The analysis of stress and strain concentrations of constructional materials subjected to bending and torsion is presented. The known methods of stress determination for any elastic-plastic material assume the linear strain concentration under bending and torsion. Determination of stress and strain concentrations and their presentation as graphs were the main aims of this paper. Five materials were tested. They had different cyclic strain curves, and were loaded by the torsional and bending moments. The results proved a linear character of the strain distribution for any material in the all applied loading range.Визначено розклад напружень і деформацій в перетині проаналізованого елемента вальців та подано графічно за допомогою комп’ютерної моделі. Аналіз п’яти матеріалів та їх кривих циклічного деформування з різними характеристиками за умов кручення та згину показав лінійний характер розкладу деформацій для кожного з них за всіх прикладених навантажень.Определено распределение напряжений и деформаций в сечении проанализированного элемента валков и представлено графически с помощью компьютерной модели. Анализ пяти материалов и их кривых циклического деформирования с разными характеристиками при кручении и изгибе показал линейный характер распределения деформаций для каждого из них при всех приложенных нагрузках

Kurek, M.

Warmuzek, P.

Łagoda, T.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

 108Ô³çèêî-õ³ì³÷íà ìåõàí³êà ìàòåð³àë³â. – 2011. – ¹ 4. – Physicochemical Mechanics of Materials  DETERMINATION OF STRESS AND STRAIN CONCENTRATIONS  IN THE ELASTIC-PLASTIC MATERIALS UNDER BENDING  AND TORSION M. KUREK, T. ŁAGODA, P. WARMUZEK Opole University of Technology, Poland  The analysis of stress and strain concentrations of constructional materials subjected to bending and torsion is presented. The known methods of stress determination for any elastic-plastic material assume the linear strain concentration under bending and torsion. Determination of stress and strain concentrations and their presentation as graphs were the main aims of this paper. Five materials were tested. They had different cyclic strain curves, and were loaded by the torsional and bending moments. The results proved a linear character of the strain distribution for any material in the all applied loading range. Keywords: stress, strain, bending, torsion, cyclic loadings. A precise determination of the fatigue life requires the determination of stress and strain distribution in the specimen section, or in the element in the regions of elastic-plas-tic strains. Thus, we obtain the stresses on the surface and distribution of stresses and strains used for the non-local models which more and more often occur at present [1, 3]. In literature the present state-of-the-art, the character of the elastic-plastic strains distri-bution under bending and torsion are not explained. The known methods of stress de-termination for elastic-plastic materials assume the linear strain distribution under ben-ding and torsion [7, 8]. In [5] it was found that the presented models of determination of stresses and strains from the known elastic stresses using the methods proposed by Neuber [10], Molski–Glinka [9], and Łagoda–Macha [6] give incorrect results for bending.  Occurrence of stress and strain gradients causes serious difficulties in determina-tion of the fatigue life under bending, torsion and combined bending and torsion [4]. Precise determination of stress and strain distribution requires numerical calculations (for example FEM – the finite element method), so the approximate analytic equations are often applied. According to [5], during determination of local stresses and strains under bending, the normal stresses and strains occurring in both elastic and elastic-plastic models must compensate the given bending moment, i.e.   ( , )gSM x y xdS= σ∫ . (1) Under torsion, like under bending, the basic relationship which must be satisfied says that the shear strain occurring in both elastic and elastic-plastic models, should compensate the given torsional model, i.e.   ( , )sSM x y xdS= τ∫ . (2) In this paper, FEM was applied for determination of stress and strain distribution in the specimens with different material characteristics under loading with torsional and bending moments. The results were applied for evaluation of the range of linear strain  Corresponding author: M. KUREK, e-mail: m.kurek@doktorant.po.edu.pl  109distributions under bending and torsion versus loading, and material character as well as the influence of the material properties on the stress distributions.  The used materials. Five materials with different cyclic strain curves were analyzed. The following parameters were taken into account while selecting the material: E (Young’s modulus), K′ (coef-ficient of cyclic strength) and n′ (expo-nent of cyclic hardening). Such selection of the materials allowed to determine the character of stress and strain courses for many materials of a similar characteristic σa – εa. Table presents properties of the con-sidered materials. The calculations were done for a circular section bar (Fig. 1) of the radius r = 5 mm and length l = 50 mm, and were realized with the program FEMAP with NX Nastran 9.3.1. The considered bar was loaded by the torsional moment, bending moment or combination of these moments. The applied loading caused in the external layers the stresses close to the maximum ones at which the specimens were tested. Under bending with torsion, the loading was divided into two equal bending and tor-sional moments. Numerical calculations of stresses and strains in the elastic-plastic range. The cal-culations were done for each material under bending, torsion and combined bending with torsion. The data in graphs were read out from the selected group of elements placed in one axis in the XZ plane. In these elements, under the bending and torsion testing a full range of the stress and strain history was observed in the specimen section. Strain and stress distributions for Ck45 steel. The maximum torsional moment applied during calculations was Ms = 120 N·m. For this moment, the maximum shear stresses in the external layers of the specimen are τmax= 330 MPa. The maximum shear strains are γmax = 0.027. The successive increments of the torsional moment show that the distribution of stresses τ is linear only for the moment equal to 12 N·m; in the case of further increments the curvature of the stress distribution can be observed. The maxi-mum shear strains have the linear distribution for each value of the moment Ms . Distri-butions of stresses and strains for pure torsion are shown in Fig. 2. The maximum ben-ding moment for which the calculations have been done, equals Mg = 90 N·m. For such a moment, the normal stresses are σz = 568 MPa, and corresponding strains εz = 0.022 correspond to the values of the cyclic hardening curve of the material. Under bending, the normal stresses are equal to the normal stresses in the Z axis. Linear stress distribu-tion occurs only in the case of low bending moments, for Mg =18 N·m we can observe a slow curving of the stress histories. As the moment increases, the curvature takes form of a part of the cyclic strain curve for the values of the maximum normal stresses. The normal strain distribution in the Z axis keeps a linear character for each loading value. Distributions of stresses and strains for pure bending are shown in Fig. 3.  Fig. 1. Isometric view of the FEM model  of a cylindrical specimen. Properties of the considered materials [2] Material E, GPa ν K′, MPa n′ Ck45 201 0.28 1414 0.231 AlCuMg1 72 0.32 617 0.066 X2CrNiMo1810 220 0.3 1977 0.335 SUS 304-HP 143 0.3 432 0.153 CuNiCr35 170 0.33 440 0.002  110 Fig. 2. Distribution of maximum shear stresses τ (a) and strains γ (b). The legend presents values of the torsional moment ( – 120 N·m;  – 96; S – 72; ± – 48; ¾ – 24; z – 12 N·m) corresponding to each curve (Ck45).   Fig. 3. Distribution of normal stresses σ (a) and strains εz (b) of the Z axis. The legend presents values of the bending moment ( – 90 N·m;  – 72; S – 54; ± – 36; ¾ – 18; z – 9 N·m) corresponding to each curve (Ck45). Distributions of stresses and strains for AlCuMg1 aluminium alloy. The maxi-mum torsional moment applied for calculations is Ms = 100 N·m. For this moment, the maximum shear stresses in the external layers of the specimen are τmax = 283 MPa, and maximum circumferential shear strains are γmax= 0.0297. The successive increments of the torsional moment show that distribution of stresses τ is linear only for the moment 40 N·m, further increments cause the curvature of the stress distribution (for the mo-ment Ms = 50 N·m curvature of the distribution is observed in the external fibbers). The maximum shear strains have a linear distribution for each Ms. Distributions of stresses and strains for pure torsion are shown in Fig. 4.   Fig. 4. Distribution of maximum shear stresses τ (a) and strains γ (b). The legend shows values of the torsional moment ( – 100 N·m;  – 80; S – 70; ± – 60; ¾ – 50; z – 40 N·m) corresponding to each curve (AlCuMg1). The maximum bending moment for which the calculations were done was Mg =  = 90 N·m and the normal stresses were σz = 472 MPa, the appropriate strains εz = 0.025 correspond to the values of the cyclic hardening curve for the material. The linear stress distribution occurs for the bending moment below Mg = 36 N·m, in the case of successive loading increments Mg = 45 N·m we can observe a slow curving of the  111stress histories. As the moment rises, it takes a shape of the part of the cyclic hardening curve up to the value of the maximum normal stresses. Distribution of the normal strains in the Z axis keeps a linear character for each loading value. Distributions of stresses and strains for pure bending are shown in Fig. 5.  Fig. 5. Distribution of normal stresses σ (a) and strains εz, (b). The legend shows values  of the bending moment ( – 90 N·m;  – 81; S – 72; ± – 63; ¾ – 45; z – 36 N·m) corresponding to each curve (AlCuMg1). Distributions of stresses and strains for X2CrNiMo1810. The maximum torsio-nal moment applied in calculations is Ms = 90 N·m. For this moment the maximum shear stresses in the specimen external layers are τmax= 275 MPa, while the maximum circumferential shear strains are γmax = 0.022. The successive increments of the torsio-nal moment show that the distribution of stresses τ is linear only for small torsional moments, above Ms = 9 N·m there is the distribution of curves and takes a form of the cyclic strain curve. The maximum shear strains are linear for each value of the moment Ms. Distributions of stresses and strains for pure torsion are presented in Fig. 6.  Fig. 6. Distribution of maximum stear stresses τ (a) and strains γ (b). The legend contains values of the torsional moment ( – 90 N·m;  – 72; S – 54; ± – 36; ¾ – 18; z – 9 N·m) corresponding to each curve (X2CrNiMo1810). The maximum bending moment Mg = 60 N·m was calculated. For this moment, the normal stresses σz = 428 MPa, and the appropriate strains εz = 0.0124 correspond to the values of the cyclic strain curves for the material. The linear distribution of stresses occurs for the bending moment below Mg= 12 N·m, for a successive loading increment Mg = 18 N·m we can observe a slow curving of the stress histories. As the moment rises, the stress course takes a shape of the part of the cyclic strain curve up to the values of the maximum normal loadings. Distribution of normal strains in the Z axis keeps the linear character for each loading value. Distributions of stresses and strains for pure bending are presented in Fig. 7. Distributions of stresses and strains for SUS 304-HP. The maximum torsional moment applied in the calculations was Ms = 50 N·m. In such a case, the maximum shear stresses in the external layers of the specimen were τmax= 140 MPa.  112 Fig. 7. Distribution of normal stresses σ (a) and  strains εz (b). The legend contains values  of the bending moment ( – 60 N·m;  – 48; S – 36; ± – 24; ¾ – 18; z – 12 N·m) corresponding to each curve (X2CrNiMo1810). The maximum circumferential shear strains were γmax = 0.027. The successive increments of the torsional moment show that the distribution of stresses τ is linear only for small torsional moments. Above Ms = 10 N·m we can observe the curvature of the distribution with a shape of the cyclic strain curve. The maximum shear strains have a linear distribution for each value of the moment Ms. Distributions of stresses and strains for pure torsion are shown in Fig. 8.  Fig. 8. Distribution of maximum shear stresses τ (a) and strains γ (b). The legend gives values  of the torsional moment ( – 50 N·m;  – 40; S – 30; ± – 20; ¾ – 15; z – 10 N·m) corresponding to each curve (SUS 304-HP ).  Fig. 9. Distribution of normal stresses σ (a) and strains εz (b). The legend presents values  of the bending moment ( – 35 N·m;  – 28; S – 21; ± – 14; ¾ – 10.5; z – 7 N·m) corresponding to each curve (SUS 304-HP). The maximum bending moment was Mg = 35 N·m, for this moment, the normal stresses were σz = 222 MPa, and the appropriate strains εz = 0.015 correspond to the values of the cyclic strain curve for the material. Linear distribution of stresses occurs for the bending moment below Mg = 7 N·m; for a successive loading increment Mg =  = 10.5 N·m we can observe a slow curving of the stress history. As the moment rises, it takes the form of a part of the cyclic strain curve for the maximum normal stresses. Di- 113stribution of normal strains in the Z axis is linear for each loading value. Distributions of stresses and strains for pure bending are presented in Fig. 9. Distributions of stresses and strains for CuNiCr35. The maximum torsional mo-ment was Ms = 75 N·m. For this moment, the maximum shear stresses in the external layers of the specimens were τmax = 266 MPa. The maximum circumferential shear strains were γmax = 0.014. Character of the cyclic strain curve for CuNiCr35 is similar to that of an elastic-perfect-plastic body. As the torsional moment rises, the stresses and strains linearly increase to the value τmax = 264 MPa next we can observe rapid breaks of the stress curves and further loading increase is accompanied by an insignificant increase of stresses. The circumferential shear strains keep their linear and constant history for any torsional moment Ms. Distributions of stresses and strains for pure tor-sion are shown in Fig. 10.  Fig. 10. Distribution of maximum shear stresses τ (a) and strains γ (b). The legend contains values of the torsional moment ( – 75 N·m;  – 67.5; S – 60; ± – 52.5;  ¾ – 45; z – 37.5 N·m) corresponding to each curve (CuNiCr35). The maximum bending moment in the calculations was Mg = 75 N·m. In this case, the normal stresses were σz = 436 MPa, and the appropriate strains εz = 0.017 corres-pond to the values of the cyclic strain curve for the material. Like in the case of torsion, as the bending moment rises, we can observe the stress increase to σz = 434 MPa. The further loading increase causes a rapid break of the stress history. From that moment, the increase is minimum in relation to the increase of the bending moment. Distribution of normal strains of the Z axis εz and the maximum shear strains γ, is linearly indepen-dent of the loading value. Distributions of stresses and strains for pure bending are shown in Fig. 11.  Fig. 11. Distribution of normal stresses σ (a) and strains εz (b). The legend contains values  of the bending moment ( – 75 N·m;  – 67.5; S – 60; ± – 52.5; ¾ – 45; z – 37.5 N·m) corresponding to each curve (CuNiCr35). CONCLUSIONS  The paper presents the analysis of the stress and strain distribution in different ma-terials and for different types of loading: pure bending, pure torsion and combined pro- 114portional bending with torsion. The following conclusions can be drawn from the re-view of the obtained results: The linear strain distribution was observed for each considered material and each loading. A diversified characteristics of the cyclic strain curve for each material allows as to think that that the strain distribution is linear for any tested material. The stress distribution has the course conforming with the cyclic strain curves applied in the materials analysis; for small loadings the stress distribution corresponds to the linear part of the curve σa – εa, when loading increases, the stress distribution takes a shape corresponding to the further part of the curve σa – εa. The non-linear increment of the stresses and strains versus loading can be seen. Each uniform loading increment causes the decrease of stresses and strains increment. The conclusions allow us to verify the known analytical methods of stress determination under pure bending, pure torsion and combined bending and torsion. РЕЗЮМЕ. Визначено розклад напружень і деформацій в перетині проаналізованого елемента вальців та подано графічно за допомогою комп’ютерної моделі. 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Determination of stress and strain concentrations in the elastic-plastic materials under bending and torsion

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Determination of stress and strain concentrations in the elastic-plastic materials under bending and torsion

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Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)