elasticity of continuous media
TRANSCRIPT
Elasticity of continuous media
Elasticity of continuous media1. deformations (strains)2. forces (stresses)
4. elastic energy3. stress–strain relation
examples
Elasticity of continuous media1. deformations (strains)2. forces (stresses)
4. elastic energy3. stress–strain relation
examples
this is a continuous body
we’ll view it as some mass points really close together
RdR
now distort these mass points
x dx
elastic energy depends on how separation behaves
expand in terms of initial coordinates
cross term square
expand in terms of initial coordinates
symmetricantisymmetric part vanishes
energy is rotational invariant
Lagrangian strain tensor(in terms of unstrained coordinates)
Eulerian strain tensor(in terms of strained coordinates)
(equation 6.5.6 misses a minus sign)
Elasticity of continuous media1. deformations (strains)2. forces (stresses)
4. elastic energy3. stress–strain relation
examples
Main idea: forces are short ranged, transmitted through the surface of a volume element
force strain
surface element
Force in direction i
Unit area in direction jDefine strain as
ij
Gauss’ theorem gives a force density
Constraint: torques should also be transmitted through surface only
expand using indices
integrate by parts
for gondor!
volume termvanishes
constraining σ to be symmetric
Elasticity of continuous media1. deformations (strains)2. forces (stresses)
4. elastic energy3. stress–strain relation
examples
1. uniform pressure
2. uniaxial tension
3. shear
Elasticity of continuous media1. deformations (strains)2. forces (stresses)
4. elastic energy3. stress–strain relation
examples
elastic constants
Hooke’s Law: response is linear
Cubic symmetry, isotropic medium: two independent elastic constants
Cubic symmetry, isotropic medium: stress–strain relation
bulk modulus B shear modulus
Cubic symmetry, isotropic medium: strain–stress relation
uniaxial tension
always positive
can be negative !
normal material
auxetic material
www.product-technik.co.uk/Auxetics/about_auxetics.htm
Stretch a piece of bungee cord, it just gets thinner
But if you wrap a cord around the bungee, it effectively gets wider
Make bundles and you’ve got an auxetic
Weave the cords and get adamantiumtastic
www.auxetix.com
ADAMANTIUMTASTIC!
Very effective at dissipating energy (car bomb test)
Elasticity of continuous media1. deformations (strains)2. forces (stresses)
4. elastic energy3. stress–strain relation
examples
displace strains by a small amount
force density is gradient of strain
symmetrise
integrate by parts, surface term vanishes for large volume
for gondor!
i and j are dummy indices, can swap them
work done by internal force
elastic energy density
Elasticity of continuous media1. deformations (strains)2. forces (stresses)
4. elastic energy3. stress–strain relation
examples
Discussion points
1. p312: Are the pre-factors of equation (6.4.17) correct?
Discussion Points
1. Are the pre-factors of equation 6.4.17 (page 312) correct?
𝐾𝐾11 = 𝐾𝐾𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 𝐾𝐾22 = 𝐾𝐾𝑦𝑦𝑦𝑦𝑦𝑦𝑦𝑦 𝐾𝐾33 = 𝐾𝐾𝑧𝑧𝑧𝑧𝑧𝑧𝑧𝑧
𝐾𝐾12 = 𝐾𝐾𝑥𝑥𝑥𝑥𝑦𝑦𝑦𝑦 𝐾𝐾13 = 𝐾𝐾𝑥𝑥𝑥𝑥𝑧𝑧𝑧𝑧 𝐾𝐾23 = 𝐾𝐾𝑦𝑦𝑦𝑦𝑧𝑧𝑧𝑧
𝐾𝐾21 = 𝐾𝐾𝑦𝑦𝑦𝑦𝑥𝑥𝑥𝑥 𝐾𝐾31 = 𝐾𝐾𝑧𝑧𝑧𝑧𝑥𝑥𝑥𝑥 𝐾𝐾32 = 𝐾𝐾𝑧𝑧𝑧𝑧𝑦𝑦𝑦𝑦
𝐾𝐾44 = 𝐾𝐾𝑦𝑦𝑧𝑧𝑦𝑦𝑧𝑧 𝐾𝐾55 = 𝐾𝐾𝑥𝑥𝑧𝑧𝑥𝑥𝑧𝑧 𝐾𝐾66 = 𝐾𝐾𝑥𝑥𝑦𝑦𝑥𝑥𝑦𝑦
𝐾𝐾44 = 𝐾𝐾𝑧𝑧𝑦𝑦𝑧𝑧𝑦𝑦 𝐾𝐾55 = 𝐾𝐾𝑧𝑧𝑥𝑥𝑧𝑧𝑥𝑥 𝐾𝐾66 = 𝐾𝐾𝑦𝑦𝑥𝑥𝑦𝑦𝑥𝑥
𝐾𝐾44 = 𝐾𝐾𝑦𝑦𝑧𝑧𝑧𝑧𝑦𝑦 𝐾𝐾55 = 𝐾𝐾𝑥𝑥𝑧𝑧𝑧𝑧𝑥𝑥 𝐾𝐾66 = 𝐾𝐾𝑥𝑥𝑦𝑦𝑦𝑦𝑥𝑥
𝐾𝐾44 = 𝐾𝐾𝑧𝑧𝑦𝑦𝑦𝑦𝑧𝑧 𝐾𝐾55 = 𝐾𝐾𝑧𝑧𝑥𝑥𝑥𝑥𝑧𝑧 𝐾𝐾66 = 𝐾𝐾𝑦𝑦𝑥𝑥𝑥𝑥𝑦𝑦
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Discussion points
2. p321:
3. p326:
4. p327:
5. p325:
How is the Hamiltonian in equation (6.4.19) calculated?
What is the physical reason for fitting different types of functions to the solid and liquid phases in Figure 6.4.4
What is Figure (6.4.5) showing us?
How are interstitials dealt with under strain?
6. p332:
7. Section 6.5.2:
“The non–linear strain field that would guarantee total rotational invariance of the solid elastic energy is the Lagrangian strain”
What is the main idea of this section?
