The Deformable Body

The flexible body The elastic energy

Kinematics

The transplacement The transplacement: The deformation gradient: Material line element The transplacement: The deformation gradient:

The deformaton gradient

The deformaton gradient

The length ratio Length: The length ratio: The strain:

The length ratio

Locally length preserving transplacement Local isometry The rigid transplacement

The shear The shear:

The shear

The volume ratio The volume: The volume ratio:

The volume ratio

The area ratio The area: The area ratio:

The normal vector

Polar factorization theorem

The displacement The displacement gradient:

Strain tensors Deformation gradient: Left Cauchy-Green strain tensor: Green-St. Venant strain tensor: Infinitesimal strain tensor:

The Green-St. Venant strain tensor

Small displacement gradient

Principal directions of strain and principal stretches

Principal directions of strain and principal stretches

Representations of basic tensors

Strain, shear, volume and area ratios

The local deformation

Velocity and acceleration

Velocity gradient Velocity gradient: Stretching: Spin:

Velocity gradient and divergence

The rigid transplacement

Summary

Mass Mass density:

Conservation of Mass Local balance of mass:

Equations of motion Euler equations: Contact force: Body force:

Cauchy’s fundamental Lemma

Cauchy’s fundamental Theorem

Equations of motion, spatial description Global equations of motion: Local equations of motion:

Gauss theorem and the divergence

Referential description

The Piola-Kirchhoff stress tensor

Equations of motion, referential description Global equations of motion: Local equations of motion:

Gauss theorem and the divergence

Summary

Kinetic energy and the Power theorem Net power: Rigid body:

Power and Energy External power: Local equation of motion:

Power and Energy

The balance of mechanical energy Specific internal energy: Internal energy: Total energy: Heat supply (”heating”): The balance of energy:

The first law of thermodynamics

The local balance of energy

The net power per unit volume Stress tensor Conjugated strain tensor The second Piola-Kirchhoff stress tensor: Relations between stress tensors:

Summary

with hyperelastic material The elastic body with hyperelastic material Elastic potentials: Constitutive equations:

Strain energy density Strain energy density:

Elastic energy Total elastic energy: The internal energy: Thermal energy neglected!

Balance of energy for an elastic body

The linear elastic body Linear elastic material: The elasticity tensor: independent elasticities

The elasticity tensor Elasticities: 36 independent elasticities

The elasticity tensor Symmetry: 21 independent elasticities Positive definiteness: Compliance tensor exists

The elastic energy Green-St. Venant strain tensor: Elastic energy:

Crystal systems There are in all 7 different Crystal systems: - these have the following unit cells with associated number of elasicities cubic (3), tetragonal (7,6), orthorombic (9), triklinic (21), hexagonal (7,6,5), rhombohedral (9), monoklinic (13),

Isotropic linear elastic material Lame moduli: Elastic potential: Stress tensor: Elastic energy:

Homogeneous isotropic linear elastic material Lame moduli and independent of Relation to Young’s modulus and Poisson’s ratio :

Example 7.2: The elastic bar Present placement Reference placement

Principal of virtual power Euler equations (Eu1, Eu2): Local equation of motion (Lem): Internal forces zero system (Int):

Principal of virtual power Virtual velocity field:

Virtual power Virtual power of external, internal and inertial forces: are linear mappings:

The principal of virtual power Note that:

Rigid virtual velocity field Note that if the internal forces constitute a zero system then :

The principal of virtual power

The principal of virtual power

The principal of virtual power in continuum mechanics (Lem 1) (Lem 2) Virtual velocity field: Virtual powers:

The principal of virtual power in continuum mechanics

The principal of virtual power Equivalences

The principal of virtual power in continuum mechanics

Exercise 2b:17

Exercise 2b:17 Solution Disc: 𝒟 Shaft: 𝒮 Ground: G

Exercise 2b:17 Solution

Exercise 2b:17 Solution

Exercise 2b:17 Solution

Exercise 2b:17 Solution

Exercise 2b:17 Solution Angular velocity of shaft: Angular velocity of disc:

Exercise 2b:17 Solution

Exercise 2b:17 Solution Free body diagrams:

Exercise 2b:17 Solution Equations of motion for Disc:

Exercise 2b:17 Solution

Moments of inertia for thin wheel

Exercise 2b:17 Solution Equations of motion for Shaft:

Exercise 2b:17 Solution Combining the equations by eliminating :

Exercise 2b:17 Solution Neglecting inertia of the shaft:

Exercise 2b:17 Solution Equations of motion in matrix representation:

Exercise 2b:17 Solution

Exercise 2b:17 Solution Equations of motion: Five equations and five unknowns:

Exercise 2b:17 Solution

Exercise 2b:17 Solution

Exercise 3:3

Exercise 3:3

Exercise 3:9

Exercise 3:12

Exercise 3:15

Exercise 3:21

Exercise 3:21, continued