Strain Chapter 12

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StrainChapter 12
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Components of homogeneous strain of a square (A)
(straight lines remain straight parallel lines
remain parallel strain the same at all
points)change in area (B) change in shape (C,
D).E and F show inhomogeneous deformation
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Relationship of s, e, and ? to the geometry of
the strain ellipseR 1sn r/R ren
(r-R)/R ?R/R ?R
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The area and volume strains - Deformation of a
unit circlee1 ? e2 ? e3 and s1 ? s2 ? s3Area
stretch sA a/A and vA v/V Area extension
eA(a-A)/A sA-1 and eV (v-V)/V sV-1 Initial
rectangle of sides L1 L3 deforms to l1 l3sA
a/A l1l3 /L1L3 s1s3 (1e1)(1e3)sV l1l2l3
/L1L2L3 s1 s2s3 (1e1)(1e2)(1e3)eA
sA-1(1e1)(1e3)1 e1e3e1e3 e1e3 (small
strain)eV sA-1 (1e1)(1e2)(1e3)1
e1e2e3 (small strain)
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Strain ellipse and Inverse strain ellipse
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Undeformed and deformed oolitic limestone
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Structures depend on the orientation of the layer
relative to the principal stretches and value of
s2
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Uniaxial strainTwo principal stretches are equal
to 1
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Pure ShearConstant volume, plane strain in which
the principal axes of strain do not rotate by
deformations1gt1 s2 1 s3lt1sV 1 s1 1/s3
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Simple ShearConstant volume, plane strain in
which the principal axes of strain rotate by
deformation. The displacement of all material
particles is strictly parallel to the shear
planes1gt1 s2 1 s3lt1sV 1 s1 1/s3
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Types of homogeneous deformation of a cube,
circle, and sphere
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Decomposition of an arbitrary homogeneous strain
into a pure strain, a rigid rotation, and a rigid
translation
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Particle motions during progressive pure shear
and simple shear
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States of strain during a steady progressive pure
shear
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States of strain during steady progressive simple
shear
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Geometry of finite and instantaneous strain
ellipses
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The histories of progressive deformation for
competent layers oriented within the different
sectors of the strain ellipse (see last slide)
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Shortened boudins
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Distributions of sectors of stretch and stretching
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Folding of a layer and simultaneous boudinage of
a perpendicular layer
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Flinn Diagram for constant volume
deformationk(a-1)/(b-1)
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Flinn Diagram plots of the line for plane strain
at various values of he volumetric stretch
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Logarithmic Flinn diagram for various amounts of
volumetric stretch sV s1s2s3 strain
eVsA-1e1e2e3
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Strain Tensor Components in 3D
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Displacement vector connects the position X of a
material particle in the undeformed state to its
position in the deformed state. (B) The vectors
for neighboring points are different
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Components of the infinitesimal strain for
two-dimensional strain