Computational Plasticity 1

Introduction to Finite Strains

Topics: Finite strain kinematics

Stress measures

Hyperelasticity

Computational Plasticity 2

Finite Strain Kinematics

Computational Plasticity 3

The deformation map

Computational Plasticity 4

Rigid deformation

Computational Plasticity 5

Motion. Time-dependent deformations

Computational Plasticity 6

Rigid motion. Rigid velocity

Computational Plasticity 7

The deformation gradient

Computational Plasticity 8

Measuring volumetric changes. The determinant of the deformation gradient

Computational Plasticity 9

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Locally isochoric (volume-preserving) deformations

Locally purely volumetric deformations

Computational Plasticity 11

Isochoric/volumetric split of the deformation gradient

Note that

Computational Plasticity 12

Polar decomposition. Local rotation and stretches

Right and left stretch tensors

Right and left Cauchy-Green strain tensors

Computational Plasticity 13

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Spectral decomposition. Principal stretches

Computational Plasticity 15

Strain measures

Green-Lagrange strain tensor

Locally rigid deformation

Computational Plasticity 16

Spectral representation of the Green-Lagrange strain

Other Lagrangian strain tensors

Computational Plasticity 17

Computational Plasticity 18

Eulerian strain measures

Computational Plasticity 19

Velocity gradient

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Rate of deformation and spin

Consider a uniform velocity gradient. We have The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again.

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or, equivalently, rigid

velocity

straining velocity

Computational Plasticity 20

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Rate of volume change

Also note that

so that, equivalently,

Computational Plasticity 22

Stress Measures

Computational Plasticity 23

The Cauchy stress tensor

Cauchys Theorem establishes that

Computational Plasticity 24

Principal Cauchy stresses

Computational Plasticity 25

Deviatoric and hydrostatic Cauchy stresses

The Kirchhoff stress tensor

Computational Plasticity 26

The First Piola-Kirchhoff (or nominal) stress tensor

measures force per unit reference area

Computational Plasticity 27

Finite Strain Hyperelasticity

Computational Plasticity 28

Thermodynamics with internal variables. Dissipative models

Free-energy function

Dissipation inequality. Second law of thermodynamics

Dissipation

Computational Plasticity 29

the dissipation inequality implies

Hyperelasticity. Definition

No dissipation !

Computational Plasticity 30

Material objectivity

Computational Plasticity 31

Isotropic hyperelasticity

the free-energy is an isotropic scalar function of a tensor argument

Computational Plasticity 32

Principal stretches representation

Principal stresses

Computational Plasticity 33

Invariant representation

Stress

Computational Plasticity 34

Regularised (compressible) Neo-Hookean model

Computational Plasticity 35

Regularised (compressible) Ogden model

Computational Plasticity 36

Hencky model (logarithmic strain-based)

Important properties

Computational Plasticity 37

Hyperelasticity boundary value problem

Linearised virtual work equation

Reference (or material) description

material tangent modulus or first elasticity tensor

Computational Plasticity 38

Spatial description

Linearised virtual work equation

spatial tangent modulus

Computational Plasticity 39

FE Equations (spatial description)

Newton-Raphson solution

Computational Plasticity 40

Example