Discussion of the Constitutive Relation on Cyclic Stress-Strain of Engineering Materials

D0I:10.13374/j.issn1001-053x.1989.06.036 第11卷第6期 北京科技大学学报 Vol.11 No.6 1989年11月 Journal of University of Science and Technology Beijing Nov.1989 Discussion of the Constitutive Relation on Cyclic Stress-Strain of Engineering Materials' Tang Junwu（唐俊武)，Wang Jianguo(王建国)， Xu Shiping(徐世平)，Wang Chen(王枨)~ ABSTRACT:Based on the experimental results and analysis of the cyclic deformation,there is an obvious yield stage on the cyclic stress-strain curve at the stage of small plastic deformation (in room or low temperatures).This phenomenon is similar to that of monotonic tensile curve case.But for the former the deformation amount at which the yield begins is much smaller than that for the latter.The cyelic stress-strain constitutive relation needs to be further studied according to the actual cyclic stress-strain curve.The conven- tional constitutive equation o=Ae"is based on the results only corresponding to the cyclic strengthening stage.It is not appropriate for the stage of the small plastic deformation and the stage of yield. KEY WORDS:cyclic stress strain,engineering material,plastic deformation The cyclic stress-strain relation and cyclic yield behaviour are very impor- tant for analyzing the stress-strain field on cyclic deformation.Many works with applicable results have been done by researchers in various countriest 1.21. But as being restricted by measuring accuracies,the cyclic stress-strain curve which formulate in a power relation o=Ae are obtained based on that the range of the total strain is beyond 0.2%and the cyclic yield strength ofis defined in the same way as the monotonic tensile yield does,that is,taking the stress corre- spendence to 0.2%cyclic plastic strain to be the cyclic yield strength gts1. Other researchers take the fatigue permanent limit o:as ot1.Our research wo- rks have been shown that for some kinds of engineering materials there is an ob- vious yield stage on the cyclic stress-strain curve at the stage of small plastic deformation (in room or low temperature).This phenomenon is similar to that of monotonic tensile curve case.But for the former the deformation amount at Manuscript Received June 17,1989 .Dept.of Mathematic and Mechanics 611

第 卷第 期 、 年 月 北 京 科 技 大 学 学 报 习 。 。 一 ’ ，二 。 唐 俊 式 ， 牙 王 建 国 ， 、 、 徐世 乎 ， 平 ， 夕 。 王 帐 ’ ， 一 。 一 主 一 一 口 。 ” · ， ， 一 手 产 一 一 · 以 ， 一 。 ” 一 七 ， ， 。 了 乙 工 〔 落 一 一 。 ， ， · 五 一 、 DOI ：10．13374／j ．issn1001－053x．1989．06．036

which the yield begins is much smaller than that for the latter.For some kinds of engineering materials we have been investlgated the whole process of cyclic toading (including 2<.3%).-The true cyclic.stress-strain relationshould:be represented properly in sectional equations according to the genuine stress-strain curves. 1 Experimental Investigation In the study of cyclic stress-strain relation and yield behaviour we need to construct the total cyclic stress-strain curves.This is the same as in that of monotonic tensile case.More significantly the hysteresis loop at the small plastic deformation stage while yield begins should also be drawn out.Our experiments are performed by using the MTS 809.30 electronic hydralic-servo materials testing system with a high accurate extensometer MTS 632.68c-01 (gage length:50mm), which can be assured to record stable hystersis loops with the strain ranged from 0.05%to 1.0%.The tests start from the begining of elastic deformatian,while the strains beyond 1.0%,the extensometer to :nother one with gage length of 25mm were changed.The materials to be tested are A3,45*and 40Cr stee's, which have different yield properties.Monotonic tensile,tensile-compressive and symmetric cyclic tests are performed in the room and low temperatures.In the cyclic tests the methed of strain control with gradu:I in:-ements of strain ampli- tude on single specimen is adopted.The results of three kinds of materials in room temperature are shown in Fig.1,2 and Fig.3.The curves of monotonic tensile, cyclic and cyclicing bilogarithm coordinates are included in the figures.The test points with two kinds of gage length (50mm and 25mm)are smoothly connected to be formed in a single continuous curve.Which shows that the testing system possesses a good stability.The results for A3 and 45#steel in low temperature (-30°Cand-50°C)are shown in Fig.4. In order to check the reliability of the test results,a method of inserting points of multi-specimen tests with stress controlled in room temperature is adopted. The results are shown in Fig.5.Fig.5(a)shows the hysteresis loop at the stage before the yield,that is,the stage of elastic deformation.In Fig-5(b)the magnitude of cyclic stress is kept at the same amount,but the cyclic strain increases along with the number of cyclic times does till the end of the yield sta- ge.This is obviously a typical physical yielding other than so-called cyclic soften- ing.In Fig-5(c)the stress is controlled at a level beyoned the yield,the hysteresis loop behaves as a cyclic softening.The strain increases somewhat,but does not so obviously as that in Fig.5(b).The mumLers of cyclic times of the three cases are the same,that is,150 times. 612

卜 。 拓 月 叮 沉 时 叮 卫 。 。 几 芦 矛 二 斗 。 二 叮 ‘ ， “ “ 直 ’ 一 杠 ‘ 一 了 一 一 一 ， 一 一 卜 ， 。 · ， 一 ， 、 卜 ， ， ， 一 一 恤 一 ， 。 ， 。 · 、 莽 一 一 。 。 ， 一 。 · · ， ， · · 杏 ， 。 ， 、 一 一 · ， 。 ， · 协 七。 万 卜 ， ， 尸 月， 口 尸

l. Cyelie 2.50 G.MPa 3C叶 MPa 200 2.7 600 Cyelie d 2.40 (BdW/p)3o1 100 400 2.6 Monotonic 2.30 200 0.000.0040.008 2.5升 0.0 0.0080.016 0,024 .20 2. -0.2 0.0 0.20.40.60.8 0.00.20.40.60.81.0. (lge)-3 (loge)-3 Fig,1 Stress-strain curve of A3 Fig,2 Stress-strain curve of 45 steel steel 32 Cyelie -50℃ -30℃ 600b ionotonie 600 4 3.0 5 P (BdW/p)3o1 400 、-Room temp. -50℃ c -30℃ 2.8 b Room temp, 2.6 200 -A3 steel 0.00.01D.020.03 .,t…45#steel 1 2.4 .1 0.5 1.0 1,5 (1oge)-3 0.0000.0050.0100.0150.020. 0 Fig.3 Stress-strain curve of 40Cr Fig.4 Stress-strain curve of A3 and 45* steel steel at low temperature 00 150 6* (a) id 00.10.2 E% Fig.5 Hysteresis loop in different stress conditions 2 Discussion The testing results in Fig.1,2,3,and Fig.4 show that for A3 steel there is a relatively longer cyclic yield stage,for 45 steel the stage is shorter,but for 40Cr there is none.It means that for those materials which have the typical physical yield behaviour under the monotonic tensile conditions,there is also an obvious physical yield behaviour under the cyclic deformation conditions.And the stress and strain of the cyclic yield are lower than those of the monotonic 613

、 亡 一。 。 ︵乙艺工己锣 。 ，、 佗 。 。 。 〕 一 一 ‘ 。 一 一 井 一 乙匕内 门曰 自 艺国 厂乡 ， 丫 一 ，了 一 一 ℃ 阴 八﹄八， 门八曰 足匕乃 一 一 杏。 笠， 二 勺山勺泪 曰 ︵芝山口宜工抽。 乙下， 一 匕 。 一 ， 。 ‘ ，已 ‘ 口 。 一一 。 赞 井 月 ￡ 一 弓 一 。 一 希 。 五 ， ， ， 五 ， 」 丁 主 ， ， 五 以 了

tensile yield.For example,the monotonic yield point of A3 steel is at e=0.3% (or ep=0.2%).However,the cyclic yield of A3 steel is at e=0.15%(or ee= 0.1%).Therefore,if the stress corresponding to ep=0.2%were defined to be the cyclic yield stresso similar to the previous definition of conditional yield stress,it will be lack of experimental basis.For those materials with no obvious cyclic physical yield phenomena,our opinion is that to define the stress correspon- ding to ep=0.1%as the cyclic yield stresso will be more reasonable. The another important conclution from our work is that the conventional constitutive equation o=Ae"is not reasonable for all the cases.Particularly,it is not appropriate for the stress-strain relation at the stage of small plastic defor- mation while yield begins.It is well known that when the load bearing engine- ering construction partially entered the plastic conditions,generally there would be a small plastic deformation.Fatigue defections also begin with plastic defor- mations,and small plastic deformations start first.That means,to do research work of small plastic deformations is very important for the related mechanical analysis and computations.So that to study the stress-strain constitutive rela- tion under the condition of cyclic deformations,we should start our work from the small plastic deformations and then choose the suitable relations respectively corresponding to the different sections of the cyclic stress-strain curves.It should include the relation at the stage of small plastic deformations (ab sections in Fig. 1,2 and 3),the relation at the stage of ideal plastic with obvious yield behaviour (bc sections in Fig.1,2 and 3)and the relation at the stage of cyclic strengthe- ning (cd sections in Fig.1,2 and 3).For example,the sections corresponding to the cyclic stress-strain curve in Fig.2 for 45*steel are: ab: 01=15100e10.s be: ideal plastic relation cd: more strictly it should be divided into two parts: 0,=202.3e9305 0s=152,6e0.340 Here is our conclusion:we should choose the suitabledifferent equations indi- vidually corresponding to different sections of the testing curve.Only in this way we can totally and precisely describe the constitutive relation of the whole stress-strain curve. 3 Conclusions The engineering materials that have typical monotonic tensile yield stage generally have obviously physical cyclic yield behaviour.Therefore,the determi- 614

七 了 ， 。 主。 。 。 。 一 ， 二 。 。 。 ， 。 ， 。 若 ， 一 ， 。 若 · 。 “ ， 一 一 ， 一 ， 。 ， 一 ， 了 一 一 ， ， ， 一 ， 一 ， 一 一 ， 口 二 君 二 口 二 。 且 · 。 ￡ · … “ ， 一 。 了 ， 血 ， 犷 ， 爪 又

nation of their cyclic yield strength could be made out directly from experiments. For the other materials,similar to the definition of conditional yield,taking the stress corresponding to 0.1%cyclic plastic strain to be defined as the cyclic yield strength will be more reasonable. The conventional constitutive equation o=Aea could not precisely describe the true cyclic stress-strain constitutive relations of the materials.Particularly, it is not suitable for those regulation for small plastic deformations of materials. However,studies of small plastic deformations are singnificant for relative mechanical analysis and computations. The cyclic stress-strain constitutive relation for some kinds of materials should be formulated in more properer equations which can be suitable to the different deformation stages corresponding to the different sections of the total curve that describes the whole process of stress-strain tesis. REFERENCES 1 Sandor B I.Fundamentals of Cyclic Stress and Strain,The University of Wisconsic Press,Wisconsin,1972,17 2 Xie Q.J.Mechanics Strength,1980;(11):75 3 Landgraf R W.ASTM,1969;467:3 4 Felther C E,Beardmore P.ASTM,1969:467:77 5 Xu H.Design of Fatigue Strength,Mechanics Industry Press,1981;17 615

、 效 士宝 五 。 。 了 二 ‘ 七 · ， ， 二 了 、 ·仙 口 君 ” 一 · ， 主 扭 ， 扭 丁 至 立 · 一 任 全 主 一 · · 也 ， ， ， ， 。 。 ， 一 ， ， ， · ， ， ， ， 声 气