Tham khảo tài liệu 'finite element analysis - thermomechanics of solids part 6', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 6 Stress-Strain Relation and the Tangent-Modulus Tensor STRESS-STRAIN BEHAVIOR CLASSICAL LINEAR ELASTICITY Under the assumption of linear strain the distinction between the Cauchy and Piola-Kirchhoff stresses vanishes. The stress is assumed to be given as a linear function of linear strain by the relation T C. E .L . ij Cijkl kl in which j are constants and are the entries of a 3 X 3 X 3 X 3 fourth-order tensor C. If T and EL were not symmetrical C might have as many as 81 distinct entries. However due to the symmetry of T and EL there are no more than 36 distinct entries. Thermodynamic arguments in subsequent sections will provide a rationale for the Maxwell relations dTjj T dEi dEj It follows that j Ckiij which implies that there are at most 21 distinct coefficients. There are no further arguments from general principles for fewer coefficients. Instead the number of distinct coefficients is specific to a material and reflects the degree of symmetry in the material. The smallest number of distinct coefficients is achieved in the case of isotropy which can be explained physically as follows. Suppose a thin plate of elastic material is tested such that thin strips are removed at several angles and then subjected to uniaxial tension. If the measured stress-strain curves are the same and independent of the orientation at which they are cut the material is isotropic. Otherwise it exhibits anisotropy but may still exhibit limited types of symmetry such as transverse isotropy or orthotropy. The notion of isotropy is illustrated in Figure . In isotropic linear-elastic materials which implies linear strain the number of distinct coefficients can be reduced to two m and l as illustrated by Lame s equation T. 2ụE i L xdps . ij ij kk ij 95 2003 by CRC CRC Press LLC 96 Finite Element Analysis Thermomechanics of Solids FIGURE Illustration of isotropy. This can be inverted to furnish E L 1 T. -ij 2ự I 1 T A ij 2ự 3Ẳ kk ij . The classical elastic .
