![Page 1: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/1.jpg)
Continuum Mechanics, part 22
Kinematics_2
• Small deformation theory• Right and left Cauchy-Green deformation tensors• Stretch, polar decomposition• Rate of polar rotation
Cm_part22
![Page 2: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/2.jpg)
![Page 3: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/3.jpg)
![Page 4: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/4.jpg)
![Page 5: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/5.jpg)
![Page 6: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/6.jpg)
![Page 7: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/7.jpg)
![Page 8: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/8.jpg)
![Page 9: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/9.jpg)
![Page 10: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/10.jpg)
![Page 11: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/11.jpg)
![Page 12: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/12.jpg)
The plane stress induced by Cauchy strain is
![Page 13: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/13.jpg)
%
r i g r o t . m
%
s t r a
% rigrot.m% strain and stress "induced" due to a rigid body rotationclearommax=6;omrange=1:ommax;ss=zeros(3,ommax); ee=zeros(3,ommax);splot=zeros(ommax,1); eplot=zeros(ommax,1);mu=0.3;E=2.1e11;const=E/(1-mu^2);d=const*[1 mu 0; mu 1 0; 0 0 (1-mu)/2];for om=omrangeom;omr=om*pi/180; e1=cos(omr)-1;eplot(om)=e1; ep=[e1 e1 0];ee(:,om)=ep'; sigma=d*ep';splot(om)=sigma(1,1); ss(:,om)=sigma;
end
figure(1)subplot(2,1,1);plot(0:ommax,[0 eplot']);title('strain11 vs. rigid body rotation [degrees]');subplot(2,1,2);plot(0:ommax,[0 splot']);title('stress11 [N/m^2] vs. rigid body rotation [degrees]')% end of rigrot
![Page 14: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/14.jpg)
False strain and stress due to rigid-body motion
![Page 15: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/15.jpg)
As before, but for smaller values of rotation angle
![Page 16: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/16.jpg)
![Page 17: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/17.jpg)
![Page 18: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/18.jpg)
![Page 19: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/19.jpg)
![Page 20: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/20.jpg)
![Page 21: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/21.jpg)
![Page 22: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/22.jpg)
![Page 23: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/23.jpg)
![Page 24: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/24.jpg)
Replaces cm0118b
So that ΛΦCΦ =T
The above equation also represents the diagonalization of C ΦCΦC T=′ with an obvious conclusion that ΛC =′ .
Now, we can transform F to the primed coordinate system ΦΦ FF T=′ and URF ′′=′
and similarly ΦΦ UU T=′ with ( )2UC ′=′
TΦΦ UU ′=⇒ but we have seen that diagonal issince Λ=′′=′ CCUTso ΦΛΦ=⇒ U , which solves the problem.
![Page 25: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/25.jpg)
{dx}=[F]{da}
In one step
Right polar decomposition F = RU
![Page 26: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/26.jpg)
Left polar decomposition F = VR
![Page 27: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/27.jpg)
![Page 28: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/28.jpg)
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−
−=
00
0
12
13
23R
ωωωωωω
Ω ij
Time derivative of a vector is the velocity of its end point
![Page 29: Continuum Mechanics, part 2 - Ústav termomechaniky AV … · Continuum Mechanics, part 22 Kinematics_2 • Small deformation theory • Right and left Cauchy-Green deformation tensors](https://reader031.vdocuments.mx/reader031/viewer/2022022516/5b0474817f8b9aba168d5396/html5/thumbnails/29.jpg)