Relationship between Local Strain Jumps and Temperature Bursts Due to the Portevin-Le Chatelier Effect in an Al-Mg Alloy

Transcription

1 Experimental Mechanics (213) 53: DOI 1.17/s Relationship between Local Strain Jumps and Temperature Bursts Due to the Portevin-Le Chatelier Effect in an Al-Mg Alloy C. Bernard & J. Coër & H. Laurent & P. Chauvelon & P.Y. Manach Received: 31 August 212 / Accepted: 22 December 212 / Published online: 19 January 213 # Society for Experimental Mechanics 213 Abstract Local strain and temperature of an AA5754-O aluminum alloy sheet have been full-field measured during monotonous tensile tests carried out at room temperature. Sharp strain increases and temperature bursts which are locally generated by the Portevin-Le Chatelier phenomenon have been measured at the same point for two strain rates: V2= s 1 and V1= s 1. A relationship, which is based on the underlying physical mechanisms, has been established between the strain and the temperature and experimentally verified for the highest strain rate V1. The discrepancy between the theoretical and experimental results for the lowest strain rate V2 suggests that the localized plastic deformations do not follow an adiabatic transformation. Such a set-up seems to offer a direct and experimental method to check the adiabatic character of localized plastic deformations. Keywords Aluminum alloys. Tension test. Portevin-Le Chatelier effect. Dynamic strain aging. Plastic deformation C. Bernard (*) : J. Coër : H. Laurent : P. Chauvelon : P.Y. Manach Laboratoire d Ingénierie des MATériaux de Bretagne (LIMATB EA 425), Université Européenne de Bretagne, Université de Bretagne-Sud, Centre de Recherche, rue de Saint Maudé, BP 92116, Lorient cedex, France cedric.bernard@univ-ubs.fr J. Coër : H. Laurent CEMUC, Departamento de Engenharia Mecânica, Universidade de Coimbra, Polo II, Coimbra, Portugal Introduction The demand for higher fuel efficiency and lightning of structures has driven many manufacturers to increasingly used aluminum alloys sheets instead of steels sheets. Their excellent strength to weight ratio allows for considerable reduction in the weight of the structure. Amongst these alloys, 5xxx series alloys (Al-Mg) are the most used due to their low cost, as stiffeners for inner body panels in the automotive industry for example. However, numerous problems still arise in the forming of such alloys at room temperature. Typically, these alloys are known to exhibit dynamic strain aging (DSA) during plastic deformation at room temperature. This phenomenon is attributed to the interaction between solute atoms and mobile dislocations and depends both on the temperature and on the strain rate. DSA is associated with the Portevin-Le Chatelier (PLC) phenomenon which is characterized by serrations in the macroscopic load/displacement tensile test curve due to the repeated propagation of localized plastic strain bands in the specimens. DSA also leads to negative strain rate sensitivity (NSRS). In the field of deep drawing, PLC effect is particularly undesirable because it reduces the ductility of the sheet and generates non-aesthetic stretcher lines on the material surface. As a consequence, the Al-Mg alloys used in the automotive industry are only incorporated into the non-visible parts of vehicles. Many studies on Al-Mg alloys have led to a better understanding of the PLC phenomenon. Considerable efforts have been made to characterize the motion and the morphology of the bands [1 4], including their type (A, B or C), onset, velocity, width, orientation and, more rarely, temperature

2 126 Exp Mech (213) 53: [5 8]; to show their dependence on test conditions [9] (temperature [1, 11], strain magnitude [6] or strain rate [12]); and to determine the relationship between DSA, PLC bands, NSRS [13 15], heat sources [5] and microstructure evolution [16 18]. Several works dealing with the PLC phenomenon in Al-Mg alloys showed a relationship between the changes in the applied strain or stress and local temperature variations [5, 6, 11]. Nevertheless, as far as we know, no experimental results were reported regarding the relationship between local temperature bursts, i.e. rapid increases in temperature, and the associated local strain increases generated by the motion of the PLC bands. Louche et al. [5], for example, have estimated the heat sources distribution associated with the motion of the PLC bands from the analysis of the temperature field. But the energy dissipated by the associated plastic deformation process, which could be calculated from stress and local strain measurements, has not been correlated with the measured amount of thermal energy. Similarly, Ait-Amokthar et al. [6], or more recently Hu et al. [19], recorded the temperature at three given points of a tensile test specimen as well as the strain field in the whole working zone. However, the relationship that Amokthar et al. established in a previous work [11] between stress, local strain rate and local temperature change was not checked to be consistent with these experimental values. The present study aims to establish and check a relationship between the rapid thermal and mechanical changes occurring at a given point during the passage of a PLC band. This paper presents, in a first part, an experimental investigation of the local temperature bursts and local strain increases generated by PLC bands by means of tensile tests on an Al-Mg alloys sheet associated with full-field measurement devices. Then, the relationship between these two key parameters is studied in light of the underlying physical mechanisms. Finally, theoretical and experimental results are compared for two strain rates and the relevance of the adiabatic transformation hypothesis with respect to the applied strain rate is discussed. Material and Experimental Methods The material is an AA5754-O (Al-Mg) aluminum alloy sheet with a thickness of 1. mm. The alloy is fully annealed and contains the following elements in weight %: Mg ( ), Mn (.5), Si (.4), Fe (.4), Cr (.3) and Cu (.1), with Al being the balance. A previous work [2] showed that the PLC effect occurs during tensile tests on this material at room temperature with a particular strain rate of s 1 but disappears at higher temperatures. The aim of the experiments was to simultaneously record the strain and temperature fields of a flat sample during a tensile test. Figure 1 presents the set-up. An electromechanical testing machine, a Gleeble 35 (Dynamics System Inc., USA), was used to apply tensile loads while the strain field was measured using a stereo-optical strain measurement system, ARAMIS (GOM, Germany) that was aimed at the upper side of the specimen (see Fig. 1 and [2] for a detailed description of the experimental set-up). The temperature field of the sample was measured with a 2D thermographic infrared camera (CEDIP: Jade III MWIR), using an image resolution of 6 pixels/mm (32 24 pixels matrix) at 2 frames/s. The noise level (NETD) was less than.2 C. The axis of the IR camera was oriented perpendicularly to the lower surface of the specimen, which was painted with a high-emissivity graphic-black coating that was assumed to be close in emissivity characteristics to a black body (.98). The 2D temperature field was extracted from the IR video with the software Altair. The strain and temperature were measured on each side of the specimen. Due to the complexity of the measurement, a unique test was performed for the two investigated strain rates. However, the amount of data provided by this unique test is large enough to be statistically representative. Indeed, a large number of local temperature bursts were observed during one test. Moreover, due to the random nature of the motion of the PLC bands, these tests are necessarily nonreproducible. Bone-shaped specimens were directly machined in the sheet to avoid any work hardening near the free edges. The dimensions of the rectangular gauge area were 4 mm 1 mm, and the length between grips was 8 mm (see Fig. 2). In this figure, the x-axis represents the tensile direction, y is the transverse direction and z is the thickness direction. Uniaxial tensile tests were performed at room temperature (2 C) with strain rates of s 1 and s 1, which are named, respectively, V1 and V2. The longitudinal components of the logarithmic strain and the Cauchy stress, σ, were calculated from the raw data (see [2] for more details). The local strain, ε loc, denotes the longitudinal logarithmic strain and is only a function of x. Considering the tensile loading applied, ε loc was supposed to remain constant with respect to y or z. The applied strain, ε app, is the longitudinal logarithmic strain applied to the whole working zone and is equal to the average of ε loc along the gauge area. Here, ε loc always denotes the local strain at a point M, located at the left side of the gauge zone, except for the curves presented in Fig. 2 where ε loc corresponds to the local strain in the middle of the specimen. This information explains why local strain reaches.25 in Fig. 2 while it does not exceed.2 for other figures of the paper. Indeed, the necking of the sample that occurs in the central zone of the specimen, leads to a higher local final strain value in the center but does not affect the strain distribution around the point M.

3 Exp Mech (213) 53: Fig. 1 (a) A view of the upper part of the set-up. The video cameras of the strain measurement system aim to the sample through an opening in the machine frame. (b) 3D overall representation of the main parts of the set-up. (c) A view of the electromechanical testing machine and the IR camera aiming to the underside of the specimen Results and Discussions The experimental results show that the PLC effect occurs for both strain rates, as the tensile curves in Fig. 2 exhibit the typical serrations. According to the classification presented in [1], the continuous propagation of the band observed in the full-field distributions (see Fig. 3(c)) reveals that type A bands are present for both strain rates. These tensile curves also slightly reveal the typical NSRS of the material in this range of strain rate and temperature. To investigate the relationship between the temperature burst and the sharp strain increase generated locally by PLC bands, the time dependence of the temperature and local strain at a given point M of the sample was extracted from the tensile test data. The position of point M is indicated in Figs. 2 and 3(c). Three local parameters were measured at point M with respect to time: the temperature elevation, θ=t T (with T being the initial temperature, i.e., room temperature), the local strain, ε loc, and the local strain rate, " loc. The results are presented in Figs. 3(a) and 4 for V2 and V1, respectively. These curves show that each time a PLC band nucleates or passes through point M, a sharp increase in the local strain value is observed, which corresponds to a peak on the strain rate vs time curve. Moreover, each strain increase is associated with a temperature burst. This phenomenon is depicted in Fig. 3(b) where the sharp strain increase occurring at t 47 s has been magnified. Moreover, for 8 time increases placed throughout this sharp increase - represented by open square symbols - the fields of the temperature, the strain and the strain rate were extracted and presented in Fig. 3(c). These fields clearly show the birth, the passage through the point M and the vanishing of the PLC band involved in this sharp strain increase. Fig. 2 Cauchy stress vs. local strain curves in the center of the specimen for strain rates of s 1 (V1) and s 1 (V2). Geometry and characteristic dimensions of the tensile specimen (L =4, b=1 and 8 mm between the two grip areas). The position of the point M is also indicated

4 128 Exp Mech (213) 53: Fig. 3 (a) Temperature, strain and strain rate time dependence at point M for V2, i.e s 1.(b) Magnification of the temperature burst, enclosed by the frame, occurring at t=47 s. (c) Temperature, strain and strain rate fields of the whole working zone for 8 time increments in the vicinity of this temperature burst. These 8 time increments are represented by open square symbols in (b) For each temperature burst presented in Figs. 3(a) and 4, the time over which the temperature increases is always shorter than that over which the temperature decreases. The increase in temperature is primarily due to the heat generated by local plastic deformation during the passage of a band. Besides, the correlation between high strain rate and temperature jumps is clearly shown in Fig. 3(c). Conversely, the decrease in temperature, which is due to Fig. 4 Temperature, strain and strain rate time dependence at point M for V1, i.e s 1

5 Exp Mech (213) 53: heat conduction in the specimen and through the outer surfaces, is a slower process. It is noteworthy that the few temperature bursts that are not associated with a sudden strain increase are caused by PLC bands that begin or stop in the vicinity of point M but do not cross it. As a result, a temperature increase caused by heat conduction can be measured at point M without any simultaneous sharp strain increase at this point. The temperature burst observed at time t=66 s (see Fig. 3(a)), for example, is due to a PLC band that appears near point M without crossing it. The thermal fields depicted in Fig. 5 clearly show this phenomenon. Figure 5 also reveals the discontinuous nature of the band motion, i.e. type B band motion, at this particular time of the test, whereas temperature bursts that are associated with a sudden strain increase are mainly due to PLC bands smooth progression, i.e. type A band motion. Note that, the two temperature curves exhibit a slight decrease (.32K in both cases) during the elastic loading in the beginning of the test. This phenomenon, due to thermoelastic coupling, was already observed by Louche et al. [5] on a similar aluminum alloy, i.e. a decrease of.4 K for an imposed strain rate of the same magnitude. References [7] and [11] also report a similar phenomenon. Figure 6 focuses on two parameters measured at point M during each passage of a PLC band; these parameters are shown with respect to the applied strain. The first parameter, ΔT, is the peak height of the temperature burst, and the second one, Δε loc, is the associated local strain increase. These two parameters are shown in Fig. 3(a) at t 47 s. Figure 6 shows that ΔT andδε loc both increase as a function of the applied strain. Kang et al. [17] observed such tendency for Δε loc on the same material for a lower strain rate, s 1. Moreover, although the temperature variation during the whole test is significantly higher for the V1 test (approximately three times higher, cf. Figs. 3(a) and 4), the applied strain rate seems to play a lesser role in the height of the temperature bursts. The same remark applies for the value of Δε loc, regarding the similarity in the magnitude of the values in Fig. 6(a) and (b). Conversely, Fig. 6 suggests a monotonic relationship between the simultaneous temperature burst and strain jump. Additionally, the experimental values in Fig. 7 clearly show an increase in ΔT withrespecttoδε loc. To understand the relationship between these two parameters of the PLC effect, an analysis of the physical processes involved has been conducted. Temperature bursts are caused by the dissipation of a quantity of power generated by the associated strain jumps (see [7, 21] for examples). The spatio-temporal temperature field obeys the subsequent general heat equation: ρc p θ ¼ P diss þ lr 2 T; ð1þ where ρ is the material density, C p is the specific heat per unit of mass and λ is the thermal conductivity (273 kg.m 3, 897 J.kg 1.K 1 and 132 W.m 1.K 1, respectively). P diss is the power dissipated by heat per unit of volume. In the framework of a thermoplastic problem, if thermoelastic coupling is ignored, this power can be expressed as: P diss ¼ b σ : " p; ð2þ where σ and " p are the stress tensor and plastic strain rate tensor, respectively. β is the Taylor-Quiney coefficient, which accounts for the fraction of the plastic power that is actually converted into heat [22, 23] and is typically assumed to be.9 for this material. Assuming that the temperature increase in a PLC band is an adiabatic phenomenon, the heat conduction term disappears, and equation (1) can be written as: ρc p θ ¼ b σ : " p: ð3þ It is noteworthy that many authors, in [8] or[19] for example, also ignored this heat conduction term for convenience of evaluation. In the uniaxial tensile case, this equation becomes: ρc p θ ¼ b σ" p; ð4þ Fig. 5 Temperature elevation fields describing the temperature burst at t=66 s for V2. The temperature elevation is not associated with a local strain jump since the PLC band nucleates and vanishes in the vicinity of point M, but do not cross it where σ is the tensile Cauchy stress, i.e., the only non-null component of the stress tensor, and ε p is the plastic part of ε loc. Finally, for the duration Δt of the sudden increase of temperature occurring at a temperature burst, equation (4) is reduced to: ρc p ΔT ¼ b σδ" loc or ΔT ¼ b σ Δ" loc ; ð5þ ρc p where ε loc = ε p, if the elastic part of the local strain ε loc is neglected in comparison with the plastic part, and ΔT and

6 13 Exp Mech (213) 53: Fig. 6 Temperature peak height and local strain jump, for each passage of a PLC band through the point M, with respect to the applied strain at V1 (a) and V2 (b). Each point corresponds to a PLC band crossing through the point M (a) Temperature peak height, T (K) Temperature peak height at V1 Local strain increase at V Local strain jump, loc Applied strain, app (b) Temperature peak height at V2 Temperature peak height, T (K) Local strain increase at V Local strain jump, loc Applied strain, app Δε loc are, respectively, the temperature and the strain increases during Δt. Note that σ is assumed to be constant during Δt (according to Fig. 2, stress jumps are less than 2.5 % in relative terms). Equation (5) gives a direct physical Fig. 7 Comparison of the measured and calculated temperature peak heights with respect to the local strain increases for the applied strain rates V1 and V2 Temperature peak height, T (K) Experimental values at V2 Calculated values at V2 Experimental values at V1 Calculated values at V1 Corresponding linear regressions T/ loc 9 K T/ loc 85 K T/ loc 31 K Local strain jump, loc

7 Exp Mech (213) 53: relationship between ΔT and Δε loc that can be compared with the experimental values presented in Fig. 6. Thus, using equation (5), a theoretical value of ΔT was calculated for each previously measured Δε loc value, and the results are presented in Fig. 7. For the tests at V1, the variation in the calculated ΔT values with respect to Δε loc is in good agreement with the experimental values. They both approximate linear functions with slopes of ΔT/Δε loc 9 K and 85 K for calculated and experimental values, respectively. Conversely, the calculated values of ΔT for V2 are three times higher than the corresponding experimental values. The trend is also linear, but with a slope of approximately 31 K. Note that, not surprisingly, the calculated values for the two applied strain rates are similar, which can be explained by the fact that the discrepancy between the σ vs. ε loc curves for these two applied strain rates is very low (a few MPa at most, see Fig. 2). One of the main hypotheses leading to equation (5) is that the process does not involve heat conduction. Such an assumption is valid if the power lost through heat transfer (λ 2 T) is negligible in comparison with the plastic power dissipated in heat ðb σ : " pþ, according to equations (1) and (2). In other words, considering the area of the specimen which is travelled by the PLC band between two successive recorded images, the quantity of thermal energy stored by the material in this area is larger than the one which is conducted in the rest of the specimen. The Fourier number, F o,isa dimensionless number usually used to characterize these typical transient conduction problems. Its accounts for the ratio of the heat conduction rate to the rate of thermal energy storage in the material. F o is defined by the subsequent equation: F o ¼ at=d 2 ð6þ Where α=λ/(ρcp)= m 2.s 1 is the thermal diffusivity of the alloy, τ is the characteristic time of the conduction and d is a characteristic length of the system. The thermodynamical evolution is usually considered as adiabatic if F o << 1. Considering the frequency of the DIC camera, τ has been set to the time step of image acquisition: τ=.2 s. The characteristic length of our problem is the distance travelled by a band during τ, which depends on the velocity of the bands. As already observed for these family of alloys in other works [1, 3, 5], the velocity of type A PLC bands decreases with increasing the applied strain. In order to compare the band motions for the two test rates V2 and V1, the distance d was determined for the two tests at the same applied strain, arbitrary set to ε.1, by measuring the displacement of a band between two successive images. The values of d, the corresponding band velocities and the resulting Fourier numbers (resulting from equation (6)) are summarized in Table 1. Thelowvalue of F o at V1 suggests that the deformation process is actually Table 1 Fourier number for the two test rates investigated and the parameters involved in its calculation adiabatic for these test conditions, which is consistent with the previous observations of ΔT infig.7. Conversely, a Fourier number above 1 for the test rate V2 invalidates the adiabatic hypothesis and thus, heat conduction seems to be not negligible. Besides, the discrepancy between the slopes of the numerical and experimental curves in Fig. 7 suggests that heat conduction accounts for approximately 66 % of the total plastic work that is converted to heat. Consequently, the thermomechanical investigation of PLC bands at V2 rate needs to consider the thermal conduction in the heat equation. This term is often ignored for sake of simplicity, regardless of the strain rate applied to the specimen. A part of the discrepancy for test V1 could also be attributed to a change in the value of the β coefficient. As mentioned earlier, β was assumed to be constant for both tests, while it may be affected by the large difference in the local strain rate. Chrysochoos and Louche [21] developed numerical models considering the different parts of the heat equation, including the conduction phenomenon, that reconstructs heat source maps from the measurements of 2D temperature fields. It would be interesting to apply such models to the case of local temperature burst and strain increase due to PLC effect in order to evaluate numerically the part of plastic work converted into heat conduction. Therefore, it seems that such an analysis of both temperature and strain fields with respect to time is a suitable method to check the adiabatic nature of a localized plastic band occurring in a metallic sheet without any further thermomechanical numerical simulation. Conclusion Characteristic time, τ (s) Characteristic length, d (mm) Band velocity at ε.1 (mm/s) Test rate V2 Test rate V Fourier number, F o By means of full-field measurements of both temperature and strain, the relationship between local strain jumps, Δε loc, and the associated temperature bursts, ΔT, that are caused by the motion of PLC bands in tensile specimens has been investigated for an Al-Mg alloy sheet. The same trend of these two parameters to increase linearly with respect to the applied strain has been observed for the two applied strain rates, suggesting a very low sensitivity to this last parameter. Moreover, a direct linear relationship between

8 132 Exp Mech (213) 53: ΔT and Δε loc based on the underlying thermo-mechanical phenomena has been used and validated for V1. The discrepancy observed between the theoretical and experimental results for V2 suggests that heat conduction is not negligible in this case, which is consistent with the high value of the Fourier number associated with these test conditions. Finally, the investigation of the time dependence of both temperature and strain fields at a single point in tensile specimens seems to be a promising method to check experimentally the adiabatic character of localized plastic deformation, such as PLC bands, that occurs in a metallic sheet without any further thermomechanical numerical simulation. Acknowledgments Authors are grateful to J. Costa for is help during the experimental part of the study. The Gleeble 35 machine at Université de Bretagne-Sud was co-financed by the European Regional Development Fund (ERDF). This work was funded by the Brittany Region (France), the Portuguese Foundation for Science and Technology via the project PTDC/EME-TME/1335/28 and by FEDER via the program FCOMP FEDER-131. References 1. Jiang H, Zhang Q, Chen X, Chen Z, Jiang Z, Wu X, Fan J (27) Three types of Portevin-Le Chatelier effects: experiment and modelling. Acta Mater 55(7): Casarotto L, Dierke H, Tutsch R, Neuhäuser H (29) On nucleation and propagation of PLC bands in an Al 3 Mg alloy. Mater Sci Eng A 527(1 2): Ait-Amokhtar H, Vacher P, Boudrahem S (26) Kinematics fields and spatial activity of Portevin-Le Chatelier bands using the digital image correlation method. Acta Mater 54(16): Louche H, Bouabdallah K, Vacher P, Coudert T, Balland P (28) Kinematic Fields and Acoustic Emission Observations Associated with the Portevin Le Châtelier Effect on an Al Mg Alloy. Exp Mech 48(6): Louche H, Vacher P, Arrieux R (25) Thermal observations associated with the Portevin-Le Châtelier effect in an Al-Mg alloy. Mater Sci Eng A 44(1 2): Ait-Amokhtar H, Fressengeas C (21) Crossover from continuous to discontinuous propagation in the Portevin-Le Chatelier effect. Acta Mater 58(4): Ranc N, Wagner D (25) Some aspects of Portevin-Le Chatelier plastic instabilities investigated by infrared pyrometry. Mater Sci Eng A 394(1 2): Benallal A, Berstad T, Borvik T, Hopperstad OS, Koutiri I, de Codes RN (28) An experimental and numerical investigation of the behaviour of AA583 aluminium alloy in presence of the Portevin-Le Chatelier effect. Int J Plast 24(1): doi:1.116/j.ijplas Halim H, Wilkinsona DS, Niewczas M (27) The Portevin-Le Chatelier (PLC) effect and shear band formation in an AA5754 alloy. Acta Mater 55(12): Picu RC, Vincze G, Ozturk F, Gracio JJ, Barlat F, Maniatty AM (25) Strain rate sensitivity of the commercial aluminum alloy AA5182. Mater Sci Eng A 39: Ait-Amokhtar H, Fressengeas C, Boudrahem S (28) The dynamics of Portevin-Le Chatelier bands in an Al-Mg alloy from infrared thermography. Mater Sci Eng A 488(1 2): Ait-Amokhtar H, Boudrahem S, Fressengeas C (26) Spatiotemporal aspects of jerky flow in Al-Mg alloys, in relation with the Mg content. Scr Mater 54(12): Wowk D, Pilkey K (29) Effect of prestrain with a path change on the strain rate sensitivity of AA5754 sheet. Mater Sci Eng A 52 (1 2): van den Boogaard AH (22) Thermally Enhanced Forming of Aluminium Sheet. Modelling and experiments. PhD Thesis, Twente University, Wageningen 15. Rusinek A, Rodríguez-Martínez JA (29) Thermo-viscoplastic constitutive relation for aluminium alloys, modeling of negative strain rate sensitivity and viscous drag effects. Mater Des 3 (1): Kang J, Wilkinson DS, Embury JD, Jain M, Beaudoin AJ (25) Effect of type-b Portevin-Le Chatelier bands on the onset of necking in uniaxial tension of strip cast AA5754 sheets. Scr Mater 53(5): Kang J, Wilkinson D, Jain M, Embury J, Beaudoin A, Kim S, Mishira R, Sachdev A (26) On the sequence of inhomogeneous deformation processes occurring during tensile deformation of strip cast AA5754. Acta Mater 54(1): doi:1.116/ j.actamat Cheng XM, Morris JG (2) The anisotropy of the portevin-le chatelier effect in aluminum alloys. Scr Mater 43(7): Hu Q, Zhang Q, Cao P, Fu S (212) Thermal analyses and simulations of the type A and type B Portevin Le Chatelier effects in an Al Mg alloy. Acta Mater 6(4): doi:1.116/ j.actamat Coër J, Bernard C, Laurent H, Andrade-Campos A, Thuillier S (211) The effect of temperature on anisotropy properties of an aluminium alloy. Exp Mech 51(7): Chrysochoos A, Louche H (2) An infrared image processing to analyse the calorific effects accompanying strain localisation. Int J Eng Sci 38(16): Rittel D (1999) On the conversion of plastic work to heat during high strain rate deformation of glassy polymers. Mech Mater 31 (2): Saai A, Louche H, Tabourot L, Chang HJ (21) Experimental and numerical study of the thermo-mechanical behavior of Al bi-crystal in tension using full field measurements and micromechanical modeling. Mech Mater 42(3):