Macro Mechanics

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Macro Mechanics

• ME257 Composite Materials
• James Iatridis

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Macro 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.
• Macromechanics will put several plies together in
  order to evaluate stresses and strains within
  each ply when a composite laminate is subject to
  force or displacement boundary conditions

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Plate Assumptions

• Flat plate of several individual layers
• Layers are perfectly bonded together, therefore
  displacements are continuous across boundaries
• Kirchoff-Love hypothesis Normals to the center
  line remain normal to center line after
  deformation -- this neglects through the
  thickness shears

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Stress Profile for Homogeneous Material

• Continuous stress and strain

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Stress and Strain Profiles for Laminates

• Continuous strain
• Discontinuous stress

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Deformations in a Plate
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Displacements

• u(x,y,z)u0(x,y)-zfx(x,y)
• v(x,y,z)v0(x,y)-zfy(x,y)
• if plate is thin
• assuming
• length of A-D is constant ezz0
• fx and fx are very small
• w(x,y,z)w0(x,y) 

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Strain-Displacement Relations

• The strains at any point in a plate are

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Strains and Curvatures

• Plugging plate deformation equations into the
  strain-displacement relations and simplifying
  yields
• Strains in terms of midplane strains and
  curvatures

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Midsurface Strains and Curvatures

• Midsurface Strains
• Curvatures

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Force and Moment Resultants
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Stress Resultants for a Single Ply

• Stress resultants
• Moment Resultants
• What are the units?

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Geometry
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Stress Resultants for a Laminate

• Simply sum over the number of layers
• Stress resultants
• Moment Resultants

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Plate Stiffness and Compliance

• We already know the stress strain relationships
  for a single ply

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Laminate Stiffness and Compliance

• Plugging plate stiffness relationships into
  laminate stress and moment resultant equations in
  terms of strains and curvatures

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ABD matrices

• where i,j 1,2,6
• zk is the coordinate of the top and bottom of ply
  surface
• 18 Constants

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ABD Matrices (part 2)

• Coefficients Aij, Bij, Dij are functions of
  thickness, orientation, stacking sequence and
  material properties of each layer
• A in-plane stiffness matrix
• D bending stiffness matrix
• B bending-extension coupling matrix
• B0 if laminate is symmetric wrt mid-plane

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Coupling
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Coupling - Constants
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ABD matrices (part 3)

• We can also take the inverse

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Future Directions

• Describe laminates
• Define special laminates
• Determine how to calculate stresses