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Stress and Strain Stress and strain Stress Lecture 2 – Loads, traction, stress Mechanical Engineering Design - N.Bonora 2018 Stress and strain Introduction • One important step in mechanical design is the determination of the internal stresses, once the external load are assigned, and to assess that they do not exceed material allowables. • Internal stress - that differ from surface or contact stresses that are generated where the load are applied - are those associated with the internal forces that are created by external loads for a body in equilibrium. • The same concept holds for complex geometry and loads. To evaluate these stresses is not a a slender block of material; (a) under the action of external straightforward matter, suffice to say here that forces F, (b) internal normal stress σ, (c) internal normal and they will invariably be non-uniform over a shear stress surface, that is, the stress at some particle will differ from the stress at a neighbouring particle Mechanical Engineering Design - N.Bonora 2018 Stress and strain Tractions and the Physical Meaning of Internal Stress • All materials have a complex molecular microstructure and each molecule exerts a force on each of its neighbors. The complex interaction of countless molecular forces maintains a body in equilibrium in its unstressed state. • When the body is disturbed and deformed into a new equilibrium position, net forces act. An imaginary plane can be drawn through the material • The force exerted by the molecules above the plane on the material below the plane and can be attractive or repulsive. Different planes can be taken through the same portion of material and, in general, a different force will act on the plane. Mechanical Engineering Design - N.Bonora 2018 Stress and strain Tractions and the Physical Meaning of Internal Stress The traction at some particular point in a material is defined as follows: • take a plane of surface area S through the point, on which acts a force F. • shrink the plane – as it shrinks in size both S and F get smaller, and the direction in which the force acts may change, but eventually the ratio F / S will remain constant and the force will act in a particular direction. • limiting value of this ratio of force over surface area is defined as the traction vector (or stress vector) An infinite number of traction vectors act at any single point, since an infinite number of different planes pass through a point For this reason the plane on which the traction vector acts must be specified; this can be done by specifying the normal n to the surface on which the traction acts Mechanical Engineering Design - N.Bonora 2018 Stress and strain Tractions and the Physical Meaning of Internal Stress The traction vector can be decomposed into components which act normal and parallel to the surface upon which it acts. These components are called the stress components, or simply stresses, and are denoted by the symbol s ; subscripts are added to signify the surface on which the stresses act and the directions in which the stresses act. Sign Convention for Stress Components The following convention is used: • the stress is positive when the direction of the normal and the direction of the stress component are both positive or both negative • the stress is negative when one of the directions is positive and the other is negative Mechanical Engineering Design - N.Bonora 2018 Stress and strain Type of stress Normal stress. The resisting area is normal to the internal force. • Tensile stress. Is the stress induced in a body when subjected to two equal and opposite pulls (tensile force) as a result of which there is the tendency to increase in length • Compressive stress. Stress induced in a body when subjected to equal and opposite pushes as a a result of which there is a tendency of decrease in lenght Mechanical Engineering Design - N.Bonora 2018 Stress and strain Type of stress Combined stress. A condition of stress that cannot be represented by a single resultant stress • Shear stress. Forces parallel to the area resisting the force cause shear stress. It differ from tensile and compressive stresses. Known also as tangential stress • Torsional stress. The stress and deformation induced in a circular shaft by a twisting moment. Mechanical Engineering Design - N.Bonora 2018 Stress and strain Simple stress In some situations, the stress within a body may adequately be described by a single number, or by a single vector (a number and a direction). Three such simple stress situations, that are often encountered in engineering design, are: • Uniaxial normal stress • Simple shear stress • Isotropic normal stress Mechanical Engineering Design - N.Bonora 2018 Stress and strain Simple stress • Uniaxial normal stress This stress state occurs in a straight rod of uniform section and homogeneous material subjected to tension by opposite forces of magnitude F along its axis. If the system is in equilibrium and not changing with time, and the weight of the bar can be neglected, then through each transversal section of the bar the top part must pull on the bottom part with the same force, F with continuity through the full cross-sectional area, A. 퐹 휎 = Therefore, the stress, σ throughout the bar, 퐴 across any horizontal surface, is expressed by: Mechanical Engineering Design - N.Bonora 2018 Stress and strain Simple stress • Simple shear stress This type of stress occurs when a uniformly thick layer of elastic material is firmly attached to two stiff bodies that are pulled in opposite directions by forces parallel to the layer Let F be the magnitude of those forces, and M be the midplane of that layer. Just as in the normal stress case, the part of the layer on one side of M must pull the other part with the same force F. Assuming that the direction of the forces is known, the stress across M can be expressed simply by the single number τ , calculated simply with the magnitude of those forces, F and the cross sectional area, A. 퐹 For any plane S that is perpendicular to the layer, the net internal 휏 = force across S, and hence the stress, will be zero. 퐴 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Simple stress • Isotropic normal stress (hydrostatic stress) Another simple type of stress occurs when the material body is under equal compression or tension in all directions. This is the case, for example, in a portion of liquid or gas at rest, whether enclosed in some container or as part of a larger mass of fluid; or inside a cube of elastic material that is being pressed or pulled on all six faces by equal perpendicular forces — provided, in both cases, that the material is homogeneous, without built-in stress, and that the effect of gravity and other external forces can be neglected. The force across any section S of the cube must balance the forces applied below the section. In the three sections shown, 퐹 the forces are 퐹 (top right), 퐹 2 (bottom left), and 퐹 3/2 휎 = 퐴 while the area of S is 퐴, 퐴 2 and 퐴 3/2, respectively. So the stress across S is F/A in all three cases. Mechanical Engineering Design - N.Bonora 2018 Stress and strain Representation of a three-dimensional stress state STRESS TENSOR. A tridimensional stress state is described by the Cauchy stress tensor σ, (or true stress tensor) The tensor consists of nine components 휎11 휎12 휎13 휎푥 휏푥푦 휏푥푧 휏 휎 휏 휎푖푗 = 휎21 휎22 휎23 ≡ 푦푥 푦 푦푧 휎31 휎32 휎33 휏푧푥 휏푧푦 휎푧 that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector n to the stress vector T(n) across an imaginary surface perpendicular to n. The SI units of both stress tensor and stress vector are N/m2, corresponding to the stress scalar. The unit vector is dimensionless. Mechanical Engineering Design - N.Bonora 2018 Stress and strain Representation of a three-dimensional stress state STRESS DEVIATOR TENSOR. The stress tensor can be always expressed as the sum of two terms: 1. a mean hydrostatic stress tensor or volumetric stress tensor or mean normal stress tensor, which is responsible for the change the volume of the stressed body; 2. a deviatoric component called the stress deviator tensor, which is responsible for the shape change at constant volume 1 휎 = 푠 + 휎 훿 푖푗 푖푗 3 푘푘 푖푗 Where 휋 = 휎푘푘 3 is the pressure. 푠11 푠12 푠13 휎11 − 휋 휎12 휎13 The deviatoric stress tensor can be obtained by 푠푖푗 = 푠21 푠22 푠23 ≡ 휎21 휎푦 − 휋 휎23 subtracting the hydrostatic stress tensor from the 푠31 푠32 푠33 휎31 휎32 휎푧 − 휋 Cauchy stress tensor: 푠푖푗 = 휎푖푗 − 휋훿푖푗 Mechanical Engineering Design - N.Bonora 2018 Stress and strain Representation of a three-dimensional stress state Principal stresses and stress invariants. At every point in a stressed body there are at least three planes, called principal 휎11 휎12 휎13 휎1 0 0 planes, with normal vectors, called principal directions, 휎푖푗 = 휎21 휎22 휎23 ≡ 0 휎2 0 where the corresponding stress vector is perpendicular to 휎31 휎32 휎33 0 0 휎3 the plane and where there are no shear stresses. The three stresses normal to these principal planes are called principal stresses. The Cauchy stress tensor obeys the tensor transformation law under a change in the system of coordinates. The components of the stress tensor σij depend on the orientation of the coordinate system at the point under consideration. However, the stress tensor itself is a physical 퐼1 = 휎11 + 휎22 + 휎33 quantity and as such, it is independent of the coordinate 2 2 2 system chosen to represent it.
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