Modulus Hooke

The surface energy of a solid is roughly proportional to its Young s modulus. Hooke s law is really an approximation which arises from the character of the chemical bond between atoms. Thus the same interatomic forces give rise to both and to y. 

Young s Modulus Hook s law assumes perfect elasticity in a material body. Young s modulus, E, may be written... 

It is important to differentiate between brittie and plastic deformations within materials. With brittie materials, the behavior is predominantiy elastic until the yield point is reached, at which breakage occurs. When fracture occurs as a result of a time-dependent strain, the material behaves in an inelastic manner. Most materials tend to be inelastic. Figure 1 shows a typical stress—strain diagram. The section A—B is the elastic region where the material obeys Hooke s law, and the slope of the line is Young s modulus. C is the yield point, where plastic deformation begins. The difference in strain between the yield point C and the ultimate yield point D gives a measure of the brittieness of the material, ie, the less difference in strain, the more brittie the material. 

When the magnitude of deformation is not too great, viscoelastic behavior of plastics is often observed to be linear, i.e., the elastic part of the response is Hookean and the viscous part is Newtonian. Hookean response relates to the modulus of elasticity where the ratio of normal stress to corresponding strain occurs below the proportional limit of the material where it follows Hooke s law. Newtonian response is where the stress-strain curve is a straight line. 

Modulus of elasticity Most materials, including plastics and metals, have deformation proportional to their loads below the proportional limit. Since stress is proportional to load and strain to deformation, this implies that stress is proportional to strain. Hooke s Law, developed in 1676, follows that this straight line (Fig. 2-2) of proportionality is calculated as ... 

We introduce the most basic aspects of elasticity. We begin with Hookes law the change in length of a strut is proportional to the applied force, or 6L = FL/EA. Note that this is a linear relationship. Restated in a normalized way, a = Ee, where a is the stress (Pa or N/m ), E is Young s modulus (Pa) a property of the material, and e is the strain (6LjL) a dimensionless quantity. 

The variation in wall thickness and the development of cell wall rigidity (stiffness) with time have significant consequences when considering the flow sensitivity of biomaterials in suspension. For an elastic material, stiffness can be characterised by an elastic constant, for example, by Young s modulus of elasticity (E) or shear modulus of elasticity (G). For a material that obeys Hooke s law,for example, a simple linear relationship exists between stress, , and strain, a, and the ratio of the two uniquely determines the value of the Young s modulus of the material. Furthermore, the (strain) energy associated with elastic de-... 

Here E is Young modulus. Comparison with Equation (3.95) clearly shows that the parameter k, usually called spring stiffness, is inversely proportional to its length. Sometimes k is also called the elastic constant but it may easily cause confusion because of its dependence on length. By definition, Hooke s law is valid when there is a linear relationship between the stress and the strain. Equation (3.97). For instance, if /q = 0.1 m then an extension (/ — /q) cannot usually exceed 1 mm. After this introduction let us write down the condition when all elements of the system mass-spring are at the rest (equilibrium) ... 

Most engineering materials, particularly metals, follow Hooke s law by which it is meant that they exhibit a linear relationship between elastic stress and strain. This linear relationship can be expressed as o = E where E is known as the modulus of elasticity. The value of E, which is given by the slope of the stress-strain plot, is a characteristic of the material being considered and changes from material to material. 

The important elastic properties of a material undergoing deformation under static tension are stiffness, elastic strength and resilience. For a material obeying Hooke s law, the modulus of elasticity, E (= o/e), can be taken to be a measure of its stiffness. The elastic... 

The modulus of elasticity of a material it is the ratio of the stress to the strain produced by the stress in the material. Hooke s law is obeyed by metals but mbber obeys Hooke s law only at small strains in shear. At low strains up to about 15% the stress-strain curve is almost linear, but above 15% the stress and strain are no longer proportional. See Modulus. 

The elastic modulus is constant at small stresses and strains. This linearity gives us Hooke s Law1, which states that the stress is directly proportional to the strain. 

Consider the situation shown in Figure 2.4 where a mass m is caused to oscillate by an initial displacement up to an amount oq at t = 0. The amplitude a would have to be smaller than shown for simple harmonic motion as a real spring would only obey Hooke s law over a limited strain amplitude. However the assumption is that Hooke s law is obeyed and the restoring force from both spring displacements is — IJcoq where k is the force constant or elastic modulus of the spring. So we may write the force at any position as... 

As the temperature is raised, the vibrational energy increases, because it is kBT in each direction. If we have a simple cubic crystal in which the intermolecular spacing is r then the molar volume is Nar3. The Young s modulus for the crystal is Y and we assume a Hooke s law spring. We can define the local stress as the applied force per molecule, Fm, divided by r2, giving a local strain of x/r where x is the extension caused by the oscillation. Hence ... 

A similar approach can be adopted for the bulk shear modulus. When a small strain is applied to a solid, the latter is stressed, and one can measure the resulting stress. At low deformation, the bulk shear stress, t, is proportional to the strain, r, following Hooke s law ... 

In the initial stages of the extension the graph is sometimes linear and obeys Hooke s law. The slope of this section is called Young s modulus. This portion of the curve is reversible. Because many polymers do not obey Hooke s law the modulus is frequently expressed as pounds per square inch at a certain elongation or extensibility. The 2% modulus is a common quotation. Some elastomers are better described as a 100% modulus. The stiffer the polymer is that is tested, the higher will be the modulus value that is recorded. 

Finally, the modulus of elasticity E (Young s modulus), which is a measure of the stiffness of the polymer, can be calculated from the stress-strain diagram. According to Hooke s law there is a linear relation between the stress o and the strain e ... 

A measure of the stiffness of a polymer is the modulus of elasticity (Young s modulus) E. It can be calculated fi om the stress-strain curve as the slope in the linear region of Hooke s law. It should be considered that due to the definition E = o/e for rubberlike materials which show a rather large extension e at quite... 

Recall from Eq. (5.10) that the shear strain can be related to the shear stress through the shear modulus, G, according to Hooke s Law, where we now add subscripts to differentiate the elastic quantities from the viscous quantities ... 

Composites provide an atPactive alternative to the various metal-, polymer- and ceramic-based biomaterials, which all have some mismatch with natural bone properties. A comparison of modulus and fracture toughness values for natural bone provide a basis for the approximate mechanical compatibility required for arUficial bone in an exact structural replacement, or to stabilize a bone-implant interface. A precise matching requires a comparison of all the elastic stiffness coefficients (see the generalized Hooke s Law in Section 5.4.3.1). From Table 5.15 it can be seen that a possible approach to the development of a mechanically compatible artificial bone material... 

Figure 3.3 shows representative stress-strain curves for a variety of polymeric materials. At normal use temperatures, such as room temperature, rigid polymers such as polystyrene (PS) exhibit a rapid increase in stress with increasing strain until sample failure. This behavior is typical of brittle polymers with weak interchain secondary bonding. As shown in the top curve in Figure 3.3, the initial stress-strain relation in such polymers is approximately linear and can be described in terms of Hooke s law, i.e., S = Ee, where E is Young s modulus, typically defined as the slope of the stress-strain plot. At higher stresses, the plot becomes nonlinear. The point at which this occurs is called the proportional limit. 

In engineering design. Yuung s modulus is used for tension and compression and the rigidity modulus lor shear, as in lorsion springs. According to Hooke s Law, Ihe stress set up within an elastic body is proportional 10 the strain lo which the body is subjected by the applied load. 

A typical curve has an essentially linear portion (OA) in which the deformation is proportional to the applied load, See Fig. 1. It follows that the unit stress iload divided by original area) is proportional to the unit strain (deformation divided by original gage length) in accordance willi Hooke s Law. The numerical value of this ratio (e,g in psi) is known as Young s Modulus or Modulus of Elasticity,... 

The first part of the graph is linear for most materials and is used to determine the elastic modulus or E-modulus according to Hooke s law in formula ... 

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