# Young's Modulus

Read here about a property of materials that is significant in tensegrity research: the Young's modulus.

## Contents

# Overview

In solid mechanics, Young's modulus, also known as the tensile modulus, is a measure of the stiffness of an isotropic elastic material. It is defined as the ratio of the uniaxial stress over the uniaxial strain in the range of stress in which Hooke's Law holds.[1] This can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a sample of the material. It is also commonly, but incorrectly, called the elastic modulus or modulus of elasticity, because Young's modulus is the most common elastic modulus used, however there are other elastic moduli, such as the bulk modulus and the shear modulus.

The Young's modulus allows the behavior of a bar made of an isotropic elastic material to be calculated under tensile or compressive loads. For instance, it can be used to predict the amount a wire will extend under tension or buckle under compression. Some calculations also require the use of other material properties, such as the shear modulus, density, or Poisson's ratio.

Young's modulus is named after Thomas Young, the 19th century British scientist. However, the concept was developed in 1727 by Leonhard Euler, and the first experiments that used the concept of Young's modulus in its current form were performed by the Italian scientist Giordano Riccati in 1782 — predating Young's work by 25 years.

# Stiffness as a critical factor of Tensegrity Struts

Stiffness is a critical factor in determining the integrity of a tensegrity structure. Ramar and Guest 2010 expoored the calculation of self-weight deflection of a prestressed tensegrity structure. They found that the Young's modulus is critical. The image below shows an Ashby bubble chart comparing Young’s Modulus E and Density D for different materials. In order to maximise the ratio E/D we seek materials towards the top -left corner of the plot. Thus steel and aluminium are likely to have a similar performance, as would bamboo, whereas carbon fibre should give reduced self -weight deflection. The plot also suggests that technical ceramics might prove to be an optimal material choice for the struts.

# Young's Modulus Units are Pressure Units

Young's modulus is the ratio of stress, which has units of pressure, to strain, which is dimensionless; therefore Young's modulus itself has units of pressure. The SI unit of modulus of elasticity (E, or less commonly Y) is the pascal (Pa or N/m²); the practical units are megapascals (MPa or N/mm²) or gigapascals (GPa or kN/mm²). In United States customary units, it is expressed as pounds (force) per square inch (psi).

# Samples of Young's Modulus in Tensegrity Research

## Cellular Elasticity Based On Tensegrity

A quantitative model of cellular elasticity based on tensegrity.

https://www.ncbi.nlm.nih.gov/pubmed/10790828

A tensegrity structure composed of six struts interconnected with 24 elastic cables is used as a quantitative model of the steady-state elastic response of cells, with the struts and cables representing microtubules and actin filaments, respectively. The model is stretched uniaxially and the Young's modulus (E0) is obtained from the initial slope of the stress versus strain curve of an equivalent continuum. It is found that E0 is directly proportional to the pre-existing tension in the cables (or compression in the struts) and inversely proportional to the cable (or strut) length square.

## Cytoskeleton Mechanics

Young's modulus is a critical paramater in the analysis of cell stress.

http://biomechanics.stanford.edu/me239_12/me239_n09.pdf

http://biomechanics.stanford.edu/Mechanics_of_the_cell_07

# Optimization Of Tensegrity Structures

Optimization of tensegrity structures by Masica, Skelton, Gill

The material Young’s module y and the yield strain σ/y are varied to investigate their impact on the parameters that characterize overall shape of the optimal design of a tensegrity beam.

https://www.sciencedirect.com/science/article/pii/S0020768305004932

## The Stiffness of Tensegrity Structures

The Stiffness of Tensegrity Structures by S.D. Guest, 2010

Material properties of tendons are compared and considered. The Young’s Modulus is a critical parameter in comparing the stiffness of steel as compared with other materials. of a steel will be the order of a hundred times stiffer than the Young’s Modulus of an elastomer.

http://www2.eng.cam.ac.uk/~sdg/preprint/tensstiff.pdf

## Minimizing the Self-weight Deflection of Tensegrity Structures

Minimizing the Self-weight Deflection of Tensegrity Structures by Ramar, Guest 2011

http://www2.eng.cam.ac.uk/~sdg/preprint/TensegrityDeflection.pdf