By Stephen Bechtel, Robert Lowe
ISBN-10: 012394600X
ISBN-13: 9780123946003
Fundamentals of Continuum Mechanics offers a transparent and rigorous presentation of continuum mechanics for engineers, physicists, utilized mathematicians, and fabrics scientists. This booklet emphasizes the position of thermodynamics in constitutive modeling, with designated software to nonlinear elastic solids, viscous fluids, and glossy shrewdpermanent fabrics. whereas emphasizing complex fabric modeling, detailed recognition is usually dedicated to constructing novel theories for incompressible and thermally increasing fabrics. A wealth of conscientiously selected examples and workouts light up the subject material and facilitate self-study.
- Uses direct notation for a transparent and easy presentation of the math, resulting in a greater figuring out of the underlying physics
- Covers high-interest examine parts corresponding to small- and large-deformation continuum electrodynamics, with program to shrewdpermanent fabrics utilized in clever platforms and structures
- Offers a different method of modeling incompressibility and thermal growth, according to the authors’ personal research
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Additional info for Fundamentals of Continuum Mechanics: With Applications to Mechanical, Thermomechanical, and Smart Materials
Sample text
2 Tensor algebra (c) δik Tkj = Tij . (d) δij Tij = Tii . Solution (a) Summing over the repeated subscript i, we obtain 3 δii = δii = δ11 + δ22 + δ33 = 1 + 1 + 1 = 3. i=1 (b) Summing over the repeated subscript j for each value of i, we obtain 3 (i = 1) δ1j vj = δ1j vj = δ11 v1 + δ12 v2 + δ13 v3 = v1 , j=1 3 (i = 2) δ2j vj = δ2j vj = δ21 v1 + δ22 v2 + δ23 v3 = v2 , j=1 3 (i = 3) δ3j vj = δ3j vj = δ31 v1 + δ32 v2 + δ33 v3 = v3 , j=1 where we have used δ11 = δ22 = δ33 = 1, with all other permutations vanishing.
5 Tensor calculus grad 1 φ =− 1 grad φ, φ2 div (φ A) = φ div A + A grad φ, div (v ⊗ w) = v div w + (grad v) w. The curl of a vector v is defined by (curl v) × a = grad v − (grad v)T a. , div curl v = 0. 101) curl (v × w) = (grad v) w − (grad w) v + v (div w) − w (div v). 103a) div (A + B) = div A + div B. 104) where dv is the volume element of R, da is the area element of ∂ R, and n is the outward unit normal on ∂ R. 52) v · A n da = ∂R (A · grad v + v · div A) dv. 49 Prove in direct notation that grad (φ v) = φ grad v + v ⊗ grad φ .
E 3 . 41 Show that u × v = ijk ui vj ek . Solution u × v = ui ei × vj ej = ui vj (ei × ej ) = ijk ui vj ek . 42 Verify in Cartesian component notation that the set of all symmetric tensors is a six-dimensional inner product space E 6 , and the set of all skew tensors is a threedimensional inner product space E 3 . Solution Consider an arbitrary symmetric tensor D. The Cartesian component form of D is D = Dij ei ⊗ ej = D11 e1 ⊗ e1 + D12 e1 ⊗ e2 + D13 e1 ⊗ e3 + D21 e2 ⊗ e1 + D22 e2 ⊗ e2 + D23 e2 ⊗ e3 + D31 e3 ⊗ e1 + D32 e3 ⊗ e2 + D33 e3 ⊗ e3 .
Fundamentals of Continuum Mechanics: With Applications to Mechanical, Thermomechanical, and Smart Materials by Stephen Bechtel, Robert Lowe
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