Development of a combined tension–torsion experiment for calibration of ductile fracture models under conditions of low triaxiality

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Abstract

Developments in computational mechanics have given engineers tools to predict the evolution of damage in complex structures. Damage models have been developed that relate failure strain to stress triaxiality and Lode angle. Calibration of these models has traditionally relied on specimens that exhibit high triaxiality and limited Lode angle. This paper presents a specimen that can be tested in combined tension and torsion to achieve low triaxiality over a range of Lode angle. Numerical analysis of the specimen shows that it exhibits uniformity of stress–strain and stable values of triaxiality and Lode angle as plastic strain develops, both of which are desirable characteristics for calibration of ductile failure models. The design of a new displacement and rotation gage is presented that allows non-contact measurement at the gage section. Experimental results are used to develop the failure surface for 5083 aluminum.

Highlights

► Computational mechanics allows engineers to predict damage evolution in structures. ► Damage models that include effects of stress triaxiality and Lode angle are required. ► Method is presented for calibrating models at low triaxiality over range of Lode angle. ► Numerical analysis shows that specimen used in the method has desirable characteristics. ► Experimental results used to develop the failure surface for 5083 aluminum.

Introduction

Developments in computational mechanics have enabled engineers to conduct stress analysis and perform failure predictions on complex structures. When stresses exceed the yield point of the material in a structure, predictions of stress, strain, and damage will be highly dependent on the constitutive model used to characterize plasticity and damage. Therefore, accurate predictions of structural integrity depend on accurate material models. Development of these models requires constitutive theories that properly account for the influence of stress on yielding and plastic flow, and on experimental methods to calibrate the models. Calibration for parts of a structure that exhibit high constraint is readily done using simple tensile type specimen geometries. However, it is a particular challenge for low constraint areas such as thin sections, regions under predominant shear, and at surfaces because the specimen geometry and loading required to achieve low triaxiality are more complex. Since failure in structures often initiates in thin sections or at surfaces, calibration could significantly influence the accuracy of failure predictions.

The most widely used approach for characterizing plasticity of ductile metals has been the classical J2 plasticity theory. In this theory the yield surface depends only on the second invariant of the stress deviator, J2. However, it has been known for some time that yielding in geomaterials is sensitive to not only the hydrostatic stress, but also to the Lode angle, which is related to the third invariant of the deviator stress tensor [1], [2], [3], [4], [5]. Recently the influence of hydrostatic stress and Lode angle on yielding in ductile metals has been investigated [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. Damage models have been developed [6], [7], [8] that relate failure strain to the stress triaxiality, which is defined as the ratio of the hydrostatic stress to equivalent stress. These models have recently been extended to include the influence of Lode angle [9], [11], [12], [13], [14].

Before continuing this review of past work, it is useful to define the various parameters used in ductile fracture models and to show how they are related.

Triaxiality, T, is universally recognized as the ratio of the mean stress over the von Mises equivalent stress, although the symbol used to represent it varies.T=η=σmσe

The mean stress is defined asσm=13(σ1+σ2+σ3)=I13σ1, σ2, and σ3 are the principal stresses and I1 is the first invariant of the stress tensor.

The Von Mises equivalent stress isσe=12(σ1σ2)2+(σ2σ3)2+(σ3σ1)2=3J2

There are many different variations on the definition of the Lode angle, θ, and related parameters in the literature. In the early work of Lode [16], the Lode angle is defined as the angle measured counter-clockwise from the projection of the σ1 axis to a point on a deviatoric plane (constant hydrostatic stress), as shown in Fig. 1. σ1′, σ2′, and σ3′ are the projections of the principal stress axes on a deviatoric plane.

P(ξ, ρ, θ) are the Haigh-Westergaard cylindrical coordinates.ξ=I13=3σmρ=2J2=23σecos(3θ)=27J32J23/2=27J32σe3

For an isotropic material the ordering of the principal stresses does not influence the material behavior, therefore the stress space has three symmetry axes, as shown in Fig. 1, and the Lode angle is limited to 0θ(π/3). Xue [17] defined an alternative Lode angle, θL, relative to the shear meridian axis [18], as shown in Fig. 2. The shear meridian axis (θ=30° or π/6) represents all states of stress that can be formed by combining a state of pure shear with a hydrostatic stress. There is also a tensile meridian axis (θ=0°) representing states of stress formed by combining uniaxial tension with hydrostatic stress, and a compression meridian axis (θ=60° or π/3) formed by combining uniaxial compression with a hydrostatic stress.

The two angles are related by θL=θ(π/6) and (π/6)θL(π/6). This angle is related to the principal stresses by the following equation, where s1, s2, s3 are the principal values of the deviatoric stress tensor (si=σiσm).θL=tan1[13(2s2s1s3s1s3)]

Xue also defines a parameter that is the relative ratio of the principal deviatoric stresses.χ=s2s3s1s3

The values of these parameters for common specimen types are compared in Table 1.

Barsoum and Faleskog [13] use the parameter μ to describe Lode angle dependence.μ=2σ2σ1σ3σ1σ3

This parameter is equivalent to the quantity in parentheses in Eq. (7), thereforeθL=tan1[μ3]

Bai and Wierzbicki [14] define a normalized third deviatoric stress invariant, ξ, which is not the same as the Haigh–Westergaard coordinate.ξ=27J32σe3=cos(3θ)where the range of the Lode angle is 0θ(π/3) corresponding to 1ξ1.

They also introduce a normalized Lode angle.θ¯=16θπ=12πcos1ξ

This Lode angle parameter has the same range as ξ.

Xue and Wierzbicki [19] introduce another variation on the Lode angle that is anti-symmetric about the tensile meridian axis.tan(θL')=μ3

The prime has been added to differentiate it from the Barsoum and Faleskog θL.

The parameters presented thus far are based on the ordering of principal stresses σ1σ2σ3. Gao et al. [10] modify the definition of ξ based on the ordering σ2σ3σ1 as presented in [20].ξg=cos(3θg+π2)=cos3(θg+π6)where θg is measured from the shear meridian axis, which is now 30° (π/6) counter-clockwise from the σ2 axis, in a manner similar to θL. Adding π/6 makes the angle anti-symmetric about the shear meridian axis, as can be seen in Table 1.

In a study by Gao et al. [11], notched round and flat tensile specimens were used to investigate the influence of hydrostatic stress and Lode angle on ductile failure in DH36 steel. Varying the notch radius allowed them to obtain a range of triaxiality. For the round tensiles, notch radii ranging from 1.12 to 8.94 mm yielded triaxiality of 1.3 to 0.9, respectively, for DH36 steel. The triaxiality depends not only on the specimen geometry, but also on the plastic behavior of the material. In a different study on 5083 aluminum [10], the same specimen geometries yielded triaxiality ranging from about 1.6 to 0.8. However, the axisymmetry of a notched tensile specimen causes it to have the same Lode angle (θ=0°, ξ=+1), regardless of the notch geometry and material. Consequently, it does not allow investigation of Lode angle effects. Notched flat tensile specimens, referred to as plane strain specimens, were used to achieve triaxiality of 0.8–1.1 (depending on the notch radius) for DH36 steel at a Lode angle of θ=30° (ξ=0 ). They point out that alternative specimen designs are required to more accurately calibrate the failure surface at Lode angles between 0 and 1 because curvature of the failure surface is greater at low triaxiality, so failure strain becomes more sensitive to ξ [11]. Wierzbicki et al. [21] developed a unique butterfly shaped specimen that is tested in combined tension and shear in order to investigate failure at low triaxiality. This specimen achieves triaxiality over the range of −0.191 to 1.01 and −0.503≤ξ≤0.858; however, it requires some complex machining and a unique apparatus to perform the tests. The variation in triaxiality and Lode angle during evolution of plastic deformation required that average values of triaxiality and ξ be calculated by integrating over strain.

It is also possible to achieve low triaxiality and a range of Lode angle using a simple notched tubular specimen tested in combined tension and torsion [13]. Faleskog et al. were able to achieve triaxiality from about 0.3–1.3 and Lode parameter 1μ0, which from Table 1 corresponds to 0ξ1, for a high yield strength and low strain hardening steel.

Gao et al. [22] developed an alternative specimen design that is similar to the notched tubular specimen used by Faleskog et al. Their specimen is a modified version of a specimen used by Lindholm et al. [23] for conducting high-rate torsion tests. The objective of this paper is to investigate the distribution of stresses, triaxiality and Load angle in the gage section of the modified Lindholm specimen for use in calibrating ductile fracture models under conditions of low triaxiality and varying Lode angle.

Section snippets

Numerical analysis

The ideal specimen for calibration of ductile failure models would have three characteristics: (1) uniform stress and strain in the gage section, (2) ability to achieve a range of triaxiality and Lode angles that fall between the tension and shear cases, and (3) stationary triaxiality and Lode angle as plasticity evolves up to failure. The rationale behind each of these characteristics, along with numerical analysis to characterize the modified Lindholm specimen, will be presented in the

Pure torsion

Fig. 7 compares the through-thickness distribution of T, μ and ε¯p for pure torsion loading (ψ=0) at the time when the equivalent plastic strain, ε¯p, reaches a level of about 0.37. For pure torsion, T and μ should both be zero. For the NT specimen, rm represents the mid-radius in the gage section (12 mm) and tn is the half net thickness in the narrowest part of the notch (0.6 mm). For the Lindholm specimen, rinner represents the inner radius (6.54 mm) and tn is the wall thickness in the gage

Material and experimental methods

Experiments were conducted using the modified Lindholm specimen over a range of tension–torsion ratios to calibrate a constitutive model for plasticity, and thereby validate the finite element model for future prediction of stresses and strains at ductile fracture.

The tests were conducted in a two-axis servo-hydraulic load frame with axial and rotational actuators and hydraulic collet grips. The grips were carefully aligned to prevent buckling and development of asymmetric stresses in the

Experimental results

For these experiments the specimens were machined from a 25.4 mm (1 in.) thick plate of 5083-H116 aluminum in the T-orientation. The chemical composition of the plate is shown in Table 4. This alloy has a yield strength of 266 MPa (38.6 ksi) a tensile strength of 321 MPa (46.6 ksi), and percent elongation of 11–15%. The effective stress (Mises)—effective plastic strain curves for uniaxial tension and pure torsion are shown in Fig. 15. For this alloy the stress state clearly effects the development of

Conclusions

The numerical analysis revealed that the modified Lindholm specimen has three desirable characteristics for calibration of ductile failure models at low triaxiality. It has relatively uniform stress and strain in the gage section, it provides a range of triaxiality and Lode angles that fall between the tension and shear cases as the tension–torsion ratio is changed, and both triaxiality and Lode angle remain relatively constant as plasticity evolves up to failure. These characteristics obviate

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Role of the funding source: this work was funded by the Advanced Combatant Materials Program through the Naval Sea Systems Command (NAVSEA) and the Metals Division of the Survivability, Structures, and Materials Department at the Naval Surface Warfare Center, Carderock Division (NSWCCD). NAVSEA had no direct involvement in the collection, analysis and interpretation of data, in writing the report, or in the decision to submit the paper for publication.

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