General Flashcards
What is elasticity?
Elasticity is the response of a material to a force of which it acts elastic.
What is a continuum?
A continuum is a describtion of a space, often in refference to a material.
Describe local and global quantities.
Local quantities describe something local to a continuum, meaning a infinitesimal piece of a continuum.
Global quantities describe something of the whole continuum, for example the elongation of the entirety of a rod.
What is a tensor?
“[a] mathematical objects that can be used to describe physical properties”.
Tensors have orders. Name the three orders we use the most by name and order. Name the highest order we have used.
n-order up to n=4 with 3^n quantaties.
n=0 -> scalar,
n=1 -> vector,
n=2 -> dyadic
The 4th order was the constitutive equations. (elastic constants of E_ijkl) (see Hooke’s law)
-> sigma_ij = E_ijkl eps_kl
What is The Einstein summation convention?
A term with repeated indices is to be summed over all indices. Example:
A_iB_i = A_1B_1 + A_2B_2 + A_3B_3
How do you find the eigenvalues and eigenvectors?
Eigenvalues:
det(A-lambdaI) = 0
Eigenvectors:
(A-lamda_iI)v_i = 0 or Av_i = lambda_i*v_i
What is the eigenproblem in context of Theory of Elasticity?
The eigenvalues and eigenvectors of for example a Cuachy stress tensor, is the principal stresses and the principal direction of the stress tensor.
What is a rotation matrix and what does it look like?
A rotation matrix is used to transform a tensor from one basis to another.
The rotation matrix is nxn, where n is the order of the tensor.
Vector:
x’ = Rx <=> A’_i = a_ijA_j
Dyadic:
A’ = RAR^T <=> A’_ij = a_ika_jl*A_kl
What is the rotation matrix for principal direction (n=2)?
R = [v_1, v_2, v_3]^T, where v_i is the eigenvectors on column form.
The rotation matrix applied:
A’ = RAR^T
What is a traction vector (T_n)?
Also known as a stress vector. It is a vector which describes the stresses in three directions on a given surface, i.e. the normal and two shear stresses.
For the cube at some point (gives three vectors): T_i = sigma_ije_j
For a surface (one vector): T = sigma_ijn_i*e_j
What is the Cauchy stress tensor?
The Cauchy stress tensor is an array of all the stresses of a given “cube”. It is therefore three faces and their respective normal stress and shear stresses. It is therefore also a combination of three traction vectors.
Why is the Cauchy stress tensor symmetrical?
Due to the conservation of angular momentum (The parallelogram where the arrows point towards each other).
How are forces handled by a solid?
The solid will be put under stress and will deform and therefore strain.
What are body forces?
Body forces are forces that acts throughtout the volume of the body, e.g. gravity. It is given as a vector: f = f_i*e_i
What are the equilibrium (balance) equations?
For static problem the r.h.s is zero. This means that the traction with or without body forces must be zero.
This becomes: sigma_ij,j + f_i = 0, which is three equations.
What is the divergence theorem?
The divergence theorem relates a volume integral to a surface integral:
int_V( div(G) )dV = int_A( n*G ) dA
where G is a dyadic and n is a normal vector to the surface.
It is used for the equilibrium equations (sigma_ij,j + f_i = 0).
What is the principle of angular momentum?
It states that if there is no moment acting on the system the angular momentum is conserved. This makes the Cuachy stress tensor symmetric
What describes deformation of a continuum.
The change of internal points in respect to each other, i.e. strain.
It is given as:
eps_ij = 1/2(u_i,j + u_j,i +- u_k,i u_k,j). (plus and minus depending on either the Lagrangian strain (+) tensor or Eulerian strian tensor (-).
but more often well approximated to:
eps_ij = 1/2(u_i,j + u_j,i)
What are the displacement gradients?
The displacement gradients are u_k,i and u_k,j, and are the nonlinear term of the strain.
They are usually very small compared to unity (one) and the product of them are then negligible.
Maybe more physical
Define displacement.
Displacement can be both relative motion/distortion within the body, i.e. strain and/or rigid body motion of the whole body.
Give a physical interpretation of the strain tensor (two types).
Extensional strain (relative elongation) and shearing strain (relative change of angles)
Given as, respectively:
eps_ij[i=j] = (dS - ds) / ds -> dL/L
eps_ij[i/=j] = 1/2 beta = 1/2(u_i,j + u_j,i)
Where beta is the changed angles summed, for example:
eps_13 = 1/2 beta = 1/2(beta^(1) + beta^(3))
Name the four special states of stress.
Plane stress:
Stress only in two dimensions.
Linear stress:
Stress only in one dimension.
Pure shear:
Only shear stresses for any coordinate system -> No normal stresses.
Hydrostatic stress:
All principal stresses are equal and no shear stresses. Principal stresses are equal in any direction; (A-lambda_i*I)v_i = 0 always, as A = 0 for all i).
What and how do you find principal strain?
Principal strain is a eigenproblem and is the highest strain in the system in the principal direction.
det(A-lambdaI) = 0
-> (A-lambda_iI)v_i = 0
How is change in size and shape described?
Strain can be described of a combination of change of size (dilatation) and shape (detrusion):
eps = eps_M + eps_D.
Change in size is described by the mean normal strain:
eps_M=1/3(eps_11+eps_22+eps_33).
Change in shape is described by the strain deviator:
eps_D=eps-eps_M.
What is the compatibility equations
The compatibility equations insures that one can go from strain to displacement.
As strain_ij = 1/2(u_i,j + u_j,i) we go from one known to two unknowns and from there 6 equations to 3 unknowns.
The six compatibility eqs. link the strains such that one can find the unique displacement field.
What are the constitutive equations?
The constitutive equations is defined from the material. It tells how a material will strain when put under stress. We have used the generalised Hooke’s law which assumes that the material is isotropic and linearly elastic.
It is denoted with a C_ij or E_ijkl.
There is 21 components and 6 eqs.
What assumptions does Hooke’s law include?
Isoptropic and linearly elastic. Results in 21 components and 6 eqs.
What are thermal strains and how does it behave?
Thermal strains is strains induced by thermal energy. Heating something up means increasing the distance between the atoms such that the solid increases in size.
What is the De Saint Venant problem?
The De Saint Venant problem is the case of the rod where load is applied at the ends.
The stresses will only be on the third plane; sigma_3i.
From there the traction or stress vector is derived.
What is the De Saint Venant principle?
The De Saint Venant principle states the stress and strain will be the same throughout each fiber except at the for small regions at the bases.
Describe the simple bending derivation using the De Saint Venant principle.
By assuming the rod only has stress in the x_3-plane; sigma_3i, the distance from the barycenter crossed with the derived traction vector can be used to get the resultant moment vector;
M = int(S) ( (x-x_G) x T_e_3 )dS
This then becomes (with int):
M’2 = -sigma_33x_1,
M’_1 = sigma_33x_2.
And
sigma_33(x(1or2) = alpha*x_(1or2).
When integrating M’_1 and M’_2 one gets:
M_2 = -alphaI_2,
M_1 = alphaI_1.
Which then becomes:
sigma(x_1,x_2) = N/A - M_2/I_2x1 + M_1/I_1x2
Describe the torsion derivation using the De Saint Venant principle.
By assuming the rod only has stress in the x_3-plane; sigma_3i, the distance from the barycenter crossed with the derived traction vector can be used to get the resultant moment vector;
M = int(S) ( (x-x_G) x T_e_3 )dS
This then becomes (with int):
x_1tau_32 - x_2tau_31
tau_31 and tau_32 can be derived from the displacement field to be:
tau_31 = -Gtheta’x_2
tau_32 = Gtheta’x_1
where:
r_31 = -theta’x_2
r_32 = theta’x_1
Inserting in the equation from before:
M_3 = Gtheta’ * int(A)( (x_1^2 + x_2^2 )
= Gtheta’ * int(A)( r^2 )
= GJtheta’
Which can be isolated for the torsional rigidity:
=> C_1 = GJ = M_3/theta’
How is Young’s modulus meassured?
Young’s modulus is meassured by elongating a object and meassuring how how the object has strained devided by how much stress has been induced:
E = sigma_11/eps_11 = (F/A)/(dL/L) = (FL)/(AdL).
How is Poisson ratio meassured?
If stress is applied in direction of x_1, then the Poisson ratio is the negative ratio of the strain eps_22 or eps_33 devided by eps_11:
v = -eps_22/eps_11 = -eps_33/eps_11.
What is the rotation tensor?
The rotation tensor is skew-symmetric with 0 for i = j. It is defined as:
omega_ij = 1/2(u_i,j - u_j,i).
It can be interpreted as the average rotation of all fibers around a given axis. Example:
omega_13 = (beta^(1) - beta(3))/2.
When can’t the compatibility equations be applied?
Strain fields due to nonmechanical loads such as thermal and plastic deformation.
What is the barycenter?
Also known as the center of mass. Every particle around it is distributed such that all particles’ gravity equals out.
It can be found by:
x_G = int(A) ( x ) dA
What is Kronecker delta, delta_ij?
It is a operation used of which when i = j the Kronecker delta is equal to 1. When i =/ j Kronecker delta is equal to 0.
It is used to change subscripts of a tensor.
What is the permutation tensor, eps_ijk?
It is a tensor of which is equal to 0, 1, or -1. It is useful for for example cross-product.
When ijk is like 1,0,1 it is equal to 0.
When ijk counts forward like 1,2,3 or 3,1,2 it is equal to 1.
When ijk counts backwards like 3,2,1 or 2,1,3 it equal to -1.
Also:
eps_ijk = 1/2(i-j)(j-k)(k-i)
What is a surface force?
A surface force is a force applied to the surface of a material.
What is the difference between the Lagrangian strain tensor and the Eulerian strian tensor?
The Lagrangian strain tensor is derived from the basis of the initial coordinates x_i which gives a plus:
eps_ij^L = 1/2(u_i,j + u_j,i + u_k,i u_k,j).
The Eulerian strain tensor is dervied from the basis of the final coordinates X_i which gives a minus:
eps_ij^E = 1/2(u_i,j + u_j,i - u_k,i u_k,j).
In material science the initial body is usually known so the Lagrangian is used. In fluids the Eulerian is used.
