CH.3. COMPATIBILITY EQUATIONS

1 COMPATIBILITY EQUATIONS Multimedia Course on Continuum Mechanics Overview Introduction COMPATIBILITY Conditions COMPATIBILITY EQUATIONS of a Potential Vector Field COMPATIBILITY Conditions for Infinitesimal Strains Integration of the Infinitesimal Strain Tensor Integration of the Deformation Rate Tensor 2 Lecture 1 Lecture 2 Lecture 3 Lecture 4 Lecture 5 Lecture 6 COMPATIBILITY EQUATIONS COMPATIBILITY Conditions 3 Introduction Given a displacement field, the corresponding strain field is found: Is the inverse possible? ( ),tUX( ),tux1,{1, 2 , 3}2jikkijjiijUUUUEijX X XX = ++ 1,{1, 2 , 3}2jiijjiuuijxx =+ ( ),tx ( ),tux4 COMPATIBILITY Conditions Given an (arbitrary) symmetric second order tensor field, , a displacement field, , fulfilling cannot always be obtained: For to match a symmetric strain tensor: It must be integrable.

2 There must exist a displacement field from which it comes from.( ),tx ( ),tux( )( ,),stt=uxx 1,{1, 2 , 3}2jiijjiuuijxx =+ 6 PDEs 3 unknowns OVERDETERMINED SYSTEM ( ),tx COMPATIBILITY CONDITIONS must be satisfied REMARK Given , there will always exist an associated strain tensor, , obtainable through differentiation, which will automatically satisfy the COMPATIBILITY conditions. ( ),tx ( ),tux5 COMPATIBILITY Conditions The COMPATIBILITY conditions are the conditions a symmetric 2nd order tensor must satisfy in order to be a strain tensor and, thus, exist a displacement field which satisfies: They guarantee the continuity of the continuous medium during thedeformation process.

3 ( )t,XEIncompatible strain field 1,{1, 2 , 3}2jiijjiuuijxx =+ 6 COMPATIBILITY EQUATIONS COMPATIBILITY EQUATIONS of a Potential Vector Field 7 Preliminary example: Potential Vector Field A vector field will be a potential vector field if there exists a scalar function (named potential function) such that: Given a continuous scalar function there will always exist a potential vector field . Is the inverse true?( ),tvx( ),t x( )( )( )( ){ },,,v,1, 2 , 3iittttix= = vxxxx ( ),t x( ),tvx( ),tvx( ),t xsuch that ( ) ( ),,tt =xvx 8 Potential Field In component form, Differentiating once these expressions with respect to : ( ),tvx( ),t xsuch that ( ) ( ),,tt =xvx ( )( )( )( ){ },,v,v,01,2,3iiiittttixx = = xxxx3 eqns.

4 1 unknown OVERDETERMINED SYSTEM jx( ){ }2,v,1, 2 , 3xijjitijxxx = 9 eqns. 9 Schwartz Theorem The Schwartz Theorem about symmetry of second partial derivatives guarantees that, given a continuous function with continuous derivatives, the following holds true: ()12, ,..,nxxx 22,i jjiijxxx x = 10 COMPATIBILITY EQUATIONS Considering the Schwartz Theorem, In this system of 9 EQUATIONS , only 6 different 2nd derivatives of theunknown appear: They can be eliminated and the following identities are obtained:2 22222 222 222vvvvvvvvvxxxyyyzzzxxyxyzxzxyxyyzyzxzx yzyzz === === === vvvvvvyyxxzzyxzxzy === ( ),t x22x 22y 22z yx 2zx 2zy 2, , , , and 11 COMPATIBILITY EQUATIONS A scalar function which satisfies will exist if the vector field verifies.

5 ( ),t x( ),tvx( ) ( ),,tt =xvx yzvv0vv0vv0defxzdefxzydefyxSxySzxSyz == == == 123 vvvxyzxyzSSxyzS eeeSv where { }vv0,1, 2 , 3jijiijxx = = v0 INTEGRABILITY ( COMPATIBILITY ) EQUATIONS of a potential vector field REMARK A functional relation can be established between these three EQUATIONS . ()0 =v 12 COMPATIBILITY EQUATIONS COMPATIBILITY Conditions for Infinitesimal Strains 13 Infinitesimal strains case The infinitesimal strain field can be written as: 112212yxxxzxxxyxzyyzxyyyyzxzyzzzzuuuuuxy xzxuuuyzyusymmetricalz ++ == + 6 PDEs 3 unknowns 14 The infinitesimal strain field can be written as: 6 PDEs 3 unknowns 100210021002yxxxxxyyxzyyxzyzzzzyzuuuxyxu uuyzxuuuzzy = + = = + = = + = The system will have a solution only if certain COMPATIBILITY conditions are satisfied.

6 15 Infinitesimal strains case COMPATIBILITY Conditions The COMPATIBILITY conditions for the infinitesimal strain field are obtained through double differentiation (single differentiation is not enough). 22 2222 22,,,,,12,,,,,xxxyzyzuxxyzxy xz yzuuzyxyzxy xz yz = + = 6 EQUATIONS 6 EQUATIONS 6x6=36 EQUATIONS 16 COMPATIBILITY Conditions The COMPATIBILITY conditions for the infinitesimal strain field are obtained through: 232332322 22323322223232332 yxy ==+ ==+ ==+ == 32232332223233221212zyzyxxxzyzyxxxzuzxyy xuuuxzx zxzz xyxzuuuyzxyzyzz yy z + ==+ ==+ ,,xxyyzz 18 EQUATIONS for ,,xyxzyz 18 EQUATIONS for 17 COMPATIBILITY Conditions All the third derivatives of and appear in the EQUATIONS : which constitute 30 of the unknowns in the system of 36 EQUATIONS .

7 332 2 3 2 2 32 2332 2 3 2 2 32 2332 2 3 2 2 32 2, ,,,,,,,, , ,,,,,,,, , ,,,,,,,,xyzuxx yx zyy xy zzz xz yxyzuxx yx zyy xy zzz xz yxyzuxx yx zyy xy zzz xz yxyz = = = ,xyzuuu10 derivatives 10 derivatives 10 derivatives {}23,01, 2,.., 36ijinjklklufnxxx xx = 30 18 COMPATIBILITY EQUATIONS Eliminating the 30 unknowns , , 6 EQUATIONS (involving only strain derivatives) are obtained: 222222222222222222020200defyyyzzzxxdefxx xzzzyydefyyxyxxzzdefyzxyxzzzxydefyyyzxxz xzSzyyzSxzxzSyxxySxyzxyzSxzyxy = + = =+ = =+ = = + + = = + + 200ydefyzxyxxxzyzzSyzxxyz = = + + + = 3ijkluxxx COMPATIBILITY EQUATIONS for the infinitesimal strain tensor ( )= =S 019 COMPATIBILITY EQUATIONS The six EQUATIONS are not functionally independent.

8 They satisfy the equation, In indicial notation:( )() = =S 0000xyxxxzxyyyyzyzxzzzSSSxyzSSSxyzSSSxyz ++= ++= ++= 20 COMPATIBILITY EQUATIONS The COMPATIBILITY EQUATIONS can be expressed in terms of the permutation operator, . Or, alternatively: ,0,1, 2 , 3mlmjq lir ij qrSml == ee ijke{}, ,, ,0, , ,1, 2 , 3ij klkl ijik jljl iki jkl + = REMARK Any linear strain tensor (1st order polynomial) with respect to the spatial variables will be compatible and, thus, integrable. 21 COMPATIBILITY EQUATIONS Integration of the Infinitesimal Strain Tensor 22 Preliminary EQUATIONS Rotation tensor : Rotation vector : ( ),t x( ),t x1( )()21,{1, 2 , 3}2jiijjiskewuuijxx = = = u u u23 []12332231313122101()020yzzxxy = == = = u Preliminary EQUATIONS Differentiating with respect to : Adding and subtracting the term.

9 Kx1122jijjiiijjikkjiuuuuxxxxxx = = 212kijuxx 221112221122ijjikkkkjiijijjjkikkikj ki i k jj iuuuuxx xxxxxxuuuux xx x xx x x = + = =+ += ( ),tx jk =ik =24 1111yzxyxzyzyzyyyzzyzzx x yzy y yzz z yz = = = = = = Preliminary EQUATIONS Using the previous results, the derivative of is obtained: 2222zxxxxzxyyzzxzxxzzzx x zxy y zxz z zx = = = = = = 3333xyxyxxxyyyxyxyyzxzx x xyy y xyz z xy = = = = = = ( ),t x25 Preliminary EQUATIONS Considering the displacement gradient tensor , Introducing the definition of , the components of are rewritten: ( ){ },11,1, 2 , 322jjiiiijijijjjijituuuuuJijxxxxx == + = =++ = + uxJx ( ),tJxij= ij =( ),t x( ),tJx3231211 231:2:3.

10 X xxxxxyxzyy yxyyyyzz zzxzyzzzu uux yzuu uxjjjiiyzu uuxziy == =+ =+= = = =+= ======26 Integration of the Strain Field The integration of the strain field is performed in two steps: of derivative of using the1st order PDE system derived for . The solution will be of the type: The integration constants can be obtained knowing the value of the rotation vector in some points of the medium (boundary conditions). and , is integrated using the 1st order PDE system derived for . The solution will be: The integration constants can be obtained knowing the value of the displacements in some point of space (boundary conditions) ( ),t x( ),t x123, and() ( ){ }, , ,1, 2 , 3iiix yzt c ti = + ( )ict( ),t x( ),t xu() ( ){ }, , ,1, 2 , 3iiiu u x yzt c ti = + ( )ict REMARK If the compati-bility EQUATIONS are satisfied, these EQUATIONS will be integra-ble.