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.