Chapter 2: Elasticity

Transcription

1 OHP 1 Mechanical Properties of Materials Chapter 2: lasticity Prof. Wenjea J. Tseng ( 曾文甲 ) Department of Materials ngineering National Chung Hsing University wenjea@dragon.nchu.edu.tw Reference: W.F. Hosford (Cambridge, 2010); N.. Dowling (Pearson, 2007); D. Roylance (Wiley, 1996) Isotropic lasticity lastic deformation is reversible. When a body deforms elastically under a load, it will revert to its original shape as soon as the load is removed. An isotropic material is one that has the same properties in all directions. In uniaxial tension, the corresponding tensile strain is e x (or x ) = x /. Since the volume is conserved in elasticity, Poisson s effect causes strain along y and z directions, i.e., e y = e z = -e x. This indicates that there exist y and z which also contribute to the e x. Taking into the Poisson contraction, the general statement of Hooke s law is For shear strain

2 Isotropic lasticity Similar expressions apply to all three directions, so that Some elastic constants,,, G (Shear modulus), and B (Bulk modulus) can be expressed as follows xample A wide sheet (in y-direction, see figure) of steel with 1mm in thickness is bent elastically to a constant radius of curvature, = 50 cm. Knowing that = 208 GPa, and = 0.29 for steel, find the stress in the surface. Assume that the steel sheet is isotropic and there is not net force in the plane of sheet. KY Solution: Z = 0 because of thin sheet (i.e., sheet buckles easily instead of building up stress in z direction), e y = 0 because of relatively wide sheet.

3 (continue) t / 2 Given that, e x Substituting into Hooke s law, Similarly, y 1 ( ) (0 ) 1 ey y z x y x 0 x 1/ ex x ( y z ) x ( y 0) x x We can hence find x since e x,, and are known. x( x (0.001)(20810 ) /( ) 227MPa MPa y x 8 ) xample A glaze is applied on a ceramic body by heating it above 600 o C, which allows it to flow over the surface. On cooling the glaze becomes rigid at 500 o C. The coefficient of thermal expansion of the glaze and the body are g = 5.5x10-6 / o C and b = 4.0x10-6 / o C. The elastic constants for the glaze are g = 70GPa and g = 0.3. Calculate the stresses in the glaze when it has cooled to 20 o C. Key e xg = e xb and e yg = e yb And, e x = e y for both glaze and ceramic body

4 Young s Modulus and CT of Crystalline Solids The Young s modulus (a bulk property) is related to inter-atomic or inter-ionic bond strength (a microscopic property), and is hence related to the slope involved in the potential energy well w.r.t. interatomic separation. This bond-strength difference also reflects to the thermal expansion coefficient (CT) of weakly bonded and strongly bonded solids. (continue) Let the direction normal to the surface be z, so that the x and y directions lie in the surface. The strains in the x and y directions must be the same, hence e xg 1 g Note that Therefore, 1 ( ) T e ( ) T xg zg xg xb g zb yg yb xg 1 g gt bt g 9 g ( b g ) T 7010 (4 5.5)(20 500) xg 72MPa g yg zg g xb b Free surface Symmetry xb Glaze film so thin that it will not affect the underlying ceramic body. Also, the ceramic body has no residual stress. b yb zb b

5 An Inter-atomic Model for and T m quilibrium separation occurs when the total potential energy of the pair of atoms is at a minimum or when no net force is acting to the atoms. A steep slope in the force-distance curve means that a greater force is required to stretch the bond; thus, the material has a high modulus of elasticity. -T m Dependence The curvature and the depth of potential well are inter-dependent, so the elastic moduli of different crystals roughly correlate with their melting points (T m ).

6 CT of Weak and Strong Interatomic (or inter-ionic) Bondings Weak bonding Strong bonding Other physical properties such the melting temperature also related to the depth of the potential well. CT of Selected Materials The slopes of these lines at any temperature are the linear coefficient of thermal expansion (). In general, increases with increasing temperature, reflecting the fact that the energy well becomes more asymmetric as one moves up the well. In addition, ceramics have lower values than metals due to strong ionic bonding.

7 -CT Correlation An inverse relationship exists between stiffness (or the modulus of elasticity) and thermal expansion for a wide range of materials. Anisotropic lasticity Few materials are really isotropic. In single crystals, elastic properties often vary with crystallographic direction. In polycrystals, anisotropic of elastic properties is caused by the presence of preferred orientations or crystallographic textures. Hooke s law for anisotropic materials can be expressed in terms of compliances, s ijmn, which is a 4 th rate tensor and relates the contributions of individual stress components to individual strain components. where summation is implied. e ij s ijmn mn

8 Anisotropic lasticity xpanding the tensor form, the Hooke s law for anisotropic material now becomes The subscripts 1,2, and 3 denote the crystallographic axes of the crystal. There are total of 36 independent compliance variables. Anisotropic lasticity Since s ij = s ji, this simplifies the matrix to There is a maximum of 21 independent compliance variables in most general case. Crystal symmetry may cause some variables to be equal or vanish.

9 Anisotropic lasticity: Orthotropic Materials With orthotropic symmetry, the 23, 31, and 12 planes are planes of mirror symmetry. The Hooke s law further simplifies to 9 compliance variables Anisotropic lasticity: Cubic Materials For cubic materials, the compliance variables further reduces to only 3 independent compliance variables (i.e., s 11, s 12, and s 44 ).

10 Anisotropic lasticity: Cubic Materials The elastic response (including Young s modulus, coefficient of thermal expansion, etc.) along any crystallographic direction can then be calculated by first resolving the stress state onto the symmetry axes, then using the matrices of elastic constants to find the strains along the symmetry axes, and finally resolving these strains onto the axes of interest. xample: Appendix: Strain Gage The term strain gage refers to a thin wire or foil, folded back and forth on itself and bonded to the specimen surface that is able to generate an electrical measure of strain in the specimen. As the wire is stretched along with the specimen, the wires electrical resistance changes, both because its length is increased and because its cross-sectional area is reduced.

11 Appendix: Other Strain-Measuring Methods Photoelasticity is able to provide the strain distribution of a deformed solid. Use of birefringence principle. Moire fringes also able to show strain distribution. Additional Freebees The Poisson s ratio ( ij ) is defined as ij = - j / i where j is the lateral strain in the j axis direction arising from a strain of i applied in a longitudinal axis direction. Most of the materials exhibit positive Poisson s ratio, since elongation along one-axis would result in contraction along its perpendicular axes, leading to a positive Poisson s ratio from its definition. Poisson s ratio normally does not vary outside the range 0 to 0.5. ceramics ~ 0.2 metals ~ 0.3 plastics ~ 0.4 rubber ~ 0.5

12 Additional Freebees. = 0.5 implies a constant volume (see derivation below), and a value larger than 0.5 implies a decrease in volume for tensile loading, which is unlikely to occur in most cases. Consider a rectangular solid where there are normal strains in three directions. x = dl/l y = dw/w z = dh/h Additional Freebees The volume, V = LWH, changes by an amount dv that is Therefore, By substituting the x, y, and z, the volumetric strain is then For an isotropic solid, dv/v = 0 for volume conservation Hence, = 0.5 (max)

13 Additional Freebees The above derivation is strictly valid for small strain (or small stress) situation, i.e., within the elastic limit of the material. Note that when the material is under plane stress, i.e., no stress component exists in the z direction (for example). The z components of stress vanish at the surface because there are no forces acting externally in that direction to balance them. However, a state of plane stress is not a state of plane strain (Definition: There is one plane (or direction) without strain). The thin plate will then experience a strain in the z direction equal to the Poisson s strain contributed from the x and y stresses Additional Freebees Some macroporous materials (e.g., foam, cork, etc.) would exhibit a very low Poisson s ratio ( ~ 0), because of the fragility of the materials. Some materials known as auxetic materials display a negative Poisson s ratio. For these materials, it is usually due to uniquely oriented, hinged molecular bonds. In order for these bonds to stretch in the longitudinal direction, the hinges must open in the transverse direction, effectively exhibiting a positive strain.

14 Summary A generalized Hooke s law is expressed in terms of - relationship with materials constants such as the Young s modulus () and Poisson s ratio () for the isotropic solids. or This - relationship is often called the Constitutive quations in lasticity. Additional Freebees (for ref. only) To get a close-form (analytical) solutions for elasticity problems, we also need quilibrium quations and Compatibility quations, plus additional Boundary conditions (if any). quilibrium equations (neglecting body forces) or Compatibility equations for strains Solve the Airy stress function () 4 = 0