Transcription of CH.3. COMPATIBILITY EQUATIONS
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CH.3. COMPATIBILITY EQUATIONS
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.
A functional relation can be established between these three equations. ∇∇⋅×= ( ) v. 0. 12 . Ch.3. Compatibility Equations ... Eliminating the 30 unknowns , , 6 equations (involving only strain derivatives) are obtained: 22 2 22 2 22 22 2 22 22 2 2. 20 20 20 0. def yy zz yz xx def zz xx xz yy def xx yy xy zz def zz yz xz xy xy def yy yz ...
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