By Antonio Romano

This book's methodological method familiarizes readers with the mathematical instruments required to properly outline and clear up difficulties in continuum mechanics. The publication covers crucial ideas and basic purposes, and offers a superior foundation for a deeper learn of more difficult and really good difficulties regarding elasticity, fluid mechanics, plasticity, fabrics with reminiscence, piezoelectricity, ferroelectricity, magneto-fluid mechanics, and nation changes.
Key themes and lines:
* Concise presentation moves a stability among basics and applications
* considered necessary mathematical history conscientiously accrued in introductory chapters and appendices
* contemporary advancements highlighted via assurance of extra major purposes to components resembling porous media, electromagnetic fields, and part transitions
Continuum Mechanics utilizing Mathematica® is geared toward complex undergraduates, graduate scholars, and researchers in utilized arithmetic, mathematical physics, and engineering. it might function a path textbook or self-study reference for someone looking an outstanding origin within the field.

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Additional resources for Continuum Mechanics using Mathematica®: Fundamentals, Applications and Scientific Computing

Example text

32), we can write the divergence of a vector or tensor ﬁeld in the following equivalent forms: 1 √ h ∇·v = √ gv ,h , g 1 √ hl ∇·T= √ ,h + Γlph T hp el . 33) Let us introduce the skew-symmetric tensor ﬁeld ∇v − (∇v)T = v,h ⊗ eh − (v,h ⊗ eh )T . 42), we have ∇v − (∇v)T = (vk ,h − Γlkh vl )ek ⊗ eh − (vk ,h − Γlkh vl )eh ⊗ ek = (vk ,h − vh ,k )ek ⊗ eh . 54 Chapter 2. 34) is called the curl of v. A simple mnemonic device for deducing the components of ∇ × v is obtained by noting that the components coincide with the algebraic complements of e1 , e2 , e3 in the matrix ⎛ ⎞ e 1 e2 e3 ⎜ ∂ ∂ ∂ ⎟ ⎜ ⎟ ⎝ ∂y 1 ∂y 2 ∂y 3 ⎠ .

96) where uiB , uiA denote the rectilinear coordinates of B and A, respectively. 97) and it assumes the Pitagoric form n (uiB − uiA )2 |AB| = i=1 when the basis (ei ) is orthonormal. 98) 32 Chapter 1. 13 Exercises 1. 64). 53) of the mixed components of a tensor. 2. Using the properties of the cross product, determine the reciprocal basis (ei ) of (ei ), i = 1, 2, 3. 19) ei · ej = δji , which constitutes a linear system of n2 equations in the n2 unknowns represented by the components of the vectors ei .

A system of applied vectors is equivalent to zero if for all P ∈ have R = MP = 0. Command Line of the Program VectorSys VectorSys[A ,V ,P ] Parameter List Input Data A = list of points of application of the vectors of Σ; V = list of components of the vectors of Σ; P = pole with respect to which the moment of Σ is evaluated. Output Data equivalent system Σ ; central axis of Σ; plot of Σ; plot of Σ ; plot of the central axis of Σ. 3 we 38 Chapter 1. Elements of Linear Algebra Worked Examples 1. Let Σ = {(Ai , vi )}i=1,···,3 be the following applied vectors system: A1 ≡ (0, 1, 0), v1 ≡ (1, 0, 1), A2 ≡ (1, 0, 0), v2 ≡ (2, 1, 0), A3 ≡ (0, 1, 2), v3 ≡ (3, 0, 0).