doubleContraction

Below is a demonstration of the features of the doubleContraction function

Contents

clear; close all; clc;

Syntax

C=doubleContraction(A,B,subContract);

Description

This function performs a double contraction of two tensors A and B to produce the output tensor C. The tensors are either both 3x3 second order tensors or A is a 4th order 3x3x3x3 tensor. The optional 3rd input subContract defines the indices to contract over for 4th order tensors e.g. subContract=[1 2] would contract over the first 2 indices.

Examples

Double contraction of a 2nd-order and a 2nd-order tensor

Creating an example tensor. Here a tensor A with all zeros except for the 3,3 component is created. Next a tensor B which, through double contraction, can "pick out" this scalar value is created. Finally the double contraction is performed illustrating this process.

%Create example tensor A
A=zeros(3,3); A(3,3)=pi; %Zeros with Azz=A33=pi
R=euler2DCM([0.25*pi 0.25*pi 0]); %Rotation matrix
A=R*A*R' %Rotated version of A

% Create 'structure' tensor example B
b=[0 0 1]*R.'; %Rotated Z vector
B=b.'*b %Rotated "structure" tensor for vector b

A =

    1.5708   -1.1107    1.1107
   -1.1107    0.7854   -0.7854
    1.1107   -0.7854    0.7854


B =

    0.5000   -0.3536    0.3536
   -0.3536    0.2500   -0.2500
    0.3536   -0.2500    0.2500

Double contraction using doubleContraction

s=doubleContraction(A,B)

s =

    3.1416

Double contraction of a 4th-order and a 2nd-order tensor

Example for Hooke's law of linear elasticity

%Creating the 4th order linear elastic stiffness tensor
I=eye(3,3); %The 2nd order identity tensor
II1=dyadicProduct(I,I,1); %4th order base tensor 1
II3=dyadicProduct(I,I,3); %4th order base tensor 3

%Parameters for Hooke's law
mu=1; %The shear modulus
lambda=5; %The lambda lame parameter

%Compose stiffness tensor
C=lambda.*II1+2.*mu.*II3;

%Create strain tensor
E=rand(3,3); E=E*E'

E =

    1.0793    0.8967    1.4848
    0.8967    0.8533    1.4394
    1.4848    1.4394    2.4337

Double contraction using doubleContraction

subContract=[3 4];
S=doubleContraction(C,E,subContract)

S =

   23.9903    1.7933    2.9697
    1.7933   23.5381    2.8788
    2.9697    2.8788   26.6990

Using symbolic variables

Example for Hooke's law of linear elasticity

%Creating the 4th order linear elastic stiffness tensor
I=eye(3,3); %The 2nd order identity tensor
II1=dyadicProduct(I,I,1); %4th order base tensor 1
II3=dyadicProduct(I,I,3); %4th order base tensor 3

%Parameters for Hooke's law
try
    syms mu lambda
catch
    mu=1; %The shear modulus
    lambda=5; %The lambda lame parameter
end
%Compose stiffness tensor
C=lambda.*II1+2.*mu.*II3;

%Display Voigt mapped version
c=voigtMap(C)

%Create strain tensor
syms E1_1 E2_2 E3_3 E1_2 E1_3 E2_3;
E=[E1_1 E1_2 E1_3;...
   E1_2 E2_2 E2_3;...
   E1_3 E2_3 E3_3]

 
c =
 
[ lambda + 2*mu,        lambda,        lambda,  0,  0,  0]
[        lambda, lambda + 2*mu,        lambda,  0,  0,  0]
[        lambda,        lambda, lambda + 2*mu,  0,  0,  0]
[             0,             0,             0, mu,  0,  0]
[             0,             0,             0,  0, mu,  0]
[             0,             0,             0,  0,  0, mu]
 
 
E =
 
[ E1_1, E1_2, E1_3]
[ E1_2, E2_2, E2_3]
[ E1_3, E2_3, E3_3]
 

Double contraction using doubleContraction

subContract=[3 4];
S=doubleContraction(C,E,subContract)

 
S =
 
[ E2_2*lambda + E3_3*lambda + E1_1*(lambda + 2*mu),                                        2*E1_2*mu,                                        2*E1_3*mu]
[                                        2*E1_2*mu, E1_1*lambda + E3_3*lambda + E2_2*(lambda + 2*mu),                                        2*E2_3*mu]
[                                        2*E1_3*mu,                                        2*E2_3*mu, E1_1*lambda + E2_2*lambda + E3_3*(lambda + 2*mu)]
 

GIBBON www.gibboncode.org

Kevin Mattheus Moerman, [email protected]

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License: https://github.com/gibbonCode/GIBBON/blob/master/LICENSE

GIBBON: The Geometry and Image-based Bioengineering add-On. A toolbox for image segmentation, image-based modeling, meshing, and finite element analysis.

Copyright (C) 2019 Kevin Mattheus Moerman

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Published with MATLAB® R2019a