lecture 2a macro mechanics stress strain relations for material types
TRANSCRIPT
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Macro Mechanical Analysis of a Composite Lamina
Chapter Objectives
Develop stressstrain relationships for different types of materials.
Develop stressstrain relationships for a unidirectional/bidirectional
lamina.
Find the engineering constants of a unidirectional/bidirectional
lamina in terms of the stiffness and compliance parameters of the
lamina.
Develop stressstrain relationships, elastic moduli, strengths, andthermal and moisture expansion coefficients of an angle ply based
on those of a unidirectional/bidirectional lamina and the angle of
the ply.
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and the angle between the axes of the 123 system and the 123
system:
Inverting Equation (2.25), the general strainstress relationship for a
three dimensional body in a 123 Orthogonal Cartesian coordinate
system would be:
2.25
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2.3.1 Anisotropic Material.
The material that has 21 independent elastic constants at a point is
called an anisotropic material. Once these constants are found at a
particular point, the stress-strain relationship can be developed at that
point. Note that these constants can vary from point to point if the
material is
non-homogeneous. Even if the material is homogeneous (orassumed to be one), one needs to find these 21 elastic constants
analytically or experimentally. However, many natural and synthetic
materials do possess material symmetry that is, elastic properties
are identical in directions of symmetry about the symmetric planes
because symmetry exists in the internal material structure.
Consequently, this material symmetry reduces the number of
independent elastic constants by zeroing out or relating some of the
constants in the 6 6 material stiffness and compliance matrices. This
i lifi th l tt t i l t i l ti hi f i t f
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C14 = 0, C15 = 0, C24 = 0, C25 = 0, C 34 = 0, C35 = 0 , C46 = 0,
and C56 = 0 (i.e. any elastic constant involving the indices 4 or 5 relatingthe shearing strain components would be zero).This is so because, the
corresponding shearing strains which reverse the sign at the symmetric
points on either side of the symmetric plane (0, 0, +z and 0, 0,-z) would violate the symmetric condition. The direction perpendicular to the
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2.3.3 Orthotropic Material
If a material has three mutually perpendicular planes of material
symmetry, then the material is referred to as Three-Dimensional
Orthotropic Material. Then the stiffness matrix would be given by:
The preceding stiffness matrix can be derived by starting from the
stiffness matrix [C] for the monoclinic material (Equation 2.35) where two
more planes of symmetry would lead to: C16 = 0, C26 = 0, C36 = 0 and
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The compliance matrix for 3-D Orthotropic Material reduces to:
2.3.4 Transversely Isotropic Material
Consider a plane of material isotropy in one of the planes of an
orthotropic body. If direction-1 is normal to that plane (23) of isotropy,
then the stiffness matrix is given by:
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The Transverse isotropy results in the following relations:
Note the five independent elastic constants. An example of this is a thin
idi ti l l i i hi h th fib d i
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2.3.5 Isotropic Material
If all planes in an orthotropic body are identical, it is an isotropic
material; then, the stiffness matrix is given by:
Isotropy results in the following additional relationships:
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The compliance matrix reduces to:
The number of independent elastic constants for various types of
materials is listed below:
1. Anisotropic Material : 21Elastic constants
2. Monoclinic Material: 13 Elastic constants
3. Orthotropic Material: 9 Elastic constants
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