A new form of equivalent stress for combined axial–torsional loading considering the tension–compression asymmetry of polymeric materials

Abstract

Although the von Mises criterion has been generally adopted to obtain the equivalent stress for materials under multiaxial loading, it does not consider the influence of the tension–compression asymmetry of pressure-sensitive materials. For polymeric materials, to include the non-negligible effect of the tension–compression asymmetry, this work proposes a new form of equivalent stress in which the tensile and compressive yield strengths are presented. This form is compared with the work of von Mises, Bai and Christensen with the support of experimental data. It was found that the proposed form is more suitable for polymeric materials under combined axial–torsional loading conditions. The implications of the present findings for designing stress-controlled combined axial–torsional loading experiments are also discussed.

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1. Introduction

With the basic assumption of isotropy, independence on both pressure and the sign of external stresses, the von Mises yield criterion has been generally accepted to describe the yielding behaviour of materials under any loading conditions using only the uniaxial tensile yield strength.¹ For many ductile metals insensitive to pressure, e.g. copper, aluminium, alloy steels, etc., the von Mises yield criterion has shown excellent agreement with experimental results.² However, for materials where the tensile yield strength is significantly less than the compressive yield strength, which is true for most polymeric materials, the form of the von Mises equivalent stress fails to describe their yield behaviours under multiaxial loading.^3,4 Substantially depending on the temperature, loading rate, pressure, etc., the yield behaviour of polymers has drawn much attention.^5–11 Bowden et al.¹² proposed modified forms of the Tresca and von Mises criterion by linearly relating the first invariant of the stress tensor, I₁, to the maximum shear stress and the square root of the second deviatoric stress invariant . Raghava et al.¹³ suggested a yield criterion in the form of a combination of I₁ and J₂ with only the tensile and compressive yield strengths as the material parameters in the yield criterion. Pae’s work¹⁴ combined with a polynomial form of I₁. Furthermore, Farrokh et al.¹⁵ introduced the dependency of strain rate to the yield criterion of polymers. Ghorbel¹⁶ considered the effect of the third invariant of the deviatoric stress tensor J₃. Altenbach et al.¹⁷ gave a detailed review on the phenomenological yield criteria.

Although a lot of work has been conducted on the yield criterion for polymeric materials, little attention has been paid to the equivalent stress for combined axial–torsional loading considering the effect of tension–compression asymmetry. The form of the von Mises equivalent stress, , was used for combined axial–torsional loading while neglecting the tension–compression asymmetry.^18–20 This is no longer valid for many polymeric materials. Since the von Mises equivalent stress overestimates the shear stress under tension–torsion loading conditions, Mittal et al.²¹ suggested an empirical equivalent stress for PMMA under monotonic tension–torsion loading conditions. Furthermore, Sittner et al.²² proposed for shape memory alloys. Mittal’s and Sittner’s work only focused on the influence of shear stress, while the tension–compression asymmetry property of polymeric materials was not included.

In this paper, a modified form of the von Mises equivalent stress considering the tension–compression asymmetry property of polymeric materials is proposed. It is compared with the work of von Mises, Bai²³ and Christensen^24,25 with the help of experimental data of three polymeric materials, which show different levels of sensitivity towards tension–compression asymmetry, i.e. polycarbonate (PC), polymethyl methacrylate (PMMA) and polystyrene (PS). As the proposed form of equivalent stress is shown to be more suitable for polymeric materials, it can be used to guide the experimental design for combined axial–torsional loading.

2. Theoretical foundation

2.1 Yield criterion considering tension–compression asymmetry

Assuming a material begins to yield when the elastic energy of distortion reaches a critical value, the von Mises criterion is written as:where σ_yld is equal to the tensile yield strength T. The compressive yield strength C is not considered in eqn (1). In other words, here T = C.

For materials sensitive to pressure, Bai et al.²³ proposed:

[small sigma, Greek, macron]_Bai = [small sigma, Greek, macron] − αI₁ = (1 − 3αη)[small sigma, Greek, macron]₀ = σ_yld. (2)

I₁ is the first invariant of the stress tensor. [small sigma, Greek, macron]₀ is the stress under zero pressure. is the normalized pressure, which is also referred to as the triaxiality parameter. J₂ is the second invariant of the deviatoric stress tensor. α is a material constant to represent the pressure sensitivity of the material. According to the literature,²⁶ α is taken in the form of:where α is a good index to represent the magnitude of the tension–compression asymmetry.

Eqn (2) can be reformed in terms of the principle stress:

The absolute value of C is adopted here. For T = C, eqn (4) reduces to the von Mises criterion.

Christensen^24,25 proposed “a two-property yield for homogeneous, isotropic materials”:

which is quite similar to Raghava’s work.¹³

2.2 Equivalent stress for combined axial–torsional loading

Although there has been extensive work on the yield criterion for polymeric materials, little attention has been paid to the equivalent stress for combined axial–torsional loading. In this section, a modified form of the equivalent stress is proposed, which considers the tension–compression asymmetry.

For combined axial–torsional experiments, the von Mises equivalent stress can be written as:

where σ is the axial stress, and τ is the torsional stress.

Bai’s work (eqn (4)) can be reformed for combined axial–torsional experiments as:

Based on eqn (5), the equivalent stress for Christensen’s work under combined axial–torsional loading is written as:

While the von Mises equivalent stress gives the simplest form of the equation, Bai’s and Christensen’s work takes into consideration the tension–compression asymmetry but with relatively complicated forms.

We can reform eqn (2) as:

Squaring both sides of eqn (9) gives:

For combined axial–torsional loading, it can be rewritten as:

For relatively ductile polymers, |C − T| ≪ |C + T|, then . Noticing that , we take the square root of both sides of eqn (11) to get:

Eqn (12) can be reduced to the form of the von Mises equivalent stress as in eqn (6) for the tension–compression symmetry case, i.e. T = C. As one can see in Table 1, the proposed equivalent stress not only takes into consideration the tension–compression asymmetry as Bai and Christensen did, but it also has a more concise form than both of them, which is quite appealing.

Table 1 Comparison of the different forms of equivalent stress

Item	Equivalent stress
von Mises
Bai
Christensen
Proposed

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3. Results and discussion

3.1 Comparison of the different forms of equivalent stress for combined axial and torsional loading

Four forms of the equivalent stress, i.e. von Mises’, Bai’s, Christensen’s and the proposed one, were compared with the experimental data of three kinds of polymer, i.e. PC, PMMA and PS, showing different levels of sensitivity towards tension–compression asymmetry.

For PC, which has a relatively low sensitivity towards tension–compression asymmetry with α_PC = 0.09, the four kinds of equivalent stress for combined axial–torsional loading are shown in Fig. 1a. The experimental results for PC were obtained from uniaxial tension, compression and a series of proportional combined axial–torsional experiments at a constant strain rate of 5 × 10⁻³ s⁻¹ at room temperature in the authors’ lab. The axial and torsional stresses are nondimensionalized by the tensile yield strength. It can be found from Fig. 1a that the von Mises equivalent stress cannot describe the tension–compression asymmetry of PC very well, while the proposed form shows a capability as good as Bai’s and Christensen’s forms.

Fig. 1 Comparison between the experimental data and the four forms of the equivalent stress, i.e. the von Mises, Bai, Christensen and proposed forms, under combined axial–torsional conditions: (a) PC, (b) PMMA (@60 °C), (c) PS (@90 °C). The experimental data for PMMA and PS are from the literature.²⁷

For PMMA (α_PMMA = 0.32) and PS (α_PS = 0.94), which are more sensitive to the tension–compression asymmetry than PC, the comparisons between the four forms of equivalent stress and the experimental data are given in Fig. 1b and c, respectively. It should be noted that PMMA and PS are brittle at room temperature, and not suitable for discussion about yield behaviour. However, at elevated temperatures, i.e. PMMA at 60 °C and PS at 90 °C, they do appear to show a certain ductility and the yield strengths are measurable. So the experimental results for PMMA (@60 °C) and PS (@90 °C) were taken from the literature²⁷ to evaluate the proposed form of the equivalent stress in this paper. For all three polymers, the von Mises equivalent stress does not consider the tension–compression asymmetry. The Christensen form gives overestimated results for the torsional stress and this is more obvious for the materials with a higher sensitivity towards tension–compression asymmetry, such as PS. Bai’s form and the proposed form are more suitable under combined axial–torsional loading.

3.2 Comparison of the different forms of equivalent stress for the plane strain compression

Fig. 2a–c illustrate the comparisons between the experimental results and the four forms of the equivalent stress in plane stress space. One can easily figure out that von Mises’ form can only be used for a certain stress status. Bai’s form, Christensen’s form and the proposed form show the same capability for PC, which is not very sensitive towards tension–compression asymmetry (α_PC = 0.09). For the materials with a higher tension–compression asymmetry sensitivity, such as PMMA and PS, Bai’s form overestimates the yield stress under the plane strain compression. The proposed form shows a good predictive ability, similar to Christensen’s form.

Fig. 2 Comparison between the experimental data and the four forms of the equivalent stress, i.e. the von Mises, Bai, Christensen and proposed forms, in plane stress space: (a) PC, (b) PMMA (@60 °C), (c) PS (@90 °C). The experimental data for PMMA and PS are from the literature.²⁷

Based on the discussions above, with a simple form, the proposed form of equivalent stress can be used for both combined axial–torsional loading and plane stress space. Only the tensile and compressive yield strengths, which can easily be obtained from uniaxial tests, are included in the proposed form of the equivalent stress, as shown in eqn (12). No curve fitting procedures are required.

The stress-controlled axial–torsional combined loading experiments of polymeric materials have drawn much attention because of their significance in safety assessments.^{18,19,28–30} Eqn (12) is very important for the design of such loading conditions for polymeric materials with a high sensitivity towards tension–compression asymmetry. The proposed equivalent stress can successfully describe the tension–compression asymmetry of polymers with a relatively simple form, which shows clear physical meaning.

4. Summary

For combined axial–torsional loading, the tension–compression asymmetry of polymeric materials cannot be ignored. A new form of equivalent stress is proposed considering the effect of tension–compression asymmetry.

1. Using tensile (T) and compressive (C) yield strengths, no curve fitting parameters are needed in the proposed form. When T = C, the proposed form reduces to the von Mises equivalent stress. When T ≠ C, it has been validated with experimental data for PC, PMMA and PS.

2. While Bai’s form overestimates the yield stress for plane strain compression and Christensen’s form overestimates the torsional stress, the proposed equivalent stress can describe the two loading conditions mentioned above well with a concise form.

3. For polymeric materials with the tension–compression asymmetry property, the proposed form of the equivalent stress is more suitable to guide the experimental design of combined axial–torsional loading.

Acknowledgements

The authors would like to thank the National Natural Science Foundation of China (11172249, 11272269) for financial support. The authors also would like to thank the partial financial support from the Ministry of Education of China (NCET-12-0938) and Science and Technology Department of Sichuan Province (2013JQ0010).

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