Axial effects investigation in fixed-end circular bars under torsion with a finite deformation model based on logarithmic strain

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Abstract

In this paper the torsion problem of a circular bar with fixed ends is solved using a finite deformation constitutive model based on the corotational rates of the logarithmic strain. The logarithmic, Green–Naghdi and Eulerian corotational rates of the logarithmic strain are used in the model. The solution is based on a von Mises type yield function that incorporates isotropic and kinematic hardenings. For the kinematic hardening, a modified Armstrong–Fredrick hardening model with the corotational rate of the logarithmic strain is used. Assuming incompressible behavior, the fixed-end torsion problem is simplified to the simple shear problem. Solving the problem, the stress components are integrated to calculate the torque and axial force. It is qualitatively shown that the results based on the logarithmic corotational rate are in good agreement with the experimental results.

Introduction

In recent years much attention has been focused on the development of constitutive laws for large deformations of solids. Analysis of the simple shear deformation has become a popular benchmark for testing the validity of finite deformation constitutive models [1]. The reasonableness and applicability of various models has been investigated using this problem [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12].

In principle, simple shear can be produced by torsion of an incompressible circular bar with the ends prevented to displace in the axial direction. Interest in this problem stems from the fact that the torsion test is perhaps the most widely used for obtaining experimental data for metals at very large strains. In addition, the nonhomogenity of this deformation process and the inherent nonproportional stressing histories involved are prime sources of interest into the basic mechanics of this problem [13]. In this problem, the axial or “Swift” effect has particular importance; i.e. the development of axial force during the fixed-end torsion [14].

Swift was the first to conduct large-strain torsion tests on solid circular bars and tubes of several metals [15]. These were free-end tests allowing length changes. He observed that, the specimens exhibited monotonic length increases. This observation, in terms of fixed-end torsion, implies that one can expect the development of an axial compressive force, which increases with twist. Also, Montheillet et al. experimentally studied the fixed-end torsion [16]. Their results show that in the torsion problem a common characteristic of the torque-twist curves is that the torque attains a single maximum at a relatively small shear strain, and thereafter it begins to drop to an almost steady state value. The axial effects seem to provide a suitable means for assessing the adequacy of constitutive models. Neale and Shrivastava [13] and Van Der Giessen et al. [1] mentioned that the prediction of the axial effects shows a remarkably strong dependence on the constitutive model adopted. These effects were studied by many researchers using different constitutive models [1], [8], [13], [14], [17], [18], [19].

Neale and Shrivastava numerically analyzed the elastic–plastic behavior of solid circular bars twisted to arbitrarily large strains [20]. They considered isotropic hardening, various kinematic hardenings, hypoelastic and hyperelastic formulations in their model. In another study, Neale and Shrivastava presented the derivation of closed-form analytical solutions of torsion problem using their model and numerous applications to a wide variety of time-independent and rate-dependent plastic constitutive laws [13]. Van Der Giessen et al. performed a detailed study of the effect of plastic spin on axial effects during large strain torsion of solid bars [1]. They presented some principal predictive features of various plastic spin constitutive laws and differences in trends between fixed-end and free-end torsion. Voyiadjis and Kattan used their earlier proposed corotational rate to solve the fixed-end and free-end torsion problem, and also solved the same problems using corotational rates proposed by other researchers [19].

Bruhns et al. studied the large torsion problem of elastoplasticity thin-walled cylindrical tubes with fixed ends by applying a self-consistent kinematic hardening model based on the logarithmic tensor rate [10]. They derived an analytical solution for the just-mentioned problem and conducted a detailed study of the coupling effect of material properties. In another study, Xiao et al. investigated the Swift effect in torsion of thin-walled cylindrical tubes [14]. They derived an analytical perturbation solution for the elastic-perfectly plastic case, and obtained a closed-form solution for the kinematic hardening plastic case. They showed that the simple idealized kinematic hardening model with logarithmic rate, which uses only two material constants, may arrive at reasonable accord with experimental observations.

Wu formulated a large strain constitutive model for anisotropic metallic materials by extending the classical flow theory of plasticity [21], [22]. Using the proposed model he solved the problem of free-end torsion of a thin-walled tube and investigated the shear stress, axial strain and back stress responses. He showed that the predicted results compare favorably with his experimental data.

In the analysis of finite plastic deformation, stress and strain rates play an important role in the constitutive equations. Among other things, stress rates must satisfy the objectivity and frame-indifference requirements. This is particularly important when formulating theories of plasticity in the Eulerian reference system where all the quantities are referred to the deformed configuration of the body. The use of an appropriate rate mainly depends on the choice of a modified spin tensor. In the past two decades, researchers in this field proposed several corotational rates. The Jaumann rate was generally used until Nagtegaal and de Jong showed that the Jaumann rate yields an oscillatory stress response in the simple shear problem which is physically impossible [23]. This result triggered a series of investigations to look for corotational stress rates appropriate for the description of metallic behavior in the finite strain range.

In an earlier study, Dienes used the Green–Naghdi (GN) corotational rate, which is defined in terms of the rotation tensor from the polar decomposition of the deformation gradient [2]. He obtained a non-oscillatory response for stress in simple shear problem. Lee et al. solved the same problem for a plastic material [24]. They found out that for simple shear problem the objective Jaumann rate gives an oscillatory solution for stress which is physically not acceptable. They attributed the stress oscillation to the use of the Jaumann rate. Based on the physical aspects of the kinematics of the simple shear problem, they proposed a “modified Jaumann rate” which eliminated the spurious stress oscillation. Sowerby and Chu assumed the corotational stress rate defined by the stretch tensor decomposed from the deformation gradient [25]. They used this rate to formulate the hypoelastic constitutive equation and obtained nonoscillatory results in simple shear problem. Dafalias defined the corotational rate by use of the plastic spin [26]. He provided explicit forms of constitutive relation for the plastic spin using the representation theorem for isotropic function. Aifantis computed the corotational rate with respect to the stress spin corresponding to the material frame whose angular velocity coincides with that of stress [17]. Reinhardt and Dubey introduced a corotational rate called D-rate and showed that the strain rate tensor is D-rate of the logarithmic strain tensor [7]. Using this corotational rate they proposed constitutive equations for hypoelastic and rigid plastic materials. Xiao et al. introduced a corotational rate based on the logarithmic spin Ωlog [27]. Utilizing this corotational rate (log-rate) they established an Eulerian large deformation constitutive model [10], [14], [28], [29].

In this paper the torsion problem of a circular bar is solved using the recently proposed constitutive model based on the corotational rates of the logarithmic strain [30]. This investigation is fulfilled on the basis of von Mises yield criterion for rigid plastic isotropic and kinematic hardening materials. Using GN, Eulerian and logarithmic corotational rates in the model the stress–strain response is obtained for the fixed-end torsion problem of an incompressible material. It is shown that the torque and axial force developed during the fixed-end torsion, predicted by the logarithmic corotational rate, are qualitatively in good agreement with the experimental data.

Section snippets

Constitutive equations

The von Mises yield criterion is generally used as the yield function in plastic constitutive modeling of initially isotropic metals;f=(S-α):(S-α)-23k2=0,where the symbol (:) stands for tensor inner product and S is deviatoric part of the Cauchy stress and α is the back stress tensor. Throughout, bold uppercase letter represents the second order tensors and the vectors are represented by bold lowercase letter. In Eq. (1), k is the yield surface size, which depends on the history of plastic

Corotational rates of the logarithmic strain

The corotational rate of the logarithmic strain associated with the body spin (Ω), the logarithmic spin (Ωlog) and the Eulerian spin (ΩE) are, respectively, expressed as follows:(lnV)GN=(lnV)·-Ω(lnV)+(lnV)Ω,(lnV)E=(lnV)·-ΩE(lnV)+(lnV)ΩE,(lnV)log=(lnV)·-Ωlog(lnV)+(lnV)Ωlog,where (lnV)· is the time derivative of logarithmic strain. (ln V)GN, (ln V)log and (ln V)E are called Green–Naghdi (GN), logarithmic (log) and Eulerian (E) corotational rates of the logarithmic strain tensor, respectively.

The

Kinematics of the fixed-end torsion

A solid circular bar with an initial radius R0 and initial length L0 is subjected to an angle of twist α produced by an applied torque T (Fig. 1-a). The end faces of the bar are fully constrained axially so that there is no axial displacement, thus allowing for the development of an axial force F. It is assumed that any cross-section of the bar remain planar and perpendicular to the axial direction. The kinematics of the problem has been established with the aid of a spatially fixed cylindrical

Application to fixed-end torsion

For the GN corotational rate using Eqs. (35), (40), the expanded form of the kinematic hardening constitutive equation, Eq. (11), in simple shear problem takes the following form:(α˙11)GN+n23γ˙q22+4q12q22(α11)GN+4γ˙q22(α12)GN+mγ˙q23[γq2-4q1]=0,(α˙12)GN+n23γ˙q22+4q12q22(α12)GN+2γ˙q22((α22)GN-(α11)GN)-mγ˙q23[2q2+2γq1]=0.Numerical solution of differential Eq. (43) yields the back stress tensor for the GN corotational rate, (α)GN. Also, for the Eulerian rate the differential equations governing the

Discussion and conclusions

In this paper large strain torsion of solid circular bars with fixed ends is investigated using three different constitutive models based on the logarithmic strain tensor. The models differ mainly in the corotational rate used and, associated with that, the flow rule and the kinematic and isotropic hardening constitutive equations. One model is based on the GN rate, another is based on the Eulerian rate and the other is based on the logarithmic rate. Assuming incompressible behavior, the

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