Mechanical Properties Session 07-14 Subject : S1014 / MECHANICS of MATERIALS Year : 2008 Mechanical Properties Session 07-14

Mechanical Properties Bina Nusantara

What is Stress ? Much Work with limited time  High Stress Bina Nusantara

What is Stress ? Less Work with long time  Low Stress Bina Nusantara

What is Stress ? stress is according to strength and failure of solids. The stress field is the distribution of internal "tractions" that balance a given set of external tractions and body forces Bina Nusantara

Stress stress is according to strength and failure of solids. The stress field is the distribution of internal "tractions" that balance a given set of external tractions and body forces Bina Nusantara

Stress Look at the external traction T that represents the force per unit area acting at a given location on the body's surface. Bina Nusantara

Stress Traction T is a bound vector, which means T cannot slide along its line of action or translate to another location and keep the same meaning. In other words, a traction vector cannot be fully described unless both the force and the surface where the force acts on has been specified. Given both DF and Ds, the traction T can be defined as                                            Bina Nusantara

Stress The internal traction within a solid, or stress, can be defined in a similar manner. Bina Nusantara

Stress Suppose an arbitrary slice is made across the solid shown in the above figure, leading to the free body diagram shown at right. Bina Nusantara

Stress Surface tractions would appear on the exposed surface, similar in form to the external tractions applied to the body's exterior surface. Bina Nusantara

Stress The stress at point P can be defined using the same equation as was used for T. Bina Nusantara

Stress Stress therefore can be interpreted as internal tractions that act on a defined internal datum plane. One cannot measure the stress without first specifying the datum plane. Bina Nusantara

Stress Surface tractions, or stresses acting on an internal datum plane, are typically decomposed into three mutually orthogonal components. One component is normal to the surface and represents direct stress. The other two components are tangential to the surface and represent shear stresses. Bina Nusantara

Stress What is the distinction between normal and tangential tractions, or equivalently, direct and shear stresses? Direct stresses tend to change the volume of the material (e.g. hydrostatic pressure) and are resisted by the body's bulk modulus (which depends on the Young's modulus and Poisson ratio). Bina Nusantara

Stress What is the distinction between normal and tangential tractions, or equivalently, direct and shear stresses? Shear stresses tend to deform the material without changing its volume, and are resisted by the body's shear modulus. Bina Nusantara

These nine components can be organized into the matrix: Stress These nine components can be organized into the matrix: Bina Nusantara

Stress where shear stresses across the diagonal are identical (sxy = syx, syz = szy, and szx = sxz) as a result of static equilibrium (no net moment). Bina Nusantara

Stress This grouping of the nine stress components is known as the stress tensor (or stress matrix). Bina Nusantara

Stress The subscript notation used for the nine stress components have the following meaning: Bina Nusantara

What is Strain? A propotional dimensional change ( intensity or degree of distortion ) Bina Nusantara

What is Strain measure? a total elongation per unit length of material due to some applied stress. Bina Nusantara

What are the types of strain ? Elastic Strain Plastic Deformation Bina Nusantara

Elastic Strain Transitory dimensional change that exists only while the initiating stress is applied and dissapears immediately upon removal of the stress. Bina Nusantara

Elastic Strain The applied stresses cause the atom are displaced the same amount and still maintain their relative geometic. When streesses are removed, all the atom return to their original positions and no permanent deformation occurs Bina Nusantara

It is usually accompanied by some elastic strain. Plastic Deformation a dimentional change that does not dissapear when the initiating stress is removed. It is usually accompanied by some elastic strain. Bina Nusantara

Elastic Strain & Plastic Deformation The phenomenon of elastic strain & plastic Deformation in a material are called elasticity & Plasticity respectively Bina Nusantara

Elastic Strain & Plastic Deformation Most of Metal material  At room temperature they have some elasticity, which manifests itself as soon as the slightest stress is applied. Usually, they are also posses some plasticity , but this may not become apparent until the stress has been raised appreciablty. Bina Nusantara

Elastic Strain & Plastic Deformation Most of Metal material  The magnitude of Plastic strain, when it does appear , is likely to be much greater than that of the elastic strain for a given stress increment Bina Nusantara

Solid material by force, F, at a point, as shown in the figure. Constitutive F Solid material by force, F, at a point, as shown in the figure. Bina Nusantara

The work done = F .d Constitutive Let the deformation at the the point be infinitesimal and be represented by vector d, as shown. The work done = F .d Bina Nusantara

W = Fx dx Constitutive For the general case: z y For the general case: W = Fx dx i.e., only the force in the direction of the deformation does work. Bina Nusantara

Constant Force W = Fx Amount of Work done If the Force is constant, the work is simply the product of the force and the displacement, W = Fx F Displacement x Bina Nusantara

Amount of Work done Linear Force: If the force is proportional to the displacement, the work is Fo F xo x Displacement Bina Nusantara

Strain Energy F x A simple spring system, subjected to a Force is proportional to displacement x; F=kx. Now determine the work done when F= Fo, from before: This energy (work) is stored in the spring and is released when the force is returned to zero Bina Nusantara

Hooke’s Law For systems that obey Hooke's law, the extension produced is directly proportional to the load: F=kx where: x = the distance that the spring has been stretched or compressed away from the equilibrium position, which is the position where the spring would naturally come to rest (usually in meters), F = the restoring force exerted by the material (usually in newtons), and K = force constant (or spring constant). The constant has units of force per unit length (usually in newtons per meter). Bina Nusantara

Hooke’s Law Bina Nusantara

Hooke’s Law Bina Nusantara

Strain Energy Density Consider a cube of material acted upon by a force, Fx, creating stress sx=Fx/a2 y x a Bina Nusantara

Strain Energy Density y causing an elastic displacement, d in the x direction, and strain ex=d/a x a Fx d Where U is called the Strain Energy, and u is the Strain Energy Density. Bina Nusantara

(a) For a linear elastic material u=1/2(300)(0.0015) N.mm/mm3 =0.225 N.mm/mm3 Bina Nusantara

(b) Consider elastic-perfectly plastic u=1/2(350)(0.0018) +350(0.0022) =1.085 N.mm/mm3 Bina Nusantara

Shear Strain Energy y x a d = gxya txy gxy Consider a cube of material acted upon by a shear stress,txy causing an elastic shear strain gxy Bina Nusantara

Total Strain Energy for a Generalized State of Stress Bina Nusantara

Strain Energy for axially loaded bar F= Axial Force (Newtons, N) A = Cross-Sectional Area Perpendicular to “F” (mm2) E = Young’s Modulus of Material, MPa L = Original Length of Bar, mm Bina Nusantara

Comparison of Energy Stored in Straight and Stepped bars Da A L (a) Bina Nusantara

Comparison of Energy Stored in Straight and Stepped bars L/2 nA (b) Bina Nusantara

What is Torsion ? an external torque is applied and an internal torque, shear stress, and deformation (twist) develops in response to the externally applied torque. Bina Nusantara

What is Torsion ? For solid and hollow circular shafts, in which assume the material is homogeneous and isotropic , that the stress which develop remain within the elastic limits, and that plane sections of the shaft remain plane under the applied torque. Bina Nusantara

What is isotropic ? is properties of the materials are the same in all directions in the material Bina Nusantara

Torsion of shafts Shafts are members with length greater than the largest cross sectional dimension used in transmitting torque from one plane to another Bina Nusantara

Internal Torque Bina Nusantara

The axis of the shaft is denoted Consider circular shaft AC subjected to equal and opposite torques T and T’. A cutting plane is passed through the shaft at B. The FBD for section BC must include the applied torque and elementary shearing forces dF. These forces are perpendicular to radius of the shaft and must balance to maintain equilibrium. The axis of the shaft is denoted as r. Bina Nusantara

dF is related to the shearing stress: dF = tdA Taking moments about the Axis of the shaft results in dF is related to the shearing stress: dF = tdA So the applied torque can be related to the shearing stress as Bina Nusantara

Equation is independent of material model as it represents static equivalency between shear stress and internal torque on a cross section Bina Nusantara

Shear stress can’t exist on one plane only. The applied torque produces a shear stress to the axis of the shaft. The equilibrium require equal stresses on the faces Bina Nusantara

What is Torsion ? To obtain a formula for the relative rotation f2-f1 in terms of the internal torque T. To obtain a formula for the shear stress txq in terms of the internal torque T. f - angle of twist Bina Nusantara

Shearing Strain Bina Nusantara

Assume: Material is linearly elastic and isotropic Polar moment of inertia for the cross section Bina Nusantara

Torsion formula Circular hollow shaft with outer radius R, inner radius r Bina Nusantara

Sign Convention Bina Nusantara

Relative rotation f2-f1 in terms of the internal torque T. Bina Nusantara

Shear stress txq in terms of the internal torque T. Maximum occurs at shaft’s outer radius Bina Nusantara

Direction of Shearing Bina Nusantara

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