A Crystal Structure- -Based Based A Crystal Structure Eigentransformation and Its Work- -Conjugate Conjugate Eigentransformation and Its Work Material Stress Material Stress Chien H. Wu Chien H. Wu University of Illinois at Chicago University of Illinois at Chicago cwu cwu UIC NSF cwu cwu cwu cwu cwu cwu UIC NSF EUROMECH COLLOQUIUM 445
Eigenstrain and Eigentransformation Eigenstrain and Eigentransformation “Eigenstrain” is a generic name given by Toshio Mura to such nonelastic strains as thermal expansion, phase transformation, initial strains, plastic, misfit strains in his book Micromechanics of Defects in Solids . “Eigentransformation” is introduced as the nonlinear counterpart of eigenstrain in finite deformation. Definition clear? Yes, if you know what you are doing. UIC cwu cwu cwu cwu cwu cwu UIC cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 01
Finite Shrink Fitting of Circular Rings Finite Shrink Fitting of Circular Rings = e SF e SF r (R ) r (R ) 1 12 2 21 SF Spatial Configuration in r SF Configuration in R = e SF < < r r (R ) SF SF SF Ring 1: R R R 1 11 12 = e SF < < r r (R ) SF SF SF Ring 2: R R R 2 21 22 The stress-free (SF) configuration consists of two mismatched rings but the final configuration after elastic (e) fitting is a single composite ring without a visible geometric discontinuity. Problem solved. Where and what are the eigentransformations? What can they contribute to the understanding of material or configurational stresses? UIC cwu cwu cwu cwu cwu UIC cwu cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 02
Eigentransformations in Shrink Fitting Eigentransformations in Shrink Fitting Referential in R The disjointed SF rings may be < < = R R R R mapped from a single contiguous 11 12 21 annular referential configuration < < R R R Λ Λ ∗ ∗ , via eigenstretchratios : 21 22 1 2 = Λ ∗ < < SF R R ( R R R ) − SF Stress Free in R 1 11 12 ∗ = Λ < < SF R R ( R R R ) < < ≠ SF SF SF SF R R R R 2 21 22 11 12 21 = Since R R < < SF SF SF 12 21 R R R 21 22 Λ Λ = ∗ ∗ SF SF / R / R 1 2 12 21 Coherence in terms of material Spatial in r length parameters , l l : r < r < r = r 1 2 11 12 21 = SF SF r < r < r R / R l l / 12 21 1 2 21 22 Eigentransformations depend on the choice of referential configuration. Requirement on coherent interface. UIC cwu cwu cwu cwu cwu UIC cwu cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 03
The Shrink Fitting Experience The Shrink Fitting Experience Referential in R Choose a contiguous referential configuration—a 3-D grid work with empty cells. Eigentransformation SF SF in R Fill the cells with component Total Transformation elements and they will combine to assume their natural stress- free cell shapes. These cells, in Elastic Transformation general, cannot be fitted together to form a stress-free body. Spatial in r Since the spatial configuration is required to be a contiguous body, an elastic transformation must be superimposed on each SF cell. UIC cwu cwu cwu cwu cwu UIC cwu cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 04
An N- -Component system Component system An N Partial Molar Density (Molar Concentration) for Component m ∑ C Per Unit Referential Volume in X : C = C m m ∑ c Per Unit Spatial Volume in x : c = c m m ∑ SF C SF SF Per Unit Stress-Free Volume : C = C m m SF SF x = C /C = c /c = C /C Mole Fraction (Composition): m m m m { J = dv/dV = C/c = C /c m m Jacobians of e SF SF SF J = dv/dV = C /c = C /c m m Transformation * SF SF SF J = dV /dV = C/C = C /C m m UIC cwu cwu cwu cwu cwu cwu UIC cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 05
An N- -Component Alloy Crystal Component Alloy Crystal An N α β γ A unit cell of edge lengths (a, b, c) and interaxial angles ( , , ) c α b β γ a ≡ α β γ The six lattice parameters, or lattice constants p (a, b, c; , , ), for a unit cell are, in general, functions of the composition, i.e. ≡ ⋅⋅⋅ p = p(x) p(x x ) , , 1 N-1 The total number of atoms per cell is N . cell UIC cwu cwu cwu cwu cwu cwu UIC cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 06
A Referential Simple Cubic Cell A Referential Simple Cubic Cell ≡ α β γ A simple cubic cell of lattice constants p (a, b, c; , , ) � � � = (a , a , a ; , , ) is used to define a referential configuration 90 90 90 o o o c α b β a γ o a o a o a � � � The mapping from the simple cubic cell p = (a , a , a ; , , ) 90 90 90 o o o ≡ α β γ to the alloy crystal cell p(x) (a, b, c; , , )(x) gives the desired ∗ eigentransformation F , which, together with p(x), varies a s a function of time via its dependance on the composition x. UIC cwu cwu cwu cwu cwu cwu UIC cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 07
A collection of Nonuniform SF Cells A collection of Nonuniform SF Cells Referential in X Fill each cell with N atoms in cell a accordance with the non- o a o Eigentransformation a uniform composition x. o SF SF in X The N atoms will combine Total Transformation cell c into a SF crystal cell via eigen- α b β γ ∗ transformation F . Elastic Transformation a Spatial in x e An elastic transformation F is then developed in response ∗ to the nonuniform F . ∗ = e The total transformation is just F F F . UIC cwu cwu cwu cwu cwu cwu UIC cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 08
The Helmholtz Free Energy The Helmholtz Free Energy Helmholtz per unit Molar Molar Piola . Deformation referential volume Helmholtz Gibbs Gradient ∗ = = + ⋅ = e A( , , F T C ) C A( , ,x) F T C G( , ,x) S T S F and F F F m = A( , 1 T ,x ) 0 as a reference o o ∗ = ≡ ∗ SF A( F , ,x) T G ( ,x) T G( , ,x) S T , F stress - free eigentransformation = S 0 = ∗ = ∗ + ∗ − ∗ e e A( , ,x) F T A( F F , ,x) T A( F , ,x) T [ A( F F , ,x) T A( F , ,x)] T ≡ ∗ − ∗ = SF e e SF Define : W ( , ) A( , ,x) A( , ,x) , W ( , ) F T F F T F T 1 T 0 The dependence of elasticity on composition is usually insignificant. c α b a β o γ a o a o a Spatial in x SF SF in X Referential in X UIC cwu cwu cwu cwu cwu cwu UIC cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 09
The Helmholtz Free Energy-- --Conclusion Conclusion The Helmholtz Free Energy = ∗ = ∗ + e SF e A( , , F T C ) C A( F F , ,x) T C [A( F , ,x) T W ( F , )], or T m Stress-free Stress-free strain energy molar Gibbs C ∗ = + SF SF e = SF + SF e A C G( , ,x) 0 T [ C W ( F , T ) ] C G ( ,x) T J W ( F , T ) SF C = + ∗ SF SF e A C G ( ,x) T J W ( F , ) T SF CG ( ,x) T ∗ SF e J W ( F , ) T c α b a β o γ a o a o a Spatial in x SF SF in X Referential in X UIC cwu cwu cwu cwu cwu cwu UIC cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 10
Piola Stresses Piola Stresses HELMHOLTZ ENERGY strain energy density per Stress-free unit stress-free volume molar Gibbs = + ∗ SF SF e A( , , F T C ) C G ( ,x) T J W ( F , ) T �� � ��� � m strain energy density per W unit referential volume ∂ ∂ ∂ SF A W W = = = J ∗ ∗ ≡ ∗ = ∗ e T e -1 Piola Stress: ( ) where , ( ) S S f S f F ∂ ∂ ∂ e F F F SF CG ( ,x) T ∗ SF e J W ( F , ) T c α b a β o γ a o a o a Spatial in x SF SF in X Referential in X UIC cwu cwu cwu cwu cwu cwu UIC cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 11
Chemical Potential and Eshelby Stress Chemical Potential and Eshelby Stress HELMHOLTZ FREE ENERGY ∗ = SF + SF e A( , , F T C ) C G ( ,x) T J W ( F , ) T m CHEMICAL POTENTIAL ∂ ∂ ∂ A = µ = ∗ = ∗ SF SF e e -1 CG (T,x)+ J W ( F ,T), F F ( F ) m ∂ ∂ ∂ C C C m m m ∗ ∗ ∂ ∂ F F µ = ∗ ⋅ ∗ ⋅ ∗ SF e T G (T ,x)+ J C ( f ) or C f m ∂ ∂ C C m m PARTIAL MOLAR GIBBS ENERGY ∂ SF SF G (T,x) = CG (T,x) ∂ C m GENERALIZED MATERIAL (or ESHELBY or CONFIGURATIONAL) STRESS = − e SF e T e C W 1 ( F ) S : Relative to Stress -Free Configuration J ∗ = SF − T W : Relative to Referential Configuration C 1 F S UIC cwu cwu cwu cwu cwu UIC cwu cwu cwu UIC UIC EUROMECH COLLOQUIUM 445 12
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