By Antonio Romano
This e-book examines mathematical instruments, ideas, and basic functions of continuum mechanics, offering a great foundation for a deeper examine of more difficult difficulties in elasticity, fluid mechanics, plasticity, piezoelectricity, ferroelectricity, magneto-fluid mechanics, and kingdom adjustments. The paintings is acceptable for complicated undergraduates, graduate scholars, and researchers in utilized arithmetic, mathematical physics, and engineering.
Read Online or Download Continuum Mechanics using Mathematica®: Fundamentals, Applications and Scientific Computing (Modeling and Simulation in Science, Engineering and Technology) PDF
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Extra info for Continuum Mechanics using Mathematica®: Fundamentals, Applications and Scientific Computing (Modeling and Simulation in Science, Engineering and Technology)
The following square-root theorem is important in ﬁnite deformation theory (see Chapter 3): 24 Chapter 1. 8 If T is a positive deﬁnite symmetric tensor, then one and only one positive deﬁnite symmetric tensor U exists such that U2 = T. 73) PROOF Since T is symmetric and positive deﬁnite from the previous theorem it follows that 3 λi ui ⊗ ui , T= i=1 where λi > 0 and (ui ) is an orthonormal basis. Consequently, we can deﬁne the symmetric positive deﬁnite tensor 3 λi ui ⊗ ui . 74) i=1 In order to verify that U2 = T, it will be suﬃcient to prove that they coincide when applied to eigenvectors.
3 Proceeding in the same way for λ = 4 and λ = 6, we obtain the components of the other two eigenvectors: 1 1 2 −√ , √ , √ 6 6 6 , 1 1 √ , √ ,0 . 2 2 From the symmetry of T, the three eigenvectors are orthogonal (verify). 34 Chapter 1. Elements of Linear Algebra 4. Let u and λ be the eigenvectors and eigenvalues of the tensor T. Determine the eigenvectors and eigenvalues of T−1 . If u is an eigenvector of T, then Tu = λu. Multiplying by T−1 , we obtain the condition T−1 u = 1 u, λ which shows that T−1 and T have the same eigenvectors, while the eigenvalues of T−1 are the reciprocal of the eigenvalues of T.
Moreover, there are systems of applied vectors which are equivalent to either a vector or a torque (see , ). 104) and the formula MO = MP + (P − O) × R ∀O, P ∈ 3 . 11. The Program VectorSys 37 More precisely, the following results hold: 1. If I = 0 (a) and R = 0, then the system Σ is equivalent to its resultant R applied at any point A of the central axis; (b) and R = 0, then the system Σ is equivalent to any torque having the moment MP of Σ with respect to P . 2. If I = 0, then the system Σ is equivalent to its resultant R applied at any point P and a torque with moment MP .