13. Linear Elasticity
13. Linear Elasticity
Fundamental Equations Elasticity
Stresses
s
Equilibrium
Body force b
Displacement u
Strains
e
Kinematics
Constitutive law
Differential eq.
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Constitutive relations • Constitutive relation
– Relation between stress and strain,
• Determined by measurements
• Elastic (linear, nonlinear), loading path independent
• Plastic, loading path dependent
• Time dependent, viscoelastic, viscoplastic
es ~
e
s
loading
unloading
e
s loading
unloading
loading
e
s
unloading
Linear elastic Nonlinear elastic Plastic
Hooke’s Law
• Linear elastic, 1-dim
– Hooke’s law (1676)
• Linear elastic, multi-dim
– Hooke’s generalized law
– D, a 6x6-matrix with 36 components
es E
Dεσ
loading
e
s
unloading
Linear elastic
D=DT
Scales of structures - inhomogenity
m
mm mm
km 100m
Naturally optimized properties - example bone
Structure Principal Stresses Culmann crane (1866)
Constitutive relations • Inhomogeneous material, D=D(x,y,z)
• Homogeneous material, D, indep. of coordinates
– Characteristic length, d
– Assume homogeneous material if size of body > ~5d
Concrete
d
Wood
Strain energy
• Path independent material (elastic) => D = DT
– 21 independent components in D
• Strain energy, W [Nm/m3]
• Hooke’s law gives
Strain energy • Since W is a scalar we have
• Using Hooke’s law and that D = DT we get
• Adding the expressions above yields
• Integration from zero to the current strain state
• And we can conclude that
Strain energy
• Since the strain energy always is positive
(see Eq. 2.66 in book, quadratic forms)
• We conclude that D is always positive definite and
• D is positive definite and is invertible
• C is the material compliance matrix
Symmetry properties
• If one symmetry plane exists in the material => 13 mtrl const.
• Other symmetry cases
Symmetry properties - otrhotropic
• Orthotropic three symmetry planes, 9 mtrl constants,
• 3 E, 3 n, 3 G
example: wood, fibre reinforced plastics
Symmetry properties - transverse isotropic
• Transverse isotropic, 5 mtrl. constants, 2 E, 2 n, 1 G
example: Sandwich materials
11
Symmetry properties - isotropic
• Isotropic material, 2 mtrl. constants, E, n
example: Metals, plastics, concrete
Isotropic material
• D positive definite only if
• Compliance matrix, C
Initial strains, e0
• Temperature expansion or shrinkage
• Total strains:
• Elastic strains:
• Hooke’s law:
• Temperature strain:
0εεε e
0εεε e
)( 0εεDDεσ e
a : thermal expansion coefficient
Initial strains, e0
• Isotropic material
• where
Plane Stress
• Thin bodies.
• All forces and stresses are located in the plane
Plane Stress
• Plane stress state: Stress only in the xy-plane, szz=sxz=syz=0
• Stress tensor
• Traction vector
• Equilibrium equations
Plane Stress
• Isotropic material
=0
=0
=0
and
=0 =0 =0
Plane Stress
• Inversion leads to
• where
• and if thermal strains are included
Plane strain
• No forces or displacement in the z-direction
Plane Strain
• Displacements at plane strain
• The strains ezz= gxz= gyz=0
Plane Strain
• Isotropic material: Dεσ
=0
=0
=0
and
=0 =0 =0
Plane Strain
• Inversion leads to
• where
• and if thermal strains are included
Fundamental Equations Elasticity
Stresses
s
Equilibrium
Body force b
Displacement u
Strains
e
Kinematics
Constitutive law
Differential eq.
sDe 0~~
buDT