Course Outcomes. Our previous continuum mechanics theories have generally been developed for problems with length scales several orders of magnitude larger than these microstructural features. CONTINUUM MECHANICS - Introduction to tensors Tensor algebra Vectors Component representation Any vector a can be uniquely de ned with the linear combination of the basis vectors (e 1, e 2 and e 3) as a = a 1e 1 + a 2e 2 + a 3e 3; (6) where the components (a 1, a 2 and a 3) are real numbers. }�!�c�mml����2c���SU�m�m6�/�fg�f��K�����죡���á������m ��M�U�{i:��[;���s{�D;C�aE����a�W�!ﵗ����6>3�v�/q2�����֯�j�[�&烪{��s>�'�R�P*" �H��R��C����~0�?ZH�}/Ճ31�E�~�iwb���/S�.�s����[��镨�t��e�ĭ����MD�G��r}��>���/q�['��,����Z99ʎ�o"�W���ثH���@�ډܷ�=H��|���8�1e�uj�-E�έ�~�%�b41>Q��R��w$��f>>I_��5�q�1�Ռ�|v��� G�˸g�)��u�k�2�M��$q'��|X��H�2��$�1 IR�`0&^ �# I���BPR)]R�T$`x��,�b �6E To this end, this chapter is devoted to the description of various numerical techniques that are robust and efficient. 800 0 obj <>stream We use cookies to help provide and enhance our service and tailor content and ads.
Paulino, ... P.S. Other commonly used functions include linear (e.g., Marur and Tippur, 1998; Kim and Paulino, 2002a,b) and hyperbolic–tangent (e.g., Eischen, 1987a). Ã0D¯27¾¶!è&´'èªûÒûS+ÐB¡ZéÁÌÐÀ6f<. Extensive discussions about these algorithms can be found in the literature. Figure 29 shows the energy gap vs. the twist κ (angle of rotation per unit length). Now consider quasistatic, two-dimensional (2D) deformations of a nonhomogeneous material with the following properties: where µ is the space-dependent shear modulus, and µ0, (ν)0, β, and γ are material constants.
Figure 29. Fundamentals of the Mechanics of Multiphase Materials, George Chatzigeorgiou, ... Fodil Meraghni, in, Thermomechanical Behavior of Dissipative Composite Materials, Continuous models for analyzing the mechanical behavior of reinforcements in composites, Composite Reinforcements for Optimum Performance, G.H. The next example will show the more typical, and much more complex case of performing a polar decomposition when shear is present in the problem. f0ñ6f´ÆÏ£àó,ø>ÏÏ«àó*ø¼ 747 0 obj <>/Filter/FlateDecode/ID[<300DB490D1D521434972DB183458AC9A><16163FCF99C93D4AA65148FFBC07D4B4>]/Index[736 65]/Info 735 0 R/Length 85/Prev 298475/Root 737 0 R/Size 801/Type/XRef/W[1 3 1]>>stream This is consistent with the Tombler et al. continuum mechanics. 736 0 obj <> endobj endstream endobj startxref %PDF-1.5 %���� G.H. The energy gap is zero for the undeformed carbon nanotube (i.e., κ=0), which indicates that the armchair carbon nanotube is like a metallic tube. 17.37 into 17.36 and applying the Cayley-Hamilton theorem, we obtain the following expression for the Cauchy stress tensor σ: where I is the second order unit tensor and B is the left Cauchy–Green tensor B = FFT. Continuum mechanics hinges on the concept of a representative volume element (RVE) playing the role of a mathematical point of a continuum field approximating the true material microstructure. We have used the continuum mechanics theory together with a semi-empirical tight-binding method to investigate the electrical property change of a carbon nanotube upon mechanical deformation. After completing ENGN2210 you should . Example Constitutive Equations 21 1.8. Rama Sreekanth, in, Reference Module in Materials Science and Materials Engineering. This work focuses on FGMs undergoing elastic deformations governed by the equations of nonhomogeneous elastic materials (Fung, 1965).
By continuing you agree to the use of cookies. Once the displacement solution is available, stresses are found simply by using Hooke’s law. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® is a registered trademark of Elsevier B.V. URL: https://www.sciencedirect.com/science/article/pii/S0065215616300011, URL: https://www.sciencedirect.com/science/article/pii/B9780123970350000033, URL: https://www.sciencedirect.com/science/article/pii/B9781785482793500014, URL: https://www.sciencedirect.com/science/article/pii/B978008096532101102X, URL: https://www.sciencedirect.com/science/article/pii/B0080437494081489, URL: https://www.sciencedirect.com/science/article/pii/B9781845699659500177, URL: https://www.sciencedirect.com/science/article/pii/B9780128035818008754, URL: https://www.sciencedirect.com/science/article/pii/B9781785482793500038, URL: https://www.sciencedirect.com/science/article/pii/B0080437494080502. However, as the twist κ becomes nonzero, the energy gap increases quickly, which creates a significant energy barrier for electrons to move in the carbon nanotube such that the tube behaves like a semiconducting one.
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