Evaluation and visualization of multiaxial stress and strain states under non-proportional loading

This paper presents a simple method of determining stresses and strains and a severity of loading history under non-proportional loadings with defining the rotation angles of the maximum principal stress and strain in a three dimensional stress and strain space. Based on the method, non-proportional stress and strain ranges are derived and applicability of the range to the life evaluation for low carbon steel under non-proportional random strain paths are discussed. The strain range taking account of intensity of loading path reducing life can be suitable parameter for multiaxial fatigue life evaluation under multiaxial non-proportional loading. This paper also shows a method of visually presenting the principal stress and strain that assists designers to understand the loading mode whether proportional or non-proportional under 3 dimensional (6 stress/strain components) multiaxial loading.


INTRODUCTION
ost design codes use equivalent values to express the intensity of multiaxial stress or strain, like von Mises or Tresca equivalent stress and strain, and fatigue lives are usually estimated using equivalent values under multiaxial stress and strain states. The equivalent value means a scalar parameter that expresses intensity of a physical phenomenon in multiaxial stress states and should be reduced to be a uniaxial value in uniaxial stress state. Most widely used equivalent parameters are the von Mises and the Tresca equivalent stresses and strains. The von Mises equivalent stress physically expresses the intensity of shear strain energy and the Tresca equivalent stress that of the maximum shear stress. For example, ASME Section III, Division 1 NH [1] uses the von Mises equivalent strain and ASME Section VIII, Division 3 [2] the maximum shear stress. However, the von Mises equivalent stress and strain have no negative values so that they have a difficulty of expressing stress and strain ranges. The Tresca equivalent stress and strain have negative values but they also have a difficulty to put a sign to the shear stress and strain under multiaxial loading. Especially, in non-proportional loading where the principal stress and strain change their directions, giving a sign to them becomes more difficult. A simple and clear method of calculating stress and strain ranges is needed for describing multiaxial fatigue. M Low cycle fatigue (LCF) lives are reduced under strain controlled non-proportional loading accompanied by additional cyclic hardening compared with proportional loading [3][4][5][6][7][8] and an appropriate design method of evaluating the nonproportional fatigue life is needed for a reliable design and maintenance of structural components. Classical models particularly applicable in multiaxial fatigue life evaluation under proportional loadings lead to overestimate the lives under non-proportional loadings. For life evaluation under non-proportional loading, commonly proposed models are critical plane approaches that consider specific plane applied the critical damage, such as a Simith-Watoson-Topper [9] and a Fatemi-Socie [10] models. The authors also proposed a strain parameter (Itoh-Sakane model) estimating the nonproportional LCF lives for several materials under various strain histories [6,7,[11][12][13][14][15]. This parameter is the strain based model with introducing two parameters, non-proportional factor and material constant. The former one reflects the intensity of non-proportional loading reducing life and the latter one is related to the material dependence for degree of life reduction due to non-proportional loading. The Simith-Watoson-Topper, the Fatemi-Socie and the Itoh-Sakane models have been demonstrated to be applicable to life evaluation under non-proportional loading using hollow cylinder specimens in a laboratory level. That is, these models can be applicable to the life evaluation under limited non-proportional loadings such as the loadings in the plane stress state. However, many components and structures such as turbine blade, pressure vessel and pipe which receive combined thermal and mechanical loadings undergo non-proportional loading in which principal directions of stress and strain are changed into various directions under wider multiaxial stress and strain states [8,13,[16][17][18]. Therefore, a development of suitable models for the design of actual components where variation in principal directions of stress and strain vs time is changed 3-dimensionally is required. This study presents a method of evaluating the principal stress and strain ranges and the mean stress and strain, and also presents a method of calculating the non-proportional factor which expresses the severity of non-proportional loading in 3-dimantional (3D) stress and strain space (6 stress/strain components). Based on the method proposed, nonproportional strain and stress ranges are derived and applicability of the range to the life evaluation of type 304 stainless steel under 15 proportional and non-proportional strain paths are also discussed. This study also shows a method of visually presenting the stress/strain, the non-proportionality of loading and the damage evaluation.

Definition of stress and strain
ig. 1 illustrates three principal vectors, S i (t), applied to a small cube in material at time t in xyz-coordinates (spatial coordinates), where "S" is the symbol denoting either stress "" or strain "". Thus, S i (t) are the principal stress vectors for the case of stress and are the principal strain vectors for the case of strain. The subscript, i, takes 1, 2 or 3 in descending order of principal stress or strain. The maximum principal vector, SI(t), is defined as Si(t) whose absolute value takes maximum one, i.e., S I (t)=S 1 (t) when S 1 (t) takes maximum magnitude among S i (t). The maximum principal value, S I (t), is defined as the maximum absolute value of S i (t) as,  The "Max" denotes taking the larger value from the three in the bracket. The maximum value of SI(t) during a cycle is defined as the maximum principal value, SImax, at t=t0 as follows,  Fig. 2 illustrates two angles, (t)/2 and (t), to express the rotation or direction change of the maximum principal vector, S I (t), in the new coordinate system of XYZ, where XYZ-coordinates are the material coordinates taking X-axis in the direction of S I (t 0 ) with the other two axes in arbitrary directions. The two angles of (t)/2 and (t) are given by

Definition of Principal Stress and Strain Directions
where dots in Eqs (3) and (4) denote the inner product and e Y and e Z are unit vectors in Y and Z directions, respectively. S i (t) are the principal vectors of stress or strain used in Eq. (1) and the subscript i takes 1 or 3. The rotation angle of (t)/2 expresses the angle between the S I (t 0 ) and S I (t) directions and the deviation angle of (t) is the angle of S I (t) direction from the Y-axis in the X-plane. Figure  Fig. 3 shows the trajectory of SI(t) in 3D polar figure for a cycle where the radius is taken as the value of SI(t), and the angles of (t) and (t) are the angles shown in the figure. A new coordinate system is used in Fig. 3 with the three axes of S I 1 , S I 2 and S I 3 , where S I 1 -axis directs to the direction of S I (t 0 ). The rotation angle of (t) has double magnitude compared with that in the specimen shown in Fig. 2 considering the consistency of the angle between the polar figure and the physical plane presentation. The principal range, S I , is determined as the maximum projection length of S I (t) on the S I 1axis. The mean value, SImean, is given as the center of the range. SI and SImean are equated as,

Definitions of Stress and Strain in Polar
S Imin is the S I (t) to maximize the value of the bracket in Eq. (5). The sign of S Imin in the figure is set to be positive if it does not cross the SI 2 -SI 3 plane and the sign negative if it crosses the plane. The advantage of the definitions of the maximum principal range and mean value above mentioned is that the two are determinable without human judgments for any loading case in 3D stress and strain space. The range and mean value are consistent used in simple loading cases which are discussed in the case studies in the followings. S I (t) can be replaced by equivalent values of stress or strains, such as the von Mises and the Tresca, in case of necessity from user's requirement. Figure 3: Definition of principal range and mean principal value.

CASE STUDIES
his section discusses a couple of case studies for S I and S Imean under proportional and non-proportional loadings under biaxial plane stress condition. The symbol S takes () in the case studies in the followings.

Proportional loading
 is the Poisson's ratio. Note that / 3=0 in the loading stages of OA and OA' and  =0 in the loading stages of OC and OC'. The maximum principal strains (1(t)) vary along the solid lines in Fig. 4 (b) in the loading stages of OA, OB and OC, and the minimum principal strains (3(t)) along the dashed lines in the same loading stages. In these cases, the maximum principal strains all have a positive sign, whereas the minimum principal strains a negative sign.  I (t) makes the positive triangle waveforms as illustrated in Fig. 4 (c) because I(t) is either 1(t) or 3(t) taking larger absolute strain as shown in Eq. (1). The principal strain,  I (t), changes its direction by an angle of 180 (/2=90) at the origin O in the reversed Imean loading stages of AA', BB' and CC', Fig. 4 (d). Finally, the straight loading lines are obtained in the polar figure as shown in Fig. 4 (e). In the proportional loadings in Fig. 4 (a), the directions of  I (t) are fixed (=0 or 180) and =0.
The maximum strain range and mean strain are expressed as as shown in Fig. 4 (e), where k takes A, B or C and k' A', B' and C'. The results agree well with the principal strain range and mean strain we have considered by putting a sign with human operation.    In the box loading,  1 (t) takes positive and  3 (t) negative values without changing sign in a cycle as shown in Fig. 5 (b). Then, I(t) in the box loading has large mean value with a small fluctuation during a cycle, Fig. 5  The mean strain in the two cases is 0.
In the equation,  I is the principal strain range discussed previously.  is a material constant expressing the amount of additional hardening by non-proportional loading and is defined as the ratio of a saturated principal stress in the circular loading to that in uniaxial push-pull loading at the same principal strain amplitude for additional hardening material. In Eq. (13), I also can be replaced by the other equivalent stains based on the user's requirement although trends of data correlation are different between the strains used which have been discussed in other studies [6,7,[11][12][13][14]. f NP is the non-proportional factor that expresses the severity of non-proportional loading in the form as [6,7,[11][12][13][14], where T is the time for a cycle. b is a constant for making f NP =1 in the circular loading on  / 3 plot and b takes /2 when I(t) is employed [6,7]. In Eq. (14), fNP is calculated by measuring the rotation of the maximum principal strain direction and the integration the strain amplitude after the rotation. Therefore, f NP evaluates comprehensively the severity of non-proportional cyclic loading based on the amplitude of strain and the direction change of principal strain. This paper proposes a modified non-proportional factor, f NP , to be applicable for 3D stress and strain space defined in chapter 3, which is expressed as, (15) where eR is a unit vector directing to SI(t), ds the infinitesimal trajectory of the loading path shown in Fig. 3. Lpath is the whole loading path length during a cycle and "" denotes vector product. The scalars, SImax and Lpath, before the integration in Eq. (15) is set to make f NP unity in the circler loading in 3D polar figure. Integrating the product of amplitude and principal direction change of stress and strain by path length in Eq. (15) is more suitable for evaluation of fatigue damage rather than the integration by time.

XYZ-coordinate (Material coordinate)
xyz-coordinate (Spatial coordinate) Fig. 6 compares the values of fNP with those of fNP for several loading paths for the case of strain. Small difference in the value between fNP and fNP is found because of different definition between them. However, fNP has the advantages applicable to 3D stress and strain conditions. Fig. 7 shows a flow chart of calculation of stress or strain range and non-proportionality. When 6 components of stress or strain at time t as the data obtained from experiment, numerical analysis such as a finite element analysis or etc., SI and f NP are calculated based on the procedures presented in above. Then, evaluation of fatigue damage can be analyzed by user's method or Itoh-Sakane model [13].

NON-PROPORTIONAL STRESS AND STRAIN RANGES FOR LIFE EVALUATION
he authors analyzed the non-proportional LCF lives of type 304 steel tube specimens fatigued using 15 strain waveforms shown in Fig. 8 at 923 K [22]. Figs 9 (a)(c) correlate non-proportional fatigue lives with the ASME equivalent strain range ( ASME ) and two maximum stress ranges of  I e and  I .  ASME is the strain range based on Mises equivalent strain and is the strain range recommended in the Code Case of ASME, Section III, Division 1 NH [1].  I e is the principal stress range calculated from  I multiplied by Young's modulus of 145 GPa. The correlation using I e was made considering that conventional design usually uses stress ranges elastically calculated from strain ranges.  I is the maximum principal stress range experimentally obtained at 1/2N f . In the figures, heavy solid lines are drawn based on the Case 0 data and two thin solid lines show a factor of 2 scatter band. Fig. 9 (a) clearly indicates that  ASME overestimates the non-proportional fatigue lives in the cases with sever nonproportionalities. For type 304 stainless steel at room temperature, the more drastic reduction in LCF lives due to nonproportional loading beyond a factor of 10 was also reported [4,5,7,12,15]. The cause of the overestimates results from that ASME does not cover the effect of non-proportional loading on life. The correlation with I e shown in Fig. 9 (b) appears to be a similar trend to that in Fig. 9 (a).  I e has larger magnitude than experimental practically because the stress was calculated elastically even in plastic deformation occurred. In the data correlation with I, Fig. 9 (c), LCF lives under non-proportional loading are underestimated. The underestimate of the LCF lives under non-proportional loading may result from the additional hardening due to non-proportional loading, because I does not cover the additional hardening due to non-proportional loading resulting in reduction in life.   This result suggests that a suitable strain parameter for multiaxial LCF life evaluation must take account of intensity of non-proportionality of loading and additional hardening due to non-proportional loading. The former is the non- proportional factor, f NP or f NP , and the latter is the material constant, , in  NP .

Outline of Program for Visualization
ased on the definition of stress and strain mentioned above, a program is developed to analyze and visualize stress/strain and evaluate the failure life under non-proportional loading. The programming code used is C # . Fig.  11 is a flowchart of the program, which is divided into 8 categories as below.
(1) Input data The stress/strain data vs. time obtained from experimental or calculated result, denoted by S(t), is read into the program.
The input data consists of the 6 components of stress/strain and time at each column. Additional columns (temperature, pressure, etc.) can be added. As an example, for a butterfly-shape strain path under the combined push-pull and cyclic reversed torsion loadings using a hollow cylinder specimen, Figs. 12 (a) and (b) show a strain path in /2 plot and a strain path in polar figure coordinates, respectively, where  and  are the total axial and the total shear strains. In the figures, attached characters 'A-C' will be referenced later. Fig. 13 is the table of the input data formatted for the program.
(2) Calculation of SI(t) and SImax In this step, S I (t) and S Imax are calculated in accordance with the IS-method.

(3) Determination of reference axis
The reference axis for defining (t) and (t) is determined. The determination of the reference axis has three methods (Method-1, -2 and -3). In Method-1, the reference axis corresponds to the direction of S Imax , which is the maximum amplitude of stress/strain in a cycle. In Method-2, the reference direction is put into the direction perpendicular to a critical plane defined as critical plane approach [23,24]. In this study, the critical plane is determined as the plane where cumulative normal stress/strain becomes the maximum in a cycle under the random loading. In Method-3, users can define the reference axis as any direction at their request. The reference axes determined by Method-1, -2 and -3 are denoted by the characters A, B and C, respectively, in Fig. 12

. (4) Calculation of ξ(t) and ζ(t)
In accordance with the IS-method, calculations of ξ(t) and ζ(t), etc. are carried out.    illustrate the strain path in the polar coordinates, i.e., S I (t)-ξ(t)-ζ(t) and the projection of S I (t) and non-proportionality of loading as a function of time t. In Fig. 15 (a), attached characters, A, B and C, indicate the directions of the reference axes. Tab. 1 shows results as output data and they are strain ranges and non-proportional factors evaluated for each reference axis assuming the axial strain range =1.0%.

B (5) Figures of S(t)-t, SI(t)-t and SI(t)-ξ(t)-ζ(t) By the calculations and analyses in the above categories, descriptions of S(t)-t, SI(t)-t and SI(t)-ξ(t)-ζ(t)
(6) Cycle counting and analysis of S I (t) for random loading Under the random loading, waveform of stress/strain must be analyzed using a counting method, such as a rain-flow counting method, and damage also has to be evaluated by a cumulative damage rule like Miner's rule. Stress/strain waveform under non-proportional loading was already reduced to the simple waveform in the categories (1)-(5), the cycle counting and the cumulative damage rule used in uniaxial loading can be applied to the non-proportional loading.

Life Evaluation under Random Loading
One example of life evaluation under random loading is presented here. Material used is a rolled steel for general structure, type SS400, whose material properties are; Yong's modulus E=206 GPa, tensile strength σB=437 MPa. The failure life curve employed for evaluation is given by Eq. (16), which is obtained from experiment.
Figs. 16 (a) and (b) show stress waveforms of the input data for 1 block cycles and the calculated stress path in the polar coordinates, respectively. It is clear that the evaluated stress path is proportional loading since the stress path in Fig. 15 (b) is shown by a unique straight line. Counting method employed was a rain-flow method and 1 block was counted as 30,000 cycles. Tab. 2 shows the numerical values of fatigue damage and evaluated life by using a linear cumulative damage low. This analysis was carried out by using the test results under proportional loading, so more discussion about the applicability of the program to the evaluation of fatigue strength under non-proportional loading will be studied after the experimental results under non-proportional loading are obtained. Under the non-proportional loading, especially definition of non-proportionality will be required. In there, f NP will be calculated at each separate stress amplitude obtained by the counting method. Experimental life (blocks) 8

CONCLUSIONS
1. This paper proposed a simple method of determining the principal stress and strain ranges together with the mean stress and strain under proportional and non-proportional loading in 3D stress and strain space. It also proposed the method of defining the rotation and deviation angles of the maximum principal stress and strain. 2. The paper extend the non-proportional factor, fNP, from 2D to 3D stress and strain space with the consistency with the previous definition of it in the 2D space.
3. Applicability of the non-proportional strain was discussed for describing the high temperature proportional and nonproportional low cycle fatigue lives. 4. Based on the IS-method to evaluate stress/strain and failure life under non-proportional loading, this study developed a program for visually presenting the stress/strain state, the non-proportionality of loading, and the damage evaluation. It is possible for researchers and engineers to assess the fatigue strength under non-proportional loading without requiring special knowledge and judgment about multiaxial loading.