Cover Letter Resume Engineering Examples Of Poissons Ratio

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ROBYN A. MATTHEWS
15 Elm Lane
Sometown, IN 55555
Home: (555) 555-5555
rmatthews@somedomain.com

 

January 9, 2018

Mr. Andrew Smith
Director of Operations
123 Company
15 Lafayette Way
Sometown, IN 55555

Re: Mechanical Engineer Position (Ref. Code: 12345), advertised on Monster.com

Dear Mr. Smith:

Your job description for a mechanical engineer perfectly matches my qualifications, and I am very interested in the opportunity.

I have enjoyed a progressively responsible engineering career with ABC Manufacturing Co., and participated in the engineering of three major car model changes. I am experienced in all phases of new vehicle model development and implementation, from conception to production stages. Most recently, I have:
 

  • Contributed to major model changes for the 2008 Carname, 2007 Carname and 2006 Carname;
  • Developed process plans and layouts for seven additional car models;
  • Reduced ergonomic impact on production team members by designing new assist lift systems;
  • Helped create estimation and calculation tool for project budgets, greatly improving negotiating power with installation contractors; and
  • Built rapport with overseas colleagues, often traveling to Japan for production consultations.

In addition, I have served as lead engineer in establishing standards that have reduced costs, enhanced efficiency, improved production methods and simplified equipment and part needs.

Mr. Smith, I have received repeated commendations from ABC Manufacturing Co. for my work quality, revenue contributions, and commitment to achieving company goals, and I know I would be a valuable asset to your North American Division. Please feel free to call me at 555-555-5555 or send an email to rmatthews@somedomain.com to arrange a meeting. I look forward to speaking with you!

Sincerely,



Robyn A. Matthews
Enclosure: Resume

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Poisson's ratio, denoted by the Greek letter 'nu', , and named after Siméon Poisson, is the signed ratio of transverse strain to axial strain. For small values of these changes, is the amount of transversal expansion divided by the amount of axial compression.

Origin[edit]

Poisson's ratio is a measure of the Poisson effect, the phenomenon in which a material tends to expand in directions perpendicular to the direction of compression. Conversely, if the material is stretched rather than compressed, it usually tends to contract in the directions transverse to the direction of stretching. It is a common observation when a rubber band is stretched, it becomes noticeably thinner. Again, the Poisson ratio will be the ratio of relative contraction to relative expansion and will have the same value as above. In certain rare cases, a material will actually shrink in the transverse direction when compressed (or expand when stretched) which will yield a negative value of the Poisson ratio.

The Poisson's ratio of a stable, isotropic, linear elastic material will be greater than −1.0 or less than 0.5 because of the requirement for Young's modulus, the shear modulus and bulk modulus to have positive values.[1] Most materials have Poisson's ratio values ranging between 0.0 and 0.5. A perfectly incompressible material deformed elastically at small strains would have a Poisson's ratio of exactly 0.5. Most steels and rigid polymers when used within their design limits (before yield) exhibit values of about 0.3, increasing to 0.5 for post-yield deformation which occurs largely at constant volume.[2] Rubber has a Poisson ratio of nearly 0.5. Cork's Poisson ratio is close to 0, showing very little lateral expansion when compressed. Some materials, e.g. some polymer foams, origami folds,[3][4] and certain cells can exhibit negative Poisson's ratio, and are referred to as auxetic materials. If these auxetic materials are stretched in one direction, they become thicker in the perpendicular direction. In contrast, some anisotropic materials, such as carbon nanotubes, zigzag-based folded sheet materials,[5][6] and honeycomb auxetic metamaterials[7] to name a few, can exhibit one or more Poisson's ratios above 0.5 in certain directions.

Assuming that the material is stretched or compressed along the axial direction (the x axis in the diagram below):

where

is the resulting Poisson's ratio,
is transverse strain (negative for axial tension (stretching), positive for axial compression)
is axial strain (positive for axial tension, negative for axial compression).

Length change[edit]

For a cube stretched in the x-direction (see figure 1) with a length increase of in the x direction, and a length decrease of in the y and z directions, the infinitesimal diagonal strains are given by

If Poisson's ratio is constant through deformation, integrating these expressions and using the definition of Poisson's ratio gives

Solving and exponentiating, the relationship between and is then

For very small values of and , the first-order approximation yields:

Volumetric change[edit]

The relative change of volume ΔV/V of a cube due to the stretch of the material can now be calculated. Using and :

Using the above derived relationship between and :

and for very small values of and , the first-order approximation yields:

For isotropic materials we can use Lamé’s relation[8]

where is bulk modulus and is elastic modulus (or Young's modulus).[clarification needed]

Note that isotropic materials must have a Poisson's ratio of . Typical isotropic engineering materials have a Poisson's ratio of .[9]

Width change[edit]

If a rod with diameter (or width, or thickness) d and length L is subject to tension so that its length will change by ΔL then its diameter d will change by:

The above formula is true only in the case of small deformations; if deformations are large then the following (more precise) formula can be used:

where

is original diameter
is rod diameter change
is Poisson's ratio
is original length, before stretch
is the change of length.

The value is negative because it decreases with increase of length

Isotropic materials[edit]

For a linear isotropic material subjected only to compressive (i.e. normal) forces, the deformation of a material in the direction of one axis will produce a deformation of the material along the other axis in three dimensions. Thus it is possible to generalize Hooke's Law (for compressive forces) into three dimensions:

where:

, and are strain in the direction of , and axis
, and are stress in the direction of , and axis
is Young's modulus (the same in all directions: , and for isotropic materials)
is Poisson's ratio (the same in all directions: , and for isotropic materials)

these equations can be all synthesized in the following:

In the most general case, also shear stresses will hold as well as normal stresses, and the full generalization of Hooke's law is given by:

where is the Kronecker delta. The Einstein sum convention is usually adopted:

In this case the equation is simply written:

Orthotropic materials[edit]

Main article: Orthotropic material

For orthotropic materials such as wood, Hooke's law can be expressed in matrix form as[10][11]

Figure 1: A cube with sides of length L of an isotropic linearly elastic material subject to tension along the x axis, with a Poisson's ratio of 0.5. The green cube is unstrained, the red is expanded in the x direction by due to tension, and contracted in the y and z directions by .
Figure 2: Comparison between the two formulas, one for small deformations, another for large deformations

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