General Elasticity in 3D: Tensors, , and
Amorphous materials like glass or clay and materials with many small randomly oriented crystal grains (like a typical metal) have no large-scale ordered crystal structure, and so are isotropic (same in all directions).
The elasticity relations we’ve used are fine for these materials, but not for single-crystal materials, which are generally non-isotropic. Non-isotropic materials can:
Copyright By PowCoder代写 加微信 powcoder
Be more or less resistant to shear, so that (e.g., transversely isotropic materials like single-crystal silicon)
2. Have different stiffness (E, , or G) in different directions (x, y, or z) (e.g., orthotropic materials like titanium)
3. Couple axial and shear stress together (e.g., anisotropic materials like quartz)
In this lecture we’ll extend Hooke’s law to the general crystal case of any material.
Materials with complicated crystal structures can relate axial and shear stress and strain differently, for example, xy-shear leading to strain in the z-direction. The stress in any direction is in general a function of the strains in all of the directions:
Where is the stress on the i face acting in the j direction, is the [true] strain of the i face in the j direction, and is the elasticity (AKA stiffness) of the i face in the j direction in response to strains of the k face in the l direction.
Here, the subscripts are written as numbers, so etc. By convention, 1 = x, 2 = y, 3= z. That said, we’ll often write the letters rather than the numbers; . These variables are renamed somewhat from what we used before:
Axial stress in x is now
Shear stress on y in x is now .
Axial strain in x is now
True shear strain on y in x is [still] (recall that this is related to engineering shear strain by )
Review: Matrix Algebra
To multiply a matrix (e.g., 3×3 matrix ) by a vector (e.g., 3×1 column-vector ), you dot each row of the matrix by the vector to form the new vector:
In Maple, you can do this operation using the following commands:
> restart:
with(LinearAlgebra):
M:=Matrix([[1,2,3],[0,1,0],[0,0,1]]); #a matrix
r:=
M.r; #r left-multiplied by M
Using subscripts, you could describe the operation by saying the ith component of the product is:
Einstein summation is a shorthand to imply sums over repeated indices without actually writing them.
e.g., means
The key is the repeated indices in product terms:
This gives unambiguous expressions as long as you don’t write each index more than twice (doesn’t have a defined meaning) and make sure all terms in an expression have the same non-summed indices ( doesn’t make sense, but is OK).
Einstein summation notation is a time saver if you’re writing a lot of summation expressions. Feel free to use this notation to speed things up!
Example: Notation
Write the following sums in Einstein notation:
b) Total derivative: (hint: define )
b) let . Then (in Einstein summation notation)
You know how a vector is a 1-dimensional array of numbers that has certain addition and scalar multiplication operations etc. and a matrix is a 2-dimensional one? The technical term for that concept is that vectors are “rank-1 tensors” while matrices are “rank-2 tensors” (a scalar is a rank-0 tensor).
{By the way, knowing something is a tensor also tells the math aficionado things about how that thing transforms under certain operations.}
The most general linear relation of one vector to another one is through a matrix:
That is, “the most general linear mapping of one rank-1 tensor to another is through a rank-2 tensor”.
Stress and strain are both rank-2 tensors, so the elasticity C or compliance S tensors relating them are consequently rank-4 tensors:
Notice that the rank of a tensor is the number of subscripts you need to specify its components. The 3-by-3 strain matrix is mapped to the 3-by-3 stress matrix via the 3-by-3-by-3-by-3 elasticity tensor (stiffness tensor). Directly writing out a 4-dimensional array of numbers is beyond the 2d medium of writing.
Fortunately, we can use Voigt notation to avoid the problem altogether…
If we take all 9 stress & strain components and put them into 9-by-1 vectors rather than matrices, then the elasticity & compliance tensors relating them become 9-by-9 matrices. In fact, we only need 6-by-1 vectors and 6-by-6 matrices, since it’s still true that (at least locally) the shear stress & strain are symmetric under index swaps: , , , etc.
This way of writing reduced tensors as vectors is called Voigt notation.
/*** Notice that the shear strain terms are now multiplied by 2; we can see why by multiplying row 1 of the matrix equation to compare with our previous :
Now, replacing xyz with 123,
and since the shear terms are symmetric locally (i.e., and ), we have:
So it works! ***/
Since Voigt notation uses a stiffness matrix rather than rank-4 tensor, it’s more convenient to re-define stiffness in terms a 2-index set of coefficients C:
so that now Hooke’s law reads:
Clarification:
Recall that true shear strains are related to the engineering shear strain via . Since we defined the tensor version of Hooke’s law using true strain , and specified that the Voigt notation stiffness matrix C terms should equal their corresponding tensor stiffness c terms, this means that Voigt notation [the notation everyone actually uses unless perhaps they’re doing a mathematical proof] uses engineering strain:
Compliance Matrix:
Even in Voigt notation, the inverse of the stiffness matrix C is the compliance matrix S:
Example: Isotropic Stiffness Matrices
For an isotropic material, we know that:
Where . Using this information, determine the components of the 6×6 stiffness matrix in terms of E and .
From these equations, shear and axial stress & strain are not coupled for an isotropic material: axial stress depends only on axial strain and not shear strain, while each shear stress term only depends on one kind of shear strain! The stiffness matrix is more like two 3×3 matrices with zeroes around them, and the shear matrix is diagonal:
Follow-up Example: Isotropic Compliance Matrix:
For an isotropic material, we know that:
Where . Using this information, determine the components of the 6×6 compliance matrix S in terms of E and .
Similarly:
(We could also have inverted the stiffness matrix we found earlier)
Maple version of finding the compliance matrix
To find the compliance matrix and explore how stress and strain relate to eachother in Voigt notation use syntax like this:
> restart:
with(LinearAlgebra):
C11:=E*(1-nu)/((1+nu)*(1-2*nu));
C12:=E*nu/((1+nu)*(1-2*nu));
C44:=E/(2*(1+nu));
C:=Matrix([
[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]]);
epsilon:=
sigma:=C.epsilon;
S:=simplify(MatrixInverse(C));
Non-isotropic Materials
Since isotropic materials don’t couple shear stress to axial strain or axial stress to shear strain using the stress-strain tensors or Voigt notation presented here is unnecessary. Fortunately, the world is more interesting than that, and not all materials are isotropic.
One of the most abundant minerals on Earth is quartz (sand is microscopic quartz crystals), which [in its naturally occurring “alpha” crystal form] has this compliance matrix:
Look at all that beautiful anisotropy! (the matrix itself is still symmetric).
Many materials fall somewhere between being perfectly isotropic and as anisotropic as quartz. For instance, an orthotropic material is like an isotropic material but with different Young’s moduli, Poisson’s ratios, and shear moduli depending on which axis (or pairs of axes) we’re looking at:
A subset of orthotropic materials are the transversely orthotropic materials, which are symmetric in x & y but not in z:
Hexagonal close-packed crystals are transversely orthotropic. Here are some common materials that have this crystal structure:
Taken from http://solidmechanics.org/text/Chapter3_2/Chapter3_2.htm which took them from Freund and Suresh, Thin Film Materials, CUP 2003, who list the original sources for their data on page 163. z here is defined to be perpendicular to the basal (0001) crystal plane. x has no special direction besides being perpendicular to z, since it’s symmetric in the xy plane.
Note: above means that the shear moduli for is the same as for , but there’s a different one for . This is again because the z-axis is the outlier but x & y are the same.
Finally, cubic crystal materials (i.e., FCC, BCC, etc.) with the x, y, & z axes perpendicular to the crystal symmetry planes are not necessarily isotropic either because in general . These materials still have the same compliance matrix form:
And stiffness matrix form:
Where , , etc.
Here are the parameters of some common materials in single crystal form:
Taken from http://solidmechanics.org/text/Chapter3_2/Chapter3_2.htm which modified it from Simmons and Wang ‘Single Crystal Elastic Constants and Calculated Aggregate Properties’ MIT Press (1970). (dc means diamond cubic). determines how far from isotropic the material behaves. A= 1 means it’s isotropic, A < 1 means it's less resistant to shear, A > 1 means more resistant to shear.
Example: in FlexPDE
Modify one of your FlexPDE scripts to use Voigt notation to couple shear to axial stress & strain.
Using the beam extension code, the new definition section needs to include all of the C components, use Engineering strain rather than true strain, and do the 6×6 matrix multiplication as the stress definition:
C11 =E*(1-nu)/(1+nu)/(1-2*nu)
C12 = E*nu/(1+nu)/(1-2*nu)
C14=0 !isotropic material
C44=G !isotropic material
C21=C12 !matrix is symmetric
!Axial Strain
!Engineering
gxy=dx(v)+dy(u)
gyz=dy(w)+dz(v)
gxz=dz(u)+dx(w)
!!Stress via Hooke’s law:
sx = C11*ex+C12*ey+C13*ez+C14*gyz+C15*gxz+C16*gxy
sy = C21*ex+C22*ey+C23*ez+C24*gyz+C25*gxz+C26*gxy
sz = C31*ex+C32*ey+C33*ez+C34*gyz+C35*gxz+C36*gxy
syz=C41*ex+C42*ey+C43*ez+C44*gyz+C45*gxz+C46*gxy
sxz=C51*ex+C52*ey+C53*ez+C54*gyz+C55*gxz+C56*gxy
sxy=C61*ex+C62*ey+C63*ez+C64*gyz+C65*gxz+C66*gxy
Running the rest of the code, it still works!
Example: Quartz Stiffness Matrix
Above, we had the compliance matrix for quartz:
However, our FlexPDE code needs the stiffness matrix components. Use Maple to determine the stiffness matrix for quartz.
with(LinearAlgebra):
S:=Matrix([
[12.77,-1.79,-1.22,-4.5,0,0],
[-1.79,12.77,-1.22,4.5,0,0],
[-1.22,-1.22,9.6,0,0,0],
[-4.5, 4.5, 0, 20.04, 0,0],
[0,0,0,0,20.04,-9],
[0,0,0,0,-9, 29.1]]);
C:=MatrixInverse(S);
(Since S was in pm/m-Pa, C is in TPa). The “e-19” components are Maple saying “0”.
Using string concatenation and sequence looping we can create an output that makes it a lot easier to put this into FlexPDE directly too:
> restart:
with(LinearAlgebra):
S:=Matrix([
[12.77,-1.79,-1.22,-4.5,0,0],
[-1.79,12.77,-1.22,4.5,0,0],
[-1.22,-1.22,9.6,0,0,0],
[-4.5, 4.5, 0, 20.04, 0,0],
[0,0,0,0,20.04,-9],
[0,0,0,0,-9, 29.1]]); #in 1e-12/Pa
Cmatrix:=MatrixInverse(S)*1e12; #in Pa
seq(seq(cat(C,n,j)=Cmatrix[n,j],n=1..6),j=1..6); #Magic!
Effective Stiffness
Recall that we originally defined a Young’s modulus as the proportionality constant between applied stress and resulting strain (when the material is otherwise free): . For an asymmetric material, we need to be specific about which axis we’re applying this on, as we’ll find different Young’s moduli in different directions. For example, the resulting axial x-strain from an applied x-stress to an otherwise free material is:
And since we defined for this loading condition (i.e., placing a uniform axial stress on an otherwise free material), therefore .
Similarly, we’ll find stiffness in y and z are .
Meanwhile shear moduli were defined as the constant relating applied shear stress and shear strain for a material in pure shear and otherwise free: , which means .
With these definitions in mind, it’s often useful to talk about the effective stiffness in a direction for a given set of boundary conditions. e.g., when a material is fixed to have no strain in any direction except axial x-strain (perhaps it’s being compressed in x by a press that keeps the x-faces parallel AND is prevented from expanding in any y or z direction) then we’ll find that axial x stress and strain are instead related through:
That is, ” when the material is otherwise fixed.”
Concluding Remarks
Equipped with Voigt notation we’re able to deal with complicated non-isotropic materials, which substantially expands the number of different problems we can solve with the skills of this course. In particular, this equips us to understand and analyze piezoelectric materials, which we’ll return to in a big way in ENGPHYS 2CM4 to design sensors and actuators.
Despite the large number of problems we’ve solved in 2P04, this is definitely not the end of the story. One thing that we haven’t dealt with in 2P04 is materials that break Hooke’s law, including failure analysis. However, with the content of 2P04 understood, you’re better equipped than you were before to explore and investigate this and other topics in mechanics. For a brief overview, have a look at this video: https://www.youtube.com/watch?v=xkbQnBAOFEg
Thanks for watching!
111111221133112311131112
xxxxyyzzyzzyxzzxxyyx
seeeeeeeee
1111111111222211333311232332111313311112
seeeeeeeee
1111111111222211333311232311323211131311
313111121211212111
cccccccccc
seeeeeeeeee
=++++++++=
111213141516111111221133112311131112
212223242526221122222233222322132212
31323334353633113322
414243444546
515253545556
616263646566
CCCCCCcccccc
CCCCCCcccccc
3333332333133312
231123222333232323132312
131113221333132313131312
121112221233122312131212
111213141516
212223242526
313233343536
414243444546
515253545556
616263646566
111213141516
212223242526
313233343536
414243444546
515253545556
616263646566
程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com