ME 582 Advanced Materials Science. Chapter 2 Macromechanical Analysis of a Lamina (Part 2)

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1 ME 582 Advanced Materials Science Chapter 2 Macromechanical Analysis of a Lamina (Part 2) Laboratory for Composite Materials Research Department of Mechanical Engineering University of South Alabama, Mobile, AL HW # Due Day: 6:00 PM, 10/04/2006, Wednesday.

2 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 strength ( σ T 1 ) ult Longitudinal compressive strength ( σ C 1 ) ult Transverse tensile strength ( σ T 2 ) ult Transverse compressive strength ( σ C 2 ) ult In-plane shear strength ( τ 12 ) ult Maximum Stress Failure Theory The lamina is considered to be failed if Each component of stress does not interact with each other.

3 Example 2.13 Example 2.13

4 Example 2.13 Example 2.13

5 Maximum Strain Theory The lamina is considered to be failed if Maximum Stress and Strain Failure Theories 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 strain theory 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.

6 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 G 1 -G 6 of the strength criteria depend on the failure strength. Components of Tsai-Hill Failure Theory

7 Components of Tsai-Hill Failure Theory Solution: Tsai-Hill Failure Theory Plane Stress

8 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

9 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 H 1 H 66 of the failure theory are found using the five strength parameters of a unidirectional lamina. Components of Tsai-Wu Failure Theory Solution:

10 Components of Tsai-Wu Failure Theory Solution: Components of Tsai-Wu Failure Theory Solution:

11 Determination of H 12 Determination of H 12

12 Empirical Models of H 12 Example 2.19

13 Example 2.19 Example 2.19

14 Example 2.19 Example 2.19

15 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

16 Strains in the Material Axes Experimental Results and Failure Theories

17 Experimental Results and Failure Theories Experimental Results and Failure Theories

18 Experimental Results and Failure Theories Hygrothermal Stress-Strain Relationship For a unidirectional lamina Thermally induced strains: Moisture induced strains:

19 Hygrothermal Stress-Strain Relationship For a unidirectional lamina Hygrothermal Stress-Strain Relationship For an angular lamina Thermally induced strains: Moisture induced strains:

20 Transformation of CTE For an angular lamina Transformation of Coefficients of Moisture Expansion For an angular lamina

21 Example 2.20 Example 2.20

22 Example 2.20 Example 2.20

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