Elastic modulus - what is it? Determination of the modulus of elasticity for materials

The elastic modulus is a physical quantity that characterizes the elastic behavior of a material when an external force is applied to it in a specific direction. By the elastic behavior of a material is meant its deformation in the elastic region.

History of the study of the elasticity of materials

The physical theory of elastic bodies and their behavior under the action of external forces was examined and studied in detail by a nineteenth-century English scientist Thomas Jung. However, the concept of elasticity itself was developed back in 1727 by the Swiss mathematician, physicist and philosopher Leonard Euler, and the first experiments related to the elastic modulus were carried out in 1782, that is, 25 years before the work of Thomas Jung, the Venetian mathematician and philosopher Jacopo Ricatti.

The merit of Thomas Jung lies in the fact that he gave the theory of elasticity a harmonious modern look, which was subsequently framed in the form of a simple, and then generalized Hooke's law.

The physical nature of elasticity

Any body consists of atoms, between which the forces of attraction and repulsion act. The balance of these forces determines the state and parameters of the substance under given conditions. Atoms of a solid body upon application of insignificant external tensile or compressive forces begin to shift, creating a force opposite in direction and equal in magnitude, which tends to return the atoms to their initial state.

In the process of such a displacement of atoms, the energy of the entire system increases. Experiments show that at small strains, the energy is proportional to the square of the magnitude of these strains. This means that the force, being a derivative with respect to energy, turns out to be proportional to the first degree of the strain, that is, it depends linearly on it. Answering the question of what is the elastic modulus, we can say that this is the coefficient of proportionality between the force acting on the atom and the deformation that this force causes. The dimension of Young's modulus coincides with the dimension of pressure (Pascal).

Elastic limit

According to the definition, the elastic modulus shows what stress must be applied to a solid body so that its deformation is 100%. However, all solids have an elastic limit of 1% strain. This means that if you apply the appropriate force and deform the body by an amount less than 1%, then after the termination of this force, the body accurately restores its original shape and size. If too much force is applied, at which the strain exceeds 1%, after the termination of the external force, the body will not restore its original size. In the latter case, they talk about the existence of residual deformation, which is evidence of exceeding the elastic limit of this material.

Young's modulus in action

To determine the modulus of elasticity, as well as to understand how to use it, you can give a simple example with a spring. To do this, you need to take a metal spring and measure the area of ​​the circle that its coils form. This is done by the simple formula S = πr², where n is the number pi equal to 3.14, and r is the radius of the coil of the spring.

Next, measure the length of the spring l ₀ without load. If you hang any load of mass m ₁ on the spring, then it will increase its length to a certain value l ₁ . The elastic modulus E can be calculated on the basis of knowledge of Hooke's law by the formula: E = m ₁ gl ₀ / (S (l ₁ -l ₀ )), where g is the acceleration of gravity. In this case, we note that the magnitude of the deformation of the spring in the elastic region can far exceed 1%.

Knowing the Young's modulus allows predicting the magnitude of the deformation under the action of a specific stress. In this case, if we hang another mass m ₂ on the spring, we obtain the following value of relative deformation: d = m ₂ g / (SE), where d is the relative deformation in the elastic region.

Isotropy and anisotropy

The elastic modulus is a characteristic of a material that describes the strength of the bond between its atoms and molecules, however, a particular material may have several different Young moduli.

The fact is that the properties of each solid depend on its internal structure. If the properties are the same in all spatial directions, then we are talking about isotropic material. Such substances have a homogeneous structure, therefore the action of an external force in different directions on them causes the same reaction from the material. All amorphous materials are isotropic, such as rubber or glass.

Anisotropy is a phenomenon that is characterized by the direction of the physical properties of a solid or liquid. All metals and alloys based on them have one or another crystal lattice, that is, an ordered rather than chaotic arrangement of ionic cores. For such materials, the elastic modulus varies depending on the axis of action of the external stress. For example, metals with cubic symmetry, which include aluminum, copper, silver, refractory metals and others, have three different Young moduli.

Shear modulus

The description of the elastic properties of even an isotropic material is not without knowledge of one Young's modulus. Since, in addition to tension and compression, the material can be affected by shear stresses or torsion stresses. In this case, he will react to external force differently. To describe the elastic shear strain, an analogue of the Young's modulus, shear modulus, or elastic modulus of the second kind is introduced.

All materials are weaker in resisting shear stresses than in tension or compression; therefore, the value of the shear modulus for them is 2–3 times lower than the values ​​of Young's modulus. So, for titanium, whose Young's modulus is 107 GPa, the shear modulus is only 40 GPa, for steel, these figures are 210 GPa and 80 GPa, respectively.

Tree modulus

Wood refers to anisotropic materials because wood fibers are oriented along a specific direction. It is along the fibers that the elastic modulus of the wood is measured, since across the fibers it is 1-2 orders of magnitude smaller. Knowledge of the Young's modulus for wood plays an important role and is taken into account when designing structures from wooden panels.

The values ​​of the elastic modulus of wood for some types of trees are shown in the table below.

It should be noted that the values ​​given may differ by a value of the order of 1 GPa for a particular tree, since its Young's modulus is affected by the density of the wood and the growing conditions.

Shear modules for various tree species are within 1-2 GPa, for example, for pine it is 1.21 GPa, and for oak 1.38 GPa, that is, the wood practically does not resist shear stresses. This fact should be taken into account in the manufacture of wooden supporting structures that are designed so that they work only in tension or compression.

The characteristics of the elasticity of metals

If we compare wood with the Young's modulus, then the average values ​​of this quantity for metals and alloys are an order of magnitude larger, as shown in the following table.

The elastic properties of metals that have cubic syngony are described by three elastic constants. Such metals include copper, nickel, aluminum, and iron. If the metal has a hexagonal syngony, then six constants are already needed to describe its elastic characteristics.

For metal systems, Young's modulus is measured within 0.2% strain, since large values ​​can already occur in the inelastic region.