sqrt 3 f}}\left[ {\exp \left( {\frac{{f \cdot {w_0} \cdot {h_0}}}{{{h^2}}}} \right) - 1} \right]$$ 27 To verify the precision of the principal stress method, the deformation load in plane compressing is solved by the DEFORM software and the Eq. ( 27 ). The calculated results by FEM and comparison of load are shown in Fig. 6 . The material is regarded as ideal plasticity and the yield stress is 300MPa. The other parameters used in the solution are shown as follows: the initial width w 0 is 20 mm, the initial height h 0 is 16mm, the final height is 10mm and the friction coefficient f is 0.3. Because of the ideal plastic material, the equivalent stress is 300MPa in most deformation zones, and only the value of the symmetric centre is less than 300MPa. The equivalent strain distribution of plane deformation compression of rectangular workpieces is X shape, and the larger value is generated in the side flattening and central zones. The increment of the contact surface during the compression processes leads to the deformation load increasing nonlinearly. The average relative error is 8.3% between the predicted loads by FEM and PSM, and the stress and load predicted by RSM are in good agreement with the FEM. 3.5 Teaching effectiveness testing In order to test the teaching effectiveness of the PSM solving rules proposed in this article, 74 and 72 elective students were tested in 2022 and 2023 respectively. The five test questions are all typical engineering problems of long pipelines subjected to internal pressure, deep drawing of cylindrical parts, cylinder compression, wide plate bending, and bar extrusion, with a full score of 100 points. In 2022, the traditional textbook PSM analysis method was taught in the classroom, and in 2023, the PSM analysis method proposed in this article is taught. The average score distribution after two years of testing is shown in Fig. 7 . From the figure, it can be seen that when using traditional PSM methods to solve practical slip line engineering problems, due to the lack of clear analytical rules, students are prone to generate ambiguous solutions when solving typical engineering problems with a high error rate, low average score, and a high failure rate of 10.8% for students; After learning the PSM method analysis rules proposed in this article, students' knowledge application ability has been significantly improved. Most students not only master the basic principles of PSM method, but also have a high accuracy rate and a significant improvement in excellence rate when using PSM method to solve engineering problems, accounting for 62.5%. The PSM method solution rules proposed in this article can effectively improve classroom teaching effectiveness and enhance students' ability to solve engineering problems. 4 Conclusions (1) The stress assumption, yield criterion determination and boundary conditions have been investigated according to the frequently asked questions in the teaching and application of the PSM, and the rule was proposed and described in detail. The stress increment should be in the positive direction of the coordinate system, and the minus should be added to the assumed compressive stress of the micro-element before substituting it into the yield criterion. Furthermore, the negative calculated results mean that the actual value is opposite to the assumed stress direction, and the minus is only added to the assumed compressive stress to demonstrate the actual physical meaning of the final calculated results. (2) The stress distribution and load in a plane deformation of a rectangular workpiece was successfully solved using the PSM and FEM. The equivalent strain distribution of plane deformation compression of rectangular workpieces is X shape. The average relative error is 8.3% between the predicted loads by FEM and PSM, and the stress and load predicted by PSM are in good agreement with the FEM. The same calculated results of stress were obtained under the different conditions of stress assumptions and yield criterion, without generating a multivalued solution, indicating that the rule proposed has reliable and practical significance for the teaching and application of the PSM of plastic mechanics. Declarations Conflict of interest statement The authors declare that there is no conflict of interest. Acknowledgement The authors gratefully acknowledge financial support from the Program of Web-Delivery for Elaborate Course of Hebei Province (No. 2020JPKC061), Hebei Higher Education Teaching Reform Research Project (2022GJJG427) and Specialized and Innovative Integrated Courses of Hebei Province (2023CXCY263). Data Availability Statement The data that support the findings of this study are available on request from the corresponding author, please contact [email protected] References Wang ZR, Yuan SJ, Hu LX et al (2007) Fundamentals of elastic and plastic mechanics (second edition). Harbin University of Technology, Harbin Peng DS (2004) Principles of metal plastic processing. Central South University, Changsha Liang H, He Y, Mei Z et al (2008) Analysis of splitting spinning force by the PSM. 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J Mech Phys Solids, 2012(5):60 Chermette C, Unruh K, Peshekhodov I et al (2020) A new analytical method for determination of the flow curve for high-strength sheet steels using the plane strain compression test. IntJ Mater Form 13(2):269–292 Ht A, Feng ZA, Dy B et al (2022) Numerical simulation of strain localization based on Cosserat continuum theory and isogeometric analysis. Comput Geotech 151:104951–104977 Cite Share Download PDF Status: Published Journal Publication published 08 Jan, 2025 Read the published version in Journal of the Brazilian Society of Mechanical Sciences and Engineering → Version 1 posted Reviewers agreed at journal 24 Jun, 2024 Reviewers invited by journal 24 Jun, 2024 Editor assigned by journal 30 May, 2024 First submitted to journal 28 May, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. 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Mei","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA3klEQVRIiWNgGAWjYFACxgYGBoMDPAzsDVCBA0Rr4YEpJawFpkwigUgtuu3NbZ95Cu7ImEu+PfiYp4xBju9GAuPnAjxazM4cbJ7NY/CMx3J2XrIxzzkGY8kbCczSM/BpuZHYzMxjcJjH4HaOmTRvG0PihhsJbMw8+LTcfwjVcvOM+W+glnrCWm4wQrXc4DFjBmpJMCCo5UxiM+McoF8MzuQYS845J2E488zDZmm8Wo4ff8zw5s8de4PjZww/vCmzkec7nnzwMz4taIBNggESuSRoIUXxKBgFo2AUjBQAAOR4SqfsNs6mAAAAAElFTkSuQmCC","orcid":"https://orcid.org/0009-0003-9338-1497","institution":"Northeastern University","correspondingAuthor":true,"prefix":"","firstName":"Ruibin","middleName":"","lastName":"Mei","suffix":""},{"id":318464987,"identity":"a9c4d81a-acfb-4fa8-a523-bf8acdf79f73","order_by":1,"name":"Li Bao","email":"","orcid":"","institution":"Northeastern University at Qinhuangdao","correspondingAuthor":false,"prefix":"","firstName":"Li","middleName":"","lastName":"Bao","suffix":""},{"id":318464988,"identity":"ac6c5287-5aea-4e4a-80cd-5118b7592a82","order_by":2,"name":"Tanqiu Chen","email":"","orcid":"","institution":"Northeastern University at Qinhuangdao","correspondingAuthor":false,"prefix":"","firstName":"Tanqiu","middleName":"","lastName":"Chen","suffix":""},{"id":318464989,"identity":"4237bf0e-2868-4d76-94df-ed68fa275027","order_by":3,"name":"Lihao Chen","email":"","orcid":"","institution":"Northeastern University at Qinhuangdao","correspondingAuthor":false,"prefix":"","firstName":"Lihao","middleName":"","lastName":"Chen","suffix":""},{"id":318464990,"identity":"e371c52e-c9eb-4f5e-9b3d-10ac23ca1026","order_by":4,"name":"Xianghua Liu","email":"","orcid":"","institution":"Northeastern University","correspondingAuthor":false,"prefix":"","firstName":"Xianghua","middleName":"","lastName":"Liu","suffix":""}],"badges":[],"createdAt":"2024-05-28 09:06:57","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4489595/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4489595/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s40430-024-05342-7","type":"published","date":"2025-01-08T15:57:41+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":60451088,"identity":"280adb4d-6bc4-4f9f-b17b-9dfae3ac05fd","added_by":"auto","created_at":"2024-07-17 00:06:34","extension":"jpg","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":24882,"visible":true,"origin":"","legend":"\u003cp\u003eRectangular workpiece compression and plane simplification\u003c/p\u003e","description":"","filename":"1.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4489595/v1/68a795b79aa9228ffbbb5a92.jpg"},{"id":60451086,"identity":"f2a4f5b5-2d3d-45a8-bf93-c72dfb2b0ecc","added_by":"auto","created_at":"2024-07-17 00:06:34","extension":"jpg","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":93950,"visible":true,"origin":"","legend":"\u003cp\u003eSolution steps of PSM\u003c/p\u003e","description":"","filename":"2.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4489595/v1/949677115a4f825d371993eb.jpg"},{"id":60451083,"identity":"b7fef0f8-7dff-4509-825d-b8d66d18c070","added_by":"auto","created_at":"2024-07-17 00:06:34","extension":"jpg","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":23414,"visible":true,"origin":"","legend":"\u003cp\u003eThe example of stress direction and increment\u003c/p\u003e","description":"","filename":"3.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4489595/v1/0d69ac5aebe766cccff8fe38.jpg"},{"id":60451345,"identity":"d527c0fa-05cf-49b9-a7f1-6e8ad4c50c7d","added_by":"auto","created_at":"2024-07-17 00:14:34","extension":"jpg","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":56031,"visible":true,"origin":"","legend":"\u003cp\u003eThe example of the application of the known boundary condition\u003c/p\u003e","description":"","filename":"4.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4489595/v1/888e1776188d098e7fc8bf52.jpg"},{"id":60451586,"identity":"74256894-e3ca-496c-927b-c68e14957915","added_by":"auto","created_at":"2024-07-17 00:22:34","extension":"jpg","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":60274,"visible":true,"origin":"","legend":"\u003cp\u003eThe plane deformation problems of rectangular workpiece compression\u003c/p\u003e","description":"","filename":"5.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4489595/v1/fb8054964db449333979b421.jpg"},{"id":60451084,"identity":"c5688bd4-45cb-4b27-8c99-aada03ecfe9e","added_by":"auto","created_at":"2024-07-17 00:06:34","extension":"jpg","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":75775,"visible":true,"origin":"","legend":"\u003cp\u003eThe calculated results by FEM and comparison of predicted load by PSM with FEM\u003c/p\u003e","description":"","filename":"6.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4489595/v1/9386974eb0bb96052c9e3289.jpg"},{"id":60451344,"identity":"6f58cec0-6d8e-4dff-8024-22e5114be508","added_by":"auto","created_at":"2024-07-17 00:14:34","extension":"jpg","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":66370,"visible":true,"origin":"","legend":"\u003cp\u003eTest scores\u003c/p\u003e","description":"","filename":"7.jpg","url":"https://assets-eu.researchsquare.com/files/rs-4489595/v1/96496f541aabf7ecd6115b07.jpg"},{"id":73694760,"identity":"20f7c379-af08-498b-9c9c-6ae994ad1acc","added_by":"auto","created_at":"2025-01-13 16:13:58","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1133856,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4489595/v1/9fc0968f-ae53-4242-a00c-975bf139f55b.pdf"}],"financialInterests":"","formattedTitle":"Discussion on stress assumption and yield criterion determination rule in teaching of the PSM","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eThe principal stress method (PSM) is the first approximate mathematic method used in the solution of engineering problems to obtain the stress distribution and deformation load. Based on a series of simplified assumptions, the mechanical characteristics and deformation behaviour are analyzed by establishing the equilibrium equations, simultaneous yield criteria, boundary conditions, and so on \u003csup\u003e[1\u0026ndash;2]\u003c/sup\u003e. As one of the main methods to solve deformation mechanics, the PSM is widely used in stress analysis \u003csup\u003e[3\u0026ndash;4]\u003c/sup\u003e and technology optimization \u003csup\u003e[5\u0026ndash;6]\u003c/sup\u003e in metal forging, rolling, drawing, extrusion, deep drawing and other plastic processing. Therefore, it is not only an important content of the mechanics textbooks including the classical elastoplastic mechanics \u003csup\u003e[1]\u003c/sup\u003e, plastic mechanics of materials forming \u003csup\u003e[7\u0026ndash;8]\u003c/sup\u003e, material processing principles \u003csup\u003e[9\u0026ndash;10]\u003c/sup\u003e and so on, but also is the key contents in the teaching and analysis of materials forming technology \u003csup\u003e[11\u0026ndash;12]\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eWith the development of computer technology, the finite element method (FEM) has more accurate and detailed calculated results of plastic deformation, which are applied in the field of plastic processing \u003csup\u003e[13\u0026ndash;14]\u003c/sup\u003e. However, the FEM consumes much longer time and may not converge during iterations \u003csup\u003e[15\u0026ndash;16]\u003c/sup\u003e, so the PSM has the significant advantage of the rapid prediction of plastic deformation and force parameters \u003csup\u003e[17\u0026ndash;18]\u003c/sup\u003e. Bao \u003csup\u003e[19]\u003c/sup\u003e used the FE software including ANSYS, MARC, ABAQUS, DEFORM and PSM to solve the mechanical characteristics and plastic deformation of Fe-6.5%Si steel in cylinder compressing. The load solved by PSM is larger than about 10% compared with the FE software, but there is an advantage in computing time. Based on the PSM, the influence of different diameters of upper and lower rolls, different linear speeds and contact friction on the rolling force and torque during serpentine rolling was studied by Salimi \u003csup\u003e[20]\u003c/sup\u003e. The influences of rolling reduction and initial aluminium thickness on the thickness ratio of aluminium-steel in the deformation zone were investigated, and then the interfacial shear stress distribution in the deformation zone was calculated and explained under different conditions of the thickness variation of aluminium and steel layer \u003csup\u003e[21]\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003eMany researchers studied the theories and application of PSM, but the lack of rule description of stress setting and yield criterion in these papers\u003csup\u003e[1,7\u0026ndash;12]\u003c/sup\u003e leads to it being difficult for students and researchers to understand them accurately and their application. The frequently asked questions such as unclear parsing rules in teaching and application of PSM are investigated, and then the stress assumption and yield criterion determination rule are proposed and described. Finally, the contact stress and load of the rectangular workpiece in plane strain compressing were calculated under different conditions, and the unique solution was obtained significantly.\u003c/p\u003e"},{"header":"2 Multivalued questions and determination rule","content":"\u003cdiv id=\"Sec3\" class=\"Section2\"\u003e \u003ch2\u003e2.1 Solution steps and questions in teaching\u003c/h2\u003e \u003cp\u003eAccording to the simultaneous solution of the equilibrium differential equations and the yield criterion based on the simplified model, the key points of the PSM to solve engineering problems are described:\u003c/p\u003e \u003cp\u003e(1) The engineering problem should be simplified into a plane or an axisymmetric problem according to the actual deformation. The complex deformation needs to be divided into several zones that are regarded as the plane or axisymmetric problem.\u003c/p\u003e \u003cp\u003e(2) According to the metal flow tendency and the determined coordinate system, the normal stress on the contact surface is assumed to be the principal stress with a uniform distribution. A differential equation is established based on the static equilibrium condition and transformed to the ordinary differential equation subsequently.\u003c/p\u003e \u003cp\u003e(3) When the yield criterion is used, the normal stress is considered as the principal stress without considering the influence of shear stress. Integrating the ordinary differential equation, and then the integral constant can be obtained according to the boundary conditions, to obtain the stress distribution and deformation load.\u003c/p\u003e \u003cp\u003eIf a rectangular workpiece with a larger size in one direction compared with two others in the deformation zone the deformation along the direction can be ignored. Then the problem can be simplified to plane deformation with zero strain in some direction. The rectangular workpiece compression and plane deformation simplification are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. Taking the solution of rectangular compression processes as an example of a plane deformation problem to describe the solution steps and frequently asked questions. As an indispensable example in the PSM teaching of plastic mechanics \u003csup\u003e[1,7\u0026ndash;10]\u003c/sup\u003e, the principle stress solution of rectangular workpieces is significant for the application of stress and load analysis in the forging processes \u003csup\u003e[22\u0026ndash;23]\u003c/sup\u003e and plane deformation experiments \u003csup\u003e[24\u0026ndash;26]\u003c/sup\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe engineering problems should be simplified firstly to plane stress, plane strain or axial symmetry model, and the simplified plane strain model is established for the rectangular workpiece compression because the deformation in length direction can be ignored. The solution steps and main questions in the solution of rectangular workpieces by PSM can be described as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e(1) Setting the micro-element, and assuming the stress direction and increment. The setting methods of compressive stress, tensile stress and the direction of stress increment are the main questions in stress assumption because the actual stress directions are usually unknown.\u003c/p\u003e \u003cp\u003e(2) An equilibrium differential equations are established according to the force balance and stress assumption.\u003c/p\u003e \u003cp\u003e(3) The equation is transformed to the ordinary differential equation based on the yield criterion. The Mises simplified yield criterion can be described as equation \u003cem\u003eσ\u003c/em\u003e\u003csub\u003emax\u003c/sub\u003e-\u003cem\u003eσ\u003c/em\u003e\u003csub\u003emin\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003eβσ\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e, but how to determine the first and third principal stress in the yield criterion under the condition of unknown stress value and direction.\u003c/p\u003e \u003cp\u003e(4) Integrating the differential equation, and then the integral constant can be obtained according to the known boundary conditions. How to substitute the boundary stress into the integrated equation, especially if the boundary stress direction is inconsistent with the assumed stress.\u003c/p\u003e \u003cp\u003e(5) Substituting the integral constant into the integrated equation, and then the contact stress can be described.\u003c/p\u003e \u003cp\u003e(6) Finally, the deformation load can be obtained based on the definite integration of contact stress equations along the contact surface.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eTherefore, due to the unknown stress direction in actual deformation, conventional methods are easy to cause uncertainty in stress assumptions and incorrect use of yield criteria and boundary conditions, which leads to ambiguity errors in the calculation results in the solution of engineering problems by PSM.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e2.2 Determination rule description\u003c/h2\u003e \u003cdiv id=\"Sec5\" class=\"Section3\"\u003e \u003ch2\u003e2.2.1 Stress assumption\u003c/h2\u003e \u003cp\u003eThe equilibrium differential equation describes the relationship among the stress vector components, coordinates and volume forces (if it is considered). The general equilibrium differential equation in a three-dimensional rectangular coordinate system can be described as\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{c}} {\\frac{{\\partial {\\sigma _x}}}{{\\partial x}}+\\frac{{\\partial {\\tau _{yx}}}}{{\\partial y}}+\\frac{{\\partial {\\tau _{zx}}}}{{\\partial z}}+{K_x}=0} \\\\ {\\frac{{\\partial {\\tau _{xy}}}}{{\\partial x}}+\\frac{{\\partial {\\sigma _y}}}{{\\partial y}}+\\frac{{\\partial {\\tau _{zy}}}}{{\\partial z}}+{K_y}=0} \\\\ {\\frac{{\\partial {\\tau _{xz}}}}{{\\partial x}}+\\frac{{\\partial {\\tau _{yz}}}}{{\\partial y}}+\\frac{{\\partial {\\sigma _z}}}{{\\partial z}}+{K_z}=0} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere the \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e, \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eK\u003c/em\u003e\u003csub\u003e\u003cem\u003ez\u003c/em\u003e\u003c/sub\u003e are components of the volume forces in the \u003cem\u003ex\u003c/em\u003e, \u003cem\u003ey\u003c/em\u003e and \u003cem\u003ez\u003c/em\u003e direction, respectively. The volume forces are gravity, magnetic force and so on, and the value usually can be ignored in the solution of plastic deformation.\u003c/p\u003e \u003cp\u003eEquation (\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) can be used directly or reconstructed according to the deformation model in the solution of engineering problems, but the stress direction and increment of micro-element should be described obviously to use the equilibrium differential equations correctly. For the first question in the solution step one (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e), it is well known that the stress increment shows only the change of stress in a certain direction and the value can be positive or negative so that the position exchange of the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e and \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;+\u0026thinsp;d\u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e does not affect the solution result. However, the Eq.\u0026nbsp;(\u003cspan refid=\"Equ1\" class=\"InternalRef\"\u003e1\u003c/span\u003e) is obtained under the conditions of stress assumption is positive, so that the stress increment should be consistent with the positive direction of the coordinate system (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e). For the second question in the solution steps one (Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e), the compressive or tensile stress should be allowed to set because of the actual existence of tensile and compressive stress states. According to the physics theory, the positive calculated value means that the actual stress direction is consistent with the assumption, and the negative calculated result means the actual value is opposite to the assumed direction.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section3\"\u003e \u003ch2\u003e2.2.2 Yield criterion\u003c/h2\u003e \u003cp\u003eThe typical yield criteria in plastic mechanics are the Tresca yield criterion and Mises yield criterion, and the Mises yield criterion can be described as the same form as the Tresca through introducing the Lode parameter.\u003cdiv id=\"Equ2\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ2\" name=\"EquationSource\"\u003e\n$${\\sigma _{\\hbox{max} }} - {\\sigma _{\\hbox{min} }}=\\beta {\\sigma _s}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e2\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eWhere \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e is yield strength, \u003cem\u003eσ\u003c/em\u003e\u003csub\u003emax\u003c/sub\u003e and \u003cem\u003eσ\u003c/em\u003e\u003csub\u003emin\u003c/sub\u003e are the maximum and minimum of principal stress, respectively. When the first and third principal stress is known in the deformation zone the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003emax\u003c/sub\u003e and \u003cem\u003eσ\u003c/em\u003e\u003csub\u003emin\u003c/sub\u003e equal to \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e and \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e, respectively. The \u003cem\u003eβ\u003c/em\u003e is coefficient related to Lode parameter \u003cem\u003e\u0026micro;\u003c/em\u003e\u003csub\u003e\u003cem\u003ed\u003c/em\u003e\u003c/sub\u003e, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\beta =\\frac{2}{{\\sqrt {3+\\mu _{d}^{2}} }}\\)\u003c/span\u003e\u003c/span\u003e, and the value equals 1 and 2/\u0026radic;3 under the axisymmetric and plane deformation condition, respectively.\u003c/p\u003e \u003cp\u003eFor solving the question in step 3 it is not necessary to set the stress direction of the micro-element, but it has an important influence on the application of yield criterion. The inaccurate determination of maximum and minimum value and direction of stress perhaps led to the multivalued solution and incorrect results. In general, the normal stress in the main deformation direction is either the first principal stress \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e or the third principal stress \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e3\u003c/sub\u003e. The positive calculated value means that the actual stress direction is consistent with the assumption, and the negative calculated result means the actual value is opposite to the assumed direction so that a minus should be added to the assumed compressive stress (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e) in the application of the yield criterion to correctly describe the physical meaning. As an example, the main compressive deformation is generated in the \u003cem\u003ey\u003c/em\u003e and \u003cem\u003er\u003c/em\u003e direction (Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e), and then the yield criterion can be described as\u003cdiv id=\"Equ3\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ3\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{c}} {{\\sigma _x} - \\left( { - {\\sigma _y}} \\right)=\\beta {\\sigma _s} \\Rightarrow {\\sigma _x}+{\\sigma _y}=\\beta {\\sigma _s}} \\\\ {\\left( { - {\\sigma _\\theta }} \\right) - \\left( { - {\\sigma _r}} \\right)=\\beta {\\sigma _s} \\Rightarrow {\\sigma _r} - {\\sigma _\\theta }=\\beta {\\sigma _s}} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e3\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section3\"\u003e \u003ch2\u003e2.2.3 Boundary condition\u003c/h2\u003e \u003cp\u003eFor the question in solution step three, the free surface or the boundary with known stress is often used to obtain the integral constants. The stress is zero on the free surface, and then it is easy to solve the integral constant. If the boundary value is known and not equal to zero, the stress on the boundary surface should be consistent with the direction of assumed stress, and there is no need to consider the physical significance of known boundary values. As an example, the known compressive stress on the boundary surface, and then the use of boundary conditions can be described as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn summary, the rule of the stress assumption, yield criterion determination and boundary conditions through the above analysis can be described as follows:\u003c/p\u003e \u003cp\u003e \u003cb\u003e(Ⅰ)\u003c/b\u003e The stress increment should be in the positive direction of the coordinate system.\u003c/p\u003e \u003cp\u003e \u003cb\u003e(Ⅱ)\u003c/b\u003e The minus should be added to the assumed compressive stress of the micro-element before substituting it into the yield criterion.\u003c/p\u003e \u003cp\u003e \u003cb\u003e(Ⅲ)\u003c/b\u003e When the known boundary stress is inconsistent with the assumed stress direction, a minus should be added to it as the boundary value.\u003c/p\u003e \u003cp\u003e \u003cb\u003e(Ⅳ)\u003c/b\u003e The negative calculated results mean that the actual stress is opposite to the assumed value direction, and the minus is only added to the assumed compressive stress to demonstrate the actual physical meaning of the final calculated results.\u003c/p\u003e \u003c/div\u003e \u003c/div\u003e"},{"header":"3 Plane deformation solution of a rectangular workpiece","content":"\u003cp\u003eThe plane deformation problems of rectangular workpiece compression are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. Where the stress direction of the micro-element is assumed to be compressive (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.b) and tensile (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.c), and the stress value on the boundary surface equals\u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e (Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.d). According to the rule (Ⅰ), the stress increment is set in the positive direction of the coordinate system. The contact friction is assumed to be the form of sliding friction met with Colulomb\u0026rsquo;s law \u003cem\u003eτ\u0026thinsp;=\u0026thinsp;fσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Assumed compressive stress in micro-element with free surface\u003c/h2\u003e \u003cp\u003eThe micro-element and the assumed stress direction are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.b, and the equilibrium differential equation is established according to the balance of force \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\sum {{F_x}} =0\\)\u003c/span\u003e\u003c/span\u003e\u003cdiv id=\"Equ4\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ4\" name=\"EquationSource\"\u003e\n$${\\sigma _x} \\cdot h \\cdot l=\\left( {{\\sigma _x}+{\\text{d}}{\\sigma _x}} \\right) \\cdot h \\cdot l+2\\tau \\cdot {\\text{d}}x \\cdot l$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e4\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eOrganizing the Eq.\u0026nbsp;(\u003cspan refid=\"Equ4\" class=\"InternalRef\"\u003e4\u003c/span\u003e), and then the friction condition \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\tau _{\\text{f}}}=f{\\sigma _y}\\)\u003c/span\u003e\u003c/span\u003e is substituted into the equilibrium differential equation\u003cdiv id=\"Equ5\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ5\" name=\"EquationSource\"\u003e\n$$\\frac{{{\\text{d}}{\\sigma _x}}}{{{\\text{d}}x}}+\\frac{{2f{\\sigma _y}}}{h}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e5\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe main compressive deformation is generated in the height direction, and the component of stress \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e is the minimum value of stress. Subsequently, according to the determination rule (Ⅱ), the minus is added to the assumed stress in the application of the yield criterion.\u003cdiv id=\"Equ6\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ6\" name=\"EquationSource\"\u003e\n$${\\sigma _{\\hbox{max} }} - {\\sigma _{\\hbox{min} }}=\\beta {\\sigma _{\\text{s}}} \\Rightarrow \\left( { - {\\sigma _x}} \\right) - \\left( { - {\\sigma _y}} \\right)=\\beta {\\sigma _{\\text{s}}} \\Rightarrow {\\sigma _y} - {\\sigma _x}=\\beta {\\sigma _{\\text{s}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e6\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAccording to the above Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), the d\u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u003c/em\u003e\u0026thinsp;d\u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e is obtained, and then the ordinary differential equation of the component of stress \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e can be described as Eq.\u0026nbsp;(\u003cspan refid=\"Equ7\" class=\"InternalRef\"\u003e7\u003c/span\u003e).\u003cdiv id=\"Equ7\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ7\" name=\"EquationSource\"\u003e\n$$\\frac{{{\\text{d}}{\\sigma _y}}}{{{\\text{d}}x}}+\\frac{{2f{\\sigma _y}}}{h}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e7\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe stress component in the \u003cem\u003ey\u003c/em\u003e direction can be obtained subsequently by integrating the differential equation.\u003cdiv id=\"Equ8\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ8\" name=\"EquationSource\"\u003e\n$${\\sigma _y}=C\\exp \\left( { - \\frac{{2f}}{h}x} \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e8\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSubsequently, the integral constant \u003cem\u003eC\u003c/em\u003e is obtained according to the boundary conditions. The \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u003c/em\u003e\u0026thinsp;0 with the \u003cem\u003ex\u0026thinsp;=\u0026thinsp;w\u003c/em\u003e/2 so that the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u0026thinsp;βσ\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e according to the Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), and then integral constant \u003cem\u003eC\u003c/em\u003e is obtained from the above Eq.\u0026nbsp;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e8\u003c/span\u003e).\u003cdiv id=\"Equ9\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ9\" name=\"EquationSource\"\u003e\n$$C=\\beta {\\sigma _{\\text{s}}}\\exp \\left( {\\frac{{fw}}{h}} \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e9\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSubstituting the constant \u003cem\u003eC\u003c/em\u003e into the Eq.\u0026nbsp;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e8\u003c/span\u003e), and then the contact stress in the \u003cem\u003ey\u003c/em\u003e direction can be described as\u003cdiv id=\"Equ10\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ10\" name=\"EquationSource\"\u003e\n$${\\sigma _y}=\\beta {\\sigma _{\\text{s}}}\\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e10\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSubstituting Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e) into Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), and then the distribution of the \u003cem\u003ex\u003c/em\u003e component stress \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e in the deformation body is described as\u003cdiv id=\"Equ11\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ11\" name=\"EquationSource\"\u003e\n$${\\sigma _x}=\\beta {\\sigma _{\\text{s}}}\\left\\{ {\\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right] - 1} \\right\\}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e11\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe values of the Eqs.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e) and (\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e) are greater than 0 which demonstrates the assumed stress direction is consistent with that of the actual stress, and the stress state is compressive. According to the rule (Ⅳ), considering the physical meaning the distribution of actual stress can be described as\u003cdiv id=\"Equ12\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ12\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{c}} {{\\sigma _x}= - \\beta {\\sigma _{\\text{s}}}\\left\\{ {\\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right] - 1} \\right\\}} \\\\ {{\\sigma _y}= - \\beta {\\sigma _{\\text{s}}}\\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right]} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e12\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAccording to the plane deformation theory, the stress in the direction without strain can be obtained\u003cdiv id=\"Equ13\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ13\" name=\"EquationSource\"\u003e\n$${\\sigma _z}=\\frac{1}{2}\\left( {{\\sigma _x}+{\\sigma _y}} \\right)= - \\beta {\\sigma _{\\text{s}}}\\left\\{ {\\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right] - \\frac{1}{2}} \\right\\}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e13\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe plane deformation of rectangular workpiece compression is three-dimensional compressive stress states according to the Eq.\u0026nbsp;(\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e) to (\u003cspan refid=\"Equ13\" class=\"InternalRef\"\u003e13\u003c/span\u003e).\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Assumed tensile stress in micro-element with free surface\u003c/h2\u003e \u003cp\u003eWhen the assumed stress of the micro-element is tensile, for the micro-element and the assumed stress direction shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e.c, the equilibrium differential equation is established\u003cdiv id=\"Equ14\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ14\" name=\"EquationSource\"\u003e\n$${\\sigma _x} \\cdot h \\cdot l+2\\tau \\cdot {\\text{d}}x \\cdot l=\\left( {{\\sigma _x}+{\\text{d}}{\\sigma _x}} \\right) \\cdot h \\cdot l$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e14\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eOrganizing the Eq.\u0026nbsp;(\u003cspan refid=\"Equ14\" class=\"InternalRef\"\u003e14\u003c/span\u003e), and then the friction condition \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\({\\tau _{\\text{f}}}=f{\\sigma _y}\\)\u003c/span\u003e\u003c/span\u003e is also substituted into the equilibrium differential equation\u003cdiv id=\"Equ15\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ15\" name=\"EquationSource\"\u003e\n$$\\frac{{{\\text{d}}{\\sigma _x}}}{{{\\text{d}}x}} - \\frac{{2f{\\sigma _y}}}{h}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e15\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eAccording to the determination rule (Ⅱ), the minus is not needed to add to the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e in the application of the yield criterion.\u003cdiv id=\"Equ16\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ16\" name=\"EquationSource\"\u003e\n$${\\sigma _{\\hbox{max} }} - {\\sigma _{\\hbox{min} }}=\\beta {\\sigma _{\\text{s}}} \\Rightarrow {\\sigma _x} - \\left( { - {\\sigma _y}} \\right)=\\beta {\\sigma _{\\text{s}}} \\Rightarrow {\\sigma _x}+{\\sigma _y}=\\beta {\\sigma _{\\text{s}}}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e16\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIt can be seen from equations (\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e) and (\u003cspan refid=\"Equ16\" class=\"InternalRef\"\u003e16\u003c/span\u003e) that the yield criterion expressions are different with the different assumed stress directions. According to the above Eq.\u0026nbsp;(\u003cspan refid=\"Equ16\" class=\"InternalRef\"\u003e16\u003c/span\u003e), the d\u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e\u003cem\u003e=-\u003c/em\u003ed\u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e is obtained, and then the ordinary differential equation is also obtained by substituting it to the Eq.\u0026nbsp;(\u003cspan refid=\"Equ15\" class=\"InternalRef\"\u003e15\u003c/span\u003e).\u003cdiv id=\"Equ17\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ17\" name=\"EquationSource\"\u003e\n$$\\frac{{{\\text{d}}{\\sigma _y}}}{{{\\text{d}}x}}+\\frac{{2f{\\sigma _y}}}{h}=0$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e17\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe stress \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e can be obtained subsequently by integrating the differential Eq.\u0026nbsp;(\u003cspan refid=\"Equ17\" class=\"InternalRef\"\u003e17\u003c/span\u003e).\u003cdiv id=\"Equ18\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ18\" name=\"EquationSource\"\u003e\n$${\\sigma _y}=C\\exp \\left( { - \\frac{{2f}}{h}x} \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e18\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe integral constant \u003cem\u003eC\u003c/em\u003e is obtained according to the boundary conditions. The \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u003c/em\u003e\u0026thinsp;0 with the \u003cem\u003ex\u0026thinsp;=\u0026thinsp;w\u003c/em\u003e/2 so that the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u0026thinsp;βσ\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e according to the Eq.\u0026nbsp;(\u003cspan refid=\"Equ16\" class=\"InternalRef\"\u003e16\u003c/span\u003e), and then integral constant \u003cem\u003eC\u003c/em\u003e is obtained from the above Eq.\u0026nbsp;(\u003cspan refid=\"Equ18\" class=\"InternalRef\"\u003e18\u003c/span\u003e).\u003cdiv id=\"Equ19\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ19\" name=\"EquationSource\"\u003e\n$$C=\\beta {\\sigma _{\\text{s}}}\\exp \\left( {\\frac{{fw}}{h}} \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e19\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSubstituting the \u003cem\u003eC\u003c/em\u003e into the Eq.\u0026nbsp;(\u003cspan refid=\"Equ18\" class=\"InternalRef\"\u003e18\u003c/span\u003e), and then the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e can be described as the same as Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e)\u003cdiv id=\"Equ20\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ20\" name=\"EquationSource\"\u003e\n$${\\sigma _y}=\\beta {\\sigma _{\\text{s}}}\\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e20\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSubstituting the Eq.\u0026nbsp;(\u003cspan refid=\"Equ20\" class=\"InternalRef\"\u003e20\u003c/span\u003e) into the Eq.\u0026nbsp;(\u003cspan refid=\"Equ16\" class=\"InternalRef\"\u003e16\u003c/span\u003e), and then the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e in the deformation body is described as\u003cdiv id=\"Equ21\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ21\" name=\"EquationSource\"\u003e\n$${\\sigma _x}=\\beta {\\sigma _{\\text{s}}}\\left\\{ {1 - \\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right]} \\right\\}$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e21\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe value of Eq.\u0026nbsp;(\u003cspan refid=\"Equ21\" class=\"InternalRef\"\u003e21\u003c/span\u003e) is less than zero that demonstrates the assumed stress direction is inconsistent with that of the actual stress, and the actual stress state is compressive. According to the rule (Ⅳ), considering the physical meaning the distribution of actual stress can be described as\u003cdiv id=\"Equ22\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ22\" name=\"EquationSource\"\u003e\n$$\\left\\{ {\\begin{array}{*{20}{c}} {{\\sigma _x}=\\beta {\\sigma _{\\text{s}}}\\left\\{ {1 - \\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right]} \\right\\}} \\\\ {{\\sigma _y}= - \\beta {\\sigma _{\\text{s}}}\\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right]} \\end{array}} \\right.$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e22\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eIt can be seen from equations (\u003cspan refid=\"Equ12\" class=\"InternalRef\"\u003e12\u003c/span\u003e) and (\u003cspan refid=\"Equ22\" class=\"InternalRef\"\u003e22\u003c/span\u003e) that no multivalued solution of stress results is generated according to the determination rules proposed and described.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec11\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Assumed compressive stress with the known stress boundary\u003c/h2\u003e \u003cp\u003eAccording to the rule (Ⅲ), the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u0026thinsp;σ\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e with the \u003cem\u003ex\u0026thinsp;=\u0026thinsp;w\u003c/em\u003e/2 so that the \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ey\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;\u003cem\u003e=\u0026thinsp;βσ\u003c/em\u003e\u003csub\u003e\u003cem\u003es\u003c/em\u003e\u003c/sub\u003e\u0026thinsp;+\u0026thinsp;\u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ek\u003c/em\u003e\u003c/sub\u003e according to the Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), and then the integral constant \u003cem\u003eC\u003c/em\u003e can be obtained from the above Eq.\u0026nbsp;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e8\u003c/span\u003e).\u003cdiv id=\"Equ23\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ23\" name=\"EquationSource\"\u003e\n$${\\sigma _y}\\left| {_{{x=w/2}}} \\right.=C\\exp \\left( { - \\frac{{2f}}{h}x} \\right)\\left| {_{{x=w/2}}} \\right.=\\beta {\\sigma _s}+{\\sigma _k} \\Rightarrow C=\\left( {\\beta {\\sigma _s}+{\\sigma _k}} \\right)\\exp \\left( {\\frac{{fw}}{h}} \\right)$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e23\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSubstituting the \u003cem\u003eC\u003c/em\u003e into the Eq.\u0026nbsp;(\u003cspan refid=\"Equ8\" class=\"InternalRef\"\u003e8\u003c/span\u003e), and then the contact stress in the \u003cem\u003ey\u003c/em\u003e direction can be described as\u003cdiv id=\"Equ24\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ24\" name=\"EquationSource\"\u003e\n$${\\sigma _y}=\\left( {\\beta {\\sigma _{\\text{s}}}+{\\sigma _k}} \\right)\\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e24\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eSubstituting Eq.\u0026nbsp;(\u003cspan refid=\"Equ24\" class=\"InternalRef\"\u003e24\u003c/span\u003e) into the Eq.\u0026nbsp;(\u003cspan refid=\"Equ6\" class=\"InternalRef\"\u003e6\u003c/span\u003e), and then the distribution of the stress \u003cem\u003eσ\u003c/em\u003e\u003csub\u003e\u003cem\u003ex\u003c/em\u003e\u003c/sub\u003e in the deformation body is described as\u003cdiv id=\"Equ25\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ25\" name=\"EquationSource\"\u003e\n$${\\sigma _x}=\\beta {\\sigma _{\\text{s}}}\\left\\{ {\\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right] - 1} \\right\\}+{\\sigma _k} \\cdot \\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e25\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eThe values of the Eqs.\u0026nbsp;(\u003cspan refid=\"Equ24\" class=\"InternalRef\"\u003e24\u003c/span\u003e) and (\u003cspan refid=\"Equ25\" class=\"InternalRef\"\u003e25\u003c/span\u003e) are greater than 0 which demonstrates the assumed stress direction is consistent with that of the actual stress, and the stress state is compressive. Furthermore, it can be seen from equations (\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e), (\u003cspan refid=\"Equ11\" class=\"InternalRef\"\u003e11\u003c/span\u003e), (24) and (25) that the application of boundary compressive stress enhances the stress states and increases the deformation load obviously, which is in agreement with the plastic deformation theory.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec12\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Deformation load\u003c/h2\u003e \u003cp\u003eTo solve the deformation load, the Eq.\u0026nbsp;(\u003cspan refid=\"Equ10\" class=\"InternalRef\"\u003e10\u003c/span\u003e) is used to integrate, and the \u003cem\u003eβ\u003c/em\u003e equals 2/\u0026radic;3 under plane deformation condition is also substituted.\u003cdiv id=\"Equ26\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ26\" name=\"EquationSource\"\u003e\n$$P=2\\mathop \\smallint \\nolimits_{0}^{{\\frac{w}{2}}} {\\sigma _y}{\\text{d}}x=2\\mathop \\smallint \\nolimits_{0}^{{\\frac{w}{2}}} \\beta {\\sigma _{\\text{s}}}\\exp \\left[ {\\frac{{2f}}{h}\\left( {\\frac{w}{2} - x} \\right)} \\right]{\\text{d}}x{\\text{=}}\\frac{{2h{\\sigma _{\\text{s}}}}}{{\\sqrt 3 f}}\\left[ {\\exp \\left( {\\frac{{fw}}{h}} \\right) - 1} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e26\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eFurthermore, the deformation load at different times is also solved according to the constant volume of plastic deformation. Assuming the initial height and width are \u003cem\u003eh\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e and \u003cem\u003ew\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e, respectively, the volume size meets the relationship \u003cem\u003ew\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e.\u003cem\u003eh\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u0026thinsp;=\u0026thinsp;\u003cem\u003ew\u003c/em\u003e.\u003cem\u003eh\u003c/em\u003e during the compressing processes. And then the relationship between the deformation load and the height of the rectangular workpiece can be described as\u003cdiv id=\"Equ27\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ27\" name=\"EquationSource\"\u003e\n$$P=\\frac{{2h{\\sigma _{\\text{s}}}}}{{\\sqrt 3 f}}\\left[ {\\exp \\left( {\\frac{{fw}}{h}} \\right) - 1} \\right]=\\frac{{2h{\\sigma _{\\text{s}}}}}{{\\sqrt 3 f}}\\left[ {\\exp \\left( {\\frac{{f \\cdot {w_0} \\cdot {h_0}}}{{{h^2}}}} \\right) - 1} \\right]$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e27\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eTo verify the precision of the principal stress method, the deformation load in plane compressing is solved by the DEFORM software and the Eq.\u0026nbsp;(\u003cspan refid=\"Equ27\" class=\"InternalRef\"\u003e27\u003c/span\u003e). The calculated results by FEM and comparison of load are shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. The material is regarded as ideal plasticity and the yield stress is 300MPa. The other parameters used in the solution are shown as follows: the initial width \u003cem\u003ew\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e is 20 mm, the initial height \u003cem\u003eh\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e is 16mm, the final height is 10mm and the friction coefficient \u003cem\u003ef\u003c/em\u003e is 0.3. Because of the ideal plastic material, the equivalent stress is 300MPa in most deformation zones, and only the value of the symmetric centre is less than 300MPa. The equivalent strain distribution of plane deformation compression of rectangular workpieces is X shape, and the larger value is generated in the side flattening and central zones. The increment of the contact surface during the compression processes leads to the deformation load increasing nonlinearly. The average relative error is 8.3% between the predicted loads by FEM and PSM, and the stress and load predicted by RSM are in good agreement with the FEM.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec13\" class=\"Section2\"\u003e \u003ch2\u003e3.5 Teaching effectiveness testing\u003c/h2\u003e \u003cp\u003eIn order to test the teaching effectiveness of the PSM solving rules proposed in this article, 74 and 72 elective students were tested in 2022 and 2023 respectively. The five test questions are all typical engineering problems of long pipelines subjected to internal pressure, deep drawing of cylindrical parts, cylinder compression, wide plate bending, and bar extrusion, with a full score of 100 points. In 2022, the traditional textbook PSM analysis method was taught in the classroom, and in 2023, the PSM analysis method proposed in this article is taught. The average score distribution after two years of testing is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. From the figure, it can be seen that when using traditional PSM methods to solve practical slip line engineering problems, due to the lack of clear analytical rules, students are prone to generate ambiguous solutions when solving typical engineering problems with a high error rate, low average score, and a high failure rate of 10.8% for students; After learning the PSM method analysis rules proposed in this article, students' knowledge application ability has been significantly improved. Most students not only master the basic principles of PSM method, but also have a high accuracy rate and a significant improvement in excellence rate when using PSM method to solve engineering problems, accounting for 62.5%. The PSM method solution rules proposed in this article can effectively improve classroom teaching effectiveness and enhance students' ability to solve engineering problems.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4 Conclusions","content":"\u003cp\u003e(1) The stress assumption, yield criterion determination and boundary conditions have been investigated according to the frequently asked questions in the teaching and application of the PSM, and the rule was proposed and described in detail. The stress increment should be in the positive direction of the coordinate system, and the minus should be added to the assumed compressive stress of the micro-element before substituting it into the yield criterion. Furthermore, the negative calculated results mean that the actual value is opposite to the assumed stress direction, and the minus is only added to the assumed compressive stress to demonstrate the actual physical meaning of the final calculated results.\u003c/p\u003e \u003cp\u003e(2) The stress distribution and load in a plane deformation of a rectangular workpiece was successfully solved using the PSM and FEM. The equivalent strain distribution of plane deformation compression of rectangular workpieces is X shape. The average relative error is 8.3% between the predicted loads by FEM and PSM, and the stress and load predicted by PSM are in good agreement with the FEM. The same calculated results of stress were obtained under the different conditions of stress assumptions and yield criterion, without generating a multivalued solution, indicating that the rule proposed has reliable and practical significance for the teaching and application of the PSM of plastic mechanics.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eConflict of interest statement\u003c/h2\u003e \u003cp\u003eThe authors declare that there is no conflict of interest.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eAcknowledgement\u003c/h2\u003e \u003cp\u003eThe authors gratefully acknowledge financial support from the Program of Web-Delivery for Elaborate Course of Hebei Province (No. 2020JPKC061), Hebei Higher Education Teaching Reform Research Project (2022GJJG427) and Specialized and Innovative Integrated Courses of Hebei Province (2023CXCY263).\u003c/p\u003e\u003ch2\u003eData Availability Statement\u003c/h2\u003e \u003cp\u003eThe data that support the findings of this study are available on request from the corresponding author, please contact \u003cspan type=\"Underline\" class=\"Underline\" name=\"Emphasis\"\[email protected]\u003c/span\u003e\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\u003cli\u003e\u003cspan\u003eWang ZR, Yuan SJ, Hu LX et al (2007) Fundamentals of elastic and plastic mechanics (second edition). 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Comput Geotech 151:104951\u0026ndash;104977\u003c/span\u003e\u003c/li\u003e\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":true,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-the-brazilian-society-of-mechanical-sciences-and-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"bmse","sideBox":"Learn more about [Journal of the Brazilian Society of Mechanical Sciences and Engineering](http://link.springer.com/journal/40430)","snPcode":"40430","submissionUrl":"https://www.editorialmanager.com/bmse/default2.aspx","title":"Journal of the Brazilian Society of Mechanical Sciences and Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"PSM, yield criterion, plane deformation, boundary condition, stress assumption","lastPublishedDoi":"10.21203/rs.3.rs-4489595/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4489595/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eAs one of the main methods to solve deformation mechanics, the principal stress method (PSM) is widely used in stress analysis and technology optimization of metal forming processes. The stress assumption, yield criterion and boundary conditions have been investigated according to the frequently asked questions in teaching and the application of the PSM, and then the determination rule was proposed and described. The yield criterion, boundary conditions and calculated results should keep the corresponding relationship with the initial stress assumption of the micro-element. The stress distribution and load in the plane deformation of the rectangular workpiece was successfully solved using the PSM based on the rule proposed and FEM. The average relative error is 8.3% between the predicted loads by FEM and PSM, and the stress and load predicted by PSM are in good agreement with the FEM. Furthermore, the same calculated results of stress were obtained under different conditions, indicating that the rule proposed is reliable and practical significance for the teaching and application of the plastic mechanics.\u003c/p\u003e","manuscriptTitle":"Discussion on stress assumption and yield criterion determination rule in teaching of the PSM","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-07-17 00:06:29","doi":"10.21203/rs.3.rs-4489595/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"","date":"2024-06-25T02:35:37+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-06-24T18:14:01+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-05-30T07:59:14+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of the Brazilian Society of Mechanical Sciences and Engineering","date":"2024-05-28T05:05:57+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-the-brazilian-society-of-mechanical-sciences-and-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"bmse","sideBox":"Learn more about [Journal of the Brazilian Society of Mechanical Sciences and Engineering](http://link.springer.com/journal/40430)","snPcode":"40430","submissionUrl":"https://www.editorialmanager.com/bmse/default2.aspx","title":"Journal of the Brazilian Society of Mechanical Sciences and Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"fc303668-b81a-427e-af40-70a72459c863","owner":[],"postedDate":"July 17th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-01-13T16:10:22+00:00","versionOfRecord":{"articleIdentity":"rs-4489595","link":"https://doi.org/10.1007/s40430-024-05342-7","journal":{"identity":"journal-of-the-brazilian-society-of-mechanical-sciences-and-engineering","isVorOnly":false,"title":"Journal of the Brazilian Society of Mechanical Sciences and Engineering"},"publishedOn":"2025-01-08 15:57:41","publishedOnDateReadable":"January 8th, 2025"},"versionCreatedAt":"2024-07-17 00:06:29","video":"","vorDoi":"10.1007/s40430-024-05342-7","vorDoiUrl":"https://doi.org/10.1007/s40430-024-05342-7","workflowStages":[]},"version":"v1","identity":"rs-4489595","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4489595","identity":"rs-4489595","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}