Tensor Components with Respect to Rotated Base Vectors (4) 

Rotate vector components about the three axes counterclockwise:

In[2]:=

Out[2]=

Change the axis:

In[3]:=

Out[3]=

Rotate the rank-2 tensor about the three axes counterclockwise:

In[4]:=

Out[4]=

Rotate a rank-4 tensor with three symmetries about the three axes counterclockwise:

In[5]:=

Out[6]=

Transform Vector Components Regarding Another Affine Frame of Reference (2) 

The contravariant vector components with respect to covariant bases are a=v^jb_j=v^i'b_i'. The old system is Cartesian (a^j=v^j):

In[7]:=

Consider a transformation between new e_i' and old e_j covariant base vectors e_1'=2e₁+3e₂+1e₃, e_2'=1e₁+2e₂+2e₃ and e_3'=1e₁+2e₃. Here are the contravariant components v^i' of the new system:

In[8]:=

Out[8]=

They should be the same vector:

In[9]:=

In[10]:=

Out[10]=

In[11]:=

Out[11]=

The covariant vector components with respect to contravariant bases are a=v^jb_j=v_i'b^j'

In[12]:=

Out[12]=

Find related contravariant base vectors b^i' by transforming the covariant base in contravariant sense (index juggling b_i'→b^i'):

In[13]:=

Out[13]=

In[14]:=

Out[14]=

In[15]:=

Out[15]=

Note that covariant and contravariant base vectors are inverse to each other b_i'b^j'=δ_i'^j':

In[16]:=

Out[16]=

A Local Base on the Manifold of a Torus Surface (Curvilinear Coordinate System) (7) 

Consider the aⁱb_i≡c^ijkls_klt_jb_i=c^i'j'k'l's_k'l't_j'b_i'≡a^i'b_i'-invariant tensors in Cartesian frame:

In[17]:=

The transformation from Cartesian to local torus coordinates (i→i'):

In[18]:=

The torus surface at α=β=0…2π, with R and r constant:

In[19]:=

Out[19]=

Transform tensors' different ranks regarding the local torus base with respect to transformation behavior. This is faster than transformation of an entire assembled rank-7 tensor):

In[20]:=

Local covariant base vector b_i':

In[21]:=

Assemble the tensor term by contracting the right indices c^i'j'k'l's_k'l't_j'b_i≡a^i'b_i':

In[22]:=

Out[22]=

In[23]:=

Out[23]=

The result should be the same in the Cartesian system:

In[24]:=

Out[24]=

TransformationBehavior (2) 

If the base vectors are not normalized ("Normalize"→False), i.e. if they have local-dependent lengths (and are not orthogonal in general reference systems either), the corresponding tensor components are transformed according to their co- and contravariant transformation behaviors. This leads to different representations of the same tensor object.

Here is an example of all four transforming possibilities of a rank-2 tensor to a local, non-normalized cylindrical system:

In[25]:=

All coordinates transform the contravariant t^ij (default):

In[26]:=

Out[26]=

All coordinates transform the covariant t_ij:

In[27]:=

Out[27]=

A tensor object with coordinates of mixed-transformation behavior t_i^j:

In[28]:=

Out[28]=

A tensor object with coordinates of opposite mixed-transformation behavior t_jⁱ:

In[29]:=

Out[29]=

Nomenclature referencing all indices together or each index separately is interchangeable:

In[30]:=

Out[30]=

This is analogous for the covariant transformation:

In[31]:=

Out[31]=

Normalize (5) 

Transformed contravariant vector components with respect to normalized base vectors:

In[32]:=

Out[32]=

The same transformed contravariant vector components with respect to non-normalized base vectors (default):

In[33]:=

Out[33]=

Transformed covariant normalized base vectors:

In[34]:=

Out[34]=

The contraction of the assigned tensors should be produce invariance:

In[35]:=

Out[35]=

Tensor components with respect to an orthonormal cylindrical system:

In[36]:=

Out[35]=

TransformedField also assumes a normalized base:

In[37]:=

Out[37]=

In[38]:=

Out[38]=

With orthonormal reference systems (e.g. normalized cylindrical systems), no distinction between co- and contravariant transformation behavior is required:

In[39]:=

Out[39]=

Stiffness Components of the Rank-4 Elasticity Tensor with Different Material Symmetries (6) 

In[40]:=

Hook's general law describes a linear relationship between the components of the rank-2 stress tensor σ and the two-stage strain tensor ε using the rank-4 tensor C: σ_ij=C_ijklε^kl with the symmetries σ_ij=σ_ji, ε^kl=ε^lk and E_ijkl=E_jikl=E_ijlk=E_klij.

Components of the general, fully anisotropic stiffness tensor with initial symmetries:

In[41]:=

Out[37]=

The number of independent components:

In[42]:=

Out[42]=

One plane of symmetry, rotated by 180 degrees, results in the same stiffness:

In[43]:=

Out[44]=

This results in a stiffness tensor with fewer independent components:

In[45]:=

Out[20]=

Determination and comparison of different material symmetries:

In[46]:=

In[47]:=

In[48]:=

Summary:

In[49]:=

Out[49]=