1 Probability and Materials: from Nano- to Macro Scale A Workshop Sponsored by the John s Hopkins University and the NSF CMS Division January Baltimore, Maryland Sarah C. Baxter Department of Mechanical Engineering University of South Carolina Todd O. Williams Theoretical Division Los Alamos National Laboratory A stochastic micromechanical basis for the characterization of random heterogeneous materials

2 The extremes in (stochastic) microstructures typically drive important phenomena due to corresponding strong localizations - Inelastic deformations - Viscoelasticity - Viscoplasticity - Failure phenomena - Interfacial debonding - Cracking/damage in the phases 1. Modeling Considerations and Issues

3 2. Goals and Motivation Develop a general, stochastic micromechanical framework for constitutive models for heterogeneous materials –Applicable to various types of composite systems –Realistic (stochastic) microstructures –Arbitrary contrast in constituents’ elastic properties –Anisotropic local behaviors –Micro- and macro-damage –Complex rate and temperature dependent constituent behaviors Micromechanical theories : Predict the local and bulk behavior of heterogeneous materials based on a knowledge of the behavior of each of the component phases, the interfaces, and the microstructure

4 3. Concepts from Classic Micromechanics Representative Volume Fraction RVE? Maybe RVE? Probably not Similar; repeating volume fraction - periodic microstructures

5 When the macro conditions of homogeneous stress or homogeneous strain are imposed on an RVE the average stress and average strain are defined as: Average Stress Average Strain V is the volume of the RVE. Average Strain and Average Stress

6 Average Strain Theorem: The average strains in the composite are the same as the constant strains applied on the boundary. Consider a two phase RVE where homogeneous strains are applied to the boundary. Using the definition of strain, the average strain is Again, consider a two phase RVE, this time with homogeneous stresses applied to the boundary S. Equilibrium, in the absence of body forces, implies The Average Stress Theorem states that the average stresses in the composite are the same as the constant stresses applied on the boundary. Average Strain Theorem/Average Stress Theorem

7 Thus the average stress is related to the average strain through effective elastic moduli, C*. A similar argument can be used to construct effective compliances. Using formulas of linear elasticity and average strain and stress theorems defines the constitutive law Effective Elastic Properties

8 For a two phase composite with perfect bonding (c 1 and c 2 are volume fractions, (and ) Building on the idea of effective moduli, and using the average strain theorem Relationship Between Averages and

9 then and The effective moduli can be determined if the average strain in the second phase is known. Relationship Between Averages

10 In an RVE there is a unique relationship between the average strain in a phase and the overall strain in the composite, which can be expressed as A 1 and A 2 are called strain concentration matrices, c 1 A 1 +c 2 A 2 = I, where I is the unit matrix. Then the effective stiffness tensor can be written as In a two phase composite where then Concentration Matrices

11 Eshelby (1957, 1959, 1961) considered the problem of an ellipsoidal inclusion in an infinite isotropic matrix. He defined two problems which he considered should be equivalent Eshelby’s Equivalent Inclusion

12 Eshelby showed that if the eigenstrains are uniform inside an ellipsoidal domain then the total strain is uniform there too, and that the total strains are related to the eigenstrains through Eshelby’s tensor. The starred strains are eigenstrains or transformation strains, resulting from the inhomogeneity. Transformation (Eigen) strains / stresses

13  C  S  Transformation Field Theory, Dvorak,  eigenstresses and eigenstrains respectively Transformation field theory Dvorak & Benveniste, Dvorak Proc. Math. and Physical Sciences, 1992.)) with for example

14 One of the simplest models used to evaluate the effective properties of a composite, it was originally introduced to estimate the average constants of polycrystals. For this approximation it is assumed that the strain throughout the bulk material is uniform (iso-strain). This implies that A 1 = A 2 = I and so 4. Concentration Tensors in Classic Models Voigt Approximation (1889)

15 The dual assumption (to Voigt) is the Reuss Approximation which assumes that the stress is uniform (iso-stress) throughout the phases. This implies that B 1 = B 2 = I and so Under the Voigt model the implied tractions across the boundaries of the phases would violate equilibrium, and under the Reuss model the resulting strains would require debonding of the phases. Reuss Approximation (1929)

16 The Dilute Approximation models a dilute suspension of spherical elastic particles in a continuous elastic phase. It assumes that the interaction between particle can be neglected. Under the assumption of spherical symmetry, u r = u r (r), u f = 0, u q = 0 the equilibrium condition reduces to Dilute Approximation

17 One can solve for the A 2 concentration tensor, or the ratio between the strains, in the second phase and the applied strain Using this relationships the effective bulk modulus is given by Dilute Approximation

18 This is the problem of an inclusion in a medium with unknown effective properties. The factors (shear and bulk) are then the same as for dilute but with effective properties replacing those of the matrix. Effective medium Self-Consistent Scheme

19 Thus, Which gets us to which are the same as for dilute, but are now implicit relationships. Self-Consistent Scheme

20 In this method, a single particle is embedded in a sheath of matrix which in turn is embedded in an effective medium. Solving the problem under dilation with this geometry yields Matrix Effective medium Generalized Self-Consistent Scheme

21 The Mori-Tanaka method (Mori and Tanaka, 1973) was originally designed to calculate the average internal stress in the matrix of a material containing precipitates with eigenstrains. Starting with Mori-Tanaka (Benveniste) with if M-T assumes that where T is the concentration tensor from the dilute approximation. T can be defined by Eshebly’s tensor, P, as

22 Then the problems concentration tensor can be defined as Mori-Tanaka (Benveniste) and

23 5. Background for Stochastic Formulation Consider that a field, g, can be decomposed into mean and fluctuating parts, i.e, the mean part is defined by The fluctuating field is then by assumption Usually normalization condition

24 d and f are the transformation field concentration tensors and the underbar operator is defined as When phases have the constitutive form of Then the solution to the differential equations of continuum mechanics results in a relationship between local (  ) and global (  fields of ,  eigenstresses and eigenstrains respectively) Transformation field theory Dvorak & Benveniste, Dvorak Proc. Math. and Physical Sciences, ) 6. Background for Stochastic Formulation

25 7. Stochastic Formulation: Part 1 Rewriting the localization equation in terms of mean and fluctuating fields The statistics are incorporated through the overbar (mean value - mechanical concentration tensor) and underbar (transformation field concentration tensor) operations By taking the mean of both sides, it can be shown that A= I, and which implies

26 The effective constitutive equations that result are then Stochastic Formulation: Part 1

27 8. Stochastic Formulation: Part 2 Hierarchal Effects It is convenient to extend this approach by further decomposing the fluctuating fields into their phase mean and fluctuating parts. This hierarchal decomposition is based on

28 Hierarchal Effects For a two phase composite, this implies that only the mechanical concentration tensor is needed. Under the additional assumption that the fluctuating parts of the local fields are zero

29 9. Application Window sizes of 7x7 and 11x11 pixels. T300/2510 composite (graphite fibers in polymer matrix). The fibers transversely isotropic. The matrix isotropic. Moving window GMC to develop a field of concentration tensor elements. (Aboudi)

30 PDFs 7 x 7 windowing used to sample A 21 A 31 A 22 A 33 A 23 A 32 A 44 A 55 A 66

31 PDFs 11 x 11 windowing used to sample A 21 A 31 A 22 A 33 A 23 A 32 A 44 A 55 A 66

Comparisons 7 x 7 11 x 11 A 21 A 31 A 22 A 33 A 21 A 31 A 22 A 33

33 Comparisons 7 x 7 11 x 11 A 23 A 32 A 44 A 55 A 66 A 23 A 32 A 44 A 55 A 66

34 GPaReuss7x711 x 11Voigt C C C C C C C C C Stiffness Matrix - Between Bounds

35 Starting point of the analysis : Localization relations based on concentration tensors Statistics incorporated thru the concentration tensors –For 2 phase materials only need the mechanical concentration tensors Can predict the mechanical concentration tensors using only elastic properties of the phases –Statistics independent of the history-dependent models for the phases Hierarchical statistical effects –Simplifies the analysis by decoupling the governing equations 11. Summary

Future Work Generate 3D statistics –Moving windows techniques using different micromechanics models –GMC Study impact of the extremes in the PDFs on predictions of the local and bulk material behavior –Enhanced computational efficiency by simplifying PDFs appropriately Extend STFA to consider debonding and damage Start implicit implementation of STFA into ABAQUS