Ply Mechanics

1
Ply Mechanics

• ME257 -- Composite Materials
• James Iatridis

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Ply Mechanics Intro

• Micromechanics allowed us to obtain average
  material properties for a composite ply from
  matrix and fiber properties.
• Ply mechanics considers average material
  properties for a single ply with any fiber
  orientation angle.
• To study ply mechanics, we must first learn about
  coordinate transformations

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Tensor Calculus

• In its simplest sense, a tensor is a linear
  transformation.
• e.g.
• translation
• rotation
• For the remainder of this course, we will use
  tensors to transform to/from
• stress strain
• material global coordinates
• engineering strain tensorial strain

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Reduced Stiffness Matrix Q

• Q transforms to/from strain stress in material
  coordinates
• s12Qe12
• e12Ss12
• S is the compliance matrix
• SQ-1
• Note that stress-strain relationship is in terms
  of engineering strain!!!

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Compute stiffness matrix Q

• Barbero Example 5.1 (p. 115)
• Compute Q using
• E119.981 GPa n120.274
• E211.389 GPa G123.789 GPa
• Solution
• Q1120.874 GPa Q2211.898 GPa
• Q123.260 GPa Q663.789 GPa

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Compute Compliance Matrix

• Barbero Example 5.2 (p. 115)
• Remember that SQ-1
• Easy to calculate computationally
• MATLAB or Mathematica
• Try this at home!!!

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Coordinate Transformations for Stress

• sxyT -1s12
• s12Tsxy
• where mcos(q) and nsin(q)

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Coordinate Transformations for Strain 1

• Note that stress-strain relationships use
  engineering strain while transormation of
  coordinates uses tensorial strain
• eengineeringRetensorial
• e tensorial R-1e engineering

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Coordinate Transformations for Strain 2

• Remember that coordinate transformations for
  strain require that we use tensorial strains that
  will be denoted with a eT
• eT12TeTxy
• eTxyT -1eT12

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Coordinate Transformations for Strain 3

• To transform engineering strain from material to
  global coordinates
• 1) transform to tensorial strain
• 2) transform coordinate system
• 3) transform back to engineering strain
• exyRT -1R-1e12

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Compute Stress Strain Transformation

• Barbero Example 5.3 (p. 120)
• Transform stresses from material to global
• q-55 degrees
• s1100 s210 s6-5
• Solution sx34.9 s275.1 s6-40.6
• Barbero Example 5.4 (p.120)
• Transform strains from material to global
• e13.635e-3 e27.411-3 e120
• ex6.169e-3 ey4.88e-3 exy3.547e-4

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Transform global strain to stress

• Steps
• global engineering strain to global tensor strain
• global tensor strain to material tensor strain
• material tensor strain to material engng strain
• material engng strain to material stress
• material stress to global stress

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Compute global stress from strain

• Barbero example 5.6 (p. 124)
• Compute stresses in global coordinates given
• ex6.169e-3 ey4.877e-3 exy3.548e-3
• E119.981 GPa n120.274 E211.389 GPa
  G123.789 GPa
• q-55 deg
• Solution
• sx0.1 GPa sy0.1 GPa sxy0 GPa

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Thermal and moisture effects

• Polymeric composites often require incorporation
  of thermal and moisture effects
• Polymers are cured at high temps, then often used
  at ambient temperature
• Moisture can cause matrix to swell, e.g. sponge,
  plexiglass, etc.
• We incorporate Thermal and moisture effects using
  superposition

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Plane stress compliance relationship

• Use superposition to add temperature and moisture
  effects, very similar effects for both T m.
• a1,a2coefficients of thermal expansion in fiber
  and transverse directions, respectively
• b1,b2coefficients of moisture expansion in the
  fiber and transverse directions, respectively

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Plane stress stiffness relationship

• for inclusion of thermal effects only

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Thermal strains in global coordinates

• First, calculate thermal strains in global
  coordinate system
• Note that there is a shear term in global
  coordinates

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Thermal stresses

• transform global strains into global stresses
• This is just like for strains, only we use the
  apparent coefficients of thermal expansion ax,
  ax, and axy

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Compute apparent thermal coeff.

• Find apparent coefficients of thermal expansion
  for AS4/3501-6 carbon/epoxy when referred to an
  x,y coordinate system with q30 deg.
• a1-0.9e-6/deg C, a227e-6/deg C (Table 1.1)
• Note fibers expands when cooled while matrix is
  more typical and contracts as cooled
• Solution
• ax 6.07e-6
• ax20.02e-6
• axy -24.16e-6