Los materiales elásticos son el tipo más simple de sólido deformable donde las tensiones en un punto depende solo de las deformaciones coocurrentes en el mismo punto. Esa restricción hace que los materiales elásticos sean sistemas termodinámicamente reversibles donde no hay disipación. Dentro de los materiales elásticos además es frecuente la diferencia entre materiales elásticos lineales, donde la ecuación constitutiva () es una función lineal en su primer argumento y además las deformaciones sean pequeñas(. Matemáticamente los materiales elásticos lineales son fácilmente tratables y gran parte de las aplicaciones prácticas y el análsiis estructural se basan en este tipo de materiales. Sin embargo, la linealidad entre deformaciones y desplazamientos solo se da aproximadamente para pequeñas deformaciones y en general los problemas con grandes deformaciones, requieren su tratamiento mediante elasticidad no lineal. Este tratamiento es sustancialmente más complejo desde el punto de vista matemático.

Linear elasticity theory

For materials that have linear, or approximately linear, elastic behavior for small or moderate deformations. The calculation of stresses and strains can be done using the linear theory of elasticity. This theory solves the problems of solid mechanics by posing a system of partial differential equations. From the physical point of view, the various subsystems of equations that this theory includes are:

• - Internal balance equations: which relate the volumetric forces (b) with those derived from the tensions (σ) inside the solid:.

• - External equilibrium equations: Which relate the surface forces or contact forces (f) applied on the surface of the solid with the value of the stresses on the boundary of the solid:.

• - Constitutive equations or Lamé-Hooke equations. They are algebraic and linear equations that relate the value of the components of the stress tensor to the value of the strain tensor:

• - Relationship between displacements and deformations. Which relate the components of the strain tensor (ε) to the components of the displacement vector u = (u, u, u):.

• - Boundary conditions, which set the value of the displacement for some points of the external contour, normally the points that are joining points of the deformable solid to some other structure or resistant element on which it rests or anchors.

Material resistance

Certain simple problems in the mechanics of deformable solids with simple geometries can be treated by classical material strength. Especially for the calculation of beams and when the stress concentration is not particularly ordinary differential equations can be proposed in one variable for the calculation of stresses and deformations, which makes it very easy to find analytical solutions that approximate the stresses of the real three-dimensional problem.

Furthermore, many problems that are indeterminate according to the model of rigid solid mechanics (hyperstatic problems) are solvable in the model of deformable solids thanks to the use of additional equations (constitutive equation and compatibility equations). Typically these additional equations are written in terms of stresses, strains or displacements (See also: Castigliano's theorems, Navier-Bresse equations, Mohr's theorems).

One of the main applications of the mechanics of deformable solids is the calculation of structures in engineering and architecture. As a field of study, the mechanics of deformable solids is part of the mechanics of continuous media. It should be noted that the simplified methods used in resistance of materials can also be extended to materials with a certain type of plasticity or viscoelastic materials, so the resistance of materials is not strictly limited to elastic materials, although in practice the resistance of non-elastic materials is little used in practice, the use of codes based on finite elements or other computational methods and the non-simplified treatment of geometry being more common.

For a viscoelastic solid, the stress tensor can be decomposed into a linear combination of equilibrium stresses (to which the stresses would converge if the deformation remains constant) and transient stresses associated with the viscoelastic behavior itself. Using the form () for the Helmholtz free energy, the stress tensor will have the form:.

where the last term contains the non-equilibrium stresses associated with creep and relaxation behavior.

• - dislocation (metalworking) "Dislocation (crystalline defect)") and plasticity "Plasticity (solid mechanics)").

• - elasticity "Elasticity (mechanics of solids)") and resistance of materials.

• - simple material.

• - mechanics of continuous media and mechanics of the rigid solid.

• - elastic problem and elastoplastic problem.

• - stress and deformation.

• - damage mechanics.

Los materiales elásticos son el tipo más simple de sólido deformable donde las tensiones en un punto depende solo de las deformaciones coocurrentes en el mismo punto. Esa restricción hace que los materiales elásticos sean sistemas termodinámicamente reversibles donde no hay disipación. Dentro de los materiales elásticos además es frecuente la diferencia entre materiales elásticos lineales, donde la ecuación constitutiva () es una función lineal en su primer argumento y además las deformaciones sean pequeñas(. Matemáticamente los materiales elásticos lineales son fácilmente tratables y gran parte de las aplicaciones prácticas y el análsiis estructural se basan en este tipo de materiales. Sin embargo, la linealidad entre deformaciones y desplazamientos solo se da aproximadamente para pequeñas deformaciones y en general los problemas con grandes deformaciones, requieren su tratamiento mediante elasticidad no lineal. Este tratamiento es sustancialmente más complejo desde el punto de vista matemático.

Linear elasticity theory

For materials that have linear, or approximately linear, elastic behavior for small or moderate deformations. The calculation of stresses and strains can be done using the linear theory of elasticity. This theory solves the problems of solid mechanics by posing a system of partial differential equations. From the physical point of view, the various subsystems of equations that this theory includes are:

• - Internal balance equations: which relate the volumetric forces (b) with those derived from the tensions (σ) inside the solid:.

• - External equilibrium equations: Which relate the surface forces or contact forces (f) applied on the surface of the solid with the value of the stresses on the boundary of the solid:.

• - Constitutive equations or Lamé-Hooke equations. They are algebraic and linear equations that relate the value of the components of the stress tensor to the value of the strain tensor:

• - Relationship between displacements and deformations. Which relate the components of the strain tensor (ε) to the components of the displacement vector u = (u, u, u):.

• - Boundary conditions, which set the value of the displacement for some points of the external contour, normally the points that are joining points of the deformable solid to some other structure or resistant element on which it rests or anchors.

Material resistance

Certain simple problems in the mechanics of deformable solids with simple geometries can be treated by classical material strength. Especially for the calculation of beams and when the stress concentration is not particularly ordinary differential equations can be proposed in one variable for the calculation of stresses and deformations, which makes it very easy to find analytical solutions that approximate the stresses of the real three-dimensional problem.

Furthermore, many problems that are indeterminate according to the model of rigid solid mechanics (hyperstatic problems) are solvable in the model of deformable solids thanks to the use of additional equations (constitutive equation and compatibility equations). Typically these additional equations are written in terms of stresses, strains or displacements (See also: Castigliano's theorems, Navier-Bresse equations, Mohr's theorems).

One of the main applications of the mechanics of deformable solids is the calculation of structures in engineering and architecture. As a field of study, the mechanics of deformable solids is part of the mechanics of continuous media. It should be noted that the simplified methods used in resistance of materials can also be extended to materials with a certain type of plasticity or viscoelastic materials, so the resistance of materials is not strictly limited to elastic materials, although in practice the resistance of non-elastic materials is little used in practice, the use of codes based on finite elements or other computational methods and the non-simplified treatment of geometry being more common.

For a viscoelastic solid, the stress tensor can be decomposed into a linear combination of equilibrium stresses (to which the stresses would converge if the deformation remains constant) and transient stresses associated with the viscoelastic behavior itself. Using the form () for the Helmholtz free energy, the stress tensor will have the form:.

where the last term contains the non-equilibrium stresses associated with creep and relaxation behavior.

• - dislocation (metalworking) "Dislocation (crystalline defect)") and plasticity "Plasticity (solid mechanics)").

• - elasticity "Elasticity (mechanics of solids)") and resistance of materials.

• - simple material.

• - mechanics of continuous media and mechanics of the rigid solid.

• - elastic problem and elastoplastic problem.

• - stress and deformation.

• - damage mechanics.