Micromechanics- Classical Laminate Theory 2 Flashcards

1
Q

How are the 9 stress components acting on a small element of material oriented and named?

A

Consider a cube in standard view. The stress tensor is of the form σij with i and j subscript. For each component the stress acts on a plane normal to the i-axis and acts in the j direction. The planes are the faces of the cube. When i=j the stress is normal σ, if not then the stress is shear τ. The directions and axes are either 1, 2, 3. In the example 1 is up, 2 is towards you and 3 is right.

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2
Q

What happens to the stress t’endors when the body is at equilibrium?

A

σ12=σ21
σ13=σ31
σ23=σ32

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3
Q

In tensor notation what are the 3 normal stresses and 3 shear stresses?

A

Normal: σ11, σ22, σ33
Shear: τ12, τ13, τ23

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4
Q

How does contracted notation work?

A

For i=j subscript is written once so σ11=σ1
For i not j subscript is 9-i-j so σ12=σ6, σ13=5

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5
Q

What is different about tensor notation for strain tensor?

A

For shear strain ε12=γ12/2
The contracted notation has ε6/2 etc

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6
Q

Difference between simple shear and pure shear

A

Simple has on side of square element fixed with opposite side under shear stress τ. Relation is τij =G γij where γ is the engineering shear strain (angle).
Pure shear has all sides under shear stress. Stress produces an average tensor shear strain εij with the relation εij =γij/2. There are two angles of εij which are half as big as γij in simple shear

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7
Q

Contracted notation of shear strain

A

εk=2εij=γij

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8
Q

How to transform a stress to a new coordinate frame

A

Transform stress σ’kl so
σij=a(ik)a(jl)σ’kl
Where aik is the direction cosine of direction i referred to direction k.
ajl is the direction cosine of direction j referred to direction l.
Basically just trigonometry using dot product and vectors
Works the same with strain as well. ‘ means applied

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9
Q

Hooke’s law with tensor notation

A

σij=Cijkl x εkl
So stress field tensor and strain field tensor (both 2nd ranked tensor) are related by the elasticity or stiffness matter tensor C which is a 4th ranked tensor. C is stiffness tensor or stiffness matrix

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10
Q

What does each quadrant of the stiffness matrix show?

A

Top left normal normal
Bottom left shear normal
Top right normal shear
Bottom right shear shear

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11
Q

What is the compliance tensor/matrix?

A

Symbol S. Relates strain matrix to stress matrix by being multiplied by stress matrix (1x6)

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12
Q

How many distinct coefficients are required for isotropic materials?

A

Only 2. S11 and S12 ni contracted notation
S11=1/E
S12=-ν/E

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13
Q

Are long fibre composites isotropic?

A

No but often orthotropic (3 mutually perpendicular planes of symmetry) and transversely isotropic (2 equivalent directions so could rotate composite and get same results)

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14
Q

What does it mean when there is no extension-shear coupling?

A

No interaction between normal stresses and shear strains or shear stresses and normal strains

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15
Q

For a lamina loaded in plane stress, what happens to some of the stress tensors?

A

σ3=σ4=σ5=0

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16
Q

For uniaxially applied stress on a lamina what is needed to find the compliance matrix?

A

E1, E2, ν12, G12

17
Q

What is a lamina loaded along the principal material axes referred to as?

A

Specially orthotropic. Loaded in 0 and 90° directions only

18
Q

Working with the compliance matrix for specially orthotropic case

A

See slide 21/22 lecture 4

19
Q

What happens when a lamina is loaded at an arbitrary angle?

A

The lamina is ‘generally orthotropic’. Will be normal-shear interactions Si number of non-zero compliance coefficients rises

20
Q

Transformation matrix

A

In the databook. Used to express the applied stresses [σ’] in terms of stresses in the principal material axes [σ]. Multiplied by the σ’ matrix. c means cosθ and s means sinθ. Where θ is angle between applied and principal directions. Can also be used for tensorial strains but not engineering strains

21
Q

Which transformation matrix is needed for engineering strains?

A

For when γ12=2ε12. Need transformation matrix U in the databook. U multiplied by ε’ matrix

22
Q

Transformed compliance tensor

A

[S bar] used for
[ε’]=[S bar][σ’]
This for relating applied stress to applied strain at any angle
Equal to [U]^-1 [S][T]

23
Q

Individual elements of transformed compliance tensor

A

Formula for each in databook for generally orthotropic or general case

24
Q

Composite YM in longitudinal and transverse direction against loading angle

A

E’1 and E’2. E’1 starts high at E1 then curves down with soon inflexion then settles down at E2 up to 90°. E’2 is opposite shape starting at low E2 going up to high E1

25
Q

Composite shear modulus vs loading angle

A

Like a bell curve centred at 45°. Lowest points at 0 and 90 are G12

26
Q

Major and minor Poisson’s ratio vs loading angle

A

ν’12 starts medium at ν12 then up then curve down to ν21 at 90°. Opposite shape for ν’21 starting low at ν21 then going high and settling at medium ν12

27
Q

Composite interaction ratios

A

η’121=E1Sbar16
η’122=E2Sbar26
First one is ratio of shear strain caused by normal stress to normal strain caused by normal stress
Higher magnitude values mean more shear-normal interaction occurs

28
Q

Composite interaction ratios vs loading angle

A

η’122 starts 0 then up a bit then curve down lots then steeply back up to 0 at 90°.
η’121 opposite shape but still starts and ends at 0