Strain

The stress at a point is defined as the limit of the force applied to a small region on a π plane that contains the point divided by the area of ​​the region, that is, the stress is the force applied per unit area and depends on the point chosen, the tensional state of the solid and the orientation of the plane chosen to calculate the limit. It can be proven that the normal to the chosen plane n and the tension t at a point are related by:.

Where T is the so-called tension tensor, also called stress tensor, which, fixed on an orthogonal vector basis, is represented by a 3x3 symmetric matrix:.

Where the first matrix is ​​the common way of writing the stress tensor in physics and the second way uses common engineering conventions. Given a region in the shape of a cuboid with faces parallel to the coordinate axes located inside a tensioned elastic solid, the components σ, σ and σ account for changes in length in the three directions, but do not distort the angles of the cuboid, while the components σ, σ and σ are related to the angular distortion that would convert the cuboid into a parallelepiped.

Deformation

In linear theory of elasticity, the smallness of the deformations is a necessary condition to ensure that there is a linear relationship between displacements and deformation. Under these conditions the deformation can be adequately represented by the infinitesimal deformation tensor or small deformation tensor (this tensor is only valid for some situations, this being a particular case of the Cauchy-Almansy and Green-Saint-Venant tensors) which is given by:.

The components of the main diagonal contain the elongations (dilations), while the rest of the components of the tensor are the half displacements. The components are linearly related to the displacements through this relationship:

Lamé-Hooke constitutive equations

The Lamé-Hooke equations are the constitutive equations of a linear, homogeneous and isotropic elastic solid, they have the form:.

In the case of a one-dimensional problem, σ = σ, ε = ε, C = E and the above equation reduces to:.

where E is the longitudinal modulus of elasticity or Young's modulus and G the transverse modulus of elasticity. To characterize the behavior of a linear and isotropic elastic solid, in addition to Young's modulus, another elastic constant is required, called Poisson's coefficient () and the temperature coefficient (α). On the other hand, the Lamé equations for a linear and isotropic elastic solid can be deduced from the Rivlin-Ericksen theorem, which can be written in the form:.

The inverse equations of the previous ones are:

The constitutive equations for anisotropic materials are more complex and depend on more elastic constants. It should also be noted that certain materials show only approximately elastic behavior, for example showing variation in deformation with time or creep. These deformations can be permanent or after unloading the body they can disappear (partially or completely) over time (viscoplasticity, viscoelasticity). In addition, some materials may present plasticity "Plasticity (solid mechanics)") or continuous damage. In plasticity, solids acquire permanent deformations above a certain stress threshold, so in many cases the previous equations do not constitute a good approximation to the behavior of these materials. In solids with continuous damage, elastic constants can vary if certain stress thresholds are exceeded.

Equilibrium equations

When the deformations do not vary with time, the stress field given by the tension tensor represents a state of equilibrium with the volume forces b = (b,b,b) at every point of the solid, which implies that the stress field satisfies these equilibrium conditions:

In addition to the last equations, the boundary conditions must be met, on the surface "Surface (mathematical)") of the solid, which relate the normal vector to it n = (n,n,n) (directed outwards) with the forces per unit surface area that act at the same point on the surface f = (f,f,f):.

elastic problem

A linear elastic problem is defined by the geometry of the solid, the properties of said material, acting forces and boundary conditions that impose restrictions on the movement of the body. From these elements it is possible to find a field of internal stresses on the solid (which will allow identifying the points that support the most tension) and a displacement field (which will allow us to find out if the rigidity of the resistant element is appropriate for its use).

To pose the elastic problem, the notions that have been described in the previous sections are necessary, which describe the tensions, deformations and displacements of a body. All these magnitudes are described by 15 mathematical functions:

To check whether these relations, made up of 15 functions, are fulfilled, the next step is to check whether the relations described so far are sufficient to completely describe the state of a body. A necessary condition for this is that the number of equations available coincides with the number of unknowns. The available equations are:

These 15 equations exactly equal the number of unknowns. A common method is to substitute the relationships between displacements and strains into the constitutive equations, which makes the compatibility equations trivially satisfied. In turn, the result of this substitution can be introduced into the Cauchy equilibrium equations, which converts the previous system into a system of three partial differential equations and three displacements as unknowns.

In this way we arrive at a system of 15 equations with 15 unknowns. The simplest formulation to solve the elastic problem is the so-called Navier formulation, this formulation reduces the system to a system of three differential equations for displacements. This is achieved by inserting the material's own equations, the displacement equations and the deformation equations into the equilibrium equations. We can express our system of equations in a system of three partial differential equations. If we reduce it to the components of the displacement vector we arrive at the Navier equations:

That with the Nabla operator and the Laplace operator can be written as:.

Through energy considerations it can be shown that these equations have a unique solution.

Elasticity and mechanical design

In mechanical engineering it is common to pose elastic problems to decide the adequacy of a design. In certain situations of practical interest it is not necessary to solve the complete elastic problem but it is enough to propose a simplified model and apply the methods of material resistance to approximately calculate stresses and displacements. When the geometry involved in the mechanical design is complex, the resistance of materials is usually insufficient and the exact resolution of the elastic problem is unapproachable from a practical point of view. In these cases, numerical methods such as the Finite Element Method are usually used to solve the elastic problem approximately. A good design usually incorporates requirements of:.

Strain

The stress at a point is defined as the limit of the force applied to a small region on a π plane that contains the point divided by the area of ​​the region, that is, the stress is the force applied per unit area and depends on the point chosen, the tensional state of the solid and the orientation of the plane chosen to calculate the limit. It can be proven that the normal to the chosen plane n and the tension t at a point are related by:.

Where T is the so-called tension tensor, also called stress tensor, which, fixed on an orthogonal vector basis, is represented by a 3x3 symmetric matrix:.

Where the first matrix is ​​the common way of writing the stress tensor in physics and the second way uses common engineering conventions. Given a region in the shape of a cuboid with faces parallel to the coordinate axes located inside a tensioned elastic solid, the components σ, σ and σ account for changes in length in the three directions, but do not distort the angles of the cuboid, while the components σ, σ and σ are related to the angular distortion that would convert the cuboid into a parallelepiped.

Deformation

In linear theory of elasticity, the smallness of the deformations is a necessary condition to ensure that there is a linear relationship between displacements and deformation. Under these conditions the deformation can be adequately represented by the infinitesimal deformation tensor or small deformation tensor (this tensor is only valid for some situations, this being a particular case of the Cauchy-Almansy and Green-Saint-Venant tensors) which is given by:.

The components of the main diagonal contain the elongations (dilations), while the rest of the components of the tensor are the half displacements. The components are linearly related to the displacements through this relationship:

Lamé-Hooke constitutive equations

The Lamé-Hooke equations are the constitutive equations of a linear, homogeneous and isotropic elastic solid, they have the form:.

In the case of a one-dimensional problem, σ = σ, ε = ε, C = E and the above equation reduces to:.

where E is the longitudinal modulus of elasticity or Young's modulus and G the transverse modulus of elasticity. To characterize the behavior of a linear and isotropic elastic solid, in addition to Young's modulus, another elastic constant is required, called Poisson's coefficient () and the temperature coefficient (α). On the other hand, the Lamé equations for a linear and isotropic elastic solid can be deduced from the Rivlin-Ericksen theorem, which can be written in the form:.

The inverse equations of the previous ones are:

The constitutive equations for anisotropic materials are more complex and depend on more elastic constants. It should also be noted that certain materials show only approximately elastic behavior, for example showing variation in deformation with time or creep. These deformations can be permanent or after unloading the body they can disappear (partially or completely) over time (viscoplasticity, viscoelasticity). In addition, some materials may present plasticity "Plasticity (solid mechanics)") or continuous damage. In plasticity, solids acquire permanent deformations above a certain stress threshold, so in many cases the previous equations do not constitute a good approximation to the behavior of these materials. In solids with continuous damage, elastic constants can vary if certain stress thresholds are exceeded.

Equilibrium equations

When the deformations do not vary with time, the stress field given by the tension tensor represents a state of equilibrium with the volume forces b = (b,b,b) at every point of the solid, which implies that the stress field satisfies these equilibrium conditions:

In addition to the last equations, the boundary conditions must be met, on the surface "Surface (mathematical)") of the solid, which relate the normal vector to it n = (n,n,n) (directed outwards) with the forces per unit surface area that act at the same point on the surface f = (f,f,f):.

elastic problem

A linear elastic problem is defined by the geometry of the solid, the properties of said material, acting forces and boundary conditions that impose restrictions on the movement of the body. From these elements it is possible to find a field of internal stresses on the solid (which will allow identifying the points that support the most tension) and a displacement field (which will allow us to find out if the rigidity of the resistant element is appropriate for its use).

To pose the elastic problem, the notions that have been described in the previous sections are necessary, which describe the tensions, deformations and displacements of a body. All these magnitudes are described by 15 mathematical functions:

To check whether these relations, made up of 15 functions, are fulfilled, the next step is to check whether the relations described so far are sufficient to completely describe the state of a body. A necessary condition for this is that the number of equations available coincides with the number of unknowns. The available equations are:

These 15 equations exactly equal the number of unknowns. A common method is to substitute the relationships between displacements and strains into the constitutive equations, which makes the compatibility equations trivially satisfied. In turn, the result of this substitution can be introduced into the Cauchy equilibrium equations, which converts the previous system into a system of three partial differential equations and three displacements as unknowns.

In this way we arrive at a system of 15 equations with 15 unknowns. The simplest formulation to solve the elastic problem is the so-called Navier formulation, this formulation reduces the system to a system of three differential equations for displacements. This is achieved by inserting the material's own equations, the displacement equations and the deformation equations into the equilibrium equations. We can express our system of equations in a system of three partial differential equations. If we reduce it to the components of the displacement vector we arrive at the Navier equations:

That with the Nabla operator and the Laplace operator can be written as:.

Through energy considerations it can be shown that these equations have a unique solution.

Elasticity and mechanical design

In mechanical engineering it is common to pose elastic problems to decide the adequacy of a design. In certain situations of practical interest it is not necessary to solve the complete elastic problem but it is enough to propose a simplified model and apply the methods of material resistance to approximately calculate stresses and displacements. When the geometry involved in the mechanical design is complex, the resistance of materials is usually insufficient and the exact resolution of the elastic problem is unapproachable from a practical point of view. In these cases, numerical methods such as the Finite Element Method are usually used to solve the elastic problem approximately. A good design usually incorporates requirements of:.