Math 2.0

Mechanics of deformable solids Definition

Mori–Tanaka method

Builds on Effective medium approximationsEshelby's inclusion problemElasticity tensor

Definition

Let a two-phase composite consist of a matrix and of aligned ellipsoidal inclusions of the same shape. The matrix has elasticity tensor C0C_0 and volume fraction f0f_0. The inclusions have elasticity tensor C1C_1 and volume fraction f1=1−f0f_1 = 1 - f_0. Let SS be the Eshelby tensor of this ellipsoid in the matrix C0C_0, given by the solution of Eshelby's inclusion problem. Let II be the identity on symmetric matrices. The dilute strain concentration tensor is Adil=[I+SC0−1(C1−C0)]−1.A^{\mathrm{dil}} = \left[ I + S C_0^{-1} (C_1 - C_0) \right]^{-1}. The Mori–Tanaka strain concentration tensor is AMT=Adil[f0I+f1Adil]−1,A^{\mathrm{MT}} = A^{\mathrm{dil}} \left[ f_0 I + f_1 A^{\mathrm{dil}} \right]^{-1}, and the Mori–Tanaka estimate of the effective elasticity tensor is CMT=C0+f1(C1−C0)AMT.C^{\mathrm{MT}} = C_0 + f_1 (C_1 - C_0) A^{\mathrm{MT}}.

Remarks

The formula expresses that the average strain in the inclusions equals AdilA^{\mathrm{dil}} applied to the average strain in the matrix. Check of the limits: for f1=0f_1 = 0 one gets CMT=C0C^{\mathrm{MT}} = C_0, and for f1=1f_1 = 1 one gets AMT=IA^{\mathrm{MT}} = I and CMT=C1C^{\mathrm{MT}} = C_1. Take spherical inclusions and isotropic phases, with bulk moduli K0,K1K_0, K_1 and matrix shear modulus G0G_0. Then the effective bulk modulus is KMT=K0+f1(K1−K0)1+3f0(K1−K0)/(3K0+4G0).K^{\mathrm{MT}} = K_0 + \frac{f_1 (K_1 - K_0)}{1 + 3 f_0 (K_1 - K_0)/(3 K_0 + 4 G_0)}. This is the Hashin–Shtrikman bound with the matrix as reference phase. It is the lower bound when the matrix is the softer phase, and the upper bound when it is the stiffer one. Non-example: the dilute estimate C0+f1(C1−C0)AdilC_0 + f_1 (C_1 - C_0) A^{\mathrm{dil}} ignores the interaction of the inclusions and does not give C1C_1 at f1=1f_1 = 1, so it is not the Mori–Tanaka estimate.

Reading path

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Effective medium approximations, concept, no evidence levelEffective medium appr…conceptElasticity tensor, concept, no evidence levelElasticity tensorconceptLinear elasticity, concept, no evidence levelLinear elasticityconceptStrain tensor, concept, no evidence levelStrain tensorconceptEshelby's inclusion problem, problem, no evidence levelEshelby's inclusion p…problemMori–Tanaka method, definition, no evidence levelMori–Tanaka methoddefinition
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  1. Effective medium approximations concept
  2. Elasticity tensor concept
  3. Linear elasticity concept
  4. Strain tensor concept
  5. Eshelby's inclusion problem problem
  6. Mori–Tanaka method definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Effective_medium_approximations
About this page
TagPDY3N747BF
URIm2://PDY3N747BF
Kinddefinition
WritingW0 stub
MSC74Q15, 74E30, 74A60
DefinesMori–Tanaka method
UsesEffective medium approximations, Eshelby's inclusion problem, Elasticity tensor
Changed3 November 2026

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