Defining Tensorial Properties

A material property is anisotropic when its response depends on direction. A tensor stores that directional dependence and relates physical quantities. For example, elastic stiffness relates strain to stress.

The rank of a tensor is the number of indices needed to name one component. In three dimensions, a rank \(r\) tensor has \(3^r\) components before physical and crystal symmetries are applied. Rank is therefore not the size of a matrix.

MTEX represents every rank with tensor. Physically specific subclasses add units, conventions, and operations. Read Vectors first if vector components are new. Crystal Reference System explains how a crystal lattice is attached to Cartesian axes.

plottingConvention.default('y↑→x');

Scalars: rank zero

A scalar has no directional index. Temperature and density are examples. The displayed object confirms that MTEX stores one coefficient at rank zero.

M = 5;
T0 = tensor(M,'rank',0)

T0 = tensor (y↑→x)
  rank: 0 ()
 
 5

Vectors: rank one

A vector has one component for each axis of its reference frame. The column below contains the components along x, y, and z.

T1 = tensor([1;2;3],'rank',1)

T1 = tensor (y↑→x)
  rank: 1 (3)
 
 1
 2
 3

Rank one tensors and vector3d objects can be converted into each other. The output below is a vector3d rather than a tensor.

T1x = tensor(vector3d.X);
v = vector3d(T1x)

v = vector3d (y↑→x)
  x y z
  1 0 0

Matrices: rank two

A rank two tensor has two indices and is represented by a \(3\times3\) component matrix. For the stress tensor, one index selects a traction component and the other identifies the normal to the plane on which it acts.

\[ \sigma_{ij} = \left[\begin{array}{ccc} \sigma_{11} & \sigma_{12} & \sigma_{13} \\ \sigma_{21} & \sigma_{22} & \sigma_{23} \\ \sigma_{31} & \sigma_{32} & \sigma_{33} \end{array}\right]. \]

A diagonal rank two tensor has no coupling between different axes.

T2diagonal = tensor(diag([1,2,3]),'rank',2);

Off-diagonal entries describe coupling between different axes. The display reports both the rank and all nine components.

M = [1     0.75  0.5;...
     0.75  1     0.25;...
     0.5   0.25  1];

T2 = tensor(M,'rank',2)

T2 = tensor (y↑→x)
  rank: 2 (3 × 3)
 
    1 0.75  0.5
 0.75    1 0.25
  0.5 0.25    1

Piezoelectricity: rank three

The direct piezoelectric effect relates a rank two stress to a rank one electric displacement through a rank three tensor \(d\):

\[ D_i = d_{ijk}\,\sigma_{jk}. \]

A general rank three tensor has \(3^3=27\) components. Since stress is symmetric, \(d_{ijk}=d_{ikj}\) and the last two indices can be stored as six columns rather than a \(3\times3\times3\) array.

The following coefficients are a compact right-handed \(\alpha\)-quartz example. 'DoubleConvention' says that columns 4 to 6 contain twice the corresponding off-diagonal tensor components.

M = [-1.9222  1.9222    0   -0.1423     0         0;...
          0        0     0       0     0.1423    3.8444;...
          0        0     0       0       0         0];

csQuartz = crystalSymmetry('32',[4.916 4.916 5.4054],...
  'X||a','Z||c','mineral','Quartz');

d = tensor(M,csQuartz,'rank',3,'DoubleConvention',...
  'name','piezoelectric strain','unit','pC/N')

d = tensor (Quartz)
  unit            : pC/N         
  rank            : 3 (3 × 3 × 3)
  doubleConvention: true         
 
  tensor in compact matrix form:
 -1.9222  1.9222       0 -0.1423       0       0
       0       0       0       0  0.1423  3.8444
       0       0       0       0       0       0

Elastic stiffness: rank four

Linear elasticity relates the symmetric strain tensor \(\varepsilon\) to the symmetric stress tensor \(\sigma\) through the stiffness tensor \(C\). The inverse relation uses the compliance tensor \(S\):

\[ \sigma_{ij}=C_{ijkl}\,\varepsilon_{kl}, \qquad \varepsilon_{ij}=S_{ijkl}\,\sigma_{kl}. \]

Four indices give \(3^4=81\) components before symmetry is considered. Symmetric stress and strain reduce the component matrix to \(6\times6\). An elastic strain-energy function reduces the independent coefficients to 21 for triclinic symmetry, and crystal symmetry can reduce them further. A cubic stiffness has only \(C_{11}\), \(C_{12}\), and \(C_{44}\) independent.

M = [320   50  50   0   0   0;...
      50  320  50   0   0   0;...
      50   50 320   0   0   0;...
       0    0   0  64   0   0;...
       0    0   0   0  64   0;...
       0    0   0   0   0  64];

csCubic = crystalSymmetry('m-3m');
C = stiffnessTensor(M,csCubic)

C = stiffnessTensor (crystal)
  unit: GPa              
  rank: 4 (3 × 3 × 3 × 3)
 
  tensor in Voigt matrix representation:
 320  50  50   0   0   0
  50 320  50   0   0   0
  50  50 320   0   0   0
   0   0   0  64   0   0
   0   0   0   0  64   0
   0   0   0   0   0  64

A tensor needs a reference frame

A reference frame is the coordinate system in which data are expressed. It is distinct from symmetry and from the plotting convention that lays the frame out on screen.

The crystal symmetries supplied to d and C also carry crystal frames. For d, the alignment options state which Cartesian axes the published quartz coefficients use. Crystal symmetry states the point group under which a property is invariant. It is attached to a reference frame but is not the frame itself.

Units and compact-matrix conventions are equally part of the data. A plausible component matrix in the wrong frame or convention still gives a plausible but physically wrong result.

Seeing the cubic symmetry

YoungsModulus evaluates the tensile stiffness of a rod cut along every crystal direction.

E = C.YoungsModulus;
plot(E,'complete','upper');
mtexColorbar('title','Young''s modulus in GPa');

youngsModulusRange = [min(E),max(E)]

youngsModulusRange =
  166.6121  306.4820

The printed range is 166.6 to 306.5 GPa for this illustrative tensor. The plot is stiffest along the cube axes and softest along the body diagonals. Its repeated fourfold pattern makes the attached cubic symmetry visible.

Physically specific tensor classes

A typed class records what a tensor means and exposes only meaningful operations. It may also set a default unit or compact-matrix convention.

SchmidTensor is a function rather than a class. It constructs a rank two velocity-gradient tensor from a slip-plane normal and a slip direction.

A component matrix constructs a typed tensor directly. The display includes the default Lagrange strain type.

M = [0 0 0;...
     0 0 0;...
     0 0 1];

eps = strainTensor(M)

eps = strainTensor (y↑→x)
  type: Lagrange 
  rank: 2 (3 × 3)
 
 0 0 0
 0 0 0
 0 0 1

Factory methods provide common physical states. Here the displayed matrix is a unit uniaxial stress along z.

sigma = stressTensor.uniaxial(vector3d.Z)

sigma = stressTensor (y↑→x)
  rank: 2 (3 × 3)
 
 0 0 0
 0 0 0
 0 0 1

Specialized operations belong to the corresponding class. For example, calcShearStress acts on a stress tensor and a slip system.

Predefined tensors

MTEX provides constructors for arrays of ones, the rank two identity, random tensors, and the rank three Levi-Civita tensor. Random tensors are useful for numerical experiments, not as material data.

Tones = tensor.ones('rank',2);
I = tensor.eye('rank',2);
Trandom = tensor.rand('rank',2);
leviCivita = tensor.leviCivita;

The Levi-Civita components are zero when any two indices are equal. They are \(+1\) for even permutations of \((1,2,3)\) and \(-1\) for odd ones. This tensor represents the cross product in index notation.

The maths behind a change of frame

The component array is a tensor because it obeys a specific transformation law. If \(Q\) changes an orthonormal basis, a rank \(r\) tensor transforms as

\[ T'_{i_1\ldots i_r} = Q_{i_1j_1}\cdots Q_{i_rj_r}T_{j_1\ldots j_r}. \]

Repeated indices are summed. This law applies the frame change to every index and distinguishes a tensor from an arbitrary multidimensional array.

A frame change re-expresses the same physical object in a different reference frame and leaves the object itself untouched. Rotating a tensor instead moves the physical property relative to the specimen. Tensor Arithmetics demonstrates both operations and the contractions that apply a tensor to vectors or other tensors.

Next

Real coefficients usually arrive in a file. Continue with Tensor Import for units, crystal frames, and Voigt conventions. Tensor Visualization develops directional plots, while Tensor Averages combines single-crystal properties with orientations or an ODF.

Piezoelectricity continues the rank three example. Elasticity develops moduli and seismic wave velocities from rank four stiffness tensors. The ODF tutorial supplies the orientation distribution needed for aggregate averages.

Further reading