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

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 ε x in the x direction due to the stress σ x is equal to σ x /E where E is the modulus of elasticity. But we also have a strain ε x due to the stress σ y and is equal to -v σ y / E where v is the Poisson’s ratio (see section 1.5) Also revise section 3.6

7.5: Hooke’s law for plane stress Special cases of Hooke’s law - Biaxial stress: σ x = σ y = 0 - Uniaxial stress: σ y = 0 - Pure shear: σ x = σ y = 0, ε x = ε y = ε z = 0 and γ xy = τ xy / 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

7.6:Triaxial stress State of triaxial stress Since there are no shear stresses on the x,y and z faces, the stresses σ x,σ y, σ z 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

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

7.7: Plane strain If the only deformations are those in the xy plane, then three strain components may exist – the normal strain ε x in the x direction (fig 7-29b), the normal strain ε y in the y direction (fig 7-29c) and the shear strain γ xy (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 ε z in the z direction and no shear strains γ xz and γ yz in the xz and yz planes respectively The definition of plane strain is analogous to that for plane stress

7.7:Transformation equations for plane strain Expression for the normal strain in the x 1 direction in terms of the strains ε x, ε y, ε z Similarly the normal strain ε y1 in the y 1 direction is obtained from the above equation by setting θ = θ+90

7.7:Transformation equations for plane strain we also have… …which is an expression for the shear strain γ x1y1 Transformation equations for plane strain