holzapfel ʻnonlinear solid mechanics holzapfel nonlinear solid...
Transcript of holzapfel ʻnonlinear solid mechanics holzapfel nonlinear solid...
1 06 – concept of stress
06 – concept of stress!
holzapfel ‘nonlinear solid mechanics‘ [2000], chapter 3, pages 109-129
2 06 – concept of stress
06 – concept of stress!
holzapfel ‘nonlinear solid mechanics‘ [2000], chapter 3, pages 109-129
3 06 – concept of stress
me338 - syllabus
review: 04 - kinematics 4
configurations
• displacement field in the spatial description
review: 04 - kinematics 5
displacement fields
• displacement field in the material description • acceleration field in material and spatial description
review: 04 - kinematics 6
velocity and acceleration field
• velocity field in material and spatial description
and
and
7
material derivative of a spatial field
• material derivative of a spatial field
• under application of the chain rule
convective term • easier to remember
review: 04 - kinematics 8
deformation gradient
• by using the chain rule we can relate the line elements
• the deformation gradient relates vectors
and
and
review: 04 - kinematics
9 06 – concept of stress
definition of stress stress [‘stres] is a measure of the internal forces acting within a deformable body. Quantitatively, it is a measure of the average force per unit area of a surface within the body on which internal forces act. These internal forces arise as a reaction to external forces applied to the body. Because the loaded deformable body is assumed to behave as a continuum, these internal forces are distributed continuously within the volume of the material body, and result in deformation of the body's shape. • the body is in equilibrium under the external forces
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traction vector
• we cut the body along a plane through point
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traction vector
• and establish a force equilibrium again, where
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traction vector
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• we call the traction vector or surface traction or more specifically the cauchy traction vector
traction vector
• the traction vector follows newton‘s third law of action and reaction
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traction vector • lets look at the traction vector in more detail
• the vectors m and n are perpendicular to each other or mathematical speaking
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traction vector
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cauchy‘s stress theorem • for every possible plane cutting through the point x at time t there exists a traction vector t
• the infinite number of traction vectors at point x define the stress state of at that point
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• cauchy‘s stress theorem states further that
cauchy‘s tetrahedron
• where is the cauchy stress tensor
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stress components
keeping in mind:
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concept of stress – example 1 • consider the cauchy stress tensor as given below
• a) find the traction vector corresponding to the plane?
• b) what is the magnitude of the normal and the shear stress?
• c) is the normal stress tensile or compressive?
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concept of stress – example 1 • a) find the traction vector corresponding to the plane?
Don’t forget to normalize the normal vector
• a) it follows that
• b) what is the magnitude of the normal stress?
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concept of stress – example 1
• b) what is the magnitude of the shear stress?
or
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concept of stress – example 1 • c) is the normal stress tensile or compressive?
• c) since the normal stress is tensile
• they follow from the characteristic equation
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extremal stress values • principal normal stresses include the maximum and minimum normal stress among all possible directions
• where are denoted the stress invariants
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extremal stress values • principal directions are the directions associated with the principal values and follow from
with (no summation)
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concept of stress – example 2 • given the following stress tensor
• a) what are the maximal stress values?
• b) what are the principal directions?
• c) what is their significance?
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concept of stress – example 2 • a) what are the maximal stress values?
• first we derive the characteristic equation (cubic)
• and solve for the eigenvalues
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concept of stress – example 2 • b) what are the principal directions?
• and so forth to obtain the three principal directions
• c) what is their significance?
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concept of stress – research example
• first piola-kirchhoff stress tensor from follows that
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alternative stress tensors
• cauchy stress tensor from follows that
• kirchhoff stress tensor different from only by
• second piola-kirchhoff stress tensor from follows