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

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Rigid deformation

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Motion. Time-dependent deformations

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Rigid motion. Rigid velocity

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The deformation gradient

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Measuring volumetric changes. The determinant of the deformation gradient

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Locally isochoric (volume-preserving) deformations

Locally purely volumetric deformations

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Isochoric/volumetric split of the deformation gradient

Note that

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Polar decomposition. Local rotation and stretches

Right and left stretch tensors

Right and left Cauchy-Green strain tensors

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Spectral decomposition. Principal stretches

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Strain measures

Green-Lagrange strain tensor

Locally rigid deformation

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Spectral representation of the Green-Lagrange strain

Other Lagrangian strain tensors

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Eulerian strain measures

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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

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Rate of volume change

Also note that

so that, equivalently,

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Stress Measures

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The Cauchy stress tensor

Cauchy’s Theorem establishes that

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Principal Cauchy stresses

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Deviatoric and hydrostatic Cauchy stresses

The Kirchhoff stress tensor

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The First Piola-Kirchhoff (or nominal) stress tensor

measures force per unit reference area

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Finite Strain Hyperelasticity

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Thermodynamics with internal variables. Dissipative models

Free-energy function

Dissipation inequality. Second law of thermodynamics

Dissipation

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…the dissipation inequality implies

Hyperelasticity. Definition

No dissipation !

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Material objectivity

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Isotropic hyperelasticity

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

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Principal stretches representation

Principal stresses

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Invariant representation

Stress

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Regularised (compressible) Neo-Hookean model

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Regularised (compressible) Ogden model

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Hencky model (logarithmic strain-based)

Important properties

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Hyperelasticity boundary value problem

Linearised virtual work equation

Reference (or material) description

material tangent modulus or first elasticity tensor

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Spatial description

Linearised virtual work equation

spatial tangent modulus

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FE Equations (spatial description)

Newton-Raphson solution

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Example