J. Cent. South Univ. (2020) 27: 2592-2605

DOI: https://doi.org/10.1007/s11771-020-4484-x

Primary and recency effects based on loading path in classical plasticity

GAO Yue(高岳)¹, SHAO Fei(邵飞)¹, FAN Peng-xian(范鹏贤)²,XU Qian(徐倩)¹, GU Juan(顾娟)², WANG Shang-long(王尚龙)³

1. Field Engineering College, Army Engineering University of PLA, Nanjing 210007, China;

2. Defense Engineering College, Army Engineering University of PLA, Nanjing 210007, China;

3. Engineering Design Institute of the Army Research Institute, Nanjing 210018, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: We have established an elastoplastic analysis model to explore the effect of loading path in an incompressible thin-walled tube under the combined action of axial force and torque based on Mises yield condition and isotropic linear hardening assumption. Further, four stress areas (σ_x, τ_x) are divided according to the characteristics of the final stress, and the plastic stress-strain relationship of twelve stress paths in different stress areas is derived. The “primary effect” of the stress path on plastic strain is demonstrated, namely, the plastic strain caused by the pre-loaded stress in path A (tensile stress is initially applied, followed by shear stress) is always greater than that caused by the post-loaded stress in path C (shear stress is initially applied, followed by tensile stress) irrespective of the value of final stress. The “recency effect” of the strain path on the stress is also established, which indicates that the stress caused by the post-loaded strain in path A is always greater than that caused by the pre-loaded strain in path C irrespective of the value of final strain. From the perspective of deformation, the “primary effect” of the stress path on the plastic strain and the “recency effect” of the strain path on the stress are unified. These effects are succinct and universal, and they provide useful insights on the plastic stress-strain relationship under different loading paths. Furthermore, they can serve as a useful reference for optimizing the processing technologies and construction procedures.

Key words: isotropic linear hardening; stress path; strain path; primary effect; recency effect

Cite this article as: GAO Yue, SHAO Fei, FAN Peng-xian, XU Qian, GU Juan, WANG Shang-long. Primary and recency effects based on loading path in classical plasticity [J]. Journal of Central South University, 2020, 27(9): 2592-2605. DOI: https://doi.org/10.1007/s11771-020-4484-x.

1 Introduction

The primary and recency effects are crucial theories in psychology and pedagogy. The primary effect implies that in social cognition, the first information or stimulus provided to an individual leaves him/her with the most striking impression, which is also called the first impression effect. In contrast, the recency effect refers to the psychological effect of recent stimuli that promotes the formation of impressions, namely, the newly obtained information has a greater impact on individuals than the previously obtained information. Presently, investigations on primary and recency effects are limited to social sciences [1-3], and relevant studies in the fields of mechanical engineering and material engineering are lacking in the exiting literature.

According to classical mechanics and practical experience, when a material is loaded into the plastic deformation stage, the stress or strain path has a significant impact on the final plastic strain or stress state. Existing studies mainly focused on the effect of loading path in engineering materials through laboratory experiments [4-9] and numerical methods [10, 11] to establish a quantitative relationship between the plasticity behavior and loading history, thus improving the plastic constitutive model of materials [12]. Most of these studies concentrated on specific materials, especially geotechnical materials. The unloading effect [13], microstructural study [14-16], fatigue prediction [17-19], and principal stress rotation effect [20] are closely related with the loading path, which plays an important role in exploring the influence of loading history on the mechanical behavior of materials according to classical plasticity theory. The constitutive modelling based on the isotropic linear hardening hypothesis and the Mises yield criterion is a relatively mature research field. Combining finite element and mathematical methods, such models have been primarily developed for elastoplastic structural response [21], non-linear mechanical analysis [22], algorithm for elastoplastic problems [23-25], residual stress prediction [26], etc. In this study, we have investigated the influence of loading path on the final stress-strain state based on Mises yield criterion and the isotropic linear hardening hypothesis.

Classical elastoplasticity generally uses internal variables to describe the impact of loading history, but it lacks a general description for the impact of loading history. For the combined tensile and torsional loading of a classical thin-walled tube, CHEN [27] proposed an analytical solution of plastic strain under three stress paths (A: pull followed by twist; B: proportional loading; C: twist followed by pull) when the end stress was (σ_s, where σ_s stands for yield stress. XU et al [28] presented the analytical solution of stress for three strain paths (A: pull followed by twist; B: proportional loading; C: twist followed by pull) when the end plastic strain in the strain space was The analysis results under the two conditions showed that the stress and strain paths have a significant effect on the plastic strain and stress, respectively. However, these studies did not indicate the law governing the influence of loading history.

Through analysis and comparison of loading history and strain (stress) results, it is observed that the first stress loaded under different stress paths has a stronger impact on the final plastic strain, which represents the primary effect due to the stress path. Under different strain paths, the post-loaded strain has a stronger impact on the final stress, which represents the recency effect due to the strain path. However, the applicability of this law for any point in the plastic area needs to be further validated. Besides, it must be clarified that whether the primary effect due to the stress path and the recency effect due to the strain path are inherently related and unified.

Based on the classical elastoplastic mechanics, this study extends two special points in the two cases in Refs. [27, 28] to the entire stress-strain space outside the initial yield surface. Based on these cases, two universal relationships on the effect of stress-strain path with practical guidance are deduced. The primary effect of stress path on the plastic strain is theoretically proven, and the recency effect of the strain path on the stress is explained through derivation and discussion. The primary and recency effects succinctly describe the two complex rules, linking them with the two familiar effects in psychology, which is easy to comprehend. Further, through tensile testing of a thin-walled tube composed 304 stainless steel, it is verified that some engineering materials are consistent with the Mises yield criterion and the isotropic linear hardening assumption, thereby indicating the good application prospects of the two effects. Finally, based on the nominal deformation modulus, the unity of primary and recency effects is proved from the perspective of material strengthening.

2 Model construction, determination of loading, and derivation of plastic deformation increment

2.1 Basic assumptions and analysis model

It is assumed that a thin-walled circular tube made of incompressible materials (μ=0.5) is subjected to axial tensile force and torque. As shown in Figure 1, the yielding of the material conforms to the Mises yield condition and isotropic linear hardening. The stress-strain relation under uniaxial stretching is as follows:

Figure 1 Schematic of loading model

                    (1)

The following three loading paths are considered for reaching the final stress state in the plastic area, and the corresponding plastic strains and are then obtained.

1) Initial loading to σ_z=σ_x along the z axis, the shear stress is increased to τ_θz=τ_x while keeping σ_x unchanged.

2) Simple loading (proportional loading), the stress is increased to σ_z=σ_x, τ_θz=τ_x=mσ_x according to the proportion σ_z:τ_θz=1:m.

3) The shear stress is increased to τ_θz=τ_x, and then the tensile stress is increased to σ_z=σ_x while keeping τ_x unchanged.

2.2 Determination of loading

After the stress state reaches the initial yield, the occurrence of plastic deformation in the materials due to the subsequent increment in the stress depends on whether the increase in the stress causes the loading of the material, i.e., whether

The bias of stress state under each loading path can be expressed as follows:

            (2)

Hence,

                            (3)

Using the Mises loading face , it can be obtained that Consequently,

         (4)

Since σ_z, dσ_z have the same sign, and τ_θz, dτ_θz have the same sign, i.e., the material has been in the loading state after yielding.

2.3 Plastic deformation increment

Considering the orthogonal flow rule, Eqs. (2), and (4), we get:

             (5)

          (6)

3 Deduction of and under different stress paths

3.1 Space partition of (σ_x, τ_x) stress

Because the integration interval ofand is affected by the final stress values of σ_x and τ_x, the (σ_x, τ_x) space is divided into four areas by making a vertical and a horizontal line perpendicular to the coordinate axes through the two intersection points of the initial yield surface and the coordinate axes. As shown in Figure 2, after partitioning, and is calculated in the areas I-IV according to paths A, B, and C, and the impact of different stress paths on the plastic strain is analyzed.

Figure 2 Space partition of and diagram of stress path in area I

3.2 Section I (σ_x≥σ_s, , m>0)

3.2.1 Stress path A

When the applied axial stress is σ_z=σ_s, the material yields after continuing to load, resulting in an axial plastic strain. After loading to σ_x, then keeping σ_z=σ_x unchanged, i.e., dσ_z=0, we apply torque to facilitate τ_θz=τ_x.

Using Eqs. (5) and (6), we get:

               (7)

               (8)

3.2.2 Stress path B

In the loading process, dτ_θz=mdσ_z, and the material is loaded to When the material begins to yield. Using Eqs. (5) and (6), we get:

                   (9)

                 (10)

3.2.3 Stress path C

When the applied shear stress is , the material yields after continuing to load, resulting in a shear plastic strain. After loading to τ_x, then keeping unchanged, i.e., dτ_θz=0, we apply an axial force to facilitate σ_z=σ_x.

Using Eqs. (5) and (6), we get:

             (11)

           (12)

3.2.4 Determination of tensile plastic strain

We construct a function:

 (13)

Substituting τ_x=mσ_x in Eq. (13), we get:

        (14)

Taking partial derivative of Eq. (14) with respect to σ_x and m, we obtain:

                     (15)

         (16)

According to Eqs. (15) and (16), the function f ¹ is monotonically increasing for both σ_x and m. Since σ_x≥σ_s and m>0 in area I, it can be obtained that in the limit of σ_x=σ_s and m→0, f ¹ takes the

minimum value, and considering Eq. (14), . However, in the area Ⅰ, m≠0, so f ¹>0 is always valid in the entire area I, namely,

                               (17)

is always valid.

Construct a function:

 (18)

Substituting in Eq. (18), we get:

         (19)

Taking partial derivative of Eq. (19) with respect to τ_x and m, we obtain:

                  (20)

         (21)

According to Eqs. (20) and (21), the function f² is monotonically increasing for τ_x and decreasing

for m . Since and m>0 in area I, it can be obtained that in the limit of and f ² assumes the minimum value, and considering Eq. (19), However, in the area I, m cannot get positive infinity, so f²>0 is always valid in the entire area I, namely,

                              (22)

is always valid.

3.2.5 Determination of shear plastic strain

Construct a function:

                  (23)

Substituting τ_x=mσ_x in Eq. (23), we obtain:

           (24)

Taking partial derivative of Eq. (24) with respect to σ_x and m, we get:

                 (25)

         (26)

According to Eqs. (25) and (26), the function f³ is monotonically decreasing for both σ_x and m. Since σ_x≥σ_s and m>0 in the area I, it can be obtained that in the limit of σ_x=σ_s and m→0, f³ takes the maximum value, and considering Eq. (24), we get However, in the area I, m≠0; so f³<0 is always valid in the entire area I, namely,

                              (27)

is always valid.

Construct a function:

           (28)

Substituting in Eq. (28), we obtain:

      (29)

Taking partial derivative of Eq. (29) with respect to τ_x and m, we get:

                   (30)

        (31)

According to Eqs. (30) and (31), the function f⁴ is monotonically decreasing for τ_x and increasing for m. Since and m>0 in the area I, it can be obtained that in the limit of and f⁴ assumes the maximum value, and considering Eq. (29), we get . However, in the area I, m cannot get positive infinity, so f⁴<0 is always valid in the entire area I, namely,

                              (32)

is always valid.

Combining Eqs. (17), (22), (27) and (32), we obtain that

                        (33)

is always valid.

3.3 Section II (σ_x≥σ_s, , )

As shown in Figure 3, when the end point of stress “(σ_x, τ_x)” is located in the area II, the final plastic strain under the three paths is calculated according to stress paths A, B, and C.

3.3.1 Stress paths A and B

The loading conditions of paths A and B in the area II are the same as those in the area I, and all of them satisfy σ_x≥σ_s. Thus, the expression of , of path A in section II is the same as that of of path A in section I, and the expression of of path B in section II is the same as that of of path B in section I. It can be obtained that

                    (34)

3.3.2 Stress path C

When the applied shear stress is then keeping unchanged, i.e., we apply an axial force. When the applied axial force is the loading continues and the material yields until it is loaded to

Figure 3 Space partition of (σ_x, τ_x) and diagram of stress path in area II

Using Eqs. (5) and (6), we obtain:

                 (35)

                   (36)

3.3.3 Determination of tensile plastic strain

Construct a function:

        (37)

Substituting in Eq. (37), we obtain:

        (38)

Taking partial derivative of Eq. (38) with respect to τ_x, we obtain

When

When

When

Taking partial derivative of Eq. (38) with respect to m, we obtain

Construct a function so that Therefore, g₁(m) monotonically increases to m.

When g₁(m)=0, and then we obtain

When g₁(m)<0, and then we obtain

When g₁(m)>0, and then we obtain

From the above analysis, it can be concluded that the function f⁵ for τ_x and m is first decreased and then increased, and the minimum value can be obtained when (τ_x, m) satisfies Substituting in Eq. (38), we obtain

Since and i.e., Since in Section II, we obtain However, σ_x≥σ_s in Section II, indicating that (τ_x, m) cannot satisfy Thus, f⁵>0 is always valid in Section II, namely,

                              (39)

is always valid.

3.3.4 Determination of shear plastic strain

Construct a function,

                    (40)

Substituting  in Eq. (40), we get:

     (41)

Taking partial derivative of Eq. (41) with respect to τ_x, we get

Taking partial derivative of Eq. (41) with respect to m, we get

Construct a function so that Therefore, g₂(m) monotonically decreases with m.

It can be concluded that the function f⁶ for τ_x and m is first increased and then decreased,and the maximum value can be obtained when (τ_x, m) satisfies Substitutingin Eq. (41), we get

Since and τ_x=mσ_x, i.e.,Since in Section II, we obtain However, σ_x≥σ_s in section II, indicating that (τ_x, m) cannot satisfy Thus, f⁶<0 is always valid in section II, namely,

                             (42)

is always valid.

Combining Eqs. (34), (39) and (42), we obtain that

                       (43)

is always valid.

3.4 Section Ⅲ (σ_s>σ_x>0, ,

As shown in Figure 4, when the end point of stress “(σ_x, τ_x)” is located in the area III, the final plastic strain under the three paths is calculated according to the stress paths A, B, and C.

Figure 4 Space partition of (σ_x, τ_x) and diagram of stress path in area III

3.4.1 Stress paths B and C

The loading conditions of paths B and C in area III are the same as those in area I, and all of them satisfy . Thus, the expression of of path B in Section III is the same as that of of path B in Section I, and the expression of of path C in Section III is the same as that of of path C in section I. Therefore, it can be obtained that

                    (44)

is always valid.

3.4.2 Stress path A

When the applied axial stress is σ_z=σ_x, then keeping σ_z=σ_x unchanged, i.e., dσ_z=0, we apply a torque. When the applied torque is the loading is continued and the material yields until it is loaded to τ_θz=τ_x.

Using Eqs. (5) and (6), we get:

                      (45)

                (46)

3.4.3 Determination of tensile plastic strain

Construct a function:

 (47)

Substituting τ_x=mσ_x in Eq. (47), we obtain:

    (48)

Taking partial derivative of Eq. (48) with respect to σ_x, we obtain . Further, the partial derivative of Eq. (48) with respect to m is

It can be concluded that the function f⁷ for σ_x and m is first decreased and then increased,and the minimum value can be obtained when (σ_x, m) satisfies Substituting in Eq. (48), we obtain

Since and i.e., As in Section III, and monotonically increases with m, we obtain However, in Section III, indicating that (σ_x, m) cannot satisfy Thus, f⁷>0 is always valid in the entire Section III, namely,

                               (49)

is always valid.

3.4.4 Determination of shear plastic strain

Construct a function:

                  (50)

Substituting τ_x=mσ_x in Eq. (50), we get:

        (51)

Taking partial derivative of Eq. (51) with respect to σ_x, we get Similarly, the partial derivative of Eq. (51) with respect to m is

It can be concluded that the function f⁸ for σ_x and m is first increased and then decreased,and the maximum value can be obtained when (σ_x, m) satisfies Substituting in Eq. (51), we get

Since and i.e., As in Section III and is monotonically increasing for m, we obtain . Further, in Section III, indicating that (σ_x, m) cannot satisfy Thus, f⁸<0 is always valid in the entire Section III, namely,

                              (52)

is always valid.

Combining Eq. (44), (49) and (52), we obtain that

                        (53)

is always valid.

3.5 Section IV (σ_s>σ_x>0, , m>0)

As shown in Figure 5, when the end point of stress “(σ_x, τ_x)” is located in the area IV, the final plastic strain under the three paths is calculated according to the stress paths A, B, and C.

The loading conditions of paths A and B in Section IV are the same as those in Section III, and all of them satisfy σ_s>σ_x>0. Therefore, the expression of of path A in Section IV is the same as that of of path A in Section III, and the expression of of path B in Section IV is the same as that of of path B in Section III. It can be directly obtained that:

                    (54)

is always valid.

Figure 5 Space partition of (σ_x, τ_x) and diagram of stress path in area IV

The loading conditions of paths B and C in Section IV are the same as those in Section II. Moreover, all of them satisfy Therefore, the expression of of path B in Section IV is the same as that of of path B in Section II, and the expression of of path C in Section IV is the same as that of of path C in Section II, respectively. It can be directly obtained that

                    (55)

is always valid.

Combining Eqs. (54) and (55), we infer that

                       (56)

is always valid.

3.6 Correlation of and under different stress paths

Combining Eqs. (33), (43), (53) and (56), it can be inferred that the following expressions of and under different stress paths are always valid:

                        (57)

According to Eq. (57), under different stress paths, if tensile stress is applied first followed by shear stress (stress path A), the corresponding tensile strain is the largest strain among the three stress paths. If shear stress is applied first followed by tensile stress (stress path C), the corresponding shear strain is the largest strain among the three stress paths. When proportional loading (stress path B) is implemented, the final tensile strain and shear strain lie between the strain results of paths A and C. Irrespective of the position of the final stress state in the plastic area, under the combined action of tension and torsion, the order of stress application has a significant impact on the composition of the final plastic strain, and the effect of first applied stress is more obvious in the final plastic strain, which is similar to the “preconceived” effect in life. This implies that the stress path has a primary effect on the plastic strain. The primary effect is not only applicable to specific stress states in Ref. [27] but also to all the stress states located in the plastic area.

4 Discussion

4.1 Recency effect based on strain path

XU et al [28] proposed an analytical solution for stress after reaching a specific strain state through three specific strain paths. It can be found that under different strain paths, if tensile strain is applied first, followed by shear strain (strain path A), the corresponding shear stress is the largest stress among the three strain paths; if shear strain is applied first, followed by tensile strain (strain path C), the corresponding tensile stress is the largest stress among the three strain paths. Under proportional loading (strain path B), the final tensile stress and shear stress lie between the stress results of paths A and C.

To verify the applicability of this relation in the entire plastic area, it is assumed that the strain state (ε_x, γ_x) of the plastic area is reached through three strain paths, which are as follows: A is first loaded with tensile strain and then with shear strain, B undergoes proportional loading, and C is first loaded with shear strain and then with tensile strain. Through stringent calculations, it can be obtained that σ and τ satisfy the following condition under different strain paths:

                          (58)

The rule: “post-loaded strain has a stronger effect on the final stress under different strain paths” is generally applicable, which is similar to the “recency effect” in psychology.

4.2 Consistency of primary and recency effects

For isotropic linearly hardened materials that satisfy the Mises yield condition, stress path has the primary effect on plastic strain; while strain path exhibits recency effect on the stress results. Consequently, it must be investigated whether the primary effect is contradictory to the recency effect.

First, stress path A (tensile stress is applied first, followed by shear stress) is analyzed, and the nominal deformation modulus is defined as E=σ/ε. Because final tensile stress σ is constant and the primary effect exists in this case, the tensile strain ε^p1 corresponding to the first applied tensile stress is the maximum among the three stress paths, so the corresponding nominal deformation modulus E is the minimum. This implies that the first applied stress has a smaller nominal deformation modulus. For strain path A (tensile strain is applied first, followed by shear strain), since the final tensile strain ε is fixed and the recency effect exists in this case, the tensile stress σ¹ corresponding to the first applied tensile strain is the minimum among the three strain paths, so the corresponding nominal deformation modulus E is also the minimum. From the perspective of the deformation modulus, the effects of stress path A and strain path A on the deformation are similar.

The stress path C (shear stress is applied first, followed by tensile stress) is analyzed, and the nominal shear deformation modulus is defined as G=τ/γ. Because the final shear stress τ is fixed and the primary effect exists, the shear strain γ^p3 corresponding to the first applied shear stress is the maximum among the three stress paths, so the corresponding nominal shear deformation modulus G is the minimum. This implies that the first applied stress has a smaller nominal shear deformation modulus. For strain path C (shear strain is applied first, followed by tensile strain), since the end shear strain γ is constant and the recency effect exists, the corresponding shear stress τ³ of the first applied shear strain is the smallest in the three strain paths, so the corresponding nominal shear deformation modulus G is the also the minimum. From the perspective of shear deformation modulus, the effects of stress path C and strain path C on deformation are consistent.

For proportional loading (stress path B and strain path B), the nominal deformation modulus lies between loading paths A and C.

To recapitulate, from the perspective of deformation, the primary effect of stress path and the recency effect of strain path are dialectically unified. The pre-loaded stress or strain has a corresponding smaller deformation (shear) modulus, while the post-loaded stress or strain has a larger deformation (shear) modulus. Further, the effects of stress path and strain path can be unified as the recency effect of loading path on the nominal deformation modulus. This implies that irrespective of whether it is stress loading or strain loading, the material is less resistant to the pre-loaded stress or strain, and is more prone to deformation, while it is more resistant to post-load stress or strain. Based on the physical mechanism, both the primary and recency effects are direct results of material strengthening.

4.3 Potential applications of primary and recency effects

Facilitated by the psychological terms, the complex loading path effect is closely related to the familiar primary and recency effects, which play a crucial role in the teaching practice of elastoplasticity. This simplification of complex problems can help the students to quickly judge the accuracy of the results of stress path or strain path, thereby strengthening their understanding of the impact of loading path on plastic deformation.

Examples of loading paths with a significant impact on a project are often encountered. For isotropic Mises materials, the recency effect is both simple and universal, and is a highly generalized law. The rules of primary and recency effects can be generalized to many engineering materials and have good application prospects.

To prove these application prospects, we designed a thin-walled tube for tensile test, as shown in Figure 6. This thin-walled tube was composed of 304 stainless steel, which is a common engineering material. The stress-strain curve obtained through the tensile test is shown in Figure 7. This curve is consistent with the “bilinear isotropic hardening” in classical theory, and the linear strengthening characteristic is obvious. This indicates that the classic Mises yield criterion and isotropic linear hardening assumption are consistent with the materials used in engineering practices. Therefore, the primary and recency effects based on the classic Mises yield criterion and the isotropic linear hardening hypothesis are not overly idealized theories but can be applied to practical engineering materials and have good application prospects. However, it must be studied whether this law can be extended to more general situations, such as non-linearly hardened materials or even more complex geotechnical materials.

Figure 6 Tensile test of a thin-walled tube composed of 304 stainless steel

Figure 7 Stress-strain curve of thin-walled tube composed of 304 stainless steel

For more complicated engineering problems, such as foundation pit excavation, pit side loading, tunneling in high-stressed rock mass, dynamic response to earthquake, the behavior of materials shows even more obvious loading path effect. However, the existence of primary and recency effects for these problems needs further exploration.

5 Conclusions

We analyzed the effect of loading path in the elastoplastic tensile and torsional problem of thin-walled circular tube in an isotropic linearly hardened material based on the classical elastoplasticity. Further, the plastic stress-strain relation under different loading paths was deduced, and the impact of loading path on deformation was examined. The main results of the study can be summarized as follows:

1) Under different stress paths, the corresponding strain caused by the pre-loaded stress is the largest in the final plastic strain state, and the stress path has the primary effect on the plastic strain.

2) Under different strain paths, the corresponding stress caused by the post-loaded strain is the largest in the final stress result, and the strain path has the recency effect on the stress results.

3) From the perspective of deformation, the primary effect of stress path on the plastic strain and the recency effect of strain path on the stress result are unified, which can be uniformly expressed as the recency effect of the loading path on the nominal deformation modulus. This implies that the material is more resistant to post-loaded stress or strain.

4) The primary and recency effects of the loading path exhibit immense potential for obtaining a deep understanding of the stress-strain relation in classical plasticity. Further, in the engineering field, they can serve as a useful reference for optimizing the design scheme and for improving the construction process.

Nomenclature

Axial plastic strain

Tangential plastic strain

Axial plastic strain of path B in area I

Tangential plastic strain of path C in area II

ε^p

Cumulative plastic strain

σ_z

Normal stress, MPa

τ_θz

Shear stress, MPa

σ_x

Normal stress at the end point of stress, MPa

τ_x

Shear stress at the end point of stress, MPa

σ_s

Yield stress, MPa

σ

Equivalent stress, MPa

s_x/y/x

Bias of the stress state, MPa

J₂

Second invariant of bias stress, MPa²

h

Plastic modulus, GPa

E

Elastic modulus, GPa

G

Shear modulus, GPa

μ

Poisson ratio

Contributors

GAO Yue provided the concept and wrote the first draft of the manuscript. SHAO Fei edited the draft of manuscript and provided funding for the research. FAN Peng-xian derived part of the formula of the article and edited the draft of manuscript. XU Qian edited the draft of manuscript.

Conflict of interest

GAO Yue, SHAO Fei, FAN Peng-xian and XU Qian declare that they have no conflict of interest.

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(Edited by ZHENG Yu-tong)

中文导读

经典塑性力学中加载路径影响的“首因效应”和“近因效应”

摘要：为探索加载路径的影响规律，基于Mises屈服条件和等向线性硬化假设，建立了轴向力和扭矩联合作用下不可压缩薄壁圆管的弹塑性分析模型，根据终点应力的特点划分了4个(σ_x，τ_x)应力区域，推导了不同应力区域中12条应力路径的塑性应力-应变关系，证明了应力路径对塑性应变影响的“首因效应”，即：无论终点应力处于何处，首先作用的应力产生的塑性变形总是更大。基于经典弹塑性理论分析了应变路径的影响，发现了应变路径对应力结果影响的“近因效应”，即：后加载的应变在最终的应力状态中产生的影响更大。从变形的角度，应力路径对塑性应变影响的“首因效应”和应变路径对应力结果影响的“近因效应”是统一的。该效应兼具简洁性与普适性，对于学习理解不同加载路径下的塑性应力-应变关系大有裨益，对于优化加工工艺和施工工序也具有一定的指导作用。

关键词：等向线性硬化；应力路径；应变路径；首因效应；近因效应

Foundation item: Project(51979280) supported by the National Natural Science Foundation of China; Projects(2016M602972, 2018M643852) supported by the Postdoctoral Science Foundation of China

Received date: 2020-03-05; Accepted date: 2020-06-28

Corresponding author: SHAO Fei, PhD, Professor; Tel: +86-13951798458; E-mail: shaofei@seu.edu.cn; ORCID: https://orcid.org/0000- 0002-7165-9967; FAN Peng-xian, PhD, Associate Professor; Tel: +86-18936890687; E-mail: fan-px@139.com; ORCID: https://orcid.org/0000-0002-1195-4884