La ley de Hooke usada para materiales elásticos reversibles y lineales es una ecuación constitutiva en que las tensiones se describen como el producto de componentes tensoriales del tensor de constantes elásticas por las componentes del tensor deformación. En dicha ley las tensiones son combinaciones lineales de las deformaciones, y no existe disipación de energía y por tanto irreversibilidad. Por esas razones no pueden describir la plasticidad. De hecho, la descripción matemática de la plasticidad debe incluir tanto la irreversibilidad o disipación de energía como la no linealidad de las expresiones que relacionan tensiones y deformaciones. Existe un buen número de modelos matemáticos de plasticidad con estas características. En todos los modelos de plasticidad la relación entre tensiones y deformaciones es del tipo:.

Donde en la ecuación anterior y en las siguientes se usa el convenio de sumación de Einstein sobre índices repetidos, y donde además:.

La diferencia básica entre los diversos modelos de plasticidad es la superficie de fluencia y por tanto la manera en que se computan las deformaciones plásticas, además de las posibles variaciones en la componente viscoplástica. De hecho, un modelo de plasticidad además de la ecuación () necesita especificar dos relaciones más:.

Donde.

Si se derivan las ecuaciones () y () respecto al tiempo y se añade la ecuación () se tiene un sistema de tres ecuaciones diferenciales ordinarias respecto al tiempo, que junto con las correspondientes ecuaciones de contorno describiendo las cargas, los valores iniciales y otras restricciones forman un problema elastoplástico cuya solución es única en el caso lineal. En el caso no lineal no considerado aquí no se ha demostrado la unicidad "Unicidad").

J2 plasticity model

This is an isotropic elasto-plastic model without viscosity or hardening and is one of the simplest elasto-plastic models. The stress at each instant is given by a purely elastic stress independent of the deformation rate:

Where the yield surface and the plastic zone are given by the second invariant or quadratic invariant #Explicit_form_of_the_basic_invariants "Algebraic invariant (linear algebra)") of the deviatoric tensor.

In the previous equations and in what follows, the Einstein summation convention is used with respect to repeated indices. The additional basic equations of the time evolution of yield stress and elastic deformation are:.

The function is the ramp function. This last expression indicates that the plastic deformation tensor is proportional to the deviatoric stress tensor.

Hydrodynamic elastoplastic model

This model attributes elastic behavior to the material below the yield point and attributes increases in plastic deformation above it. Warp rate plays no role within it. The relationships between stress and deformation are of the form:.

Where the are used to calculate Jaumann's objective rate of change and where the yield surface and the area where plastic deformations occur is the same as in the J plasticity model, which will mean that there will be an increase in plastic deformation as long as:

Additional equations for the temporal evolution of yield stress and plastic deformation are:.

Where the initial instant has been taken before plasticization appeared.

Krieg-Key visco-elastoplastic model

This model is an elasto-plastic model with kinematic hardening, once the material's yield point has passed. The relationship between stresses and deformations is given by an elastic contribution plus a plastic contribution.

In the isotropic case the yield surface is taken as the geometric locus:[1]​.

Where:.

The isotropic version of this model contains 7 material constants: two elastic constants, two plasticity parameters, two viscoelastic parameters and the hardening parameter.

J2 plasticity model with hardening

This is an isotropic elasto-plastic model without viscosity that generalizes the J model without hardening. In this model, the evolution equations of the plastic deformation tensor are replaced by more complicated ones and the following internal variables are added. Plastic deformation evolves according to the equation:.

Where:.

While the evolution equations of the internal variables are given by:.

En los metales, la plasticidad frecuentemente aparece relacionada con el desplazamiento de dislocaciones en el interior del material. Los metales usualmente están formados por cristales con planos razonablemente bien alineados dentro de cada cristal, aunque siempre existen algunas dislocaciones y planos atómicos incompletos. A partir de un cierto valor de la tensión esas dislocaciones sufren desplazamientos, que constituyen transformaciones irreversibles que absorben energía y cuyas deformaciones asociadas no se recuperan cuando desaparece el esfuerzo.

Plastic calculation in metal structure

Plastic calculation refers to the calculation of stresses, tensions and deformations in structural engineering of elements that have plastic behavior. Unlike mechanisms that must operate reversibly, static structures can be designed to work above the elastic domain, thereby achieving a more complete use of their resistant capacity. This is because, once the elastic domain of reversibility has been exceeded, some construction materials continue to have the capacity to resist greater stresses, due to kinematic hardening, even at the cost of undergoing irreversible internal transformations.

In metal structures, plastic calculation basically consists of identifying the points of appearance of plastic hinges or regions of plasticization that, once completely plasticized, become joints, called "plasticization hinges." To find for what value of the load a plastic hinge is formed, the structure is represented by a linear elastic structure where all the plasticizing hinges already formed have been replaced by joints. The appearance of plasticizing hinges reduces the degree of hyperstaticity by expanding the number of degrees of freedom "Degrees of freedom (engineering)"). When a sufficient number of plastic hinges appear, the structure becomes a mechanism, and its configuration gives the structure's collapse mechanism. Plastic calculation is especially useful in hyperstatic structures with redundant bond conditions. The plastic calculation includes the identification of the collapse modes due to the formation of plastic hinges, and the load necessary for the plasticization of all hinges. The ultimate plastic load is the value from which the structure is converted into a mechanism by plasticization of the last hinge.

In a structure with a single quasi-statically applied load, the first yielding hinge will have finished forming when the maximum moment equals the plastic moment. To calculate it, an arbitrary test load applied at the same point as the original load is considered and the bending moments at all points are calculated based on said load, then the formation load of the first hinge P is calculated simply as:

Where:.

Once the first hinge has been identified, we continue calculating a structure like the original but in which the point of formation of the plasticizing hinge has been replaced by a joint, a new test load is considered, it is seen at which other point the maximum moment now occurs and it is determined what load is needed for the new point, taking into account the total bending moment that it already had in the previous phase, so that the moment equals the plastic moment:

The previous procedure is generalizable to the case of several charges P, ...,P that increase quasi-statically in a uniparametric manner P = P(λ). In the more general case in which each load varies independently, the final state will depend on which loads increase more rapidly, so the ultimate resistance in the plastic regime can only be determined if the variation of all loads in time is specified: P = P(t).

Plastic calculation in reinforced concrete

Also in the calculation of reinforced concrete structures, it is accepted that steel bars subjected to tension acquire plastic deformations, since steel has a plastic behavior with hardening, and when it exceeds its elastic limit it hardens and can withstand higher stresses than before acquiring plastic deformations. This hardening or increase in the resistant capacity of steel in tension allows for savings and the construction of structures with a smaller amount of steel.

In the case of some humid soils, plasticity is the property that allows them to be molded by applying external forces, and to maintain the acquired shapes, even when the humidity and external forces disappear. According to Atterberg[2]​ two limits of plasticity can be defined,[3]​ the maximum and the minimum. With a humidity percentage above the maximum plasticity limit, the earthen mass becomes fluid and loses its ability to maintain its shape, and if the soil has a humidity percentage below the minimum plasticity limit, the earthen mass becomes brittle, and cannot be molded.[4] It is evident that not all soils have the same plasticity; Sands and silts have low or very low plasticity, while soils with high clay content have higher plasticity. In general it can be stated that soils with a clay content of less than 15% are not plastic.[5]​.

For each of the plasticity limits, the maximum and the minimum, a percentage of humidity corresponds, depending on the terrain, the difference between the two percentages of humidity flame limits number or plasticity index. Both the plasticity limits and the corresponding plasticity number or plasticity index vary, obviously from terrain to terrain, depending mainly on the texture and more precisely on the content of inorganic colloids.

Another important factor that influences plasticity is the type of cations available.[6]​ Generally the K+ ion decreases both plasticity limits and the plasticity index, while the Na+ ion decreases the plasticity limits, but increases the plasticity index; Mg++ and Ca++ cations increase plasticity, but soils saturated with them require a high amount of water to reach the state of plasticity, unlike those saturated with K+ cations.

The hydration and dispersion effect of Na+ determine a plasticity of soils saturated with this cation greater than that achieved by soils saturated with divalent cations.

Generally, the influence of various cations on plasticity varies with the quality and nature of the clay.

The organic matter contained in the soil also has an important effect on the plasticity of soils.[7]​ In general, the upper strata of the soil have a greater plasticity than the deeper strata. This can be attributed to the greater presence of organic material in the upper layers of the soil.

La ley de Hooke usada para materiales elásticos reversibles y lineales es una ecuación constitutiva en que las tensiones se describen como el producto de componentes tensoriales del tensor de constantes elásticas por las componentes del tensor deformación. En dicha ley las tensiones son combinaciones lineales de las deformaciones, y no existe disipación de energía y por tanto irreversibilidad. Por esas razones no pueden describir la plasticidad. De hecho, la descripción matemática de la plasticidad debe incluir tanto la irreversibilidad o disipación de energía como la no linealidad de las expresiones que relacionan tensiones y deformaciones. Existe un buen número de modelos matemáticos de plasticidad con estas características. En todos los modelos de plasticidad la relación entre tensiones y deformaciones es del tipo:.

Donde en la ecuación anterior y en las siguientes se usa el convenio de sumación de Einstein sobre índices repetidos, y donde además:.

La diferencia básica entre los diversos modelos de plasticidad es la superficie de fluencia y por tanto la manera en que se computan las deformaciones plásticas, además de las posibles variaciones en la componente viscoplástica. De hecho, un modelo de plasticidad además de la ecuación () necesita especificar dos relaciones más:.

Donde.

Si se derivan las ecuaciones () y () respecto al tiempo y se añade la ecuación () se tiene un sistema de tres ecuaciones diferenciales ordinarias respecto al tiempo, que junto con las correspondientes ecuaciones de contorno describiendo las cargas, los valores iniciales y otras restricciones forman un problema elastoplástico cuya solución es única en el caso lineal. En el caso no lineal no considerado aquí no se ha demostrado la unicidad "Unicidad").

J2 plasticity model

This is an isotropic elasto-plastic model without viscosity or hardening and is one of the simplest elasto-plastic models. The stress at each instant is given by a purely elastic stress independent of the deformation rate:

Where the yield surface and the plastic zone are given by the second invariant or quadratic invariant #Explicit_form_of_the_basic_invariants "Algebraic invariant (linear algebra)") of the deviatoric tensor.

In the previous equations and in what follows, the Einstein summation convention is used with respect to repeated indices. The additional basic equations of the time evolution of yield stress and elastic deformation are:.

The function is the ramp function. This last expression indicates that the plastic deformation tensor is proportional to the deviatoric stress tensor.

Hydrodynamic elastoplastic model

This model attributes elastic behavior to the material below the yield point and attributes increases in plastic deformation above it. Warp rate plays no role within it. The relationships between stress and deformation are of the form:.

Where the are used to calculate Jaumann's objective rate of change and where the yield surface and the area where plastic deformations occur is the same as in the J plasticity model, which will mean that there will be an increase in plastic deformation as long as:

Additional equations for the temporal evolution of yield stress and plastic deformation are:.

Where the initial instant has been taken before plasticization appeared.

Krieg-Key visco-elastoplastic model

This model is an elasto-plastic model with kinematic hardening, once the material's yield point has passed. The relationship between stresses and deformations is given by an elastic contribution plus a plastic contribution.

In the isotropic case the yield surface is taken as the geometric locus:[1]​.

Where:.

The isotropic version of this model contains 7 material constants: two elastic constants, two plasticity parameters, two viscoelastic parameters and the hardening parameter.

J2 plasticity model with hardening

This is an isotropic elasto-plastic model without viscosity that generalizes the J model without hardening. In this model, the evolution equations of the plastic deformation tensor are replaced by more complicated ones and the following internal variables are added. Plastic deformation evolves according to the equation:.

Where:.

While the evolution equations of the internal variables are given by:.

En los metales, la plasticidad frecuentemente aparece relacionada con el desplazamiento de dislocaciones en el interior del material. Los metales usualmente están formados por cristales con planos razonablemente bien alineados dentro de cada cristal, aunque siempre existen algunas dislocaciones y planos atómicos incompletos. A partir de un cierto valor de la tensión esas dislocaciones sufren desplazamientos, que constituyen transformaciones irreversibles que absorben energía y cuyas deformaciones asociadas no se recuperan cuando desaparece el esfuerzo.

Plastic calculation in metal structure

Plastic calculation refers to the calculation of stresses, tensions and deformations in structural engineering of elements that have plastic behavior. Unlike mechanisms that must operate reversibly, static structures can be designed to work above the elastic domain, thereby achieving a more complete use of their resistant capacity. This is because, once the elastic domain of reversibility has been exceeded, some construction materials continue to have the capacity to resist greater stresses, due to kinematic hardening, even at the cost of undergoing irreversible internal transformations.

In metal structures, plastic calculation basically consists of identifying the points of appearance of plastic hinges or regions of plasticization that, once completely plasticized, become joints, called "plasticization hinges." To find for what value of the load a plastic hinge is formed, the structure is represented by a linear elastic structure where all the plasticizing hinges already formed have been replaced by joints. The appearance of plasticizing hinges reduces the degree of hyperstaticity by expanding the number of degrees of freedom "Degrees of freedom (engineering)"). When a sufficient number of plastic hinges appear, the structure becomes a mechanism, and its configuration gives the structure's collapse mechanism. Plastic calculation is especially useful in hyperstatic structures with redundant bond conditions. The plastic calculation includes the identification of the collapse modes due to the formation of plastic hinges, and the load necessary for the plasticization of all hinges. The ultimate plastic load is the value from which the structure is converted into a mechanism by plasticization of the last hinge.

In a structure with a single quasi-statically applied load, the first yielding hinge will have finished forming when the maximum moment equals the plastic moment. To calculate it, an arbitrary test load applied at the same point as the original load is considered and the bending moments at all points are calculated based on said load, then the formation load of the first hinge P is calculated simply as:

Where:.

Once the first hinge has been identified, we continue calculating a structure like the original but in which the point of formation of the plasticizing hinge has been replaced by a joint, a new test load is considered, it is seen at which other point the maximum moment now occurs and it is determined what load is needed for the new point, taking into account the total bending moment that it already had in the previous phase, so that the moment equals the plastic moment:

The previous procedure is generalizable to the case of several charges P, ...,P that increase quasi-statically in a uniparametric manner P = P(λ). In the more general case in which each load varies independently, the final state will depend on which loads increase more rapidly, so the ultimate resistance in the plastic regime can only be determined if the variation of all loads in time is specified: P = P(t).

Plastic calculation in reinforced concrete

Also in the calculation of reinforced concrete structures, it is accepted that steel bars subjected to tension acquire plastic deformations, since steel has a plastic behavior with hardening, and when it exceeds its elastic limit it hardens and can withstand higher stresses than before acquiring plastic deformations. This hardening or increase in the resistant capacity of steel in tension allows for savings and the construction of structures with a smaller amount of steel.

In the case of some humid soils, plasticity is the property that allows them to be molded by applying external forces, and to maintain the acquired shapes, even when the humidity and external forces disappear. According to Atterberg[2]​ two limits of plasticity can be defined,[3]​ the maximum and the minimum. With a humidity percentage above the maximum plasticity limit, the earthen mass becomes fluid and loses its ability to maintain its shape, and if the soil has a humidity percentage below the minimum plasticity limit, the earthen mass becomes brittle, and cannot be molded.[4] It is evident that not all soils have the same plasticity; Sands and silts have low or very low plasticity, while soils with high clay content have higher plasticity. In general it can be stated that soils with a clay content of less than 15% are not plastic.[5]​.

For each of the plasticity limits, the maximum and the minimum, a percentage of humidity corresponds, depending on the terrain, the difference between the two percentages of humidity flame limits number or plasticity index. Both the plasticity limits and the corresponding plasticity number or plasticity index vary, obviously from terrain to terrain, depending mainly on the texture and more precisely on the content of inorganic colloids.

Another important factor that influences plasticity is the type of cations available.[6]​ Generally the K+ ion decreases both plasticity limits and the plasticity index, while the Na+ ion decreases the plasticity limits, but increases the plasticity index; Mg++ and Ca++ cations increase plasticity, but soils saturated with them require a high amount of water to reach the state of plasticity, unlike those saturated with K+ cations.

The hydration and dispersion effect of Na+ determine a plasticity of soils saturated with this cation greater than that achieved by soils saturated with divalent cations.

Generally, the influence of various cations on plasticity varies with the quality and nature of the clay.

The organic matter contained in the soil also has an important effect on the plasticity of soils.[7]​ In general, the upper strata of the soil have a greater plasticity than the deeper strata. This can be attributed to the greater presence of organic material in the upper layers of the soil.