Strain Energy Projectby Marina Gandelsman. Strain energy is one of fundamental concepts in mechanics and its principles are widely. Strain Energy in Uniaxial Loads.
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Contact mechanics is the study of the deformation of solids that touch each other at one or more points. The physical and mathematical formulation of the subject is built upon the mechanics of materials and continuum mechanics.
Consider a prismatic bar of length L subjected to a tensile force P. The load is. applied slowly, so there are no effects due to motion. Such loads are called static loads. As the load reaches its full value P, the bar gradually elongates to L + d. During this process, the load P gradually moves over the length d and does a certain amount of work.
From physics we recall that W = F * d. However, in this case, the force varies in magnitude (from F=0 to F=P). To find the value. The work done by the load is equal to the area under the curve.
As the. load is applied, strains are produced and their presence increases the energy of our bar. This strain energy is the energy absorbed by the bar as a result of its deformation under. From the principle of conservation of energy we know that this energy is equal to. U = W = SL P(x) dx. Sometimes this energy is referred to as internal work, to distinguish it from work done. The unit of strain energy is the same as work - J (SI) and ft- lb (British). If the force P is gradually removed, the bar will shorten and at least a portion of the.
If the material has not exceeded its. L, otherwise a permanent set. If the material of the bar follows Hooke's Law, the load- displacement.
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P = kd, and the strain energy stored. U = W = kd. 2/2 = Pd/2. The complimentary energy in the preceding picture is used with Castigliano's theorems. Complementary. Strain Energy theorems. The total strain energy in a bar composed of several sections is equal to the sum of. The total strain energy determined from a load- deformation curve is not really.
In. order to eliminate size as a factor, we consider the strain energy per unit volume (also. Since P = s. A and d = e. L, (2) can be rewritten as follows: U = (se/2)*ALu = U/ALu = se/2 = s. E. = Ee. 2/2 (3) The unit of strain energy density are J/m. SI) and in- lb/in. British). The area under a complete stress- strain diagram gives a measure of a. The larger the. area under the diagram, the tougher the material.
A high modulus of toughness is important. In the inelastic range, only a small part of. Most of the energy is dissipated in. The energy that may be recovered when a specimen has been stressed to. A is represented by triangle ABC.
AB is parallel to OD since all materials. The area OABO represents the. The strain- energy density of the material when it is. D on the diagram) is called modulus of resilience. It. is found by substituting the proportional limit spl.
EResilience represents the ability of the material to absorb and release energy within the. Strain Energy in Torsion. Consider a prismatic bar AB in pure torsion under the action of torque T. When the load is applied statically, the bar twists and the free end rotates through an. Again, assume the material is linearly elastic and follows Hooke's Law. The. relationship between T and f will also be linear. From this, we determine that.
U = W = Tf/2. Using the equation f = TL/GIp, we can. U = SL T(x)2/2. GIp. T2. L/2. GIp = GIpf. L If the bar is subjected to non- uniform torsion, the total strain energy is equal to the. Strain Energy in Pure Shear.
Consider an element of dimensions x, y, and z subjected to shear load t. As this element is deformed, the force on top plane reaches a. The total displacement of this force for a.
Therefore, W = U = 1/2 t xz * gy = 1/2 tg V = 1/2 t. V/GThe strain- energy density in this case isu = 1/2 tg = 1/2 t. GStrain Energy in Bending. Consider a beam in pure bending by couples of moment M. Its material follows Hooke's Law. The normal stress varies linearly from the neutral axis and s = - My/I. From U = sІ/2. E and the.
U = M2. L/2. EISubstituting q = ML/EI, we get. U = EIq. 2/2. LIf the bending moment in a beam varies along its length, we can obtain the total strain. From dq = 1/r. dx = d. U = SL M(x)2/2. EI. SL EI/2 (d. 2v/dx. The previous equations only consider the effect on bending on the beam.
If shear forces. are also present, additional strain energy will be stored in the beam, however, this. L > > t. If a beam supports a single load (either. P or moment M0, we can determine either deflection d (for P) or angle of rotation q (for M0). The deflection is measured along the line of action of the load and is. The angle of rotation is the angle of rotation of. We can obtain the following. U = W = Pd/2. U = W = M0q/2.
This method is limited in its applications because only one deflection (or angle) can be. The equations demonstrated here are fairly basic and simplified in several ways. Only the. perfectly elastic bodies do not dissipate any energy and thus can store all of the work as. Most real- life materials are not elastic. A lot of research is. Bibliography. Mechanics of Materials, Gere & Timoshenko, 4th edition, PWS Publishing.
Engineering Mechanics of Materials, B. B. Muvdi and J. W. Mc. Nabb, Macmillan. Publishing Co. Engineering Mechanics of Solids, E. Popov, 2nd edition, Prentice Hall.
Engineering Mechanics of Deformable Bodies, E. F. Byars and R. D. Snyder. 3rd edition, Intext Educational Publishers© 1. Marina Gandelsman. Distribute freely (I borrowed quite liberally from my.