1、 Material Nonlinear Analysis 1 Material Nonlinear Analysis A fundamental difference between elastic and plastic material behaviors is that no permanent deformations occur in the structure in elastic behavior,whereas permanent or irreversible deformations occur in the structure in plastic behavior.Pl
2、asticity theory The components of static plastic strain are constituted by the following assumptions:?Constitutive response is independent of the rate of deformation.?Elastic response is not influenced by plastic deformation.?Additive strain decomposition into elastic and plastic parts is defined by
3、 =+?ep (1)where,?:total strains?e:elastic strains?p:plastic strains And the following basic concepts are used to formulate the equations:?Yield criteria to define the initiation of plastic deformation?Flow rule to define the plastic straining?Hardening rule to define the evolution of the yield surfa
4、ce with plastic straining Yield criteria The yield function(or loading function),F,which defines the limit for the range of elastic response,is as follows(Fig.2.16):(,)(,)()0ppepF =?(2)ANAYSIS&DESIGN 2 where,?:current stresses e:equivalent or effective stress :hardening parameter which is a function
5、 of p?p:equivalent plastic strain In classical plasticity theory,a state of stress at which the value of the yield function becomes positive is not admissible.When yielding occurs,the state of stress is corrected by scaling plastic strains until the yield function is reduced to zero.This process is
6、known as the plastic corrector phase or return mapping.Fig 2.16 Geometric illustration of associated flow rule and singularity Flow rule The flow rule defines the plastic straining,which is expressed as follows(Fig 2.16):?pda Smooth Plastic potential=g()F()0?Corner?a?,b=?pfd?pd?d Material Nonlinear
7、Analysis 3 =b?pgddd (3)where,?g:the direction of plastic straining d:plastic modulus which identifies the magnitude of plastic straining The function g is termed as the plastic potential function,which is generally defined in terms of stress invariants.If g=F,it is termed as associated flow rule,and
8、 if gF,it is referred to as non-associative flow rule.ANAYSIS&DESIGN 4 The associated flow rule is adopted for all the yield criteria of MIDAS programs.As the direction of the plastic strain vector is normal to the yield surface,the above equation can be expressed as follows:=a?pFddd (4)The corner o
9、r the flat surface in Fig 2.16 represents a singular point,which can not uniquely determine the direction of plastic flow.These points require special consideration.Hardening rule The hardening rule defines the expansion and translation of the yield surface with plastic straining as the material yie
10、lds.Depending on the method of defining the effective plastic strain,the hardening rule is classified into strain hardening and work hardening.The strain hardening is defined by the hypothesis of plastic incompressibility,and as such it is appropriate for a material model,which is not influenced by
11、hydrostatic stress.Accordingly,work hardening,which is defined by plastic work,is more generally applicable than strain hardening.Also,depending on the type of change of yield surface,the hardening rule is classified into isotropic hardening,kinematic hardening and mixed hardening(Fig.2.17).Initial
12、Yield Surface O 21O 21Subsequent Yield Surface Initial Yield Surface O1 (a)Isotropic hardening rule(b)Kinematic hardening rule Subsequent Yield Surface Material Nonlinear Analysis 5 Fig 2.17 Mixed hardening with kinematic hardening Classification by the method of defining the effective plastic strai
13、n 1.Strain hardening The effective plastic strain in strain hardening is defined as follows:()2233=a a?TppTpdddd (5)The effective plastic strain is derived from transforming the norm of plastic strains to conform to uniaxial strain with the assumption that there is no volumetric plastic deformation.
14、Although this is applicable in principle only to Tresca or von Mises,it is often applied to other cases because of numerical convenience.Mixed Translation and Expansion Initial Yield Surface Translation only 12 ANAYSIS&DESIGN 6 2.Work hardening The increment of plastic work is as follows:=a?TpTpdWdd
15、 (6)In the case of uniaxial strain,the increment of the plastic work is expressed as,11=pepdWdd (7)Hence the effective plastic strain pertaining to work hardening is defined as follows:=a?Tpedd (8)Classification by the types of change of yield surface 1.Perfectly plastic A perfectly plastic material
16、 does not change the yield surface even after plastic deformation has taken place.The yield function then can be expressed as follows:()(),=?eF (9)where,:constant 2.Isotropic hardening In the case of isotropic hardening,the yield surface expands uniformly as shown in Fig.2.18(a).The yield function c
17、an be expressed as follows:()()(),=?epF (10)3.Kinematic hardening In the case of kinematic hardening,the size of the yield surface remains unchanged and the center location of the yield surface is shifted as shown in Fig.Material Nonlinear Analysis 7 2.18(b).The yield function can be expressed as fo
18、llows:()(),=?eF (11)where,?:the center coordinates of yield surface :constant (a)Isotropic hardening (b)Kinematic hardening Fig.2.18 Hardening rule in 1-dimension In kinematic hardening,it becomes important to determine the center coordinates of the subsequent yield surface,?.In order to determine t
19、he“kinematic shift”,?,there exist Pragers hardening rule,Zieglers hardening rule,etc.The Pragers hardening rule can be expressed as,=a?pppdC dCd (12)where,pC:Pragers hardening coefficient This method may present some problems when it is used in the sub space of A O C A B B A B O C A B a a ANAYSIS&DE
20、SIGN 8 stress.For example,?d may not be 0 even any component of stresses is 0,which may not only present translation of the yield surface.The Zieglers hardening rule on the other hand assumes that the rate of translation of the center,?d,takes place in the direction of the reduced-stress vector,?.He
21、nce,it presents no such problem.This hardening rule is expressed as follows:()()=?zpddC d (13)where,zC:Zieglers hardening coefficient 4.Mixed hardening Mixed hardening is a hardening type,which represents the mix of isotropic hardening and kinematic hardening,which is expressed as follows:()()(),=?e
22、pF (14)Constitutive equations Standard plastic constitutive equations are formulated as below.Stress increments are determined by the elastic part of the strain increments.That is,()()=DDa?epeddddd (15)where,e?D:elastic constitutive matrix In order to always maintain the stresses on the yield surfac
23、e,the following consistency condition needs to be satisfied.()0=+=+=a Da D a?TpTeTepFFFdFddddh d (16)where,h:plastic hardening modulus()epdd=Material Nonlinear Analysis 9 Accordingly,the rate of infinitesimal stress increments can be obtained as follows:eeddd=DD a?=+D aa DDa D a?eTeTeTeddh (17)When
24、the full Newton-Raphson iteration procedure is used and if a consistent stiffness matrix is used,a much faster convergence can be achieved due to the second-order convergence characteristic of the Newton-Raphson iteration procedure.eeedddd=aDD aD?=+Raa RRa Ra?TTTddh (18)where,()11eeeedd=+=+aRIDDID A
25、D?Stress integration The following two methods can be used for the integration of stresses:?Explicit forward Euler algorithm with sub-incrementation(Fig.2.19&20)?Implicit backward Euler algorithm(Fig.2.21)ANAYSIS&DESIGN 10 (a)Locating intersection point A (b)Moving tangentially from A to C and subse
26、quently correcting to D Fig.2.19 Explicit forward-Euler procedure eABXX:final stress status at the previous step A:intersection point of stress increment and yield surface B:stress vector assuming elastic strainingABXCDeC:stress state after correction D:stress state after returning the stress to the
27、 yield surface artificially?eD Material Nonlinear Analysis 11 Fig.2.20 Sub-incrementation in Explicit forward-Euler procedure Fig.2.21 Implicit backward-Euler procedure ABCDEA,B,C,D:stress state at each sub-increment after correction E:stress state after returning the stress to the yield surface art
28、ificiallyBC XX:final stress status at the previous step B:stress vector assuming elastic strainingC:final unknown stress state xc1 ANAYSIS&DESIGN 12 In the Forward-Euler algorithm,the hardening data and the direction of plastic flow are calculated at the intersection point,where elastic stress incre
29、ments cross the yield surface(at point A in Fig.2.20).Whereas in the Backward-Euler algorithm,they are calculated at the final stress point(at point B in Fig.2.21).The Forward-Euler algorithm is relatively simple,and the stresses are directly integrated.That is,it need not iterate at the Gauss point
30、s,but presents the following drawbacks:?It is conditionally stable.?Sub-increments are required while correcting the stresses to obtain allowable accuracy.?An artificial returning scheme is required to correct the stress state for drift from the yield surface.Also,this method does not permit formula
31、ting a consistent stiffness matrix.The Implicit Backward-Euler algorithm is unconditionally stable and accurate without sub-increments or artificial returning.However for general yield criteria,iterations are required at the Gauss points.Because a consistent stiffness matrix can be formulated using
32、this method,even if iterations are performed at the Gauss points,it is more efficient if the Newton-Raphson iteration procedure is used.Steps for applying the Explicit forward-Euler procedure 1.Calculate strain increments.=B u?dd (19)where,B?:strain-displacement relation matrix d?:the changes of dis
33、placements 2.Calculate elastic stresses assuming elastic straining(at point B in Fig.2.19(a).Material Nonlinear Analysis 13 =+D?eBXddd (20)The Fig.2.19 should be referenced for the subscripts in the equations above and below.3.If the calculated stresses remain on the yield surface,stress correcting
34、is completed.If the stresses exist beyond the yield surface,the stresses are returned to the yield surface by plastic straining.4.Subsequently,the stresses at the intersection point are calculated.Elastic stress increments are divided into allowable stress increments and unallowable stress increment
35、s;whereas,stresses at the intersection point are calculated by the following expressions(point A in Fig.2.19(a):()()10+=?XBBXFr dFrFF (21)5.Further straining would cause the stress location to traverse the yield surface.This is approximated by sub-dividing the unallowable stress increments,rd?,into
36、the m number of small stress increments(Fig.2.20).The number of sub-increments,m is directly related to the magnitude of the error resulted from a one step return,which is calculated as,()()INT 81=+eBeAeAm (22)6.If the final stress state does not lie on the yield surface,the following method of arti
37、ficial returning is used to return the stress to the yield stress(point E in Fig.2.20).=+=a D aD a?CCTeCCeDCCCFh (23)ANAYSIS&DESIGN 14 Notes?The shape of the yield surface is corrected using the hardening rule at the end of each sub-increment.?Unloading is assumed to be elastic.Steps for applying th
38、e Implicit backward-Euler procedure The final stress in the Backward-Euler algorithm is calculated by the following equation:=D a?eCBCd (24)The Fig.2.21 should be referenced for the subscripts.Since the point C in the equation(24)is unknown,the Newton iteration is used to evaluate the unknowns.Accor
39、dingly,a vector,r?,is set up to represent the difference between the current stresses and the backward-Euler stresses.()=rD a?eCBCd (25)Now,iterations are introduced in order to reduce r?to 0 while the final stresses should satisfy the yield criterion,f=0.Using assumed elastic stresses,a truncated T
40、aylor expansion is applied to the equation(25)to produce a new residual,=+rrD a?eno (26)where,?:the change in?:the change in d Setting the above equation to 0,and solving it for?,we obtain the following:=rD a?eo (27)Similarly,a truncated Taylor expansion is applied to the yield function,which Materi
41、al Nonlinear Analysis 15 results in the following:0=+=+=a?TTCnCopCoCpFFFFFh (28)where,p:effective plastic strain Hence,?is obtained,and the final stress values can be obtained as well.=+a ra D a?TooTeFh (29)Plastic material models The following 4 types of general plastic models are used:?Tresca&von
42、Mises suitable for ductile materials such as metals,which exhibit plastic incompressibility(Fig.2.22).?Mohr-Coulomb&Drucker-Prager suitable for materials such as concrete,rock and soils,which exhibit volumetric plastic deformations(Fig.2.23).ANAYSIS&DESIGN 16 Fig 2.22 Tresca&von Mises yield criteria
43、 Fig 2.23 Mohr-Coulomb&Drucker-Prager yield criteria Hydrostatic axisvon Mises yield surface Tresca yield surface 312123Drucker-Prager Mohr-Coulomb 123 Material Nonlinear Analysis 17 Tresca criterion The Tresca yield criterion is suitable for ductile materials such as metals,which exhibit little vol
44、umetric plastic deformations.The yielding of a material begins when the maximum shear stress reaches a specified value.So if the principal stresses are()123123,the yield function becomes the equation(30).()()13,=?pF (30)Numerical problems arise when the stress point lies at a singular point on the y
45、ield surface,which occurs when the lode angle approaches 30.In such cases,the stress integration scheme must be corrected.ANAYSIS&DESIGN 18 Von Mises criterion The Von Mises criterion is a most widely used yield criterion for metallic materials.It is based on distortional strain energy,and the yield
46、 function is expressed as follows:()()2,3 =?pFJ (31)where,J2:second deviatoric stress invariant Mohr-Coulomb criterion The Mohr-Coulomb criterion is suitable for such materials as concrete,rock and soils,which exhibit volumetric plastic deformations.The Mohr-Coulomb yield criterion is a generalizati
47、on of the Coulombs friction rule,which is defined by,()(),tan=?nFc (32)where,:the magnitude of shearing stress n:normal stress c:cohesion :internal friction angle The cohesion,c,and the internal friction angle,are dependent upon the strain hardening parameter,.Similar to the Tresca criterion,numeric
48、al problems occur when the stress point lies at a singular point on the yield surface.For the Mohr-Coulomb criterion,such numerical problems occur as the lode angle,approaches 30 or at the apex points.Hence,the stress integration scheme must be corrected for the two cases.Material Nonlinear Analysis
49、 19 Drucker-Prager criterion The Drucker-Prager criterion is suitable for such materials as soils,concrete and rock,which exhibit volumetric plastic deformations.This criterion is a smooth approximation of the Mohr-Coulomb criterion and is an expansion of the von Mises criterion.The yield function includes the effect of hydrostatic stress,which is defined as follows:()()()122sin6 cos,3 3sin3 3sin=+?cFIJ (33)where,I:first stress invariant For the Drucker-Prager criterion,Numerical problems occur when the stress point lies at the apex points of the yield surface.
