The key to the overall macroscopic elastic and plastic properties of a polycrystal is the precise knowledge about the local stress and strain in its individual constituent single crystals. As the first step, The averaged stress or strain and elastic strain energy or plastic flow potential of the polycrystal are estimated using the Reuss/Voigt and Sachs/Taylor models to obtain its overall elastic and plastic responses. In the future work, linear finite element methods will be applied to analyze the elastic deformation of a representative volume element of the polycrystal made of hundreds and thousands of single crystal grains. Nonlinear finite element methods will be developed and applied to the same polycrystal to investigate its overall plastic deformation behavior. Results of both finite element analyses will be compared with the lower and upper bounds estimates presented here.
1. Introduction to computational homogenization of polycrystals
1. Introduction to computational homogenization of polycrystals
Engineering metals and alloys are polycrystalline solids. Mechanics (macro-mechanics) and micro-mechanics of polycrystalline materials are analogous to the two subjects of thermodynamics and statistical mechanics in physics that deal with macroscopic and microscopic worlds respectively. Macroscopic mechanical response of a polycrystalline material is intrinsically linked to the collective behaviors of its micro scale single crystal constituents. This module teaches about the idea of homogenization and how to obtain overall estimates of effective properties of a polycrystal from its constituent single crystals.
1.1 The homogenization problem
1.1 The homogenization problem
One is often not concerned with the local stress and strain in detail at all but only asks for the overall effective or macroscopic response of the material which is much more predictive. In this case, micro-mechanics of polycrystal is significantly simplified. The task is narrowed down towards developing some analytical, semi-analytical, or computational homogenization methods (also called mean-field theories or theories of effective properties) to obtain various estimates of macroscopic properties of a polycrystalline material.
1.2 The representative volume element (RVE) and Hill’s lemma
1.2 The representative volume element (RVE) and Hill’s lemma
The starting point of micro-mechanics modeling of a polycrystal is to introduce the so-called representative volume element (RVE) of the material which will be modeled as a continuum element at the macro scale. What is a representative volume element then? According to Hill [1]:
“This phrase will be used when referring to a sample that (a) is structurally entirely typical of the whole mixture on average, and (b) contains a sufficient number of inclusions for the apparent overall moduli to be effectively independent of the surface values of traction and displacement, so long as these values are ‘macroscopically uniform.’ That is, they fluctuate about a mean with a wavelength small compared with the dimensions of the sample, and the effects of such fluctuations become insignificant within a few wavelengths of the surface. The contribution of this surface layer to any average can be made negligible by taking the
sample large enough.”
sample large enough.”
To derive the overall stress-strain response of a composite by homogenization using any particular volume-averaging method, one relies on Hill’s lemma or strain energy condition [1], also called the macro-homogeneity or Hill–Mandel condition. It is one of the basic principles in micro-mechanics modeling of statistically representative volume of a composite material.
1.3 Lower and upper bound estimates
1.3 Lower and upper bound estimates
One of the central problems in micro-mechanics of polycrystals has been to find better estimates of effective macroscopic response of a given micro-structured material. This problem can be reformulated as a problem of finding better upper and lower bounds of the estimates between which the actual material response sits. That is, narrower the difference between upper and lower bound estimates is, better the modeling results are.
Two simplest homogenization models of a polycrystal are 1) the uniform stress or lower bound model which assumes that the stress is the same everywhere throughout the volume of a material element and 2) the uniform strain or upper bound model which assumes that the strain (rate) is the same everywhere throughout the volume of a material element.
Increasingly, computational micro-mechanics or homogenization is becoming a powerful and valuable tool for possible better estimates of the effective properties of a polycrystal by obtaining a more detailed full-field solutions of elastic-plastic deformation of a RVE with microscale heterogeneities in both stress and strain.
1.4 Teaching and learning objectives
1.4 Teaching and learning objectives
This module aims to teach students about the concept of homogenization in materials modeling of polycrystals. The specific learning objectives are for the students to learn 1) the difference of describing the elastic-plastic properties in terms of the local single crystal lattice frame versus the global polycrystal texture frame, 2) how to obtain the lower and upper bound estimates on the elastic moduli of a polycrystal, 3) how to obtain the lower and upper bound estimates on the yield strength of a polycrystal, and 4) possible ways to refine those estimates using finite element based virtual testing of the polycrystals.
2. Reuss and Voigt linear elasticity models of a polycrystal
2. Reuss and Voigt linear elasticity models of a polycrystal
This part of the teaching module on computational micro-mechanics of composites will a computational supplementary material to Chapter 8 of a graduate course titled “advanced mechanical behaviors of materials” at SMU. The result will illustrate how heterogeneous elastic properties of a polycrystal under iso-stress and iso-strain conditions can be calculated and averaged to obtain its overall response.
This part of the teaching module on computational micro-mechanics of composites will a computational supplementary material to Chapter 8 of a graduate course titled “advanced mechanical behaviors of materials” at SMU. The result will illustrate how heterogeneous elastic properties of a polycrystal under iso-stress and iso-strain conditions can be calculated and averaged to obtain its overall response.
Basic steps:
1. Construct a model of a polycrystal microstructure with a given orientation distribution (e.g., a random polycrystal).
2. For each single grain of the polycrystal, compute its coordinate transformation matrix from the global (polycrystal) texture frame to the local single crystal lattice frame based on either Miller indices or Euler angles.
3. Define the quadratic elastic energy function of the single crystal in terms of stress (Reuss model) or strain (Voigt model) in the global coordinates
4. Obtain the quadratic elastic energy function of the polycrystal as the volume weighted averaging of the elastic energy of all single crystal grains
5. Extract the coefficients of the polycrystalline quadratic elastic energy function to form both its stiffness matrix and compliance matrix.
6. Compute the axial strain of the polycrystal under uniaxial tension and thus its Young’s modulus for both Reuss and Voigt models.
1. Construct a model of a polycrystal microstructure with a given orientation distribution (e.g., a random polycrystal).
2. For each single grain of the polycrystal, compute its coordinate transformation matrix from the global (polycrystal) texture frame to the local single crystal lattice frame based on either Miller indices or Euler angles.
3. Define the quadratic elastic energy function of the single crystal in terms of stress (Reuss model) or strain (Voigt model) in the global coordinates
4. Obtain the quadratic elastic energy function of the polycrystal as the volume weighted averaging of the elastic energy of all single crystal grains
5. Extract the coefficients of the polycrystalline quadratic elastic energy function to form both its stiffness matrix and compliance matrix.
6. Compute the axial strain of the polycrystal under uniaxial tension and thus its Young’s modulus for both Reuss and Voigt models.
2.1 Generation of grain orientations of a polycrystal
2.1 Generation of grain orientations of a polycrystal
Per convention, an ideal texture component (or a disoriented crystal) is specified in terms of the Miller indices for both ND or normal direction and RD or rolling direction (z -axis and x -axis) of a polycrystalline sheet metal as {hkl}<uvw>. As a rolled sheet metal is typically orthotropic so each component will have four symmetry variants: (hkl)[uvw], (-h-k-l)[uvw], (hkl)[-u-v- w], and (-h-k-l))[-u-v-w].
Alternatively, the orientation of each grain in a crystalline aggregate can also be specified in terms of three Euler angles (there are several equivalent variants of Euler angles such as Bunge or Matthies types). In the following, the Miller indices will be used for specifying the single crystal orientations.
Alternatively, the orientation of each grain in a crystalline aggregate can also be specified in terms of three Euler angles (there are several equivalent variants of Euler angles such as Bunge or Matthies types). In the following, the Miller indices will be used for specifying the single crystal orientations.
2.2 Coordinate transformation matrix for each single crystal grain
2.2 Coordinate transformation matrix for each single crystal grain
There are some major texture components commonly observed in FCC polycrystals such as aluminum and copper. Four of them are
1) Goss texture is defined as {110}<001>: G1 variant x=(0,0,1), z=(1,1,0), y=z cross-product x, then G2, G3, G4
1) Goss texture is defined as {110}<001>: G1 variant x=(0,0,1), z=(1,1,0), y=z cross-product x, then G2, G3, G4
In[]:=
ez={1,1,0}/Sqrt[2];ex={0,0,1};ey=Cross[ez,ex];Goss1=Transpose[{ex,ey,ez}];
In[]:=
ez={-1,-1,0}/Sqrt[2];ex={0,0,1};ey=Cross[ez,ex];Goss2=Transpose[{ex,ey,ez}];
In[]:=
ez={1,1,0}/Sqrt[2];ex={0,0,-1};ey=Cross[ez,ex];Goss3=Transpose[{ex,ey,ez}];
In[]:=
ez={-1,-1,0}/Sqrt[2];ex={0,0,-1};ey=Cross[ez,ex];Goss4=Transpose[{ex,ey,ez}];
In[]:=
{Goss1//MatrixForm,Goss2//MatrixForm,Goss3//MatrixForm,Goss4//MatrixForm}
Out[]=
,
,
,
0 | 1 2 | 1 2 |
0 | - 1 2 | 1 2 |
1 | 0 | 0 |
0 | - 1 2 | - 1 2 |
0 | 1 2 | - 1 2 |
1 | 0 | 0 |
0 | - 1 2 | 1 2 |
0 | 1 2 | 1 2 |
-1 | 0 | 0 |
0 | 1 2 | - 1 2 |
0 | - 1 2 | - 1 2 |
-1 | 0 | 0 |
2) copper texture {112}<111>: C1 variant: (112)[11-1] so x=(1,1,-1), z=(1,1,2), y=z cross-product x, then C2, C3, C4
In[]:=
ez={1,1,2}/Sqrt[6];ex={1,1,-1}/Sqrt[3];ey=Cross[ez,ex];Copper1=Transpose[{ex,ey,ez}];
In[]:=
ez={-1,-1,-2}/Sqrt[6];ex={1,1,-1}/Sqrt[3];ey=Cross[ez,ex];Copper2=Transpose[{ex,ey,ez}];
In[]:=
ez={1,1,2}/Sqrt[6];ex={-1,-1,1}/Sqrt[3];ey=Cross[ez,ex];Copper3=Transpose[{ex,ey,ez}];
In[]:=
ez={-1,-1,-2}/Sqrt[6];ex={-1,-1,1}/Sqrt[3];ey=Cross[ez,ex];Copper4=Transpose[{ex,ey,ez}];
In[]:=
{Copper1//MatrixForm,Copper2//MatrixForm,Copper3//MatrixForm,Copper4//MatrixForm}
Out[]=
,
,
,
1 3 | - 1 2 | 1 6 |
1 3 | 1 2 | 1 6 |
- 1 3 | 0 | 2 3 |
1 3 | 1 2 | - 1 6 |
1 3 | - 1 2 | - 1 6 |
- 1 3 | 0 | - 2 3 |
- 1 3 | 1 2 | 1 6 |
- 1 3 | - 1 2 | 1 6 |
1 3 | 0 | 2 3 |
- 1 3 | - 1 2 | - 1 6 |
- 1 3 | 1 2 | - 1 6 |
1 3 | 0 | - 2 3 |
3) brass texture {110}<-112>: B1 variant =(110)[-112] so x=[-1,1,2], y=[-1,1,-1], z=[1,1,0], then B2, B3, B4
In[]:=
ez={1,1,0}/Sqrt[2];ex={-1,1,2}/Sqrt[6];ey=Cross[ez,ex];Brass1=Transpose[{ex,ey,ez}];
In[]:=
ez={-1,-1,0}/Sqrt[2];ex={-1,1,2}/Sqrt[6];ey=Cross[ez,ex];Brass2=Transpose[{ex,ey,ez}];
In[]:=
ez={1,1,0}/Sqrt[2];ex={1,-1,-2}/Sqrt[6];ey=Cross[ez,ex];Brass3=Transpose[{ex,ey,ez}];
In[]:=
ez={-1,-1,0}/Sqrt[2];ex={1,-1,-2}/Sqrt[6];ey=Cross[ez,ex];Brass4=Transpose[{ex,ey,ez}];
In[]:=
{Brass1//MatrixForm,Brass2//MatrixForm,Brass3//MatrixForm,Brass4//MatrixForm}
Out[]=
,
,
,
- 1 6 | 1 3 | 1 2 |
1 6 | - 1 3 | 1 2 |
2 3 | 1 3 | 0 |
- 1 6 | - 1 3 | - 1 2 |
1 6 | 1 3 | - 1 2 |
2 3 | - 1 3 | 0 |
1 6 | - 1 3 | 1 2 |
- 1 6 | 1 3 | 1 2 |
- 2 3 | - 1 3 | 0 |
1 6 | 1 3 | - 1 2 |
- 1 6 | - 1 3 | - 1 2 |
- 2 3 | 1 3 | 0 |
4) strange type 1 texture {123}<63-4>: S1=(123)[63-4] so x=[6,3,-4], y=[-17,22,-9], z=[1,2,3], then S2, S3, S4
In[]:=
ez={1,2,3}/Sqrt[14];ex={6,3,-4}/Sqrt[36+9+16];ey=Cross[ez,ex];Strange1=Transpose[{ex,ey,ez}];
In[]:=
ez={-1,-2,-3}/Sqrt[14];ex={6,3,-4}/Sqrt[36+9+16];ey=Cross[ez,ex];Strange2=Transpose[{ex,ey,ez}];
In[]:=
ez={1,2,3}/Sqrt[14];ex={-6,-3,4}/Sqrt[36+9+16];ey=Cross[ez,ex];Strange3=Transpose[{ex,ey,ez}];
In[]:=
ez={-1,-2,-3}/Sqrt[14];ex={-6,-3,4}/Sqrt[36+9+16];ey=Cross[ez,ex];Strange4=Transpose[{ex,ey,ez}];
In[]:=
{Strange1//MatrixForm,Strange2//MatrixForm,Strange3//MatrixForm,Strange4//MatrixForm}
Out[]=
,
,
,
6 61 | - 17 854 | 1 14 |
3 61 | 11 2 427 | 2 7 |
- 4 61 | - 9 854 | 3 14 |
6 61 | 17 854 | - 1 14 |
3 61 | -11 2 427 | - 2 7 |
- 4 61 | 9 854 | - 3 14 |
- 6 61 | 17 854 | 1 14 |
- 3 61 | -11 2 427 | 2 7 |
4 61 | 9 854 | 3 14 |
- 6 61 | - 17 854 | - 1 14 |
- 3 61 | 11 2 427 | - 2 7 |
4 61 | - 9 854 | - 3 14 |
2.3 Quadratic elastic energy functions of each single crystal grain
2.3 Quadratic elastic energy functions of each single crystal grain
For now, we consider only cubic single crystals such as BCC and FCC types. Its elastic strain energy function is given may be computed from the stiffness matrix as stress:strain/2 where stress = C:strain (in the Voigt matrix form for the tensors). Similar, one can express the complementary elastic energy function in terms of stress using the compliance matrix stress:strain/2 where strain=S:stress. Both the stiffness matrix C and compliance matrix S are a 6-by-6 symmetric one with three unique entries for a cubic crystal (and they are inverse to each other):
In[]:=
Clear[c11,c12,c44,s11,s12,s44];Ccubic={{c11,c12,c12,0,0,0},{c12,c11,c12,0,0,0},{c12,c12,c11,0,0,0},{0,0,0,c44,0,0},{0,0,0,0,c44,0},{0,0,0,0,0,c44}};
In[]:=
Scubic={{s11,s12,s12,0,0,0},{s12,s11,s12,0,0,0},{s12,s12,s11,0,0,0},{0,0,0,s44,0,0},{0,0,0,0,s44,0},{0,0,0,0,0,s44}};{Ccubic//MatrixForm,Scubic//MatrixForm}
Out[]=
,
c11 | c12 | c12 | 0 | 0 | 0 |
c12 | c11 | c12 | 0 | 0 | 0 |
c12 | c12 | c11 | 0 | 0 | 0 |
0 | 0 | 0 | c44 | 0 | 0 |
0 | 0 | 0 | 0 | c44 | 0 |
0 | 0 | 0 | 0 | 0 | c44 |
s11 | s12 | s12 | 0 | 0 | 0 |
s12 | s11 | s12 | 0 | 0 | 0 |
s12 | s12 | s11 | 0 | 0 | 0 |
0 | 0 | 0 | s44 | 0 | 0 |
0 | 0 | 0 | 0 | s44 | 0 |
0 | 0 | 0 | 0 | 0 | s44 |
Here are the two elastic energy functions for a cubic crystal in terms of stress and strain expressed in the cubic lattice coordinates
In[]:=
Clear[sigma,sigma11,sigma22,sigma33,sigma23,sigma31,sigma12];sigma={sigma11,sigma22,sigma33,sigma23,sigma31,sigma12};
In[]:=
ElasticEnergyS=sigma.Scubic.sigma/2;Simplify[ElasticEnergyS]
Out[]=
1
2
2
sigma12
2
sigma23
2
sigma31
2
sigma11
2
sigma22
2
sigma33
In[]:=
Clear[epsilon];epsilon={epsilon11,epsilon22,epsilon33,gamma23,gamma31,gamma12};
In[]:=
ElasticEnergyC=epsilon.Ccubic.epsilon/2;Simplify[ElasticEnergyC]
Out[]=
1
2
2
epsilon11
2
epsilon22
2
epsilon33
2
gamma12
2
gamma23
2
gamma31
2.4 Volume weighted averaged elastic energy functions of a polycrystal
2.4 Volume weighted averaged elastic energy functions of a polycrystal
Now we will consider some simple examples of textured polycrystals. For example, an aluminium alloy (Al–0.55Mn–0.5Mg) AA3104 is reported to consist of five major texture components of cube, Goss, copper, brass and strange type 1 [3]. One computes first the elastic energy function for each texture component under the applied Cauchy stress in the XYZ polycrystal frame:
In[]:=
CauchyXYZ={{Sx,Txy,Txz},{Txy,Sy,Tyz},{Txz,Tyz,Sz}};
In[]:=
sigmaCube={CauchyXYZ[[1,1]],CauchyXYZ[[2,2]],CauchyXYZ[[3,3]],CauchyXYZ[[2,3]],CauchyXYZ[[3,1]],CauchyXYZ[[1,2]]};
In[]:=
CubeElasticEnergyS=Simplify[sigmaCube.Scubic.sigmaCube/2]
Out[]=
1
2
2
Sx
2
Sy
2
Sz
2
Txy
2
Txz
2
Tyz
In[]:=
Cauchy123=Goss1.CauchyXYZ.Transpose[Goss1];sigmaGoss1={Cauchy123[[1,1]],Cauchy123[[2,2]],Cauchy123[[3,3]],Cauchy123[[2,3]],Cauchy123[[3,1]],Cauchy123[[1,2]]};
Per Reuss iso-stress polycrystal model, one can obtain the elastic energy function of stress for the AA3104 polycrystal as volume weighted average.
Per Reuss iso-stress polycrystal model, one can obtain the elastic energy function of stress for the AA3104 polycrystal as volume weighted average.
2.5 Elastic stiffness and compliance matrices of the polycrystal
2.5 Elastic stiffness and compliance matrices of the polycrystal
Each entry in the compliance matrix of the Reuss model for an orthotropic polycrystal is obtained via the second order partial derivative of the elastic energy function.
Let’s assume Their volume fractions are 12%, 10%, 20%, 25% and 33%. Elastic equations
Let’s assume Their volume fractions are 12%, 10%, 20%, 25% and 33%. Elastic equations
Not clear to me whether it is better to operate the derivatives through first, or wait and let NDSolve to it.
Young’s moduli along three axes of the orthotropic polycrystals are given as:
2.6 Numerical results for elastic Young’s modulus of the polycrystal
2.6 Numerical results for elastic Young’s modulus of the polycrystal
Let’s assume the volume fractions of five texture components are 12%, 10%, 20%, 25% and 33% (the sum of the total volume fractions should be 100%)
For an aluminum single crystal, one have the reference values for s11, s12 and s44 as (in the unit of 1/GPa) [4]
So its estimated compliance matrix is given as (in the unit of 1/GPa)
and its estimated stiffness matrix is given as (in the unit of GPa)
The Young’s moduli along three axes of the orthotropic polycrystals and the cubic single crystal are:
It is noted that the Young’s modulus of a random polycrystal is about 70 GPa which is higher than the Young’s modulus of the single crystal along its axes at about 63 GPa.
3. Sachs and Taylor quadratic plasticity models of polycrystals
3. Sachs and Taylor quadratic plasticity models of polycrystals
3.1 Quadratic plastic potential functions of each single crystal grain
3.1 Quadratic plastic potential functions of each single crystal grain
Quadratic plastic potential may be used for modeling anisotropic plasticity of cubic single crystals such as BCC and FCC types. Its plastic potential function in stress (the yield function) or its plastic potential function in plastic strain rate (the strain-rate function) are simply given as [3]:
where the first three terms are so-called deviatoric axial stresses and the parameter alpha may vary from 1/3 to 1/2.
3.2 Volume weighted averaged plastic potential functions of a polycrystal
3.2 Volume weighted averaged plastic potential functions of a polycrystal
For the aluminium alloy (Al–0.55Mn–0.5Mg) AA3104 with five major texture components of cube, Goss, copper, brass and strange type 1, one computes first the yield function for each texture component under the applied Cauchy stress in the XYZ polycrystal frame:
Per Sachs iso-stress polycrystal model, one can obtain the plastic potential function of stress for the AA3104 polycrystal as volume weighted average.
Per Sachs iso-stress polycrystal model, one can obtain the plastic potential function of stress for the AA3104 polycrystal as volume weighted average.
3.3 Yield strength of the polycrystal
3.3 Yield strength of the polycrystal
The coefficients of the quadratic yield function are
Yield strengths of the orthotropic polycrystals along x-axis and y-axis are given as:
3.4 Numerical results for the polycrystal AA3104 sheet
3.4 Numerical results for the polycrystal AA3104 sheet
Let’s assume the volume fractions of five texture components are 12%, 10%, 20%, 25% and 33% (the sum of the total volume fractions should be 100%)
Single crystal yield strength is
For an aluminum single crystal, one has alpha and polycrystal yield strength in MPa as
So its estimated yield strengths (in the unit of MPa)
For random polycrystals, the yield strength is known to be about 1.732 times of that of a single crystal. It is noted that the yield strength depends weakly on the parameter alpha and the yield strength is about the same in all three axial directions in this case.
4. Concluding remarks
4. Concluding remarks
By using the Mathematica Notebook, the topic on micro-mechanics of elastic-plastic polycrystals is presented in a computational essay format. Reuss and Voigt elasticity models as well as Sachs and Taylor plasticity models have successfully applied to, a textured rolled aluminum sheet metals. The Young’s modulus and yield strength of the sheet metal are estimated accordingly. Further improvements of the homogenized polycrystal model may be obtained by using both linear and nonlinear finite element methods to enhanced the prediction of the overall elastic-plastic behavior of polycrystals for practical engineering applications. To complete such a task inside the Mathematica Notebook, material models based on general anisotropic elasticity and plasticity for a single crystal will be developed first.
Keywords
Keywords
◼
Polycrystalline materials
◼
Textured sheet metals
◼
Homogenization methods
Acknowledgment
Acknowledgment
WT would like to thanks both Paul Abbott and W Craig Carter for the guidance and help in working through the project. Their encouragement, patience and willingness in sharing their vast experience are greatly appreciated.
References
References
◼
[1] Hill R (1963). Elastic properties of reinforced solids: some theoretical principles. Journal of Mechanics and Physics of Solids 11: 357–372.
◼
[2] Hill R (1952). The elastic behaviour of a crystalline aggregate. Proceedings of the Physical Society. Section A 65: 349–354.
◼
[3] Darrieulat M and Montheillet F (2003). A texture based continuum approach for predicting the plastic behaviour of rolled sheet. International Journal of Plasticity 19: 517–546.
◼
[4] Hosford WF (1993). The Mechanics of Crystals and Textured Polycrystals. Oxford University Press, Oxford, UK.