A New Stored Energy Model Based on Plastic Work of Back Stress during Cyclic Loading in Polycrystalline Metal

Two mesomechanics models were analyzed in an attempt to reveal the relationship between stored energy and back stress. It has been indicated that the portion of elastic stored energy due to residual microstresses (ESR) is closely related to intergranular back stress (Xinter), and the stored energy of dislocations inside grains (ESD) can be estimated with the plastic work of intragranular back stress (Xintra). Then, the evolution of back stress during cyclic loading was studied, and the plastic work of back stress (WpB) was calculated with the low cycle fatigue experimental data of Ti-6Al-4V. The result shows that WpB is partially released at every reverse loading, sufficient to reproduce the evolution of stored energy correctly under cyclic loading. The study also reveals that partially released energy is related to the decrease of Xinter at the initial state of reversal loading resulting from the reduction of the plastic strain incompatibility between grains.


Introduction
When a metal deforms plastically, most of the mechanical energy W p expended in the deformation process is converted into heat Q. The remainder is stored in the material by creating and rearranging crystal imperfections, especially dislocations, stacking faults, etc. [1][2][3]. This is known as the stored energy of cold work E S , related to the damage leading to the initiation of fatigue microcracks [4,5].
The first attempts to describe stored energy were performed by Farren and Taylor [6]. Later, Taylor and Quinney [7] introduced the concept of stored energy and carried out a thorough study. An extensive review of the early work within this field was published by Bever et al. [8]. In these early studies, heat produced during deformation was measured using thermo-junctions or calorimeters. More attempts have been performed since then and today infrared radiometers or cameras are widely used [9][10][11][12]. Although there are differences in experimental methods, operators, and materials, the general trends are as follows: 1.
The stored energy E s is only a small part of the total plastic work W p : only 1% to 15% for the pure metals [8].

2.
The ratio (E s /W p ) is quite dependent on the strain level. In some cases [9,13], the ratio shows a maximum at minor plastic strains (less than 2%). Thereafter, the ratio decreases.

3.
Under cyclic deformation, part of the stored energy is released at each reversed loading as shown in Figure 1 [14], the effect of which is believed to be closely related to the Bauschinger effect [8], or kinematic hardening [15]. The experimental results give impetus to the theoretical description of energy balance in material. One of the possible ways to calculate stored energy is based on the stressstrain curve. Martin [16], who regarded the segment of the plastic work associated with work hardening as accumulating damage, first proposed a stored energy method from the stress-strain curve, as shown in Figure 2a with the shaded area. Later, a dislocation model, on the assumption of the irreversibility of dislocation motion, was established by Tanaka [17] to prove that the shaded area in Figure 2a is the stored energy related to the dislocations of pile-up in a single crystal. However, this method is not available for polycrystalline materials, especially for low cycle fatigue.
Another stored energy method from the stress-strain curve is proposed by Skelton [18][19][20] and Aravas [3]; they suggested that the shaded area in Figure 2b is the internal energy stored in the metal, which is famously known as "complementary plastic strain energy". However, Szczepinski [21] and Oliferuk et al. [10] pointed out that this method can give us only part of the stored energy related to residual mircrostresses during nonhomogeneous plastic deformation between grains (ESR), which is much lower than that corresponding to the creation of defects (mainly dislocations) inside grains (ESD). According to Szczepinski's separation [21], the total stored energy is a sum of two parts: (a) (b)  The experimental results give impetus to the theoretical description of energy balance in material. One of the possible ways to calculate stored energy is based on the stress-strain curve. Martin [16], who regarded the segment of the plastic work associated with work hardening as accumulating damage, first proposed a stored energy method from the stressstrain curve, as shown in Figure 2a with the shaded area. Later, a dislocation model, on the assumption of the irreversibility of dislocation motion, was established by Tanaka [17] to prove that the shaded area in Figure 2a is the stored energy related to the dislocations of pile-up in a single crystal. However, this method is not available for polycrystalline materials, especially for low cycle fatigue. The experimental results give impetus to the theoretical description of energy balance in material. One of the possible ways to calculate stored energy is based on the stressstrain curve. Martin [16], who regarded the segment of the plastic work associated with work hardening as accumulating damage, first proposed a stored energy method from the stress-strain curve, as shown in Figure 2a with the shaded area. Later, a dislocation model, on the assumption of the irreversibility of dislocation motion, was established by Tanaka [17] to prove that the shaded area in Figure 2a is the stored energy related to the dislocations of pile-up in a single crystal. However, this method is not available for polycrystalline materials, especially for low cycle fatigue.
Another stored energy method from the stress-strain curve is proposed by Skelton [18][19][20] and Aravas [3]; they suggested that the shaded area in Figure 2b is the internal energy stored in the metal, which is famously known as "complementary plastic strain energy". However, Szczepinski [21] and Oliferuk et al. [10] pointed out that this method can give us only part of the stored energy related to residual mircrostresses during nonhomogeneous plastic deformation between grains (ESR), which is much lower than that corresponding to the creation of defects (mainly dislocations) inside grains (ESD). According to Szczepinski's separation [21], the total stored energy is a sum of two parts:  Another stored energy method from the stress-strain curve is proposed by Skelton [18][19][20] and Aravas [3]; they suggested that the shaded area in Figure 2b is the internal energy stored in the metal, which is famously known as "complementary plastic strain energy". However, Szczepinski [21] and Oliferuk et al. [10] pointed out that this method can give us only part of the stored energy related to residual mircrostresses during nonhomogeneous plastic deformation between grains (E SR ), which is much lower than that corresponding to the creation of defects (mainly dislocations) inside grains (E SD ). According to Szczepinski's separation [21], the total stored energy is a sum of two parts: In order to determine the evolution of the stored energy, the evolution of the defect structure needs to be predicted for the given loading conditions. Chaboche [15] and Mollica et al. [22], Kamlah [23] and Cho et al. [24] calculated the stored energy by a continuum framework with internal variables used to characterize the defect evolution. They found that back stress is an essential internal variable in establishing their stored energy models.
With the fast development of computer technology, Benzerga et al. [2] developed a discrete dislocation plasticity model to calculate the stored energy of cold work associated with defects of dislocations. They also found that most of the energy stored in the dislocations is associated with their long-range stress fields (back stress), which accounted for more than 95%. Recently, mesoscopic microstructure-based modeling approaches such as crystal plasticity finite element method (CPFEM), have been popular for calculating the stored energy in a heterogeneous polycrystalline [25]. However, those methods are too complex to be used in engineering applications.
Both of the two classical stored energy methods shown in Figure 2 are related to flow stress, which is usually composed of friction stress σ F and back stress X. The friction stress, corresponding to the stress required locally for a dislocation to move, is mainly related to the short-range obstacles [26]. Many scholars [27][28][29][30] believe that heat dissipation phenomena are related to dislocation movements in the material lattice (internal friction): overcoming short-range obstacles, while back stress refers to the stress associated with a local strain process providing long-range interactions with mobile dislocations. Thus, back stress is closely related to the stored energy, while the friction stress corresponds to the dissipated heat. In addition, back stress can be further divided into two components [31,32], namely intergranular back stress X inter , relating to the plastic strain incompatibilities between the different grains, and intragranular back stress X intra , linking to the dislocation distribution structures inside the grains. Thus, According to the definition, it is not difficult to discover that both E SR and X inter refer to the heterogeneous deformation between different microstructures, while both E SD and X intra correspond to the dislocation activities inside grains. Thus, this paper attempts to find the connection between back stress and energy storage by studying the role of back stress in constructing classical energy storage models.

Relationship between E SR and X inter
The stored energy due to the residual microstresses remaining in a metal (E SR ) has been studied by Szczepinski [21], on the basis of a simple mezomechanical model. This model represents a regular array of cuboidal A and B elements (Figure 3), based on the assumption that the elements deformed plastically without strain hardening, but each of them has a different yield stresses, σ A y and σ B y , respectively, which cause non-homogeneous plastic deformation of the composite during uniaxial tensile loading, as illustrated in Figure 4. At the point of D, when the A element begins to deform plastically, the plastic strain of B element has been ∆ε p greater than that of A element, and the plastic strain difference (∆ε p ) remains unchanged during the subsequent deformation. At the point of E, the friction stress of the composite is σ F = σ B y , and the back stress is Note that this model excludes strain hardening of the A and B elements. Hence, the back stress arising during plastic deformation could only be related to plastic strain incompatibilities between grains, namely intergranular back stress X inter . From Equation (4), the larger the difference between σ A y and σ B y , the greater value of X inter , which indicates that X inter is a macro character of the microscopically structural inhomogeneity.
Moreover, they are also equal to Xinter in Equation (4). Thus, Xinter can be used to evaluate the magnitude of residual microstresses. The elastic energy per unit volume stored in the model due to the residual stresses remained in elements is equal:    Figure 4, and they are of the same absolute value: Moreover, they are also equal to Xinter in Equation (4). Thus, Xinter can be used to evaluate the magnitude of residual microstresses. The elastic energy per unit volume stored in the model due to the residual stresses remained in elements is equal: After unloading, the residual tensile stresses σ A r remain in the A element while the residual compressive stresses σ B r remain in the B element, as shown in Figure 4, and they are of the same absolute value: Moreover, they are also equal to X inter in Equation (4). Thus, X inter can be used to evaluate the magnitude of residual microstresses. The elastic energy per unit volume stored in the model due to the residual stresses remained in elements is equal: where σ A r = −σ B r = X inter and E is the elastic modulus of elements. This energy density is equal to the complementary plastic strain energy density, E C , the shaded triangle ∆CDP in Figure 4, which can be easily estimated from the stress-strain diagram by measuring the area dashed in Figure 2b.
When the uniaxial loading direction is reversed, as shown in Figure 4b, the weaker B element first reaches its compressive yield stress −σ B y and deforms plastically along M→H. In contrast, the A element remains in the elastic state and deforms along N→H, reducing the plastic strain difference (∆ε p ) and releasing residual microstresses remaining in elements. At the point of H, where the two reversal loading curves intersect, the plastic strain difference reduces to zero, and the intergranular back stress X inter = 0, as shown in Figure 4b. After unloading, there are no residual microstresses remaining in A and B elements anymore; That is, σ A r = σ B r = X inter = 0, and no elastic energy is stored in the composite as well: E SR = 1 2E (X inter ) 2 = 0. Thus, at the initial stage of reversal loading, the release of stored energy connected to the residual microstresses is attributed to decreased plastic strain incompatibility from grain to grain.
As the reversal loading continues, the stored energy increases again, as shown in Figure 4b. From H to I, the B element deforms plastically in the reversal direction. In contrast, the A element remains in the elastic state and deforms along H→L. Thus, the degree of the plastic strain incompatibility increases again between elements, which increases the absolute value of intergranular back stress |X inter |. At the point of J, the intergranular back stress as shown in Figure 4b. After unloading, residual compressive stresses σ A r and residual tensile stresses σ B y remain in the A and B elements, respectively: Thus, during the subsequent reversal loading, the elastic energy per unit volume stored in the model due to residual stresses remaining in elements is E SR = 1 2E (X inter ) 2 equal to the stored energy during the tensile process.
Overall, in the mesomechanics model, our study on the process of loading and reversal loading reveals that the accumulation or dissipation of E SR is related to the increase or decrease of residual microstresses remaining in elements resulting from the increase or decrease of the degree of the plastic strain incompatibility between grains. While the |X inter | is a macroscopic representation of the residual microstresses, the E SR can be expressed by X inter in the formula form of Equation (6).

Relationship between E SD and X intra
In the real material, besides the energy stored due to residual microstresses (E SR ), there is also the energy connected with the creation of defects (mainly dislocations) in particular grains (E SD ). Thus, the strain-hardening effect inside grains should be accounted for. A dislocation model was introduced by Tanaka [17] to study the stored energy of dislocations within long-range stress fields inside a single crystal, as shown in Figure 5.
In this model, the dislocations generated in a most favorably oriented grain under the tensile stress are piled up against the grain boundary, as illustrated in Figure 5a, causing the long-range stress inside the grain, namely the dislocation stress τ D , corresponding to the intragranular back stress X intra . The dislocation stress τ D and the plastic displacement Φ(x) depend on the dislocation density D(x). The stored energy of dislocations in unit volume has been given by Tanaka [17].
Using the inversion formula of Muskhelishvili, Equation (9) can be simplified as: where τ is the loading of stress, k is the friction stress and γ is the plastic strain. Therefore, the amount of stored energy of dislocations (E SD ) for a single crystal is equal to plastic work done by X intra , corresponding to the segment of the plastic work associated with work hardening as shown in Figure 5b by the shaded triangles.
volume has been given by Tanaka [17].
Using the inversion formula of Muskhelishvili, Equation (9) can be simplified as: where τ is the loading of stress, k is the friction stress and γ is the plastic strain. Therefore, the amount of stored energy of dislocations (ESD) for a single crystal is equal to plastic work done by Xintra, corresponding to the segment of the plastic work associated with work hardening as shown in Figure 5b by the shaded triangles.
(a) (b) The above analysis demonstrates that both ESR and ESD can be simply expressed by Xinter and Xintra, respectively, in Equations (6) and (10). Thus, it is reasonable to assume that there is a close connection between stored energy ES and plastic work of back stress WpB during working hardening.

Plastic Work of Back Stress during Cyclic Loading
During cyclic tests, the plastic work of back stress (WpB) is dependent on the evolution of back stress.

Hysteresis Loop of Back Stress
From a mechanical point of view, the back stress, X, corresponds to the translation of the elastic domain, whereas the friction stress, σF, represents the radius of this domain. Thus, according to Kuhlmann's method [33], the back stress and friction stress can be extracted directly from the hysteresis loops, as schematically in Figure 6, where σF is determined at a reversed plastic strain offset (εoffset) within the range of 5 × 10 −6~1 0 −3 , as suggested in references [32,34,35]. The back stress X can be thus obtained by σa = X + σF. However, this method can only give out the internal stresses at the peak point.
During cyclic loading, Kuhlmann [33] also pointed out that the friction stress is approximately the same at the beginning and end of each half cycle with only slow changes The above analysis demonstrates that both E SR and E SD can be simply expressed by X inter and X intra, respectively, in Equations (6) and (10). Thus, it is reasonable to assume that there is a close connection between stored energy E S and plastic work of back stress W pB during working hardening.

Plastic Work of Back Stress during Cyclic Loading
During cyclic tests, the plastic work of back stress (W pB ) is dependent on the evolution of back stress.

Hysteresis Loop of Back Stress
From a mechanical point of view, the back stress, X, corresponds to the translation of the elastic domain, whereas the friction stress, σ F , represents the radius of this domain. Thus, according to Kuhlmann's method [33], the back stress and friction stress can be extracted directly from the hysteresis loops, as schematically in Figure 6, where σ F is determined at a reversed plastic strain offset (ε offset ) within the range of 5 × 10 −6~1 0 −3 , as suggested in references [32,34,35]. The back stress X can be thus obtained by σ a = X + σ F . However, this method can only give out the internal stresses at the peak point. from one cycle to the next, which indicate that only kinematic hardening (represented by X) is taken into account in a complete cycle. Thus, it is justifiable to assume that the size of the elastic domain remains unchanged and only the center of the yield surface travels along the loading path. In the case of kinematic hardening, the stress, at any given moment t, can be given by [36].
where 1 v = ± , according to the direction of plastic strain, as illustrated in Figure 7a at A' point. The circles in Figure 7a represent the yield surface with a fixed radius of σF. In the case of tension-going, F X σ σ = − , the back stress increases from E to G along an upward locus EFG . While the loading direction reverses, F X σ σ = + , the back stress decreases from G to E along a downward locus GHE as shown in Figure 7a with dot lines. Let the short-range interaction stress (σS) equal to the second term of the right-hand side  During cyclic loading, Kuhlmann [33] also pointed out that the friction stress is approximately the same at the beginning and end of each half cycle with only slow changes from one cycle to the next, which indicate that only kinematic hardening (represented by X) is taken into account in a complete cycle. Thus, it is justifiable to assume that the size of the elastic domain remains unchanged and only the center of the yield surface travels along the loading path. In the case of kinematic hardening, the stress, at any given moment t, can be given by [36].
where v = ±1, according to the direction of plastic strain, as illustrated in Figure 7a at A' point. The circles in Figure 7a represent the yield surface with a fixed radius of σ F . In the case of tension-going, F X σ σ = − , the back stress increases from E to G along an upward locus EFG . While the loading direction reverses, F X σ σ = + , the back stress decreases from G to E along a downward locus GHE as shown in Figure 7a with dot lines. Let the short-range interaction stress (σS) equal to the second term of the right-hand side of Equation (11) The hysteresis loop of σS turns into a rectangle, as shown in Figure 7b with dot lines, and the hysteresis loop of X turns into a fusiform loop, as shown in Figure 7b  In the case of tension-going, X = σ − σ F , the back stress increases from E to G along an upward locus EFG. While the loading direction reverses, X = σ + σ F , the back stress decreases from G to E along a downward locus GHE as shown in Figure 7a with dot lines. Let the short-range interaction stress (σ S ) equal to the second term of the right-hand side of Equation (11): σ S = vσ F , then Equation (11) can be rewritten as The hysteresis loop of σ S turns into a rectangle, as shown in Figure 7b with dot lines, and the hysteresis loop of X turns into a fusiform loop, as shown in Figure 7b with bold lines.
In Figure 7b, the total plastic work expended around the hysteresis loop is defined as 'plastic strain hysteresis energy', W p , and is given by W p = σdε p . According to Equation (12), the plastic work done by external stress can be divided into two components as follows: In the first item of Equation (13), the absolute value of σ S is equal to the friction stress σ F during cyclic plastic deformation; thus, it is defined as plastic work of friction stress, W pF .
This work, corresponding to the shaped rectangle area of the loop in Figure 7b with dot lines, is related to the dissipated work against the friction stress resulting from the movement of dislocations: overcoming short-range obstacles [29]. The second item of Equation (13) is the plastic work done by the back stress, W pB .
This work, corresponding to the shaped spindle area of the loop in Figure 7b, is associated with internal energy stored during the work-hardening process.

Accumulation and Dissipation of the Plastic Work of Back Stress
In the case of tensile-going loading, as shown in Figure 8a, the plastic work done by the back stress W pB1 can be easily integrated from X 1 -ε p curve, as given by a definite integral of ε pH ε pE X 1 dε p for curve 1 according to the Equation (15). Mathematically, the definite integral is defined as the signed area of the region in the X-ε p plane that is bounded by the graph of curve 1 and the ε p -axis and two vertical boundary lines. Thus, the area I below the ε p -axis subtracts from the total, while the area II above the ε p -axis adds to the total. However, during the compression-going loading, as shown in Figure 8b, the orientation of the definite integral of ε pE ε pH X 2 dε p for curve 2 is reversed; thus, the sign of the integral should be switched. Then, the area III above the ε p -axis subtracts from the total, while the area IV below the ε p -axis adds to the total. The negative areas I and III are related to the release of the previously cumulated plastic work of back stress when the absolute value of back stress |X| decreases to zero at each reverse loading. Finally, the total plastic work of back stress per cycle, ∆W pB , is equal to the area of the closed loop in Figure 8c.
where ∆εp is the plastic strain range. In our previous work [37], the low cycle fatigue experiments of Ti-6Al-4V alloy have already been carried out with a total strain-controlled mode, with a ratio of -1, a constant total strain rate of 4 × 10 −3 s −1 and a triangular waveform performed on a computer-controlled 250 kN MTS810 closed-loop servo-hydraulic test machine at room temperature. Strain control was achieved by extensometer (12 mm gage length) arm tips located on the gage length. The strain amplitude chosen for the present tests ranges from 0.7% to 2.0%.
The tests were continued until fracture. The correlation between the low cycle fatigue behavior of Ti-6Al-4V alloy and its microstructure evolution has been discussed in our previous work [37]. This paper will focus on the evolution of the plastic work of back stress during cyclic loading with different strain amplitudes.
Transmission electron microscopy (TEM) examinations were also carried out to study the microstructure evolutions during cyclic deformations. At the small strain amplitude of 0.7%, the heterogeneous dislocation distribution between the adjoining αp grains with different orientations is visible in Figure 9a. The grain boundary acts as a solid barrier to slips, so that the high density of dislocation lines observed in soft grain cannot cross through but only pile up at the grain boundary, which leads to a plastic strain incompatibility between microstructures and generates the long-range internal stresses between hard/soft grains, namely Xinter. When the load is removed, the residual tensile stress remains in hard grains, and residual compressive stress remains in soft grains, as illustrated in Figure 4a. Associated with these residual microstresses is the elastic energy remaining in the internal structure after plastic deformation [21], i.e., ESR. The other part of To simplify the calculation, the plastic work of back stress per cycle, ∆W pB , can also be obtained by subtracting the plastic work of friction stress per cycle, ∆W pF , from the total plastic work per cycle, ∆W p .
∆W pB = ∆W p − ∆W pF (16) in which, Since the friction stress is assumed to be a constant during a single complete cycle, ∆W pF can be simply expressed by ∆W pF = 2σ F ∆ε p (18) where ∆ε p is the plastic strain range. In our previous work [37], the low cycle fatigue experiments of Ti-6Al-4V alloy have already been carried out with a total strain-controlled mode, with a ratio of -1, a constant total strain rate of 4 × 10 −3 s −1 and a triangular waveform performed on a computercontrolled 250 kN MTS810 closed-loop servo-hydraulic test machine at room temperature. Strain control was achieved by extensometer (12 mm gage length) arm tips located on the gage length. The strain amplitude chosen for the present tests ranges from 0.7% to 2.0%. The tests were continued until fracture. The correlation between the low cycle fatigue behavior of Ti-6Al-4V alloy and its microstructure evolution has been discussed in our previous work [37]. This paper will focus on the evolution of the plastic work of back stress during cyclic loading with different strain amplitudes.
Transmission electron microscopy (TEM) examinations were also carried out to study the microstructure evolutions during cyclic deformations. At the small strain amplitude of 0.7%, the heterogeneous dislocation distribution between the adjoining α p grains with different orientations is visible in Figure 9a. The grain boundary acts as a solid barrier to slips, so that the high density of dislocation lines observed in soft grain cannot cross through but only pile up at the grain boundary, which leads to a plastic strain incompatibility between microstructures and generates the long-range internal stresses between hard/soft grains, namely X inter . When the load is removed, the residual tensile stress remains in hard grains, and residual compressive stress remains in soft grains, as illustrated in Figure 4a. Associated with these residual microstresses is the elastic energy remaining in the internal structure after plastic deformation [21], i.e., E SR . The other part of stored energy, E SD , is related to the high density of dislocation lines in the soft grain and dislocation pile-ups at the grain boundary, which produces the internal stress inside soft α p grain in Figure 9a, namely X intra . At the high strain amplitude of 2.0%, as shown in Figure 9b, homogeneous dislocation structure is observed in both of the two adjoining α p grains, which indicates that plastic deformation of the Ti-alloy tends to homogenization from grain-to-grain throughout the microstructures. Furthermore, the density of dislocation inside α p grains is larger than that at low strain amplitude. Thus, the contribution of residual microstresses will decrease with the increasing applied strain. c give out the evolution of WpB as a function of the total plastic work Wp at strain amplitudes of 0.8%, 1.2%, and 2.0%, respectively. It is observed that the WpB does not change monotonically during a half cycle, but instead decreases first and then increases in sequence. The dotted lines in Figure 10 predict very similar evolutions to the measurements made by Halford [14] as shown in Figure 1, i.e., that the stored energy is released at the very beginning of every reverse loading. According to the theoretical analysis in Section 2.1, the elastic energy connected with the residual microstresses (ESR) stored during the non-homogeneous plastic deformation is partially or completely released at every half cycle while |Xinter| decreases to zero, resulting from the decrease of the degree of the plastic strain incompatibility from grain-to-grain at the initial stage of reversal deformation, which could also be used to explain the phenomenon of the release of stored energy during cyclic loading. It is convenient to separate the plastic work of back stress changes during a half cycle into a cumulative and a released component, as illustrated in Figure 11. By doing so, the released part of the plastic work of back stress (Wrel), corresponding to the ESR, can be directly compared from test-to-test with different applied strains. Figure 12a shows the half-life plastic work of back stress per cycle ∆WpB and its released energy per cycle ∆Wrel versus strain. The ∆WpB increases linearly with increasing strain while ∆Wrel increases are gently first and then tends to be stable. Thus, as expected, the ratio of ∆Wrel/∆WpB drops exponentially from 18% to 2% for Ti-6Al-4V alloy with increasing strain (Figure 12b), which further confirmed that the proportion of ESR decreases with increasing strain when the plastic deformation between heterogeneous microstructures tends to be more homogeneous.  The back stress and friction stress have been extracted from the hysteresis loops with ε offset = 10 −4 in reference [37] according to Kuhlmann's method [33]. The hysteresis loop of back stress can be easily calculated from Equation (12), in which σ S is a constant for each half cycle. Based on the numerical integration of X-ε p cyclic loops, we can observe the evolution of W pB and its partial released for each half cycle quantitatively. Figure 10a-c give out the evolution of W pB as a function of the total plastic work W p at strain amplitudes of 0.8%, 1.2%, and 2.0%, respectively. It is observed that the W pB does not change monotonically during a half cycle, but instead decreases first and then increases in sequence. The dotted lines in Figure 10 predict very similar evolutions to the measurements made by Halford [14] as shown in Figure 1, i.e., that the stored energy is released at the very beginning of every reverse loading. According to the theoretical analysis in Section 2.1, the elastic energy connected with the residual microstresses (E SR ) stored during the non-homogeneous plastic deformation is partially or completely released at every half cycle while |X inter | decreases to zero, resulting from the decrease of the degree of the plastic strain incompatibility from grain-to-grain at the initial stage of reversal deformation, which could also be used to explain the phenomenon of the release of stored energy during cyclic loading.   It is convenient to separate the plastic work of back stress changes during a half cycle into a cumulative and a released component, as illustrated in Figure 11. By doing so, the released part of the plastic work of back stress (W rel ), corresponding to the E SR , can be directly compared from test-to-test with different applied strains. Figure 12a shows the half-life plastic work of back stress per cycle ∆W pB and its released energy per cycle ∆W rel versus strain. The ∆W pB increases linearly with increasing strain while ∆W rel increases are gently first and then tends to be stable. Thus, as expected, the ratio of ∆W rel /∆W pB drops exponentially from 18% to 2% for Ti-6Al-4V alloy with increasing strain (Figure 12b), which further confirmed that the proportion of E SR decreases with increasing strain when the plastic deformation between heterogeneous microstructures tends to be more homogeneous.   According to the Equations (1), (3), and (4), the ∆Wp, ∆WpF and ∆WpB have been calculated and displayed in Figure 13 as the function of the logarithm of the number of cycles at various applied strains. As can be noted from the comparison between Figure 13a,b, the evolution of ∆WpF are consistent with that of ∆Wp at any applied strain. Both of them increase with cyclic numbers at small strain ranges (εta ≤ 1.0~1.2%) and decrease at large strain ranges (εta > 1.0~1.2%). However, the evolution of ∆WpB has a different tendency, which stays almost constant with increasing cyclic numbers while εta < 1.8%, and shows a decreasing tendency while εta ≥ 1.8% (Figure 13c). The total hysteresis energy (or fatigue toughness) Wf, the total plastic work of friction stress ∑WpF and the total plastic work of back stress ∑WpB up to failure for all specimens tested are also present in Figure 14. All of them show a maximum value at the strain amplitude of 1.0%, which corresponds to the breaking point of the bilinear behavior observed in the C-M plot of Ti-6Al-4V Alloy [37]. It may be seen from Figure 14 that both of Wf and ∑WpF decrease exponentially with increasing applied strain at the strain range of εta ≥ 1.0%, while ∑WpB remains relatively constant over the strain range of εta ≥ 1.0%. Wong et al. [38] also found that the total stored energy is almost constant while the total hysteresis energy varies over the fatigue life range. Thus, the plastic work of back stress is sufficient to correctly reproduce and explain the evolution of stored energy during cyclic loading. According to the Equations (1), (3), and (4), the ∆W p , ∆W pF and ∆W pB have been calculated and displayed in Figure 13 as the function of the logarithm of the number of cycles at various applied strains. As can be noted from the comparison between Figure 13a,b, the evolution of ∆W pF are consistent with that of ∆W p at any applied strain. Both of them increase with cyclic numbers at small strain ranges (ε ta ≤ 1.0~1.2%) and decrease at large strain ranges (ε ta > 1.0~1.2%). However, the evolution of ∆W pB has a different tendency, which stays almost constant with increasing cyclic numbers while ε ta < 1.8%, and shows a decreasing tendency while ε ta ≥ 1.8% (Figure 13c). The total hysteresis energy (or fatigue toughness) W f , the total plastic work of friction stress ∑W pF and the total plastic work of back stress ∑W pB up to failure for all specimens tested are also present in Figure 14. All of them show a maximum value at the strain amplitude of 1.0%, which corresponds to the breaking point of the bilinear behavior observed in the C-M plot of Ti-6Al-4V Alloy [37]. It may be seen from Figure 14 that both of W f and ∑W pF decrease exponentially with increasing applied strain at the strain range of ε ta ≥ 1.0%, while ∑W pB remains relatively constant over the strain range of ε ta ≥ 1.0%. Wong et al. [38] also found that the total stored energy is almost constant while the total hysteresis energy varies over the fatigue life range. Thus, the plastic work of back stress is sufficient to correctly reproduce and explain the evolution of stored energy during cyclic loading.
breaking point of the bilinear behavior observed in the C-M plot of Ti-6Al-4V Alloy [37]. It may be seen from Figure 14 that both of Wf and ∑WpF decrease exponentially with increasing applied strain at the strain range of εta ≥ 1.0%, while ∑WpB remains relatively constant over the strain range of εta ≥ 1.0%. Wong et al. [38] also found that the total stored energy is almost constant while the total hysteresis energy varies over the fatigue life range. Thus, the plastic work of back stress is sufficient to correctly reproduce and explain the evolution of stored energy during cyclic loading.

Conclusions
In this work, two mesomechanics models were analyzed in an attempt to find the connection between back stress and stored energy. Then the evolution of back stress during cyclic loading was studied, and the plastic work of back stress (WpB) was calculated with the low cycle fatigue experimental data of Ti-6Al-4V. The main findings of this study can be summarized as follows: (1) Based on the mesomechanics model, our study on the process of cyclic loading reveals that the partial release of stored energy is related to the decrease of residual microstresses remaining in elements resulting from the decrease of the degree of the

Conclusions
In this work, two mesomechanics models were analyzed in an attempt to find the connection between back stress and stored energy. Then the evolution of back stress during cyclic loading was studied, and the plastic work of back stress (WpB) was calculated with the low cycle fatigue experimental data of Ti-6Al-4V. The main findings of this study can be summarized as follows: (1) Based on the mesomechanics model, our study on the process of cyclic loading reveals that the partial release of stored energy is related to the decrease of residual

Conclusions
In this work, two mesomechanics models were analyzed in an attempt to find the connection between back stress and stored energy. Then the evolution of back stress during cyclic loading was studied, and the plastic work of back stress (W pB ) was calculated with the low cycle fatigue experimental data of Ti-6Al-4V. The main findings of this study can be summarized as follows: (1) Based on the mesomechanics model, our study on the process of cyclic loading reveals that the partial release of stored energy is related to the decrease of residual microstresses remaining in elements resulting from the decrease of the degree of the plastic strain incompatibility between grains. Additionally, the |X inter | is a macroscopic representation of the residual microstresses. (2) In Tanaka s dislocation model, the amount of stored energy of dislocations (ESD) for a single crystal is equal to plastic work done by X intra . (3) The plastic work of back stress (W pB ) was calculated with the low cycle fatigue experimental data of Ti-6Al-4V. The results show that W pB is partially released at every reverse loading, sufficient to reproduce the evolution of stored energy correctly under cyclic loading. Furthermore, the total plastic work of back stress ∑W pB up to failure is almost constant over the strain range of ε ta ≥ 1.0%, while the fatigue toughness W f decrease exponentially with increasing applied strain. Thus, the plastic work of back stress is a more suitable damage criterion.
Is the plastic work of back stress the stored energy? Further theoretical analysis and experimental verification are still needed. The crystal plasticity finite element method (CPFEM) is a useful alternative for quantifying the stored deformation energy. It is also a great tool for investigating the relationship between plastic work of inter/intra back stress and inter/intra stored energy.