Give s simple constitutive equation for normal stress
Hooke´s law is one of the most basic constitutive equations. It relates strains to stresses via the Young´s modulus:
Give Hooke´s law for shear deformation?
For shear deformation Hooke´s law relates Shear deformation to the shear stresses via the Shear modulus:
How many discplacements are to found in a three dimensional body?
3
How many stresses are to found in a three dimensional body?
How many independent stresses are to found in a three dimensional body?
9 stresses (three normal & six shear stresses)
6 independent stresses (shear stresses are reduced to three via Moment equilibrium)
How many strains are to found in a three dimensional body?
6 (three normal strains and three shear strains)
Which and how many state variables are to found in a three dimensional body?
How many equations are needed to determine those?
15
3 displacements
6 independent stresses
6 strains
—> 15 Equations needed
Whicgh equations are needed in order to determine the state variables of a body?
3 Equlibirum condtitions
6 Kinematic equations
Hooke´s generalized law (sigma = C*epsilon) —> 6 equations
What is described by Hookes generalized law?
What is described by the Kinematic equations?
The Discplacements and Deformations are connected.
How do Kinematic and Kinetic differentiate?
Kinematic describes pure geometric behaviour, while Kinetic describes the influence of forces on the Movement of a body.
What is described by the quilibrium conditions?
The equilibrium conditions call for an equilibrium of stresses over an infinitesimal volume —> Stresses should equal out, otherwise it would move.
E.g. sigma_xx can only change over the body if shear stresses not perpendicular to x call for a deformation over the body.
They are fully coupled and statically indeterminate
What is characteristic for a beam?
b&h << l —> 1D element
Bending - transverse shear load as well as torsional moments, described by EI (yy&zz), GI and Warping stiffness EI_omega
Which justifications are to be carried out for beams?
Static justification (Bending, Torsional Moment as well as transverse shear)
Global Buckling
Local Buckling dependent on cross section
What is characteristic for a disc?
b << h&l —> 2D structural element
Load exclusively in disc plane —> Force flows: Nxx, Nyy and shear force flow Nxy
Reduction to disc middle plane
Which justifications are to be carried out for a disc?
Static
Buckling
What is characteristic for a truss?
b << h&l —> 2D element, idealized as structure of 1D elements (bars)
Load only in disc plane —> Bars only Normal stresses!
Which justifications are to be carried out for a truss?
Column buckling
Flexural-torsional buckling
local buckling
What is characteristic for a plate?
h << b&l —> Idealized as 2D structural element
Load exclusively perpendicular to plate plane —> Transverse Force Flows, Moment flows around edge axis
Reduction to plate middle plane
Which justifications are to be carried out for a plate?
What is characteristic for a shell?
Curved 2D structural element (Design Space fully 3D)
Arbitrary load —> Combined disk and plate characteristics
Reduction to middle plane
Which justifications are to be carried out for a shell?
How does Orthotropy look, and what is avoided?
How many independent Material parameters are required?
Bending extension coupling is avoided
Bending twisting coupling is avoided
9 independent material parameters are required
How does Transversal Isotropy look, and what is avoided?
5 independent material parameters are required
How many independent material parameters are required for full Isotropy/ anisotropy?
Isotropy: 2
Anisotropy: 21
Which engineering constants are required for an Orthotropic behaviour and how many are independent?
12 engineering constants:
3 Young´s moduli (E11, E22, E33)
6 Poisson´s ratios (nu12, nu13, nu23, nu32, nu21, nu31)
3 Shear moduli (G12, G13, G23)
—> Only 9 independent due to:
nu12/E11 = nu21/E22
How to draw Mohr´s Circle?
What are the characteristic stresses in Mohr´s circle and which angles lie between them?
Principal stresses sigma-1 and sigma-2, no shear stress
sigma-M and max shear stress (45° off principal stresses)
What is used to derrive the dependence of shear stresses from eacht other?
Moment equilibrium at the infitisimal element
Give the compliance matrix with the engineering constant for an orthotropic material
Under which assumption can we assume orthotropy for laminate layers?
If the fibres are parallel and evenly spread.
However more important: if we look at the material in an on axis system with the fibres (will not be the case in assembled state anymore)
Name 4 material symmetries and their required number of independent material constants. (For full picture give number of required material constants in Full anisotropy)
Describe coupling in an Anisentropic material.
A single stress component will activate all 6 strains and vice versa.
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