# ME 2570 MECHANICS OF MATERIALS

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1 ME 2570 MECHANICS OF MATERIALS Chapter III. Mechanical Properties of Materials 1

2 Tension and Compression Test The strength of a material depends on its ability to sustain a load without undue deformation or failure. The strength of a material is inherent to the material itself and needs to be determined experimentally. One of these experimental tests is a tension test. The tension (or compression) test is used to determine the relation between the average normal stress and the average normal strain. This test is applied to engineering materials such as metals, ceramics, polymers and composites. 2

3 When preparing a tension or compression test of a certain material, a specimen of the material of standard shape and size is prepared. Before the test, two punch marks are placed along the specimen s uniform length. The initial cross section area, A o, and gauge length distance, L o, between marks are measured. The specimen is mounted on a machine that applies the load. Frequent measurements of the applied load and the elongation δ = L L o of the specimen are recorded. 3

4 The Stress Strain Diagram A plot of P vs. ΔL is meaningless since specimens with different cross section areas or different gauge lengths will behave differently. Instead, in order to eliminate the dependence on the cross section the normal stress (σ = P/A o )is plotted on the ordinates axis instead of force P. And to eliminate the dependence on length, normal strain (ε = δ/l o ) is plotted instead of the change in length on the axis of the abscissae. Thus, a plot of ε vs. σ will then be substantially the same for multiple test specimens of the same material but different sizes and lengths. The resulting curve is called conventional stress strain diagram. 4

5 From the conventional stress strain diagram four different ways in which a material behaves depending on the amount of strain induced in the material. 5

6 Elastic Behavior. In general, there is an initial region which is linear. This is called the elastic or proportional region. In this region, stress and strain are related to one another by the slope of the curve which is a property called the modulus of elasticity or E. Thus, this region is governed by the relation σ = E ε. This equation is known as Hooke s Law. At the top of the proportional region is the proportional limit σ pl. Most materials exhibit the elastic region. When a material is loaded with a stress within the elastic region and then unloaded, the material will return to the original size and shape, i.e., there is no permanent deformation. 6

7 The elastic region is generally slightly larger than the proportional region, but for practical purposes can be considered to be the same. The top of the elastic region, called the elastic limit, is then basically the same as the proportional limit. Yielding. A slight increase in stress beyond the elastic limit will result in a breakdown of the material causing permanent deformation. This behavior is called yielding. Once yielding is reached large increases in strain occur with no further increase in stress. 7

8 Yielding point is generally very near the elastic limit and proportional limit. Mild steel exhibits a well defined yield point, σ y but not all materials do. For practical purposes, the yield point, elastic limit and proportional limit will be taken to be the same. Strain Hardening. Once yielding has ended an increase in load can be supported by the specimen resulting in a curve that raises continuously. This curve becomes flatter as it reaches a maximum stress known as ultimate stress, σ u. The rise in the curve in this way is called strain hardening. 8

9 Necking. Up to the ultimate stress, as the specimen elongates its cross section area decreases uniformly. However after the ultimate stress is reached the cross section area starts decreasing at a localized region. This reduction of the cross section area at a localized region results ultimately on the fracture of the specimen at the fracture stress, σ f. 9

10 Stress Strain Behavior of Ductile and Brittle Materials Depending on their stress strain characteristics a material can be Ductile or Brittle. Ductile Materials can be subjected to large strains before a fracture occurs. The ductility of a material can be defined in terms of its percent of elongation or percent of reduction in area at the moment of fracture. The final area is measured at the region of necking 10

11 Not all materials exhibit a well defined yield point. In this event, it is customary to use the 0.2% offset stress as the yield strength (offset method). Starting from a strain value of a line parallel to the proportional region is drawn until it intersects the stress vs, strain curve. 11

12 Brittle Materials exhibit little or not yielding before failure such as gray cast iron. 12

13 Hooke s Law Most engineering materials exhibit a linear relationship between stress and strain within the elastic region. E is the modulus of elasticity or Young s modulus. E represents the slope of the initial straight lined portion of the stress strain diagram. E is a mechanical property that indicates the stiffness of a material. The relation σ = Eε is valid only for the region where the material shows linear elastic behavior. If the stress in the material has exceeded the proportional limit, σ = Eε is not longer valid. 13

14 When a ductile material is loaded into the plastic region and the unloaded there will be some permanent deformation. The unloading path is nearly linear and parallel to the proportional region. Thus, there is an elastic strain which is recovered when the material is unloaded. ε E = σ / E. Subtracting the elastic strain from the total strain, the permanent or plastic strain is determined, ε P. This plastic strain is known as the permanent set. If the material is reloaded, the behavior will follow the unloading path back to the original curve. Thus, the size of the proportional region is increased raising the yield point. This phenomenon is called work or strain hardening. 14

15 Strain Energy A material that is deformed by an external loading stores energy internally throughout its volume. This energy is known as strain energy: ΔU = ½ σεδv For applications it is more convenient to specify the strain energy per unit volume known as strain energy density: u = ½ σε. For linearly elastic materials (σ = Eε): u = ½ σ 2 /E. 15

16 The Modulus of Resilience, u r, is defined as the strain energy density at the moment the stress, σ, reaches the proportional limit: u r = ½ σ pl ε pl = ½ σ 2 pl /E. u r represents the ability of the material to absorb energy without any permanent damage. The Modulus of Toughness, u t, indicates the strain energy density of the material before ir fractures. 16

17 Poisson s Ratio When a deformable body is subjected to an axial tensile force additionally to elongate it also contracts laterally. This is called the Poisson effect. Consider a bar subjected to an axial load P and as a result it elongates axially by δ and contracts radially by δ. Then Poisson s ratio is defined as: 17

18 The Shear Stress Strain Diagram The behavior of a material subjected to pure shear is studied by applying a torsional load to specimens in the shape of thin tubes. Measurements of the torsional load applied to the specimen and the angle of twist experienced are collected to create a shear stress strain diagram. The shear stress strain and normal stress strain diagrams are different but have analogous attributes. There is a proportional region for which the shear form of Hooke s Law can be written as τ = γ G. G is the shear modulus or slope of the proportional region. 18

19 The shear modulus of elasticity or modulus of rigidity, G, represents the slope of the line on the τ γ diagram. τ pl is the proportional limit. τ u is the ultimate shear stress. τ f is the shear stress at the point of fracture. The modulus of elasticity, the modulus of rigidity and the Poisson s ratio are related by. 19

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