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A brief overview of the topics that are part of the vast solid mechanics topic.It covers the basic formulas and concepts of deformation gradient
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Overview of ASM
Contents
• Basics• Strain• Stress• Constitutive Relations• Energy relations• Failure Theories
• Non Linearity• Linearity and Non‐linearity• Examples• Abaqus Model
BasicsOverview of stress, strain and other topics
Types of strain deformation
• Large strain deformation• Finite strain• Obtained by exact measures of changes in dimensions or angles
• Small strain deformation• Linearized or infinitesimal strain• Obtained by linearizing a finite strain measure w.r.to displacement gradients
Image Source:"DeformationOfRod plain" by Bbanerje ‐ Own work. Licensed under CC BY‐SA 3.0 via Wikimedia Commons ‐http://commons.wikimedia.org/wiki/File:DeformationOfRod_plain.svg#/media/File:DeformationOfRod_plain.svg"2D geometric strain" by 2D_geometric_strain.png: Sanpazderivative work: Mircalla22 (talk) ‐ 2D_geometric_strain.png. Licensed under Public Domain via Wikimedia Commons ‐http://commons.wikimedia.org/wiki/File:2D_geometric_strain.svg#/media/File:2D_geometric_strain.svg
Types of strain
• 1‐Dimensional strain
• Lagrangian strain ; 1
• Eulerian strain ; 1
• Lagrangian strains are preferred in solid and structural mechanics• Eulerian strains are preferred in fluid mechanics• For large strains difference between them is huge• For strains 1%, difference between them is insignificant
Strain
• 3D Strain‐Displacement equations
; ;
; ;
• Strain Transformation from an axis to ′
; ;
Compatibility conditions
• Needs to be satisfied by strain field in order to ensure single valued continuous displacements
; ;
2 ; 2 ; 2
Principal strains and axes• Strain at point depends on the point and the orientation of coordinate system
• There exists three orthogonal directions along which is extreme• Extreme values of normal strain are principal strains and are associated with principal axes
; ̂ ̂
0 ⇒ 0
Principal strains and axes
• , , are known as strain invariants• Two identical strain states must have same set of strain variants• To determine uniquely 1is used
;
Definitions
Reference configuration• Condition of the body in 3D space before loads have been applied
Deformed configuration• Location and shape of the body after loads have been applied
Image Source: "Displacement of a continuum" by Displacement_of_a_continuum.png: Sanpazderivative work: Nicoguaro (talk) ‐ Displacement_of_a_continuum.png. Licensed under CC BY‐SA 3.0 via Wikimedia Commons ‐http://commons.wikimedia.org/wiki/File:Displacement_of_a_continuum.svg#/media/File:Displacement_of_a_continuum.svg
Deformation gradient
Derivative of each component of the deformed vector with respect to each component of the reference vector
,
• Lagrangian strain 1• Eulerian strain 1
Stress
• Infinitesimal Stress idealization at a point• Stress on an arbitrary plane
Image Source: "Stress in a continuum" by Sanpaz ‐ Own work. Licensed under CC BY‐SA 3.0 via Wikimedia Commons ‐ http://commons.wikimedia.org/wiki/File:Stress_in_a_continuum.svg#/media/File”Stress_in_a_continuum.svg"Cauchy tetrahedron" by Sanpaz ‐ Own work. Licensed under CC BY‐SA 3.0 via Wikimedia Commons ‐ http://commons.wikimedia.org/wiki/File:Cauchy_tetrahedron.svg#/media/File:Cauchy_tetrahedron.svg
)( kjiT xzxyx
x )( kjiT yzxyx
y
)( kjiT zzyzx
z kTjTiTT
n
z
n
y
n
x
n
nTn
n . 22
2n
n
n T
Stress Transformation
′
; ;
Image Source “Stress transformation 3D" by Sanpaz ‐ Own work. Licensed under CC BY‐SA 3.0 via Wikimedia Commons ‐http://commons.wikimedia.org/wiki/File:Stress_transformation_3D.svg #/media/File:Stress_transformation_3D.svg
Principal stresses and planes
• Extreme values of normal stress are principal stresses and are associated with principal planes
; ̂ ̂
0 ⇒ 0
; I ;
• To determine uniquely 1is used
Types of stress
• Cauchy’s stress• Kirchhoff stress• First Piola‐Kirchhoff stress (PK1)• Second Piola‐Kirchhoff stress (PK2)• Plane stress
Cauchy’s Stress
• Used for stress analysis of bodies experiencing small deformations• Measure of force acting on an element area in deformed configuration• Second order tensor, Nine stress components, Symmetric tensor• The tensor relates a unit‐length direction vector n to stress vector across an imaginary surface perpendicular to n
. ;
Image Source: "Components stress tensor cartesian" by Sanpaz ‐ Own work. Licensed under CC BY‐SA 3.0 via Wikimedia Commons ‐http://commons.wikimedia.org/wiki/File:Components_stress_tensor_cartesian.svg#/media/File:Components_stress_tensor_cartesian.svg
Kirchhoff stress
• Widely used in numerical algorithms in metal plasticity (Where there is no change in volume during plastic deformation)• For large (finite) deformations, Kirchhoff stress tensor is required• Defined as the product of and
Where ; also known as jacobian
1st PK stress
• Used to measure large deformations• Relates the force in the deformed configuration to an oriented area vector in the reference configuration
• Stress in unsymmetrical and is a two point tensor (?) • Double vector – Capital indices are often used to indicate reference coordinates and lowercase for present coordinates
=
2nd PK stress
• Used as a measure for finite deformations• Defined in the reference configuration
Plane stress
• A material is said to be under plane stress if , ,perpendicular to ‐ plane are assumed to be zero
• Often stress analysis of thin plates is consideredas plane stress case
• Stress and strain at some point in material are given by00
0 0 0;
00
0 0
Image Source: "Plane stress" by Sanpaz ‐ Own work. Licensed under CC BY‐SA 3.0 via Wikimedia Commons ‐http://commons.wikimedia.org/wiki/File:Plane_stress.svg#/media/File:Plane_stress.svg
Plane strain
• A material is said to be under plane strain if , ,
perpendicular to ‐ plane are assumed to be zero • One of the dimensions is much larger than the othertwo such as thick plate, dam
• Stress and strain of the material are given by
00
0 0;
00
0 0 0Image Source: "Plane strain" by Sanpaz ‐ Own work. Licensed under CC BY‐SA 3.0 via Wikimedia Commons ‐http://commons.wikimedia.org/wiki/File:Plane_strain.svg#/media/File:Plane_strain.svg
Materials• Metals
• Alloys• Polymers – Made from long chain molecules
• Thermoplastics – May be reformed with heat Eg: PVC• Thermosetting plastics – Once formed, cannot be deformed by heat Eg: Epoxy
• Elastomers – Long chain elastic molecules Eg: Natural Rubber
• Ceramics – Made by heating together materials such as silica, chalk and clays
• Composites – Mixtures of materials which give improved properties
Metal
CeramicsPolymerComposites
Types of materials
• Linearly elastic: Material satisfies Hooke’s law• Viscoelasticity: Time‐dependent resistive contributions are large and certainly do not satisfy Hooke’s law Eg: Rubbers and plastics
• Plasticity: Applied force induces non‐linear displacements in the material
• Hyperelastic: Applied forces induces displacements in the material which follows a strain energy density function
Constitutive Relations
• In 3D there are 15 variables• Six stresses• Six strains• Three displacements
• In 3D number of available equations (6+3)• Six geometric relations• Three equilibrium equations
Six material constraints (Constitutive relations)
Constitutive Relations: Linearly elastic Material
• General stress‐strain relationship ⇒ 81material constants
; , 1 3
• Stress and strain tensor are symmetric ⇒ 36material constants
• Introduction of potential function ⇒ 21material constants12
12
12
Constitutive Relations: Linearly elastic Material
• is known as constitutive matrix, inverse of is known as compliance matrix
• For an orthotropic material (three planes of elastic symmetry) ⇒ 9material constants
zx
yz
xy
z
y
x
zx
yz
xy
z
y
x
CC
CCCCCCCCCC
66
55
44
332313
232212
131211
0000000000000000000000001
1
1
1
1
1
Constitutive Relations: Linearly elastic Material• For homogenous isotropic linearly elastic continuum⇒ 2material constants
zx
yz
xy
z
y
x
zx
yz
xy
z
y
x
E
)1(2000000)1(2000000)1(2000000100010001
1
Energy relations
• Work done by surface and body forces on an elastic solid are stored inside the body in the form of strain energy
• Strain energy density is given by12
• Strain energy can be expressed in terms of stress or straine;
• Strain energy can be decomposed into two parts• Volumetric change • Distortional energy‐change in shape
Failure Theories
• Used to determine the safe dimension of a component subjected to combined stresses due to various loads
• Various theories• Maximum principal strain theory (St. Venant’s theory)• Maximum principal stress theory (Rankine’s theory)• Maximum strain energy theory (Haigh’ theory)• Maximum distortion energy theory (Von‐mises theory)• Maximum shear stress theory(Treska’s theory)
Failure Theories
• As area bounded by curve increases• Failure stress increases• Dimensions of part decreases• Safety and cost decreases
Image source: http://www.slideshare.net/MANMEET2591/theories‐of‐failure
Linearity
• Material in the model should comply with Hooke’s law• Induced displacements during loading do not cause
• Geometric properties• Material properties• Stiffness – Extent to which it resists deformation
in response to an applied force
• Boundary conditions do not change during loading• Sub set of non linear analysis
Idealization Idealization
Obtain bending moment Obtain bending moment
Obtain stress tensorObtain stress tensor
Constitutive relationsConstitutive relations
Apply failure theoriesApply failure theories
Compare materialsCompare materials
Thank you
NonlinearityExamples and mathematical model
Nonlinearity
• Geometric nonlinearities – Effect of large displacement on overall geometric configuration of the structure
• Material nonlinearities – Material behavior is nonlinear• Nonlinear elastic – Strain produced is not proportional stress • Elastoplastic – Has both elastic and plastic properties• Viscoelastic – Has viscous (resistance to gradual shear deformation) and elastic properties
• Viscoplastic – Deformation of the material depends on the rate at which loads are applied
• Boundary nonlinearities – Displacement dependent boundary conditions
Non‐linearity
Geometric non linearityMaterial non linearity Boundary non linearity
Image Source: Types of analysis: Linear static, linear dynamic and non linear static‐ Paulo B. Lourenço
Consequences of non linearity
• Principle of superposition cannot be applied• Only one load case can be handled at a time• Sequence of application of loads is important (Eg: Plastic deformations) • Structural behavior can be markedly non‐proportional to applied load• Initial state of stress may be important
Examples ???
• Fishing rod exhibits geometric non linearity
Abaqus Model