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Mechanics of Materials – MAE 243 (Section 002) Spring 2008

Mechanics of Materials – MAE 243 (Section 002) Spring 2008. Dr. Konstantinos A. Sierros. 7.5: Hooke’s law for plane stress.

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Mechanics of Materials – MAE 243 (Section 002) Spring 2008

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  1. Mechanics of Materials – MAE 243 (Section 002) Spring 2008 Dr. Konstantinos A. Sierros

  2. 7.5: Hooke’s law for plane stress • Materials that meet two important conditions: 1) The material is uniform throughout the body and has the same properties in all directions (ie homogeneous and isotropic) and 2) The material follows Hooke’s law (ie is linearly elastic) • For example the strain εx in the x direction due to the stress σx is equal to σx/E where E is the modulus of elasticity. But we also have a strain εx due to the stress σy and is equal to -v σy / E where v is the Poisson’s ratio (see section 1.5) • Also revise section 3.6

  3. 7.5: Hooke’s law for plane stress • Special cases of Hooke’s law • Biaxial stress: σx = σy = 0 • Uniaxial stress: σy = 0 • Pure shear: σx = σy = 0, εx = εy = εz = 0 • and γxy = τxy / G • Volume change: The change in volume • can be determined if the normal strains • In the three perpendicular directions • Strain – Energy density in plane stress • Revise sections 2.7 and 3.9

  4. 7.6:Triaxial stress • State of triaxial stress • Since there are no shear stresses on the x,y and z faces, the stresses σx ,σy, σz are the principal stresses • If an inclined plane parallel to the z axis is cut through the element (fig 7-26b), the only stresses on the inclined face are the normal stress σ and shear stress τ, both of which act parallel to the xy plane

  5. 7.6:Triaxial stress • The stresses acting on elements oriented at various angles to the x, y and z axes can be visualized using the Mohr’s circle. • For elements oriented by rotations about the z axis, the corresponding circle is A • For elements oriented by rotations about the x axis, the corresponding circle is B • For elements oriented by rotations about the y axis, the corresponding circle is C

  6. 7.7: Plane strain • If the only deformations are those in the xy plane, then three strain components may exist – the normal strain εx in the x direction (fig 7-29b), the normal strain εy in the y direction (fig 7-29c) and the shear strain γxy (fig 7-29d). An element subjected to these strains (and only these strains) is said to be in a state of plane strain • It follows that an element in plane strain has no normal strain εz in the z direction and no shear strains γxz and γyz in the xz and yz planes respectively • The definition of plane strain is analogous to that for plane stress

  7. 7.7:Transformation equations for plane strain • Expression for the normal strain in the x1 direction in terms of the strains εx, εy , εz • Similarly the normal strain εy1 in the y1 direction is obtained from the above equation by setting θ = θ+90

  8. 7.7:Transformation equations for plane strain • we also have… …which is an expression for the shear strain γx1y1 Transformation equations for plane strain

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