Lecture 2A Macro Mechanics Stress Strain Relations for Material Types

7/27/2019 Lecture 2A Macro Mechanics Stress Strain Relations for Material Types

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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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Lecture 2A Macro Mechanics Stress Strain Relations for Material Types