# Stress strain relationship and elastic constants equations

### Introduction to Elasticity/Constitutive relations - Wikiversity

Jan 9, For further reading about Strain and determination of relation between Elastic Constants, please click on the link given below. complex and leads to the introduction of higher-order elastic constants. These higher- .. Stress-strain relationship for a single filament of Thornel .. Equations 11 to 15 show the relationship between the engineering and elastic compliance. Stress, Strain and Elastic Moduli: An elastic strain. We wish to understand only the basic elastic equations c ⇒ c11 elastic constant relations σ11 to ε

### Hooke's law - Wikipedia

Usage[ edit ] The Young's modulus enables the calculation of the change in the dimension of a bar made of an isotropic elastic material under tensile or compressive loads. For instance, it predicts how much a material sample extends under tension or shortens under compression. The Young's modulus directly applies to cases of uniaxial stress, that is tensile or compressive stress in one direction and no stress in the other directions.

Young's modulus is also used in order to predict the deflection that will occur in a statically determinate beam when a load is applied at a point in between the beam's supports. Other elastic calculations usually require the use of one additional elastic property, such as the shear modulusbulk modulus or Poisson's ratio. Any two of these parameters are sufficient to fully describe elasticity in an isotropic material. Linear versus non-linear[ edit ] Young's modulus represents the factor of proportionality in Hooke's lawwhich relates the stress and the strain.

However, Hooke's law is only valid under the assumption of an elastic and linear response. Shown here is bending, by a moment applied to opposite edges, of a honeycomb with hexagonal cells. Since the honeycomb is anisotropic, the Poisson's ratio need not lie within the above range. Poisson's ratio and anisotropy. In anisotropic solids including single crystals, honeycombs, and fibrous composites, physical properties, including Poisson's ratio and elastic moduli, depend on direction.

Poisson's ratio can have positive or negative values of arbitrarily large magnitude in anisotropic materials. For orthotropic materials, Poisson's ratio is bounded by the ratio of Young's moduli E as follows.

Poisson's ratio in viscoelastic materials The Poisson's ratio in a viscoelastic material is time dependent in the context of transient tests such as creep and stress relaxation.

If the deformation is sinusoidal in time, Poisson's ratio may depend on frequency, and may have an associated phase angle. Specifically, the transverse strain may be out of phase with the longitudinal strain in a viscoelastic solid.

## Elasticity

Get pdf of a research article on this. Poisson's ratio and phase transformations Poisson's ratio can vary substantially in the vicinity of a phase transformation. Typically the bulk modulus softens near a phase transformation but the shear modulus does not change much. The Poisson's ratio then decreases in the vicinity of a phase transformation and can attain negative values.

Phase transformations are discussed further on the linked page. Poisson's ratio, waves and deformation The Poisson's ratio of a material influences the speed of propagation and reflection of stress waves.

In geological applications, the ratio of compressional to shear wave speed is important in inferring the nature of the rock deep in the Earth.

### Young's modulus - Wikipedia

This wave speed ratio depends on Poisson's ratio. Poisson's ratio also affects the decay of stress with distance according to Saint Venant's principle, and the distribution of stress around holes and cracks.

Analysis of effect of Poisson's ratio on compression of a layer. What about the effect of Poisson's ratio on constrained compression in the 1 or x direction?

Constrained compression means that the Poisson effect is restrained from occurring. This could be done by side walls in an experiment. Also, compression of a thin layer by stiff surfaces is effectively constrained. Moreover, in ultrasonic testing, the wavelength of the ultrasound is usually much less than the specimen dimensions. The Poisson effect is restrained from occurring in this case as well. Actually, you may encounter a couple of additional elastic constants: Still there are only two independent elastic constants, and any one can be expressed in terms of any two of the other constants.

We have previously seen how we can determine E modulus of elasticity from a tensile test. With proper instrumentation we can also determine Poisson's ratio from the tensile test. Poisson's ratio is defined as the negative of the ratio of the lateral strain to the axial strain for a uniaxial stress state: You may also see it defined as the ratio of the absolute values of lateral and axial strain. In this experiment we shall employ either electrical resistance strain gages or a biaxial extensometer to measure the axial and lateral circumferential strains on a uniaxial tensile specimen Figure 1.

These values along with load will permit us to directly determine E and v.

The modulus of rigidity G may then be determined indirectly using the relationship between the elastic constants. Experiment Option a Set up a data sheet with space to record the two strain values for each of 10 loads.

Be sure to also record information about your specimen, the date and other pertinent information.

Option b Use automatic data acquisition equipment to collect and store data. Pertinent information about the experiment should be stored with the data set.