Warm-up Exercises: 3D stress-strain equations: σ = E * ε 2D stress-strain equations: σ = E * ε Orthotropic material: The material is described as orthotropic, meaning its mechanical properties vary with direction. This implies that there are 9 independent elastic constants (Eijkl). Stress-Strain Equations: The image shows a set of equations relating stress and strain components in a 3D material: σ₁₁ = E₁₁₁₁ε₁₁ + E₁₁₂₂ε₂₂ + E₁₁₃₃ε₃₃ σ₂₂ = E₂₁₁₁ε₁₁ + E₂₂₂₂ε₂₂ + E₂₂₃₃ε₃₃ σ₃₃ = E₃₁₁₁ε₁₁ + E₃₂₂₂ε₂₂ + E₃₃₃₃ε₃₃ σ₁₂ = 2E₁₂₁₂ε₁₂ σ₂₃ = 2E₂₃₂₃ε₂₃ σ₁₃ = 2E₁₃₁₃ε₁₃ Plane Stress Conditions: Under plane stress conditions, σ₃₃ = σ₂₃ = σ₁₃ = 0. This simplifies the equations to: E₁₁₁₁ε₁₁ + E₁₁₂₂ε₂₂ = 0 E₂₁₁₁ε₁₁ + E₂₂₂₂ε₂₂ = 0 Expanding Equations: Equations (1) and (2) from the previous image are expanded to express the stress components in terms of the strain components and elastic constants. Simplifying and Comparing Equations: By comparing equations (1) and (2) with their simplified forms, relationships between certain elastic constants are derived. These relationships are summarized as follows: E₁₁₁₁* = E₁₁₁₁ - E₁₁₃₃² / E₃₃₃₃ E₁₁₂₂* = E₁₁₂₂ - E₁₁₃₃E₂₂₃₃ / E₃₃₃₃ E₁₁₁₂ = E₁₂₁₁ E₁₁₃₃ = E₃₁₁₁ E₂₂₃₃ = E₃₂₂₂ Note: The note at the bottom emphasizes that the stress-strain relations are symmetric, meaning that Eᵢⱼₖₗ = Eⱼᵢₖₗ. Isotropic materials: The text highlights that isotropic materials have the same properties in all directions, which leads to a reduction in the number of independent elastic constants. Relationships between elastic constants: Due to the symmetry of isotropic materials, several elastic constants become equal: E₁₁₁₁ = E₂₂₂₂ = E₃₃₃₃ E₁₁₂₂ = E₂₂₁₁ = E₂₃₂₃ = E₃₂₃₂ = E₁₃₁₃ = E₃₁₁₃ E₁₂₁₂ = E₂₁₁₂ = E₁₃₁₃ = E₃₁₁₃ Independent elastic constants: This leaves us with 3 independent elastic constants. Lame constants: To further reduce the number of independent constants, the Lame constants (λ and μ) are introduced. These constants relate the elastic constants as follows: E₁₁₁₁ = λ + 2μ E₁₁₂₂ = λ E₁₂₁₂ = μ

Solutions to Home Assignment #2

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