Tham khảo tài liệu 'smart material systems and mems - vijay k. varadan part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | Introductory Concepts in Modeling 115 Equation is called Hooke s Law which states that the stress tensor is linearly related to the strain tensor. The term in Equation Cjkl is a fourth-order tensor of elastic constants which are independent of either stress or strain. The tensorial quality of the constants Cijkl follows the quotient rule according to which for a fourth-order tensor it should have 34 81 elements. Due to symmetry of the stress tensor ơij ơjí we should have Cijkl Cjikl. Furthermore since the strain tensor is also symmetric ekl elk we have Cijkl Cijlk. Under these conditions the fourth-order tensor Cijkl will have only 36 independent constants. Hence the total number of elastic constants cannot exceed 36 since the maximum independent elements in the stress and strain tensors are only 6 each. With these reductions the generalized Hook s law can be written in the matrix form as Using the Divergence Theorem the surface integral can be converted to the volume integral and the above equation becomes V j ơijd ijdV V The change in the potential energy also called the Strain Energy is given by dU j dUdV V Sxx C11 C12 C13 C14 C15 C16 Sxx s C21 C22 C23 C24 C25 C26 eyy Szz C31 C32 C33 C34 C35 C36 ezz ryz C41 C42 C43 C44 C45 C46 d rxz C51 C52 C53 C54 C55 C56 gxz txy .C61 C62 C63 C64 C65 C66 gxy where U is the potential energy per unit volume which is also called the strain energy density function . Assuming that U is a function of only deformations strains which is the basic hypothesis on which this material model is based we can write dU dU Sij d U dsij deij Using the above in Equation we get where all of the r s represent the shear stresses in their respective planes and all of the g s are the corresponding shear strains. For most elastic solids the number of elastic constants can further be reduced by exploiting the material symmetry about different reference planes. dU @UdeiidV J deij V Comparing Equations and we .
