macromechanics of a composite lamina

Atta ur Rehman Shah PhD Student Advanced Composite Materials Lab. CHANGWON NATIONAL UNIVERSITY

Macro-mechanical Behavior of a Lamina

Mechanics of Composite Materials

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Introduction A lamina is a flat (or sometimes curved) arrangement of

unidirectional (or woven) fibers suspended in a matrix material.

A lamina is generally assumed to be orthotropic, and its thickness depends on the material from which it is made.

Schematic illustration of lamina composite

2.1

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Introduction Macromechanics – Composite: Concepts, math-models and equations which are used to

transform ply properties from its material axes to composite structural axes.

2.1

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Stress-Strain Relations for Anisotropic Materials

Hooks law in contracted notations is

2.2

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Stress-Strain Relations for Anisotropic Materials

Engineering shear strain, γij, is the total angle of shearing under a state of simple shear, while tensor shear strain, εij, is half of the angle of shearing under pure shear stress.

Engineering Shear Strain Vs Tensor Shear strain

2.2

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Stress-Strain Relations for Anisotropic Materials

The stiffness matrix, Cij, has 36 constants for an anisotropic material.

An anisotropic material has no planes of symmetry for material properties.

The number of constants in the stiffness matrix can be reduced for materials with more property symmetry, shown in the next slide.

2.2

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Stress-Strain Relations for Anisotropic Materials Mono clinic, 13 independent constants Orthotropic, 9 independent constants

Transversely isotropic, 5 independent constants Isotropic, 2 independent constants

2.2

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Stress-Strain Relations for Anisotropic Materials

Similarly for strain stress relationship:

2.2

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Stress-Strain Relations for Anisotropic Materials

Various couplings are shown for an arbitrary stressed body in the fig, where the physical response of each compliance is labeled.

2.2

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Stiffnesses, Compliances, and Engineering Constants for Orthotropic Materials

Engineering constants (also known as technical constants) include Young’s moduli, Poisson’s ratios, and shear Moduli as well as some other behavioral constants (to be discussed later).

These constants can be measured in simple tests such as uniaxial tension or pure shear tests.

These components have more direct meaning than the components of compliance and stiffness matrices.

The engineering constants are generally the slop of a stress-strain curve (e.g. E= σ/ε) or the slope of a strain-strain curve (e.g. ν= -εy/εx).

2.3

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Stiffnesses, Compliances, and Engineering Constants for Orthotropic Materials

For an orthotropic material, the components of compliance matrix in terms of engineering constants are:

Where E1,E2,E3 = Young’s moduli in 1,2 and 3 directions νij = -εj/εi = Poisson’s ratio G23,G31,G12 = Shear moduli in the 2-3, 3-1, and 1-2 planes

2.3

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Since compliance matrix for an orthotropic material is symmetric, i.e.

Sij = Sji

When engineering constants are substituted in the above equation; we get a reciprocal relation,

Thus three relations must be satisfied for an orthotropic material.

Stiffnesses, Compliances, and Engineering Constants for Orthotropic Materials 2.3

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Since stiffness and compliance matrices are mutually inverse, thus by matrix algebra the components of a stiffness matrix will be:

In terms of engineering constants, the stiffness matrix components can be written as:

Stiffnesses, Compliances, and Engineering Constants for Orthotropic Materials 2.3

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Restrictions on Engineering Constants

For isotropic materials certain relations between engineering constants must be satisfied. e.g. G in terms of E and v as

In order that E and G always be positive. Similarly, if an isotropic body is subjected to hydrostatic pressure, p,

i.e,, σx=σy=σz=-p, then the volumetric strain becomes:

2.4 2.4.1: Isotropic Materials

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For orthotropic materials, the relations between engineering constants are more complex.

The product of stress and the corresponding strain represent work done by the stress.

The sum of the work done by all the stresses must be positive, hence matrices relating stress to strain must be positive-definite. i.e. the diagonal elements must be positive.

Also

hence

Restrictions on Engineering Constants 2.4 2.4.2: Orthotropic Materials

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Stress-Strain relations for Plane Stress in an Orthotropic Material

For a unidirectionally reinforced or woven lamina in the 1-2 plane, a plane stress state is defined by

2.5

Since a lamina can not withstand high stresses in any direction other than that of fibers, so it can not be subjected to unnatural stresses as σ3.

Thus plane stress condition is a practical approximation