Abstract
The mechanics of a deformable body treated here is based on Newton’s laws of motion and the laws of thermodynamics.
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Notes
- 1.
In the second law (2.2) if we set \({ f}=\,{ 0}\) and solve the differential equation, we obtain \({ v}\,=\,\text{ constant}\) since m = { constant}, which suggests that the first law is included in the second law. This apparent contradiction results from the misinterpretation of the first law.
- 2.
‘Configuration’ is defined as an invertible continuous function that maps every material point \(X \in \mathfrak{B}\) to a point \({z}\) in a subset of the n-dimensional real number space \({\mathbb{R}}^{n}\). A time-dependent motion is considered; therefore the configuration is a function of the material point X and time t. The configuration at a given time t 0 is set as a reference configuration κ, and the point \({X} \in {\mathbb{R}}^{n}\) corresponding to a material point X is written as \({X}\,=\,\kappa (X),X\,=\,{\kappa }^{-1}({X})\) where κ − 1 is an inverse mapping of κ. The current configuration χ at time t maps X to \({x} \in {\mathbb{R}}^{n}\) as \({x}\,=\,\chi (X,t),X\,=\,{\chi }^{-1}({X}\!,t)\). The composite function \({\chi }_{\kappa }\,=\,\chi \circ {\kappa }^{-1}\) is introduced as \({x}\,=\,\chi ({\kappa }^{-1}({X}),t)\,=\,{\chi }_{\kappa }\,=\,\chi \circ {\kappa }^{-1}({X}\!,t)\,=\,{\chi }_{\kappa }({X}\!,t)\). The function χκ gives a mapping between the position vector \({X}\) in the reference configuration and the position vector \({x}\) in the current configuration. Since this formal procedure is complicated, the above simplified descriptions are employed.
- 3.
If the gradient is used with respect to Eulerian coordinates with the basis {\({{e\!}}_{i}\}\), it is denoted as (2.14). If we explicitly explain the gradient with respect to the Eulerian system, it is denoted as \(\text{ grad} = {\Delta }_{x} ={ {e\!}}_{i} \dfrac{\partial \ } {\partial {x}_{i}}.\) If the gradient is operated with respect to Lagrangian coordinates {\({{E\!}}_{I}\}\), it is represented as \(\text{ Grad} = {\Delta }_{X} ={ {E\!}}_{I} \dfrac{\partial \ } {\partial {X}_{I}}.\)
- 4.
Here we deal with a general case in which two coordinate systems may not be inertial systems. If both are inertial systems, \({Q}\) is time-independent as given by (2.4). Therefore, we have \({{x}}^{{_\ast}} = {Q}\,{x} + {V }t,\quad {{x}}_{0}^{{_\ast}} = {Q}\,{{x}}_{0} + {V }t\quad \Rightarrow \quad {u} ={ {x}}^{{_\ast}}-{{x}}_{0}^{{_\ast}} = {Q}({x}-{{x}}_{0}),\) which shows that the two-point vector \({u}\) is frame indifferent.
- 5.
The convected derivatives of a vector \({v}\) are sometimes written as \({\delta }^{c}{v}/\delta t ={ \vartriangleleft \mathbf{v}},\:{\delta }_{c}{v}/\delta t ={ \vartriangleright \mathbf{v}}\). For the second-order tensor \({T}\) these are \({\delta }^{cc}{T}/\delta t ={ \vartriangleleft \mathbf{T}},\:{\delta }_{cc}{T}/\delta t ={ \vartriangleright \mathbf{T}}\).
- 6.
The inner product of the second-order tensors \({A},\ {B}\) is introduced by \({A} : {B} = \mathrm{tr}\,({{A}}^{T}{B}) = {A}_{\mathit{ij}}\,{B}_{\mathit{ij}}\). \({A}\) and \({B}\) are orthogonal if \({A} : {B} = 0\).
- 7.
The definition of the δ-function is given by \({\int \nolimits \nolimits }_{\Omega }dy\,\delta (y - x)f(y) = f(x)\).
- 8.
Leibnitz rule: If we have an integral of a continuous function f such as
$$\qquad \phi (x) ={ \int \nolimits \nolimits }_{{h}_{0}(x)}^{{h}_{1}(x)}f(x,\xi )\,d\xi,$$and if h 1(x) and h 0(x) are continuous on R = { (x, ξ) : a ≤ x ≤ b, c ≤ ξ ≤ d }, then
$$\qquad \frac{d\phi (x)} {dx} = f\left (x,{h}_{1}(x)\right )\frac{d{h}_{1}(x)} {dx} - f\left (x,{h}_{0}(x)\right )\frac{d{h}_{0}(x)} {dx} +{ \int \nolimits \nolimits }_{{h}_{0}(x)}^{{h}_{1}(x)}\frac{\partial f(x,\xi )} {\partial x} \,d\xi $$(see, e.g., Protter and Morrey 1977, pp. 284).
- 9.
Note that the differentiation implies \({\overline{E}}^{(k)}(k\tau ) = \dfrac{{d}^{k}\overline{E}(k\tau )} {d{(k\tau )}^{k}}\). Others are the same.
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© 2012 Springer-Verlag Berlin Heidelberg
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Ichikawa, Y., Selvadurai, A.P.S. (2012). Introduction to Continuum Mechanics. In: Transport Phenomena in Porous Media. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25333-1_2
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DOI: https://doi.org/10.1007/978-3-642-25333-1_2
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