T-II-2new - Revision history

Rosso: /* Displacement fields and strain tensor */

2019-11-16T11:29:00Z

Displacement fields and strain tensor

← Older revision		Revision as of 13:29, 16 November 2019
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	Here we review the basic objects of elastic theory that you have done last year (e.g. see pages 62-66 of '' Mécanique du solide et des matérieaux, partie 1, Pascal Kurowski'').		Here we review the basic objects of elastic theory that you have done last year (e.g. see pages 62-66 of '' Mécanique du solide et des matérieaux, partie 1, Pascal Kurowski'').


	Consider a perfectly isotropic continuum medium with density mass <math>\rho_0</math> at equilibrium:. Assume that at time <math> t=0 </math> a body is at rest and no external forces are applied. Each point of the body can be identified with its postion in the space:		Consider a perfectly isotropic continuum medium with density mass <math>\rho_0</math> at equilibrium:. Assume that at time <math> t=0 </math> a body is at rest and no external forces are applied. Each point of the body can be identified with its postion in the space:

Rosso: /* Displacement fields and strain tensor */

2019-11-16T11:28:37Z

Displacement fields and strain tensor

← Older revision		Revision as of 13:28, 16 November 2019
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	Here we review the basic objects of elastic theory that you have done last year (e.g. see pages 62-66 of '' Mécanique du solide et des matérieaux, partie 1, Pascal Kurowski''}		Here we review the basic objects of elastic theory that you have done last year (e.g. see pages 62-66 of '' Mécanique du solide et des matérieaux, partie 1, Pascal Kurowski'').
	Consider a perfectly isotropic continuum medium with density mass <math>\rho_0</math> at equilibrium:. Assume that at time <math> t=0 </math> a body is at rest and no external forces are applied. Each point of the body can be identified with its postion in the space:		Consider a perfectly isotropic continuum medium with density mass <math>\rho_0</math> at equilibrium:. Assume that at time <math> t=0 </math> a body is at rest and no external forces are applied. Each point of the body can be identified with its postion in the space:

Rosso: /* Displacement fields and strain tensor */

2019-11-16T11:27:32Z

Displacement fields and strain tensor

← Older revision		Revision as of 13:27, 16 November 2019
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	Here we review the basic objects of elastic theory that you have done last year (e.g. see pages 62-66 of '' ~~M\'ecanique~~ du solide et des ~~mat\'erieaux~~, partie 1, Pascal Kurowski''}		Here we review the basic objects of elastic theory that you have done last year (e.g. see pages 62-66 of '' Mécanique du solide et des matérieaux, partie 1, Pascal Kurowski''}
	Consider a perfectly isotropic continuum medium with density mass <math>\rho_0</math> at equilibrium:. Assume that at time <math> t=0 </math> a body is at rest and no external forces are applied. Each point of the body can be identified with its postion in the space:		Consider a perfectly isotropic continuum medium with density mass <math>\rho_0</math> at equilibrium:. Assume that at time <math> t=0 </math> a body is at rest and no external forces are applied. Each point of the body can be identified with its postion in the space:

Rosso: /* Compression: the (relative) density field */

2019-11-16T11:26:45Z

Compression: the (relative) density field

← Older revision		Revision as of 13:26, 16 November 2019
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	<center><math> d V_{\text{new}}\approx d V \left(1+ \text{Tr}(\epsilon)\right) \quad </math></center>.		<center><math> d V_{\text{new}}\approx d V \left(1+ \text{Tr}(\epsilon)\right) \quad </math></center>.

	Tip: it is convenient to write the element volume in the principal coordinate around the point $\vec r$. There the symmetric deformation tensor is diagonal.		Tip: it is convenient to write the element volume in the principal coordinate around the point <math>\vec r</math>. There the symmetric deformation tensor is diagonal.

	Conclude that		Conclude that

Rosso: /* Displacement fields and strain tensor */

2019-11-16T11:26:00Z

Displacement fields and strain tensor

← Older revision		Revision as of 13:26, 16 November 2019
Line 33:		Line 33:


			Here we review the basic objects of elastic theory that you have done last year (e.g. see pages 62-66 of '' M\'ecanique du solide et des mat\'erieaux, partie 1, Pascal Kurowski''}
	Consider a perfectly isotropic continuum medium with density mass <math>\rho_0</math> at equilibrium:. Assume that at time <math> t=0 </math> a body is at rest and no external forces are applied. Each point of the body can be identified with its postion in the space:		Consider a perfectly isotropic continuum medium with density mass <math>\rho_0</math> at equilibrium:. Assume that at time <math> t=0 </math> a body is at rest and no external forces are applied. Each point of the body can be identified with its postion in the space:

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	'''Q1:''' Take two points at ~~an infinitesimal~~ distance <math> d \ell </math> and compute the distance <math> d \ell_{\text{new}} </math> after a small deformation. Show that at the lowest order it can be written as		'''Q1:''' Take two points at a distance <math> d \ell </math> and compute the distance <math> d \ell_{\text{new}} </math> after a small deformation. Show that at the lowest order in the deformation it can be written as
	<center><math> d \ell_{\text{new}}^2 \approx d \ell^2 + 2 \sum_{i,k} \epsilon_{ik} d x_i dx_k </math></center>		<center><math> d \ell_{\text{new}}^2 \approx d \ell^2 + 2 \sum_{i,k} \epsilon_{ik} d x_i dx_k </math></center>

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	'''Q2:''' Show that at the first order in perturbation one has		'''Q2:''' Show that at the first order in perturbation one has

	<center><math> d V_{\text{new}}\approx d V \left(1+ \text{Tr}(\epsilon)\right) \quad </math></center>		<center><math> d V_{\text{new}}\approx d V \left(1+ \text{Tr}(\epsilon)\right) \quad </math></center>.

			Tip: it is convenient to write the element volume in the principal coordinate around the point $\vec r$. There the symmetric deformation tensor is diagonal.

	~~and conclude~~ that		Conclude that
	<center><math>\sigma(\vec r,t) =-\text{div}(\vec u) </math> </center>		<center><math>\sigma(\vec r,t) =-\text{div}(\vec u) </math> </center>

Rosso at 09:19, 13 November 2019

2019-11-13T09:19:14Z

← Older revision		Revision as of 11:19, 13 November 2019
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	'''Q5:''' Using the variational methods show that the equation of motion for a given component <math> u_i(\vec{r},t)</math> of the displacement writes		'''Q5:''' Using the variational methods show that the equation of motion for a given component <math> u_i(\vec{r},t)</math> of the displacement writes
	<center><math> \rho_0 \ddot u_i = - \sum_k		<center><math> \rho_0 \ddot u_i = \sum_k
	\frac{d}{d x_k} \frac{d}{d (\partial_k u_i)} \mathcal{E}_{\text{el}} </math></center>		\frac{d}{d x_k} \frac{d}{d (\partial_k u_i)} \mathcal{E}_{\text{el}} </math></center>
	'''Q6:''' Show that the 3 equations of motion take the form		'''Q6:''' Show that the 3 equations of motion take the form

Rosso: /* Sound speed in perfect gas */

2019-11-11T14:53:56Z

Sound speed in perfect gas

← Older revision		Revision as of 16:53, 11 November 2019
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	<center><math>c_l =\sqrt{\frac{K + (4 \mu)/3}{\rho_0}}</math> </center>		<center><math>c_l =\sqrt{\frac{K + (4 \mu)/3}{\rho_0}}</math> </center>

	the constant K is much bigger in solid then liquids and much bigger in liquids then gases. For this reason sound propagates much faster in solids (6 km /s in steal) then in water (1 km /s) or, even worst, in air (331 m /s at $T=273$ K).		the constant K is much bigger in solid then liquids and much bigger in liquids then gases. For this reason sound propagates much faster in solids (6 km /s in steal) then in water (1 km /s) or, even worst, in air (331 m /s at T=273 K).

Rosso: /* Sound speed in perfect gas */

2019-11-11T14:53:36Z

Sound speed in perfect gas

← Older revision		Revision as of 16:53, 11 November 2019
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	<center><math>c_l =\sqrt{\frac{K + (4 \mu)/3}{\rho_0}}</math> </center>		<center><math>c_l =\sqrt{\frac{K + (4 \mu)/3}{\rho_0}}</math> </center>

	the constant K is much bigger in solid then liquids and much bigger in liquids then gases. For this reason sound propagates much faster in solids (6 km /s in steal) then in water (1 km /s) or, even worst, in air (~~343~~ m /s).		the constant K is much bigger in solid then liquids and much bigger in liquids then gases. For this reason sound propagates much faster in solids (6 km /s in steal) then in water (1 km /s) or, even worst, in air (331 m /s at $T=273$ K).


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	'''Q12:''' Show that in this case		'''Q12:''' Show that in this case
	<center><math> K= P_0 </math> </center>		<center><math> K= P_0 </math> </center>
	and using that at 1 atmosphere (<math>101325</math> Pa) the density of air is <math>1.293</math> kg/m<math>^3</math> compute the sound speed predicted by Newton.		and using that at 1 atmosphere (<math>101325</math> Pa) the density of air is <math>1.293</math> kg/m<math>^3</math> (at T=273 K)) compute the sound speed predicted by Newton.

Rosso: /* Equation of motion with the variational principle */

2019-11-11T14:45:50Z

Equation of motion with the variational principle

← Older revision		Revision as of 16:45, 11 November 2019
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	'''Q5:''' Using the variational methods show that the equation of motion for a given component <math> u_i(\vec{r},t)</math> of the displacement writes		'''Q5:''' Using the variational methods show that the equation of motion for a given component <math> u_i(\vec{r},t)</math> of the displacement writes
	<center><math> \rho_0 \ddot u_i = \sum_k		<center><math> \rho_0 \ddot u_i = - \sum_k
	\frac{d}{d x_k} \frac{d}{d (\partial_k u_i)} \mathcal{E}_{\text{el}} </math></center>		\frac{d}{d x_k} \frac{d}{d (\partial_k u_i)} \mathcal{E}_{\text{el}} </math></center>
	'''Q6:''' Show that the 3 equations of motion take the form		'''Q6:''' Show that the 3 equations of motion take the form

Rosso: /* Sound speed in perfect gas */

2019-11-11T14:23:08Z

Sound speed in perfect gas

← Older revision		Revision as of 16:23, 11 November 2019
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	Generally speaking the ~~sond~~ speed (the transverse one for solids) takes the form		Generally speaking the sound speed (the transverse one for solids) takes the form

	<center><math>c_l =\sqrt{\frac{K + (4 \mu)/3}{\rho_0}}</math> </center>		<center><math>c_l =\sqrt{\frac{K + (4 \mu)/3}{\rho_0}}</math> </center>