ME 582 Advanced Materials Science Chapter 2 Macromechanical …web.iitd.ac.in/~sroy/CourseMaterials/Constitutive... · 2012. 3. 1. · ME 582 Advanced Materials Science Chapter 2

ME 582 Advanced Materials Science • Department of Mechanical Engineering Dr. Jan Gou

ME 582 Advanced Materials Science

Chapter 2 Macromechanical Analysis of a Lamina(Part 2)

Dr. Jan GouLaboratory for Composite Materials Research

Department of Mechanical EngineeringUniversity of South Alabama, Mobile, AL 36688


HW #3

2.35

2.38

2.43

Due Day: 6:00 PM, 10/04/2006, Wednesday.


Strength Failure Theories of an Angle Lamina

The failure theories are generally based on the normal and shear strengths of a unidirectional lamina.

An isotropic material generally has two strength parameters: normal strength and shear strength.

In the case of a unidirectional lamina, the five strength parameters are

Longitudinal tensile strengthLongitudinal compressive strengthTransverse tensile strengthTransverse compressive strengthIn-plane shear strength

ultT )( 1σ

ultC )( 1σ ultC )( 1σ

ultC )( 2σ

ultT )( 2σ

ult)( 12τ


Maximum Stress Failure Theory

The lamina is considered to be failed if

Each component of stress does not interact with each other.


Example 2.13


Example 2.13


Example 2.13


Example 2.13


Maximum Strain Theory

The lamina is considered to be failed if


The ultimate strains can be found from the ultimate strength parameters and the elastic moduli, assuming the stress-strain response is linear until failure.

For the maximum strain failure theory, no interactions occurs between various components of strain.

The maximum stress failure theory and the maximum failure straintheory give different results because the local strains in a lamina include the Poisson’s ratio.

If the Poisson’s ratio is zero in the unidirectional lamina, the two failure theories will give identical results.

Maximum Stress and Strain Failure Theories


Tsai-Hill Failure Theory

Based on the distorsion energy theory, they proposed that a lamina has failed if

This theory is based on the interaction failure theory.

The components G1 - G6 of the strength criteria depend on the failure strength.


Components of Tsai-Hill Failure Theory


Components of Tsai-Hill Failure Theory

Solution:


Tsai-Hill Failure Theory – Plane Stress


Tsai-Hill Failure Theory

Unlike the maximum strain and maximum stress failure theories, the Tsai-Hill failure theory considers the interaction among the three unidirectional lamina strength parameter.

The Tsai-Hill failure theory does not distinguish between the compressive and tensile strengths in its equation. This can result in underestimation of the maximum loads that can be applied when compared to other failure theory.

Tsai-Hill failure theory underestimates the failure stress because the transverse strength of a unidirectional lamina is generally much less than its transverse compressive strength.


Modified Tsai-Hill Failure Theory


Tsai-Wu Failure Theory

Tsai-Wu applied the failure theory to a lamina in plane stress. A lamina is considered to be failed if

The components H1 – H66 of the failure theory are found using the five strength parameters of a unidirectional lamina.


Components of Tsai-Wu Failure Theory

Solution:



Solution:



Solution:


Determination of H12


Determination of H12


Empirical Models of H12


Example 2.19


Example 2.19


Example 2.19


Example 2.19


Example 2.19


Experimental Results and Failure Theories

Tsai-Wu compared the results from various failure theories to some experimental results. He considered an angle lamina subjected to a uniaxial load in the x-direction.

The failure stresses were obtained experimentally for tensile and compressive stresses for various angles of the lamina.


Stresses in the Material Axes


Strains in the Material Axes










Hygrothermal Stress-Strain Relationship

For a unidirectional lamina

Thermally induced strains:

Moisture induced strains:


For a unidirectional lamina



For an angular lamina


Thermally induced strains:

Moisture induced strains:


Transformation of CTE



Transformation of Coefficients of Moisture Expansion



Example 2.20


Example 2.20


Example 2.20


Example 2.20

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ME 582 Advanced Materials Science Chapter 2 Macromechanical …web.iitd.ac.in/~sroy/CourseMaterials/Constitutive... · 2012. 3. 1. · ME 582 Advanced Materials Science Chapter 2