Mechanics of Materials: Strain » Mechanics of Slender Structures | Boston University
This allows Young's modulus E, Poisson's ratio ν and Lame's coefficients µ and λ Relationship between Young's modulus and Lame's coefficient µ is linear. As a student you ought to be able to make a start on this problem, eg by defining each modulus. Please show your working so far. – sammy. alent Young's modulus and Poisson's ratio for realizing the finite element analysis. Difficulties of the tensile tests exist in sample handling, manipulation and “bulk” elasticity in the finite-thickness shell, the relationship between the surface.Relation among Young's modulus , Bulk modulus and Poisson's ratio - in HINDI - EduPoint
When a force acts parallel to the surface of an object, it exerts a shear stress. Let's consider a rod under uniaxial tension. The rod elongates under this tension to a new length, and the normal strain is a ratio of this small deformation to the rod's original length.
Strain is a unitless measure of how much an object gets bigger or smaller from an applied load. Shear strain occurs when the deformation of an object is response to a shear stress i. Mechanical Behavior of Materials Clearly, stress and strain are related. Stress and strain are related by a constitutive law, and we can determine their relationship experimentally by measuring how much stress is required to stretch a material. This measurement can be done using a tensile test.
In the simplest case, the more you pull on an object, the more it deforms, and for small values of strain this relationship is linear. This linear, elastic relationship between stress and strain is known as Hooke's Law. If you plot stress versus strain, for small strains this graph will be linear, and the slope of the line will be a property of the material known as Young's Elastic Modulus.
This value can vary greatly from 1 kPa for Jello to GPa for steel. In this course, we will focus only on materials that are linear elastic i. From Hooke's law and our definitions of stress and strain, we can easily get a simple relationship for the deformation of a material. Intuitively, this exam makes a bit of sense: If the structure changes shape, or material, or is loaded differently at various points, then we can split up these multiple loadings using the principle of superposition.
Generalized Hooke's Law In the last lesson, we began to learn about how stress and strain are related — through Hooke's law. But, up until this point we've only considered a very simplified version of Hooke's law: In this lesson, we're going to consider the generalized Hooke's law for homogenousisotropicand elastic materials being exposed to forces on more than one axis. First things first, even just pulling or pushing on most materials in one direction actually causes deformation in all three orthogonal directions.
Let's go back to that first illustration of strain. This time, we will account for the fact that pulling on an object axially causes it to compress laterally in the transverse directions: This property of a material is known as Poisson's ratio, and it is denoted by the Greek letter nu, and is defined as: Or, more mathematically, using the axial load shown in the above image, we can write this out as an equation: Since Poisson's ratio is a ratio of two strains, and strain is dimensionless, Poisson's ratio is also unitless.
Poisson's ratio is a material property. Poisson's ratio can range from a value of -1 to 0. For most engineering materials, for example steel or aluminum have a Poisson's ratio around 0. Incompressible simply means that any amount you compress it in one direction, it will expand the same amount in it's other directions — hence, its volume will not change.
Physically, this means that when you pull on the material in one direction it expands in all directions and vice versa: We can in turn relate this back to stress through Hooke's law. This is an important note: In reality, structures can be simultaneously loaded in multiple directions, causing stress in those directions. A helpful way to understand this is to imagine a very tiny "cube" of material within an object.
That cube can have stresses that are normal to each surface, like this: There is a change in dimensions, or deformation elongation, DL as a result of a tensile or compressive stress.
To enable comparison with specimens of different length, the elongation is also normalized, this time to the length L. This is called strain, e.
Poisson's Ratio: Examples based on longitudinal and lateral strains
One could divide force by the actual area, this is called true stress see Sec. For torsional or shear stresses, the deformation is the angle of twist, q Fig. When the stress is removed, the material returns to the dimension it had before the load was applied. Valid for small strains except the case of rubbers.
Problems on Poisson’s Ratio
Deformation is reversible, non permanent Plastic deformation. When the stress is removed, the material does not return to its previous dimension but there is a permanent, irreversible deformation.
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- Young's modulus
In tensile tests, if the deformation is elastic, the stress-strain relationship is called Hooke's law: E is Young's modulus or modulus of elasticity. In some cases, the relationship is not linear so that E can be defined alternatively as the local slope: Elastic moduli measure the stiffness of the material.
material science - Derive equation for shear modulus $G=E/(1+2v)$ - Physics Stack Exchange
They are related to the second derivative of the interatomic potential, or the first derivative of the force vs. By examining these curves we can tell which material has a higher modulus. Due to thermal vibrations the elastic modulus decreases with temperature. E is large for ceramics stronger ionic bond and small for polymers weak covalent bond. Since the interatomic distances depend on direction in the crystal, E depends on direction i. For randomly oriented policrystals, E is isotropic.
Anelasticity Here the behavior is elastic but not the stress-strain curve is not immediately reversible. It takes a while for the strain to return to zero. The effect is normally small for metals but can be significant for polymers. Elastic Properties of Materials Materials subject to tension shrink laterally.
Those subject to compression, bulge. The ratio of lateral and axial strains is called the Poisson's ratio n. If the stress is too large, the strain deviates from being proportional to the stress. The point at which this happens is the yield point because there the material yields, deforming permanently plastically. Hooke's law is not valid beyond the yield point.
The stress at the yield point is called yield stress, and is an important measure of the mechanical properties of materials. In practice, the yield stress is chosen as that causing a permanent strain of 0. The yield stress measures the resistance to plastic deformation. The reason for plastic deformation, in normal materials, is not that the atomic bond is stretched beyond repair, but the motion of dislocations, which involves breaking and reforming bonds.
Plastic deformation is caused by the motion of dislocations.
When stress continues in the plastic regime, the stress-strain passes through a maximum, called the tensile strength sTSand then falls as the material starts to develop a neck and it finally breaks at the fracture point Fig. Note that it is called strength, not stress, but the units are the same, MPa.