Strength and Stiffness

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1 Strength and Stiffness Stress = load/area is applied to a material by loading it Strain = deformation/length a change of shape (dimensions and twist angles) is its response Stiffness = stress/strain is the resistance to change of shape that is elastic the material will return to its original shape when unloaded Strength = maximum stress of yielding is the resistance to permanent deformation (damages) or even final failure.

2 Material Properties Stress and strain are not material properties they describe a stimulus and a response Stiffness and strength are material properties which are measured by the elastic modulus (E), elastic limit (σ y ), and tensile strength (σ ts ) Stiffness, strength, and density are three material properties central to mechanical design Material Properties Stress and strain are not material properties they describe a stimulus and a response Stiffness and strength are material properties which are measured by the elastic modulus (E), elastic limit (σ y ), and tensile strength (σ ts ) Stiffness, strength, and density are three material properties central to mechanical design Elastic deformation Plastic deformation

3 Density Mass per unit volume kg/m 3 or lb/in 3 Figure 4.1 Vol. of the material = ρ fl /(m 1 -m 2 ) (Archimedes' principle) Double-weighing method for calculating density 1 MPa = 145 psi = 10 atm. pressure 1 N/m 2 = 1 Pascal (Pa) 1 lb/in 2 = 1 psi 10 6 Pa = 1 MPa 10 3 psi = 1 ksi Stress (a) Force applied normal to surface Positive F indicates tension Negative F indicates compression (b) Force applied parallel to surface Shaded plane carries the shear stress (c) Equally applied tensile and compressive forces on all six sides of a cubic element Hydrostatic pressure Figure 4.3

4 Strain Strain is the ratio of two lengths and is therefore dimensionless Figure 4.3 (a) Tensile stress lengthens the element causing a tensile strain (+) Compressive stress shortens the element causing a compressive strain (-) (b) Shear strain is defined as the change in twist angle Stress-Strain Curves Initial portion of curve is approximately linear and is elastic the material returns to its original shape once the stress is removed Within the linear elastic region, strain is proportional to stress E: Young s modulus G: shear modulus K: bulk modulus Figure 4.4

5 Stress-Strain Curve Brittle Response Entire response is elastic no plastic deformation Yield strength not reached before failure Young s modulus determined by calculating the slope of this region Ductile Response Tensile strength is maximum stress on the curve Yield strength determined by standard offset methods Permanent deformation occurs at stresses beyond the yield strength material will not return to its original shape past this point

6 Ductile Response Yield strength is strength at which metal or alloy show significant amount of plastic deformation. 0.2% offset yield strength is that strength at which 0.2% plastic deformation takes place. Construction line, starting at 0.2% strain and parallel to elastic region is drawn to fiend 0.2% offset yield strength. Figure 5.23 Ductile Response Ultimate tensile strength (UTS) is the maximum strength reached by the engineering stress strain curve. Necking starts after UTS is reached. More ductile the metal is, more is the necking before failure. S T R E S S Mpa Stress increases till failure. Drop in stress strain curve is due to stress calculation based on original area. Al 2024-Tempered Necking Point Al 2024-Annealed Strain

7 True Stress & True Strain True stress and true strain are based upon instantaneous cross-sectional area and length. True Stress = σ t = F A i (instantaneous area) True Strain = ε t = li l0 dl li = Ln l l A0 = Ln A True stress is always greater than engineering stress. 0 i Poisson s Ratio Negative of the ratio of transverse strain to axial strain in tensile loading Relates the Young s modulus, shear modulus, and bulk modulus to one another Materials of loose structures (e.g. foams) have ν ~ 0 Materials of dense structures (e.g. rubber materials) have ν ~ 0.5 Usually, ν is between 0 to 0.5 (for metals, ν ~ 0.3)

8 Poisson s Ratio - Elastomers Rubber is easy to stretch in tension, but becomes very stiff when constrained from changing shape or loaded hydrostatically Relationship among G, K, and E

9 Stress-Free Strain In certain situations, strain is not caused by stress; however, stresses can develop if the body suffering the strain is constrained Figure 4.5 Material-Property Charts: Modulus - Density Identifies materials that are both stiff and light Critical for material selection of stiffness-limited designs Figure 4.6

10 Modulus Relative Cost Identifies materials that are both stiff and cheap Useful when the objective is minimizing cost Figure 4.7 Anisotropy The properties of many materials glasses, ceramics, polymers and metals do not depend on the direction in which they are measured across the material Certain materials are considered anisotropic meaning their properties are dependant upon which direction in the material they are being measured Woods are stiffer along the grain than with it; fiber composites are stronger and stiffer parallel to the direction of the fibers than perpendicular to them

11 What Determines Density Density is mostly dependant on atomic weight Metals are dense because their atoms are heavy iron has an atomic weight of 56 Polymers have low densities because they are made of light atoms carbon has an atomic weight of 12 while hydrogen has an atomic weight of 1 The size of atoms and the way in which they are packed also influence density, but to a much lesser degree Atomic Packing Most materials are crystalline have a regularly repeating pattern of structural units Atoms often behave as if they are hard and spherical Layer A represents the close-packed layer there is no way to pack the atoms more closely than this

12 Atomic structures are close-packed in three dimension Hexagonal close-packed : ABABAB stacking sequence Face-centered cubic: ABCABC stacking sequence Packing fraction for HCP (hexagonal close packed) and FCC (Face centered cubic) structures is 0.74 meaning spheres occupy 74% of all available space Figure 4.8 FCC HCP FCC HCP

13 Volume density of metal = ρ Mass/Unit cell = v = Volume/Unit cell Example:- Copper (FCC) has atomic mass of g/mol and atomic radius of nm. a= 4 R = nm = nm 2 2 Volume of unit cell = V= a 3 = (0.361nm) 3 = 4.7 x m 3 FCC unit cell has 4 atoms. 6 (4atoms)(63.54g / mol ) Mass of unit cell = m = 10 Mg = 4.22 x Mg atmos / mol g (ton) 28 m Mg Mg g ρ v = = = 8.98 = V m m cm Example: 3.72 Non Close-Packed Structures Figure 4.9 Body-centered cubic: ABABAB packing sequence Packing fraction = 0.68 A BCC cell has 2 atoms

14 Non Close-Packed Structures Amorphous structure: Packing fraction 0.64 Figure 4.10 Unit Cell Red lines define the cell while spheres represent individual atoms Shaded regions represent close or closest packed plane Figure 4.11

15 Atomic Packing in Ceramics Figure 4.13 (a): Hexagonal unit cell with a W-C atom pair associated with each lattice point (b): Cubic unit cell with a Si-C atom pair associated with each lattice point Atomic Packing in Glasses Amorphous silica is the bases of most glasses Rapid cooling allows material to maintain amorphous structure achieved after melting Figure 4.14

16 Atomic Packing in Polymers Figure 4.15 Polymers have a carbon-carbon backbone with varying side-groups Figure 4.16 Figure 4.17 Polymer chains bond to each other through weak hydrogen bonds Red lines indicate strong cross-linked carbon-carbon bonds

17 Polymer Structure Figure 4.18 (a): No regular repeating pattern of polymer chains results in a glassy or amorphous structure (b): Regions in which polymer chains line up and register forms crystalline patches (c): Occasional cross-linking allowing they polymer to stretch typical of elastomers (d): Heavily cross-linked polymers exhibit chain sliding typical of epoxy Elastic Moduli of Elastomers Undeformed polymer chains has high randomness (entropy) Stretched polymer chains resemble more of a crystalline structure and has a lower entropy Figure 4.20 Moduli of elastomers is generally low and unlike metals, increases with temperature

18 Cohesive Energy Atoms are held together by bonds that behave like springs Cohesive energy is a measure of the strength of the bonds Bond Stiffness Figure 4.19 Table 4.1 Bond stiffness largely determines the value of the modulus - E Bond stiffness Bond length at eq m (atomic size)

19 Rule of Mixtures Modifying the modulus and density is most effective when done at a macro scale such as creating a hybrid rather than a micro scale such as alloying a metal Density of solid solution or hybrid materials ~ ρ f ρ A ρ B volume fraction of material or element A density of material or element A density of material or element B Composites Density and Modulus Polymer matrix composite (PMC) Ceramic matrix composite (CMC) Metal matrix composite (MMC) Modulus can be altered by combining stiff fibers with a less-stiff matrix Figure 4.21

20 ρ r density of reinforcement ρ m density of matrix Modulus of composite bracketed by two bounds: Upper bound: assumes that, on loading, both components strain by the same amount, like springs in parallel Lower Bound: assumes that, on loading, each component carries the same stress, like springs in series Range of modulus and density properties for composites with a ceramic reinforcement and polymeric matrix Figure 4.22

21 Foams Density and Modulus Figure 4.23 ρ s and E s are the density and modulus of the solid from which the foam is made Modulus and density range for foams made from elastomers and polymers foaming lowers both of these properties Figure 4.24

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