p03 basics of finite strains
DESCRIPTION
Finite Strain PlasticityTRANSCRIPT
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Computational Plasticity 1
Introduction to Finite Strains
Topics: Finite strain kinematics
Stress measures
Hyperelasticity
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Computational Plasticity 2
Finite Strain Kinematics
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Computational Plasticity 3
The deformation map
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Computational Plasticity 4
Rigid deformation
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Computational Plasticity 5
Motion. Time-dependent deformations
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Computational Plasticity 6
Rigid motion. Rigid velocity
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Computational Plasticity 7
The deformation gradient
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Computational Plasticity 8
Measuring volumetric changes. The determinant of the deformation gradient
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Computational Plasticity 9
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Computational Plasticity 10
Locally isochoric (volume-preserving) deformations
Locally purely volumetric deformations
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Computational Plasticity 11
Isochoric/volumetric split of the deformation gradient
Note that
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Computational Plasticity 12
Polar decomposition. Local rotation and stretches
Right and left stretch tensors
Right and left Cauchy-Green strain tensors
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Computational Plasticity 13
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Computational Plasticity 14
Spectral decomposition. Principal stretches
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Computational Plasticity 15
Strain measures
Green-Lagrange strain tensor
Locally rigid deformation
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Computational Plasticity 16
Spectral representation of the Green-Lagrange strain
Other Lagrangian strain tensors
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Computational Plasticity 17
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Computational Plasticity 18
Eulerian strain measures
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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
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Computational Plasticity 20
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Computational Plasticity 21
Rate of volume change
Also note that
so that, equivalently,
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Computational Plasticity 22
Stress Measures
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Computational Plasticity 23
The Cauchy stress tensor
Cauchys Theorem establishes that
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Computational Plasticity 24
Principal Cauchy stresses
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Computational Plasticity 25
Deviatoric and hydrostatic Cauchy stresses
The Kirchhoff stress tensor
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Computational Plasticity 26
The First Piola-Kirchhoff (or nominal) stress tensor
measures force per unit reference area
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Computational Plasticity 27
Finite Strain Hyperelasticity
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Computational Plasticity 28
Thermodynamics with internal variables. Dissipative models
Free-energy function
Dissipation inequality. Second law of thermodynamics
Dissipation
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Computational Plasticity 29
the dissipation inequality implies
Hyperelasticity. Definition
No dissipation !
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Computational Plasticity 30
Material objectivity
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Computational Plasticity 31
Isotropic hyperelasticity
the free-energy is an isotropic scalar function of a tensor argument
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Computational Plasticity 32
Principal stretches representation
Principal stresses
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Computational Plasticity 33
Invariant representation
Stress
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Computational Plasticity 34
Regularised (compressible) Neo-Hookean model
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Computational Plasticity 35
Regularised (compressible) Ogden model
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Computational Plasticity 36
Hencky model (logarithmic strain-based)
Important properties
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Computational Plasticity 37
Hyperelasticity boundary value problem
Linearised virtual work equation
Reference (or material) description
material tangent modulus or first elasticity tensor
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Computational Plasticity 38
Spatial description
Linearised virtual work equation
spatial tangent modulus
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Computational Plasticity 39
FE Equations (spatial description)
Newton-Raphson solution
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Computational Plasticity 40
Example