# Linear isotropic stress strain relationship

### Linear Elasticity Equations - MATLAB & Simulink

However, using the relationships we previously discussed, loads and Note: Hooke's Law describes only the initial linear portion of the stress-strain curve for Isotropic – Isotropic materials have elastic properties that are independent of. Characterizing the stress-strain relation of the material thus becomes a Linear elastic constitutive relations model reversible behavior of a material The material is isotropic its stress-strain response is independent of material orientation. 5. Stress strain relations for isotropic, linear elastic materials. Young's Modulus, Poissons ratio and the Thermal Expansion Coefficient. Before writing down.

The general parameters identified here can also be viewed as a flexible basis for coupling elastic responses in multi-scale processes, where an open challenge is the transfer of meaningful information between scales. Introduction An elastic body or material is linear elastic or Hookean if the force needed to extend or compress it by some distance is proportional to that distance [ 2 ].

The mechanical response of a homogeneous isotropic linearly elastic material is fully characterized by two physical constants that can be derived by simple experiments. Any other linear elastic parameter can then be obtained from these two constants [ 3 ].

The assumption that, under the small strain regime, materials are linearly elastic with possibly a geometrically nonlinear behaviour is successfully used in many engineering applications. However, many modern applications and biological materials involve large strains, whereby the deformations are inherently nonlinear and the corresponding stresses depend on the underlying material properties.

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Biological and bioinspired materials are the subject of continuous intensive research efforts in biomedical applications, and can also be found in everyday life as well as in several industrial areas, e. For these complex materials, reliable models supported by rigorous mechanical analysis are needed and can also open the way to new applications [ 4 — 12 ].

### Hooke's Law for Isotropic Materials

Here, we concentrate on the nonlinear elastic response of materials and do not discuss possible viscoelastic behaviours which may be relevant in many biological systems.

In general, the mechanical responses of nonlinear elastic materials cannot be represented by constants but are described by parameters which are scalar functions of the deformation.

The complexity of defining such functions comes from the fact that there are multiple ways to define strains and stresses in nonlinear deformations, giving rise to multiple nonlinear functions corresponding to the same linear parameter.

Heavily cross-linked polymers elastomers are the most likely to show ideal rubbery behavior. Features of the behavior of a solid rubber: The material is close to ideally elastic. The material strongly resists volume changes.

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The bulk modulus the ratio of volume change to hydrostatic component of stress is comparable to that of metals or covalently bonded solids; 3. The shear modulus is temperature dependent: When stretched, the material gives off heat.

They are close to reversible, and show little rate or history dependence. This time, we will account for the fact that pulling on an object axially causes it to compress laterally in the transverse directions: This property of a material is known as Poisson's ratio, and it is denoted by the Greek letter nu, and is defined as: Or, more mathematically, using the axial load shown in the above image, we can write this out as an equation: Since Poisson's ratio is a ratio of two strains, and strain is dimensionless, Poisson's ratio is also unitless.

Poisson's ratio is a material property. Poisson's ratio can range from a value of -1 to 0. For most engineering materials, for example steel or aluminum have a Poisson's ratio around 0.

**08.4 Generalized Hooke's Law**

Incompressible simply means that any amount you compress it in one direction, it will expand the same amount in it's other directions — hence, its volume will not change. Physically, this means that when you pull on the material in one direction it expands in all directions and vice versa: We can in turn relate this back to stress through Hooke's law. This is an important note: In reality, structures can be simultaneously loaded in multiple directions, causing stress in those directions.

A helpful way to understand this is to imagine a very tiny "cube" of material within an object. That cube can have stresses that are normal to each surface, like this: And, as we now known, stress in one direction causes strain in all three directions.

### Mechanics of Materials: Strain » Mechanics of Slender Structures | Boston University

So now we incorporate this idea into Hooke's law, and write down equations for the strain in each direction as: These equations look harder than they really are: Now we equations for how an object will change shape in three orthogonal directions.

Well, if an object changes shape in all three directions, that means it will change its volume. A simple measure for this volume change can be found by adding up the three normal components of strain: Now that we have an equation for volume change, or dilation, in terms of normal strains, we can rewrite it in terms of normal stresses. A very common type of stress that causes dilation is known as hydrostatic stress. Since it is acting equally, that means: