Continuum Mechanics using Mathematica®: Fundamentals, by Antonio Romano

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.

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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 [8], [9]). 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 .

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