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Mechanics of heterogeneous media Method of inclusions and its applications for random fibre composites Stepan V. Lomov S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 1 F rom E shelbyprinciple to equivalent stiffness


  1. Mechanics of heterogeneous media Method of inclusions and its applications for random fibre composites Stepan V. Lomov S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 1

  2. F rom E shelbyprinciple to equivalent stiffness of an inclusion assembly… � � ij � � D D D � �� D � � � � � � ij � � � eff � C m m ijkl � � S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 2

  3. … applied to random fibre reinforced composites … � � eff C m � ijkl S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 3

  4. … and to textile composites � � eff C m � ijkl S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 4

  5. G eneral scheme of application of the inclusion method 1. The heterogeneous medium should constitute a homogeneous matrix with a second (discontinuous) phase, or more phases of reinforcement embedded in it 2. Build a geometrical model of the RVE of the reinforcement 3. Subdivide the reinforcement into elements, which somehow could be represented as ellipsoids. 4. Consider the assembly of the ellipsoidal inclusions in the matrix 5. Using properties of the reinforcement, assign stiffness tensors to the inclusions (micro-homogenisation may be performed on this step) 6. Apply the inclusion theory to calculate the equivalent stiffness of the RVE � � eff C m ijkl � S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 5

  6. Method of inclusions • Eshelby transformation principle for an assembly of inclusions • Mori-Tanaka algorithm • Self-consistent algorithm S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 6

  7. R eminder: E shelbytransformation principle for O N E inclusion The solution (disturbance fields) � � for the elasic problem � � � � � � � � � � ij ij ij ij of an anisotropic ellipsoidal inclusion C � � � � � C � ijkl � m ijkl in an anisotropic matrix � C � m ijkl � with the strain at infinity � is given by � ij m C � � x : � x ; � S �� const � � �� � � � � ij ijkl kl ijkl ij � x : � D �� x � � � � � �� � � � � ij ijkl kl where S and D are Eshelby tensors and �� �� � � � ijkl ijkl � � � m � � � � � � C � S � � C S � � � � � � � � � � � � � � � � � kl klmn mn kl klmn mn kl ij ijkl ijkl � 1 � � m m � C S I C S � C � C � � �� � � � � � � � � � � � � � � � � � m m C � ijkl 1 m x C � x G x x G x x d x � � � � � � � � � � � � � � � � � � � � � � � � � ij kl mn mn ik lj , jk li , � � 2 � � � ij 1 m C � G x x G x x d x � � � � � � � � � � � � � � � � � klmn mn ik lj , jk li , � � 2 � S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 7

  8. � � ✁ � � ✁ E shelbytransformation principle for an assembly of inclusions Consider disturbance field produced by inclusion � (or by the source domain ): � 1 m � x , � C � G x x G x x d x � � � � � � � C � � � � � � � � � � � � � � � � ij klmn mn ik lj , jk li , � � 2 ijkl m � � The average disturbance in induced by the inclusion : � � � m � C � � � 1 ijkl x , d x � � � � � � � � � � � ij � � � ij ij V � � � � � 1 1 � � � � � � � m C G x x G x x d x d x � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ij klmn ik lj , jk li , mn � � 2 V � � � � � � � � � � � � m or m � � � C � ij ijkl � ; � � S �� � � � � � � � � � ij ijkl kl � � 1 ij m S C G x x G x x d x d x �� � � � � � � � � � � � � � � � � � � � � � � � � ijkl mnkl ik lj , jk li , � � 2 V � � � � 1... M � � � � For : S is Eshelby tensor for an isolated inclusion. �� � � � ijkl NOTE: From now on we consider averaged strains in inclusions and the matrix. S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 8

  9. ✂ ✂ ✂ ✄ ✂ E quations for disturbance strains and eigenstrains Summing up disturbance strains for all the inclusions, M M 1 1 x d x x , d x � � � � � � � � � � � � � � � � � � � � � � � ij ij ij ij V V � � � � � � � � 1 1 � � � � � � � C � � or in tensor notation ijkl m � M S � S � , 1... M (1) � �� � �� � � �� � � �� � m � � � � � C � � � 1, � � � � � ijkl ij Stresses i n the inclusions m C � C � � , 1... M (2) � � � � � � � � � � �� � � �� � � � � � � � � � � � � � Total disturbance strains ij � � M m c c � =0 (3) m � � � � m � � m � � � C 1 � � ij ijkl m where is the average disturbance strai n in the matrix: � � � � � 1 ij m x d x � � � � � � ij ij M M V V � � � � � � V 1... M � � � � � 1 1 � � c c , are the volume fractions of the matrix and the inclusions: m � M V � � � � � V � � � M � 1 c = , 1... M c ; 1 � ; c c 1 � � � � � � � � m m � V V � 1 � � S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 9

  10. Image strain and the mean field assumption M S � S � � �� � �� � , 1... M � �� � � �� � � � � � � C � 1, � � � � � � ijkl m The second term is called image strain and accounts � for the additional (in comparison with the isolated m � C � inclusion case) strain, that the inclusion � receives � � ijkl ij due to interaction with other inclusions. Mean field assumption � � � ij Image strain could be approximated by its mean value � � which is the same everywhere – in all the inclusions m m � � � � C and in the matrix ij ijkl � M � � � im � � � S S S � �� � �� � �� � � �� � � �� � �� � � � � � � ij 1, � � � � � 1... M � � m im � � � � � � � im m � S � S � � �� � �� � � � � �� � � �� � � � � � � Pedersen, O.B. Thermoelasticity and � � plasticity of composites - I. Mean field 1 � m � S (4) � �� � � � � � theory Acta Metallurgica Materials � �� � � � � � � 31 (11): 1983 1795-1808. S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 10

  11. S train concentration factors and homogenised stiffness Strain concentration tensors A relate full strains in the inclusions with the applied strain: � A � � (*) � � � � �� � Dilute strain concentration tensors A relate full strains in the inclusion s with the matrix strain � m m � A � (**) � �� � � m � � � � � � � � � � � 1 � M � � A A c I c A (5) � � � A � � � � � � � � m m m � � � A � 1 � � � � ij m Proof: m m � � � � � � � M m (3) c � c � =0 � � � � � � � � � � � � � � m m � � � � 1 � � M M m � c c 1 c c � � � � � � � � � � � m m � � � � 1 1 � � � � 1 � M M � � 1 � m m m (*),(**) A A c c A A A c I c A , QED � � � � � � � � � � � � � �� �� � � � �� � � � � � � � � � m m m m m m � � � � � � � 1 1 � � � � � � S.V. Lomov - Mechanics of heterogeneous media - 3. Method of inclusions 11

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