Question

A thin plate is subjected to a biaxial stress field of Oxx = 1 GPa and Oyy = 0.5 GPa. Calculate the strains in the xy directions if the plate is made of (a) steel, (b) a 0° unidirectional boron-epoxy composite, and (c) a 45° unidirectional boron-epoxy composite. Use the elastic properties of the boron-epoxy composite given in Table 1. (5) Table 1: Typical Mechanical Properties of unidirectional continuous fiber composites Boron- Epoxy Property 1.99 1585 (230) 62.7 (9.1) 207 (30) 19 (2.7) 0.21 Density, g/cm Tensile properties Strength, MPa (ksi) 0° 90° Modulus GPa (Msi) 0° 90° Major Poisson's ratio Compressive properties Strength, MPa (ksi), 0° Modulus, GPa (Msi), 0° Flexural properties Strength, MPa (ksi), 0° Modulus, GPa (Msi), 0° In-plane shear properties Strength, MPa (ksi) Modulus, GPa (Msi) Interlaminar shear strength, MPa (ksi) 0° 2481.5 (360) 221 (32) 131 (19) 6.4 (0.93) 110 (16)

Answer 1

To calculate strain, use generalized Hooke's law for the given biaxial stress field. For isotropic materials like steel, strain is a function of stress, Young's modulus, and Poisson's ratio. For anisotropic materials, such as the boron-epoxy composite, fiber orientation affects strain calculations and may require complex composite theories.

Step-by-step explanation:

To calculate the strains in the xy directions for the given biaxial stress for different materials, we use the generalized Hooke's law. The stresses σxx and σyy are given as 1 GPa and 0.5 GPa respectively. For steel and isotropic materials, strain can be calculated using the formula:

ε = σ / E - ν(σ

perpendicular

/ E)

where ε is the strain, σ is the stress, E is Young's modulus, and ν is Poisson's ratio. The strain for a boron-epoxy composite material is different and depends on the fiber orientation relative to the stress direction.

For a 0° unidirectional boron-epoxy composite (along the fiber direction), the modulus is 207 GPa and Poisson's ratio is 0.21. For a 45° unidirectional boron-epoxy composite, the calculation is more complex and requires the transformation of stresses and calculation of strains using composite material theories, such as the Classical Laminate Theory. Here, strain is affected by material properties in both the fiber direction and perpendicular to it, due to the anisotropic nature of composites.

Stress-strain relations, modulus of elasticity, and Poisson's ratio are essential for these calculations, and the values should be taken from reliable data sources like engineering data tables.