弹塑性变形中的弹性效应_英文_

弹塑性变形中的弹性效应_英文_ Article ID: 1007- 7294( 2007) 03- 0415- 12 Elastic Defor mation in Yield Zones for the Elastic- Plastic Plane Str ain Pr oblems 12TIAN Chang- lu , NA Ri- su (1 Dept. of Mechanics, Southern Yangze University, Wuxi 214122, China; 2 Dept. of Mechanics, Inner Mongolia Polytechnic University, Hohhot 010062, China) Abstr act: In the solution procedures of elastic - plastic problems the elastic deformation in yield zones was generally omitted due to the mathematical difficulty,and some simplified constitutive e- quations such as the incompressible model are widely employed in the classical plasticity. The anal- ogy results derived in this work are compared with those of the common elastic- plastic methods, and the merit of the analogy results and the improvements in some respects are presented. Based on the basic stress element,the effects of elasticity in different stress states of non - linear elastic - plastic problems are evaluated in detail. Key wor ds: elastic effect; elastic- plastic defor mation; constitutive equations CLC number : TB301 U661.4 Document code: A 1 Intr oduction The deformation theory and the incremental theory are the two applicable hypotheses in plasticity. However, it could be very difficulty to obtain analytical solutions for the general e- lastic- plastic problems if the volume deformation related to hydrostatic stress were not omit- ted. One common way in the classical plastic theory is to employ simplified constitutive equa- tions of the incompressible model. Because of its mathematical difficulty, the most analytical solutions in the elastic - plastic analyses are still limited to plane stress problems.For plane !"!"++!!!!strain problems,it is generally assumed of !/ =0.5 in yield zones,and !/ 1122 1122 33 33 = in elastic zones (where is out- of- plane stress, , are in- plane stresses, and is!!!!! 33 11 22 Poisson’s ratio).The metal experiments have also proved that the volume deformation is only related to elasticity,and the incompressible model omits the elastic deformation in the yield zones. However, the incompressible solutions are the limit cases, and it is applicable only on the condition that mean stresses are comparatively very low.For the plane strain problems, the fields with comparatively high mean stresses occur frequently, and the hydrostatic stresses are important to mechanism of material failure process. Another common way to solve the elastic- plastic problems is the asymptotic method. But the high- order term asymptotic results are essentially the proximate solutions,and many terms [1] have to be employed to ensure the precision. Our recent work reported that the assumption of separable functions employed in the asymptotic solutions could lead to the unconvincing Received date: 2006- 06- 13 Biography: TIAN Chang- lu(1963- ), male, Ph. D., professor of Southern Yangze University, E- mail address: cltian197@sina.com. results. An alternative way of analogy method was proposed to solve the elastic- plastic crack- tip fields in reference[1], and the details of asymptotic results in elastic- plastic crack tip so- lutions have been discussed.It is reported that the elastic deformation developed in the yield zones of crack- tip fields is important,and the asymptotic solutions are not reasonable in many elastic - cases.The present work focuses on evaluating the effects of elasticity in the general plastic problems of different stress states, and the analogy results are compared with the com- mon elastic- plastic results. 2 The analogy elastic- plastic equations [1] For convenience, the basic equations documented in our previous work related to the present subject of interest are repeated in this section.The elastic - plastic problems under plane strain conditions are considered,and it is limited to the deformation theory.For simplifi- cation,the bi - linear elastic - plastic model is studied as shown in Fig.1.!is the initial yield stress and "=!/E is 0 00 the corresponding strain,Edenotes the tangent modulus of p elastic- plastic stress- strain curve (i.e., the slope of the line for !>!), respectively. 0 Fig.1 Elastic- plastic stress- For the deformation theory, the stress- strain relation in strain relation multiaxial stress state is in the form of 1 1 - 1 3 1+# 1- 2$ - #(1) +%+ SS!"="!" ! 1- !i j kki j i j i j e E E E 3E 2p where $ is Poisson’s ratio,the strain components "are related to the deviation stress compo- i j #nents Sof stress tensor , S=- %. And =/, all parameters with overbars are!!!!!! ee 0 i ji j i j i j mi j !dimensionless ones throughout this paper. The effective stress is in the form of e 1 2 2 2 2 1 2 != ?(2) "%- &!!!!"!"- - ! !!!!"+ + +6!e33 1 1 22 22 33 12 11 2$ The first two terms of Eq.(1) are related to elasticity, and the linear elastic stress- strain relation is 1+$ 1- 2$ (3) +S"=!% i j i j kki jE 3E The deformation theory in plasticity could be considered as the non- linear elastic theory without unloading,the stress - strain relation of elastic portion in the constitution of Eq. (1)is linear, while the relation of plastic portion is non- linear.So it is very difficulty to find a stress function that could satisfy the compatibility equation for elastic- plastic (i.e.non- linear elastic) problems if both elastic and plastic strains are considered,especially in triaxial stress state. The plane strain problems are actually in triaxial stress state,and the constitution for power hardening materials is more complex than the bi- linear one of Eq.(1). As a result, the consti- tutive equation has generally been simplified so that the many reported analytical solutions are obtained. There are several well - known simplified hypotheses, such as the incompress - !"+!!ible assumption of ! / =0.5,and the full plastic assumption that further omits first 11 22 33 two elastic part of Eq.(1),or employs equation of =/??.These assumptions are wide- !!!ee 0 ly employed in traditional elastic- plastic analyses (such as in the plasticity textbooks) as well [2- 6]as in the crack- tip analyses. [1]As an alternative way to make further research on the elastic effects, Tian and Gao rewrote Eq.(1) as 1+#1- 2$ epep(4) +S"=!% i j i j kki jE 3E ep ep Eq.(4) has the similar form of linear elastic relation as Eq.(3).The elastic- plastic parameter $ epis determined by the analogy requirement, - 1 ! " E 1- ! e- 1 $+ !" E 2 P (5) $= ep - 1 E - 1 1+ !"!" 1- !Ee P and Eis not an independent variable, ep !"1- 2$ ep E=E? (6) ep!" 1- 2$ For plane strain problem ("=0), Eq.(4) leads to z ! 33$= (7) ep !" !+! 11 22 It is noted that $was introduced and defined by the analogy at the beginning,but Eq.(7) ep indicates that $also represents the ratio of out- of- plane to in- plane stresses, so it could ep be named as elastic - plastic Poisson’s ratio.The values of $ is not only dependent on the ep materials constants $ and E/E, but also dependent on the normalized effective stress. So $ P epis a stress- dependent elastic- plastic parameter, its value changes from $ to 0.5, and is in the range of E 1 - 1 %!+ !"& E 2 P$?$<$=(8) ep max E E P The values of $represent the ratio of elastic to plastic deformation, and it also provides ep a convenient way to analyze the deformation combined by elastic and plastic strains. 3 The str ess and str ain expr essions in elastic- plastic str ess states A typical problem, a homogeneous stress field or a plane strain element in biaxial tension, is shown in Fig.2. The stress components in linear elastic condition (i.e. = / <1.0) for!!! e e 0 this element is # q = !11 % !0 % % q =k (9) !$!" 22 ! 0 % % q %"!=" !k+1 !" 33 ! & 0 (a) Homogeneous stress field (b) Elastic element (c) Elastic- plastic element Fig.2 Typical stress states under plane strain condition with =0.Because of the analogy between linear elastic Eq.(3) and elastic- plastic Eq.(4), we! 12 suppose that there would be the analogy stress field in elastic- plastic condition (i.e. =/!!! ee 0 ?1.0), that is # q != 11 % !0 % % q (10) =k !$!" 22 ! 0 % % q %"!!="k+1 "! 33 ep !0 & here k=/(by assuming ?) is constant and its different values present different!!!! 22 11 11 22 stress states for the biaxial elements (or different boundary conditions of the homogeneous stress in Fig.2(a)). Generally, "in Eq.(4) is a stress- dependent elastic- plastic parameter. For this simple ep problem, it actually presents and element in biaxial tension or a uniform stress field. And !, 11 as well as and are all constants, so the values of "defined by Eq.(5) are!!!, ! 33 e ep 22 12 constants in the stress fields. Therefore, the constitutive Eq.(4) of the elastic- plastic (or non- linear elastic) problem is the same as that of the linear elastic Eq.(3) except that the meaning and values of "and Eare different. As a result, the compatibility equation can be exactly ep ep satisfied just as it is in the linear elastic condition,and therefore the analogy stress result of Eq.(10) for this uniform field is just exact solutions. 3.1 The str esses and str ains of two simplified elastic- plastic models Some common simplified cases of elastic - plastic problems referred in literature are reviewed firstly.With the assumption of !=!/!??,the elastic- plastic strain Eq.(1) is sim- ee 0 plified as 1+ 1 1 " 1- 2 3 "- (11) #=S+!$+ "S! i j i j kki j i j E E E 3E 2p This case can be named as“deep plastic model”. The first two terms of Eq.(11) are the elastic portion and the third term is the plastic portion. Eqs.(5,6) for this case become E 1- 1 #+ "$!! E 2 P(12) =! ep E E P and (13) E=E epp “”become material constants. For thedeep plastic model, both the moduli of Eand ! ep ep So the compatibility equation can be exactly satisfied and the analogy results even for the of Eq.(12) into Eq.(10), the non- homogenous stress fields are exact solutions.By submitting ! ep expressions of stress can be derived, % q = "11 ’ "0 ’ ’ q =k "22 ’ ’"0 (14) & % ) E ’ 1 - 1 ?!+ !" ’ ’’ ’ E 2 P q !"’*k+1 "=& 33 E " 0 ’ ’ ’E P( ( + By submitting stresses of Eq.(14) into Eq.(11), we have % % ) E - 1 ,!"- ! k+1 ?!+ /2 #!"-’’ e ’’E P q ’ 1- !k-? *#=& 11 E " ’ 0’’ ’ ’E P ( +’ % ) E ’- 1 #!"- ! k+1 ?!+ /2 #!"- e ’’’E P q k- !- ?#=(15) * && 22 E "’ ’ 0 ’’E ’P( + ’ . 1 E ’- 1 "! 1- 2 !!" e 2/E’ P k+1 q 2 / ??#= ’33 E "2 0 2’ / E P 30 ( %. 471 1- 2! p 1- E 3 q 582’/- 1 !" E #= ?1- k+1 !"11 582/E ’ 3 " !2p " 0E P 0693 ’ ’ .471 1- 2 ! p 1- E ’ 3 q 5 82 /- 1 ’ !" E ?k- k+1 #= !" 22 / 5 82 E 3 !" "& 2(16) p 0E 0693P ’ ’ 1.E - 1 !" 1- 2! "! ’2 / pE P k+1 q ’2 / ??#=- 33 E ’ 2 "2 0’ / 2 E P( 0 3 Moreover, if the first two elastic terms in Eq.(11) are omitted, the constitutive equation is further simplified as 1 1 3 - S= (17) !!" i ji j EE 2p eIn this case, elastic strains are zero, i.e. !=0; and plastic strains are the total strain, i.e. i j p=. Following the analogy procedure reported in reference[1], we rewrite Eq.(17) in form!! i j i j of Eq.(4), and requirement of this analogy becomes (18) "=0.5 ep 1 (19) E= ep 1 1 - " !E E P It is noted that Eq.(18) and Eq.(19) cannot be directly derived from Eq.(5) and Eq.(6). The two moduli of and are also the material constants for the present case.By E " ep ep submitting stresses of Eq.(19) into Eq.(10), the stress expressions can be derived. # q #= 11 % #0 % % q #=k (20) $!" 22 # 0 % % q 1 !"%#= k+1 !" 33 # 2& 0 By submitting stresses of Eq.(20) into Eq.(17), the strain expression is p # E 3 1- k q - 1 ! = ?" !% 11 E 22 #p 0% % p E 3 1- k q (21) $- 1 ! =- ?!" 22 E% 2 2 #p 0% p% ! =0 &33 Eqs.(14,15,16) are for the“deep plastic model”, and Eqs.(20,21) are for the incompressible model, these two simplified models are widely employed in the elastic- plastic analyses. 3.2 The str esses and str ains of gener al elastic- plastic pr oblems in var ious str ess states The general cases are to be discussed in this section. By substituting Eq.(10) into Eq.(2), the expression of effective stress #is e 2 2 q 2 !"! "(22) #= k - k+1- k+1 " !"- " eep ep ’ # 0 For plane strain problems,a general relation between #and "of Eq.(5)has already been ob- e ep tained, and it can be rewritten as E - 1 " !0.5- " "! ep E E p,"(23) #=#=?1.0 !ep " ee E E p - 1 !0.5- ""+- !""" ep ep E p By combining these two !~"relations leads to epe E - 1 "!0.5- !!" ep 2 2 2qE p!"!"k - k+1- =?1.0 (24a) " k+1 ! - ! !ep ep # " E 0- 1 "!0.5- !+!- ! !" ep ep E p so the expression for the problem showed in Fig.2 is obtained, and it can be expressed as! ep E E ! , ",k, , ! ",", (24b) !=!!"or !=!! "11 11 22 epep epep E Ep p Eq.(24)gives a general relation for different stress states.Eq.(24)means that !as well as out- ep plane stresses ,can be expressed by in - plane stresses and before solutions are """"33 e 11 22 obtained.So some application of Eq.(24) in solving elastic- plastic plane strain problems is ex- pected.For the problem showed in Fig.2,the value of !can be determined for different values ep k separately. It is noted that Eq.(10), Eq.(23) and Eq.(24) are applicable only for the case of high load level of =/?1.0,i.e. they are only applicable for the problems in yield condition; while""" e e0 Eq.(9) is applicable only for the case of low load level of "="/"<1.0, i.e. it is only applica- ee 0 ble for the problems in linear elastic condition. As mentioned above,the compatibility equation can be exactly satisfied for this simple problem, so the strain components can be determined directly. With the determined values of by Eq.(24), the elastic, plastic strain fields are obtained by substituting Eq.(10) into Eq.(1), !ep they are e ’ q !"%&1- !k- ! k+1 ! # = ? ep ) 11 "0 ) e ) q !"%& k- !- ! k+1 !?# = (25a) (ep 22 " 0 ) ) e q ""!!!- ! # ?= k+1 ) ep33 " 0 * and p !"!"!"! +1 - !! ’ k+1 3 q ep ep # =?%1-& )11 1- 2! 3 " ep 0) p ) !"!"!"+1 ! - ! ! k+1 3 q ep ep ?# %k-&= (25b) ( 22 "1- 2! 3 0ep ) ) p q !"!"!- ! # ??=- k+1 ) ep33 " * 0 Eqs. (14 - 25)give the stress and strain expressions of different models for the element showed in Fig.2.And the effects of elasticity in different stress states of non - linear elastic - plastic problems are to be evaluated in following sections. 4 Role of elasticity in elastic- plastic pr oblems of var ious str ess states 4.1 The“elastic- plastic Poisson ’s r atio”! ep The materials constants used in this paper are all the same, they are =0.3, E=200GPa, !E/E=5,20 and =/E=0.002.The curves of versus k for different stress states determined "#!p00 ep by Eq.(24) are shown in Fig.3. (a) E/E=5 (b) E/E=20 PP ’Fig.3 Distributions of elastic- plastic Poissons ratio !for different stress states of ep general elastic- plastic problem It is shown that the values of !=#/(#+#) is dependent on stress states. All the ! ep33 11 22 epvalues for two kinds of materials cannot reach the incompressible values of 0.5, especially as k=#/#?1.0. As mentioned above, Fig.3 or Eq.(24) gives the general relation between out- 22 11 plane stresses and in- plane stresses. Eq.(24) may be helpful in solving elastic- plastic plane strain problems, this relation make it possible that out- plane stresses can be determined by in- plane stresses before stress fields are obtained. The dependence of ! on the material con- ep stants for the two simplified models are showed in Fig.4.It is shown that the values of ! for incom- ep pressible model are constant; and the values of ! ep for“deep plastic model”are still dependent on the modulus ratio of the elastic to plastics (i.e., E/E). P In a word,the ratio of !=#/(#+#) for ep 33 11 22 general elastic- plastic problem is a variable, and its values dependent on stress states; while the ratio of Fig.4 Distributions of elastic- plastic Poi- #/(#+#) for the two simplified models are con- sson’s ratio for different stress !ep 33 11 22 states of two simplified models stants, and no longer stress- dependent. str esses 4.2 The out- plane The values of out- of- plane stress !are considerably overestimated by the incompress- 33 ible assumption because the values of "have been overestimated.By denoting the two differ- ep Real Incomp !"!"++!!!!ent out- of- plane stresses as !="and !=0.5 for the real elastic- 11 22 11 22 33 ep 33 by the in- plastic fields and incompressible fields respectively,the deviations of ! caused 33 compressible assumption for different stress states can be evaluated by Incomp! 0.533 (26) = Real" ep! 33 with the values in Fig.3, this deflection distributions are shown in Fig.5. " ep (a) E/E=5 (b) E/E=20 PP Fig.5 The deflections of out- of- plane stress caused by the incompressible assumption It is shown that the deflections caused by the incompressible assumption are consider- able, especially as k=!/!?1 for two different materials.The values of out- of- plane stress 22 11 could be overestimated above 50% at most, so the revision made by this analogy method is re- markable. “”Accordingly, the deviations of caused by thedeep plastic modelfor different stress! 33 states can be evaluated by E E Deep- 1 ?$"+0.5!"% !EE 33pp= (27) Real" ep! 33 Deep ! 33For two typical cases of E/E=5, 20, there are =0.46, 0.49/". The distributions are PepReal ! 33 similar to those in Fig.5, and they are omitted here. The differences between incompressible and“deep plastic”results are Incompr! 0.5 33= (28) Deep E E !- 1 33?#+0.5$%&"!" E E pp This equation is shown in Fig.6. It is seen that the differences between these two simplified models increase as the values of E/E de- P crease, i.e., the differences increase as the plastic modu- lus approach to the elastic modulus. Substituting Eqs. (18,12,24) into Eq. (22) separately, Fig.6 The differences of out- of we get the distributions of effective stresses for three - plane stress of the two different models. For a typical case, k=/=1.0, E/E!! 22 11 Psimplified models =5,20,they are plotted in Fig.7. Fig.7 shows that the deviations of effective stresses !caused e by the incompressible assumption are also remarkable.The values of of the incompressible !e model keeps zero. Substituting Eq.(18) into Eq.(22), the expression of is !e !! 11 e 3’ !" (29) = 1- k ?!" !2! 00 Fig.7 Comparing of effective stresses given by analogy results and by the results of the two simplified models Generally,the values of !are dependent on material constants E, Eas well as load lev- e P “”els of /.Theincompressible materialswould never yield in the case of k=/=1.0,!!!! 11 0 22 11 and neither plastic nor elastic deformation would develop. Obviously, this is not true for real materials. 4.3 The elastic and plastic str ains With the values of "for different values of k in Fig.3, elastic, plastic strain components ep eare obtained by Eq.(25). As a typical parameter, the distributions of elastic strain #and the 11 e pratio of elastic to plastic strain #/#are plotted in Fig.8. 11 11 (a) Elastic strain (b) The ratio of elastic to plastic strain Fig.8 Distributions of elastic- plastic strain for different stress states of "!E/E=20 general elastic- plastic problem P e p It is shown that the ratio of !/!increases as k="/"?1.0,so the elastic strain in 11 11 22 11 yield zones in such cases is remarkable. 5 Concluding r emar ks The fundamental problems in non- linear elastic- plastic analyses are discussed, and it is limited to the deformation plasticity theory (or the non- linear elasticity). It is noted that it is nearly impossible to obtain the analytical solutions for the general elastic- plastic problems if the volume deformation related to hydrostatic stress were not omitted in its solution procedure, even for very simple problems. Because of the mathematical difficulty, the simplified constitu- tive equations or asymptotic method as well as other simplifications are employed, but the re- sults are not reasonable in many cases. We also note that the most common simplified way of elastic- plastic analyses in litera- tures as well as in plasticity textbooks is the incompressible assumption. The incompressible results are the limit cases of the elastic- plastic solutions, and it is only applicable on the con- dition that the mean stresses are comparatively very low. For plane strain problems, especial- ly, the fields with high mean stresses occur quite often, such as the fields near a hole, a notch or crack tip, etc. When the mean stresses are not very low, especially as k= /?1, the"" 22 11 deflections caused by the incompressible assumption are remarkable,and all the values of out- of- plane stresses such as ,as well as strains need reconsideration. ""e 33 Based on the element of different stress states,a general relation between out - plane stresses and in- plane stresses for elastic- plastic plane strain problems is derived. And this relation may be helpful in solving elastic- plastic plane strain problems because it is made possible that out- plane stresses ,can be determined and expressed by in- plane stresses!! e 33 at the beginning of solution procedures. The dependences of the out- plane stress on !,! 11 22 the stress states for general elastic- plastic problems under plane strain condition is examined carefully.The present results reveal some basic facts such as the elastic deformation in the yield zones, and the elastic effects on the elastic- plastic fields. It is concluded that the elastic portion developed in the elastic- plastic deformation may be significant, and a more reason- able scheme that could properly consider out - plane stresses for plane strain elastic - plastic problems is expected. 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Asymptotic deformation and stress fields at the tip of a sharp notch in an elastic- plastic material[J]. International Journal of Fracture, 1992,54:211- 224. 弹塑性变形中的弹性效应 12 田常录 那日苏 , 江南大学机械学院江苏 无锡 内蒙古工业大学机械学院内蒙古 呼和浩特 ( 1 , 214122; 2 , 010062) 摘要由于数学上的困难弹塑性问题分析中一般忽略弹性变形而且求解时对本构方程的一些简化方法如不可压缩: , 。, 假设等被广泛采用本文分析了一种比拟解答的优点和其在某些方面的改进并以平面应变下的弹塑性单元体为例。, , 详细计算和分析了不同应力状态下弹塑性变形过程中的弹性效应。 关键词弹性效应弹塑性变形本构方程: ; ; 中图分类号文献标识码: TB301 U661.4 : A 作者简介田常录男江南大学机械学院教授博士生导师: ( 1963- ) , , , ; 那日苏女内蒙古工业大学机械学院讲师( 1976- ) , , 。