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Chapter 4 Material Behavior – Linear Elastic Solids

Chapter 4 Material Behavior – Linear Elastic Solids. Linear Elastic Constitutive Solid Model Develop Force-Deformation Constitutive Equation in the Form of Stress-Strain Relations Under the Assumptions: Solid Recovers Original Configuration When Loads Are Removed

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Chapter 4 Material Behavior – Linear Elastic Solids

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  1. Chapter 4 Material Behavior – Linear Elastic Solids • Linear Elastic Constitutive Solid ModelDevelop Force-Deformation Constitutive Equation in the Form of Stress-Strain Relations Under the Assumptions: • Solid Recovers Original Configuration When Loads Are Removed • Linear Relation Between Stress and Strain • Neglect Rate and History Dependent Behavior • Include Only Mechanical Loadings • Thermal, Electrical, Pore-Pressure, and Other Loadings Can Also Be Included As Special Cases

  2. Typical One-Dimensional Stress-Strain Behavior  Steel Tensile Sample Cast Iron Aluminum  = E Applicable Region for Linear Elastic Behavior

  3. Linear Elastic Material ModelGeneralized Hooke’s Law with 36 Independent Elastic Constants or

  4. (Body-Centered Crystal) (Hexagonal Crystal) (Fiber Reinforced Composite) Anisotropy and Nonhomogeneity Anisotropy - Differences in material properties under different directions. Materials like wood, crystalline minerals, fiber-reinforced composites have such behavior. Typical Wood Structure Note Particular Material Symmetries Indicated by the Arrows Nonhomogeneity - Spatial differences in material properties. Soil materials in the earth vary with depth, and new functionally graded materials (FGM’s) are now being developed with deliberate spatial variation in elastic properties to produce desirable behaviors. Gradation Direction

  5. Isotropic MaterialsAlthough many materials exhibit non-homogeneous and anisotropic behavior, we will primarily restrict our study to isotropic solids. For this case, material response is independent of coordinate rotation Generalized Hooke’s Law  - Lamé’sconstant  - shear modulus or modulus of rigidity

  6. Isotropic MaterialsInverted Form - Strain in Terms of Stress

  7.    p  p  p Physical Meaning of Elastic Moduli Hydrostatic Compression Pure Shear Simple Tension

  8. Relations Among Elastic Constants

  9. Typical Values of Elastic Moduli for Common Engineering Materials

  10. Hooke’s Law in Cylindrical Coordinates x3 z z z r rz r  x2 d r x1  dr

  11. Hooke’s Law in Spherical Coordinates x3 R R R   R   x2  x1

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