13. Linear Elasticity

Fundamental Equations Elasticity

Stresses

s

Equilibrium

Body force b

Displacement u

Strains

e

Kinematics

Constitutive law

Differential eq.

? ?

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