Human beings have known the tensions inside materials since ancient times. Until the 19th century, this knowledge was largely intuitive and empirical, although this did not prevent the development of relatively advanced technologies such as the compound bow and glass blowing.[2]​.

Over several millennia, architects and builders in particular learned to join wooden beams and carefully shaped blocks of stone to support, transmit and distribute stress most effectively, with ingenious devices such as capitals, arches, domes, trusses and the buttresses of Gothic cathedrals.

Ancient and medieval architects developed some geometric methods and simple formulas to calculate the proper size of pillars and beams, but the scientific understanding of stresses was only possible after the invention of the necessary tools in the centuries: the rigorous experimental method of Galileo Galilei, the coordinates and analytical geometry of René Descartes, and the laws of motion and equilibrium and the calculus of infinitesimals of Newton.[3]​ With these tools, Augustin-Louis Cauchy was able to give the first mathematical model. rigorous and general of a deformed elastic body introducing the notions of stress and strain.[4] Cauchy observed that the force through an imaginary surface was a linear function of its normal vector; and, furthermore, that it had to be a symmetric function (with zero total moment).

Understanding stress in liquids began with Newton, who provided a differential formula for friction forces (shear stress) in parallel laminar flow.

Stress is defined as the force across a small boundary per unit area of ​​that boundary, for all orientations of the boundary. Derived from a fundamental physical quantity (force) and a purely geometric quantity (area), stress is also a fundamental quantity, such as speed, torque, or energy, that can be quantified and analyzed without explicit consideration of the nature of the material or its physical causes.

Following the basic premises of continuum mechanics, tension is a macroscopic concept. That is, the particles considered in their definition and analysis must be small enough to be treated as homogeneous in composition and state, but large enough to ignore the effects of quantum and the detailed motions of the molecules. Thus, the force between two particles is actually the average of a large number of atomic forces between their molecules; and physical quantities such as mass, velocity, and forces acting through the mass of three-dimensional bodies, such as gravity, are assumed to be distributed smoothly over them.[6] Depending on the context, the particles may also be assumed to be large enough to allow averaging of other microscopic features, such as the grains of a bar of metal or the fibers of a piece of wood.

Quantitatively, the tension is expressed by the Cauchy traction vector T defined as the traction force F between adjacent parts of the material through an imaginary separation surface S, divided by the area of ​​S.[7]​ In a fluid at rest the force is perpendicular to the surface, and is the known pressure. In a solid, or in a viscous liquid flow, the force F may not be perpendicular to S; Therefore, the stress across a surface should be considered a vector quantity, not a scalar one. Furthermore, the direction and magnitude generally depend on the orientation of S. Thus, the stress state of the material must be described by a tensor, called stress tensor; which is a linear function relating the normal vector n of a surface S to the traction vector T through S. With respect to any chosen coordinates, the Cauchy stress tensor can be represented as a symmetric matrix of 3x3 real numbers. Even within a homogeneous body, the stress tensor can vary from place to place, and can change with time; therefore, the stress within a material is, in general, a varying tensor field over time.

If a body is considered subject to a system of forces and moments of force, the action of mechanical stresses can be observed if a cut is imagined through an imaginary plane π that divides the body into two parts. For each part to be in mechanical balance, the interaction exerted by the other part of the body would have to be restored on the cutting surface of each part. Thus, on each element of the surface (dS), an elemental force (dF) must act, from which a tension vector (t) is defined as the result of dividing said elemental force by the surface of the element.

This tension vector depends on the internal tension state of the body, the coordinates of the chosen point and the unit vector normal to the π plane (n). It can be proven that t and n are related by a linear map T or tensor field called tension tensor:.

Mechanical stress is expressed in units of pressure, that is, force divided by area. In the International System, the unit of mechanical stress is the pascal "Pascal (unit)") (1 Pa = 1 N/m²). However, in engineering it is also common to express other units such as kg/cm² or kg/mm², where "kg" refers to kilopond or kilogram-force, not to the mass unit kilogram.

Let it be a deformed continuous medium, then in each subdomain there is a vector field, called the stress field, such that the volume forces and the stress field satisfy the following equilibrium equations.

This principle was stated by Augustin Louis Cauchy in its most general form, although Leonhard Euler had previously made a less general formulation. From this principle, the theorem due to Cauchy for the tension tensor can be proven, which postulates that Cauchy's principle is equivalent to the existence of a linear application, called a tension tensor, with the following properties:

With the principle, he also stated the two postulates that define the action of vectors on a surface.

If we consider a specific point of a deformable solid subjected to tension and a cut is chosen through an imaginary plane π that divides the solid in two, a tension vector t is defined that depends on the internal tension state of the body, the coordinates of the chosen point and the normal unit vector n to the plane π defined by the tension tensor:.

Usually this vector can be decomposed into two components that physically produce different effects depending on whether the material is more ductile or more brittle. These two components are called intrinsic components of the stress vector with respect to the π plane and are called normal stress or perpendicular to the plane and tangential stress or flush with the plane, these components are given by:.

Similarly, when there are two solids in contact and the tensions between two points of the two solids are examined, the previous decomposition of the contact tension can be done according to the plane tangent to the surfaces of both solids, in that case the normal tension has to do with the pressure perpendicular to the surface and the tangential tension has to do with the friction forces between the two.

A particular case is that of uniaxial tension, which is defined in a situation in which force F is applied uniformly distributed over an area A. In that case the uniaxial mechanical stress is represented by a scalar designated with the Greek letter σ (sigma) and is given by:.

The concept of longitudinal stress is based on two simple observations about the behavior of cables under tension:

These observations show that the fundamental characteristic that affects the deformation and resistant failure of materials is the magnitude σ, called mechanical stress or tension. More precise measurements show that the proportionality between the stress and the elongation is not exact because during the stretching of the cable its cross section experiences a narrowing, so A decreases slightly. However, if the true stress is defined as σ = F/A' where A' now represents the true area under load, then correct proportionality is observed for small values ​​of F.

Poisson's ratio was introduced to account for the relationship between the initial area A and the deformed area A' . The introduction of Poisson's ratio in the calculations correctly estimated the stress by taking into account that the force F was distributed in an area somewhat smaller than the initial section, which makes σ > s.

Human beings have known the tensions inside materials since ancient times. Until the 19th century, this knowledge was largely intuitive and empirical, although this did not prevent the development of relatively advanced technologies such as the compound bow and glass blowing.[2]​.

Over several millennia, architects and builders in particular learned to join wooden beams and carefully shaped blocks of stone to support, transmit and distribute stress most effectively, with ingenious devices such as capitals, arches, domes, trusses and the buttresses of Gothic cathedrals.

Ancient and medieval architects developed some geometric methods and simple formulas to calculate the proper size of pillars and beams, but the scientific understanding of stresses was only possible after the invention of the necessary tools in the centuries: the rigorous experimental method of Galileo Galilei, the coordinates and analytical geometry of René Descartes, and the laws of motion and equilibrium and the calculus of infinitesimals of Newton.[3]​ With these tools, Augustin-Louis Cauchy was able to give the first mathematical model. rigorous and general of a deformed elastic body introducing the notions of stress and strain.[4] Cauchy observed that the force through an imaginary surface was a linear function of its normal vector; and, furthermore, that it had to be a symmetric function (with zero total moment).

Understanding stress in liquids began with Newton, who provided a differential formula for friction forces (shear stress) in parallel laminar flow.

Stress is defined as the force across a small boundary per unit area of ​​that boundary, for all orientations of the boundary. Derived from a fundamental physical quantity (force) and a purely geometric quantity (area), stress is also a fundamental quantity, such as speed, torque, or energy, that can be quantified and analyzed without explicit consideration of the nature of the material or its physical causes.

Following the basic premises of continuum mechanics, tension is a macroscopic concept. That is, the particles considered in their definition and analysis must be small enough to be treated as homogeneous in composition and state, but large enough to ignore the effects of quantum and the detailed motions of the molecules. Thus, the force between two particles is actually the average of a large number of atomic forces between their molecules; and physical quantities such as mass, velocity, and forces acting through the mass of three-dimensional bodies, such as gravity, are assumed to be distributed smoothly over them.[6] Depending on the context, the particles may also be assumed to be large enough to allow averaging of other microscopic features, such as the grains of a bar of metal or the fibers of a piece of wood.

Quantitatively, the tension is expressed by the Cauchy traction vector T defined as the traction force F between adjacent parts of the material through an imaginary separation surface S, divided by the area of ​​S.[7]​ In a fluid at rest the force is perpendicular to the surface, and is the known pressure. In a solid, or in a viscous liquid flow, the force F may not be perpendicular to S; Therefore, the stress across a surface should be considered a vector quantity, not a scalar one. Furthermore, the direction and magnitude generally depend on the orientation of S. Thus, the stress state of the material must be described by a tensor, called stress tensor; which is a linear function relating the normal vector n of a surface S to the traction vector T through S. With respect to any chosen coordinates, the Cauchy stress tensor can be represented as a symmetric matrix of 3x3 real numbers. Even within a homogeneous body, the stress tensor can vary from place to place, and can change with time; therefore, the stress within a material is, in general, a varying tensor field over time.

If a body is considered subject to a system of forces and moments of force, the action of mechanical stresses can be observed if a cut is imagined through an imaginary plane π that divides the body into two parts. For each part to be in mechanical balance, the interaction exerted by the other part of the body would have to be restored on the cutting surface of each part. Thus, on each element of the surface (dS), an elemental force (dF) must act, from which a tension vector (t) is defined as the result of dividing said elemental force by the surface of the element.

This tension vector depends on the internal tension state of the body, the coordinates of the chosen point and the unit vector normal to the π plane (n). It can be proven that t and n are related by a linear map T or tensor field called tension tensor:.

Mechanical stress is expressed in units of pressure, that is, force divided by area. In the International System, the unit of mechanical stress is the pascal "Pascal (unit)") (1 Pa = 1 N/m²). However, in engineering it is also common to express other units such as kg/cm² or kg/mm², where "kg" refers to kilopond or kilogram-force, not to the mass unit kilogram.

Let it be a deformed continuous medium, then in each subdomain there is a vector field, called the stress field, such that the volume forces and the stress field satisfy the following equilibrium equations.

This principle was stated by Augustin Louis Cauchy in its most general form, although Leonhard Euler had previously made a less general formulation. From this principle, the theorem due to Cauchy for the tension tensor can be proven, which postulates that Cauchy's principle is equivalent to the existence of a linear application, called a tension tensor, with the following properties:

With the principle, he also stated the two postulates that define the action of vectors on a surface.

If we consider a specific point of a deformable solid subjected to tension and a cut is chosen through an imaginary plane π that divides the solid in two, a tension vector t is defined that depends on the internal tension state of the body, the coordinates of the chosen point and the normal unit vector n to the plane π defined by the tension tensor:.

Usually this vector can be decomposed into two components that physically produce different effects depending on whether the material is more ductile or more brittle. These two components are called intrinsic components of the stress vector with respect to the π plane and are called normal stress or perpendicular to the plane and tangential stress or flush with the plane, these components are given by:.

Similarly, when there are two solids in contact and the tensions between two points of the two solids are examined, the previous decomposition of the contact tension can be done according to the plane tangent to the surfaces of both solids, in that case the normal tension has to do with the pressure perpendicular to the surface and the tangential tension has to do with the friction forces between the two.

A particular case is that of uniaxial tension, which is defined in a situation in which force F is applied uniformly distributed over an area A. In that case the uniaxial mechanical stress is represented by a scalar designated with the Greek letter σ (sigma) and is given by:.

The concept of longitudinal stress is based on two simple observations about the behavior of cables under tension:

These observations show that the fundamental characteristic that affects the deformation and resistant failure of materials is the magnitude σ, called mechanical stress or tension. More precise measurements show that the proportionality between the stress and the elongation is not exact because during the stretching of the cable its cross section experiences a narrowing, so A decreases slightly. However, if the true stress is defined as σ = F/A' where A' now represents the true area under load, then correct proportionality is observed for small values ​​of F.

Poisson's ratio was introduced to account for the relationship between the initial area A and the deformed area A' . The introduction of Poisson's ratio in the calculations correctly estimated the stress by taking into account that the force F was distributed in an area somewhat smaller than the initial section, which makes σ > s.