# Elasticity theory

The theory of elasticity deals with elastic bodies like the disk in the picture and how their properties can be represented with a material model.

Elasticity ( ancient Greek ελαστικός elastikos , "adaptable") is the property of a body under force action of its shape to change ( deformation ), and as spring back in the animation on discontinuation of the force acting in the original shape. All materials have a more or less pronounced elastic area, even ceramic , water or air . Here the two main branches of elasticity theory announce themselves:

Possible causes of elasticity are:

Real materials have an elastic limit within which they deform elastically and beyond which dissipative processes such as viscous or plastic flow, creep or fracture occur. Real liquids, gases and some solids (such as iron and glass ) are elastic to a good approximation in the event of rapid, slight changes in volume (e.g. sound waves ). In the case of solids, the elastic limit can be adhered to in the case of slow and sufficiently small deformations that are present in many applications, especially in the technical field. Directional dependencies of the material such as the orthotropy of wood or material constraints such as incompressibility occur in elasticity, but also in other material behavior.

The laws of mechanics and thermodynamics provide a framework in which real bodies move. The mathematical equations of these laws do not make any statements about the individual properties of the body and are therefore not sufficient to clearly determine the movements of the body. This requires constitutive equations that describe the material-specific response of the body to an external force. B. flows away or just impresses.

The theory of elasticity deals with the mathematical formulation of this relationship in elastic bodies. In addition to the theory of the linear-viscous fluid, it forms the basis of the classical material theory on which other theories for plasticity and viscoplasticity are based.

## Macroscopic behavior

The following properties can be observed macroscopically on an elastic body:

• With a given deformation (fluids: volume change) the reaction forces (the pressure) always have the same value regardless of the previous history.
• The material behavior does not depend on the rate of deformation (fluids: the change in volume); this speed therefore has no influence on the resistance (pressure) that the body opposes to the deformation.

These two features characterize elasticity as a time-independent material property ; together with the following two, they make up Cauchy elasticity .

• If the initial state is unloaded, it is resumed after any deformation when the loads are removed. In the case of elastic liquids and gases, the state is determined by the volume occupied, which is always the same under the same conditions.
• In the uniaxial tensile test , loading and unloading always take place along the same path as in the adjacent picture. With liquids and gases this corresponds to a compression and expansion test.

If the following property is also present, the material is hyperelastic :

With sufficiently small deformations, the force-displacement relationship in solids is linear and the elasticity can be described in terms of modulus. These material properties quantify the relationship between the stresses (force per effective area) and the elongations (deformation path per dimension):

The full description

• the isotropic linear elasticity requires two of the named quantities (a modulus of elasticity and a Poisson's ratio)
• the cubic anisotropy requires three quantities (a modulus of elasticity, a Poisson's ratio and a modulus of shear)
• the transversal isotropy already requires five quantities (two moduli of elasticity, two Poisson's ratio and one modulus of shear)
• the orthotropy needed nine sizes (three elastic moduli, Querdehnzahlen and shear moduli).

A maximum of 21 parameters are required to describe a real linear elastic material, see the section #Material models of hyperelasticity .

## Continuum mechanical theory

### Cauchy elasticity

The four properties mentioned first in the previous chapter determine the Cauchy elasticity . The tensions depend on her, i. H. the resistance to deformation depends exclusively on the current deformation and any residual stresses that have occurred from the beginning, but not on the history or the speed of the deformation.

In addition, with Cauchy elasticity, the deformations (within the elastic limit) are reversible; H. the body can be deformed by a force, but when the force is removed it springs back to its original state.

In the case of general, anisotropic, linear elasticity, the relationship between the six stresses and the six elongations can be represented with a maximum of 36  proportionality constants .

#### Reference system invariance

A moving observer always measures the same material behavior as a stationary one, which is reflected in the principle of material objectivity . At the Cauchy elastic material can already the conditions are found under which material equations respect system invariant or, more precisely, invariant with respect to a Euclidean transformation of the reference system of an observer are. Material equations for elastic fluids are automatically reference system invariant. In the case of solids, this requirement is met by setting up the material equations between stresses and strains in the Lagrangian version .

#### Elastic fluids

From a continuum mechanical point of view, fluids differ from solids in that the state of stress in them does not change in the event of any volume-preserving deformation (their symmetry group is formed by the unimodular tensors from the special linear group ). In elastic fluids there is only one tension component, pressure; On the other hand, shear stresses, such as can occur in viscous fluids or solids, are excluded or negligibly small in them.

The elastic fluids include the ideal liquid , the ideal gas, and the frictionless real gas . Many material equations of elastic gases are called equations of state , which underlines that the pressure in them is always the same under the same conditions and that they are therefore Cauchy elastic. Kinematically, the pressure depends only on the instantaneous volume expansion or density .

An important special case are the barotropic fluids, in which the density is exclusively a function of the pressure. The barotropic, elastic fluids modeled in this way are automatically isotropic, invariant in the reference system and conservative or - in other words - hyperelastic .

##### Equations of motion

The equations of motion of elastic fluids are Euler's equations of fluid mechanics .

An important special case is when the liquid is barotropic, the volume force (including gravity ) is conservative and the velocity field is stationary . Then the integration of the Euler equations along a streamline leads to Bernoulli's energy equation , which describes technical pipe flows well.

If the velocity field is also rotation-free , then there is a potential flow in which Bernoulli's energy equation not only applies along streamlines, but also applies. Potential flows can be calculated mathematically exactly using analytical means.

#### Thermodynamic consistency

Although the reaction forces are not influenced in a Cauchy-elastic material on the distance deformation in solids, the work done on different deformation routes (with the same start and end points) can change in shape of work vary in size. In the absence of a dissipation mechanism, this is in contradiction to thermodynamic principles.

In contrast, the independence of the deformation work also leads to thermodynamically consistent hyperelasticity, a special case of Cauchy elasticity.

### Hyperelasticity

Hyperelastic fabrics are Cauchy elastic and also conservative. The work of deformation is path-independent in the case of hyperelasticity, and the stresses have a potential relationship to the elongations. In the case of solids, the potential is Helmholtz's free energy , from which, according to the Clausius-Duhem inequality in isothermal processes, the stresses are calculated by deriving them according to the strains.

It can be shown that hyperelastic materials are isotropic and reference system invariant if and only if the Helmholtz free energy is a function of the change of material volume, surface and line elements during a deformation.

#### Conservatism

The path-independence of the deformation work is expressed in that the deformation work only depends on the start and end point of the deformation path, but not on its course. In the special case of the coincidence of the start and end point, the following results: No work is performed or energy consumed along a closed deformation path; Expended work is completely returned by the body until the return to the starting point. The conservatism here also follows from the fact that the deformation performance is exactly the rate of the deformation energy , i.e. the work involved is converted completely (without dissipation) into deformation energy.

#### Material models of hyperelasticity

A number of material models are available for isotropic solids with which real, reversible and large deformations can be reproduced in good approximation. The simplest of these models is Hooke's law for linear elasticity, which approximates any material model of hyperelasticity for small deformations in the first order. A second order approximation for incompressible material is the Mooney-Rivlin model. The Neo-Hooke model, a special case of this model, appropriately generalizes Hooke's law to large deformations for which it is otherwise unsuitable.

The elasticity tensor results in the hyperelasticity from the second derivative of the deformation energy after the strains. Because the order of the derivatives is interchangeable, the elasticity tensor is symmetrical and of the 36 material parameters in the linear Cauchy elasticity only 21 in the hyperelasticity are independent; a linear-hyperelastic material can therefore be described with a maximum of 21 parameters.

### Linear isotropic Hooke's elasticity of solids

In this section, in addition to linear elasticity, kinematic linearity is also assumed, which is the case with small deformations of solids.

#### Navier-Cauchy equations

The local momentum balance is an equation in which only the stresses, acceleration and gravity appear. Now the stresses can be expressed using Hooke's law with the strains and these in turn with the displacements , which leads to the Navier-Cauchy equations. These contain wave equations as a solution for longitudinal , primary because faster moving P-waves and transverse , secondary because slower moving S-waves . In the case of harmonic gravity, the displacement field is a biharmonic function .

#### Clapeyron's Theorem

• The work ${\ displaystyle \ int _ {a} {\ vec {t}} \ cdot {\ vec {u}} \, \ mathrm {d} a}$ • that attacking the surface a of a body
• Powers ${\ displaystyle {\ vec {t}}}$ plus

• the work ${\ displaystyle \ int _ {v} {\ vec {b}} \ cdot {\ vec {u}} \, \ mathrm {d} v}$ • those acting in the volume v of the body
• Volume force ,${\ displaystyle {\ vec {b}}}$ respectively at the displacement field , is equal to ${\ displaystyle {\ vec {u}}}$ • work ${\ displaystyle \ int _ {v} {\ boldsymbol {\ sigma}}: {\ boldsymbol {\ varepsilon}} \, \ mathrm {d} v}$ • the tension field that fulfills the equilibrium condition,${\ displaystyle {\ boldsymbol {\ sigma}}}$ • on the distortions resulting from the displacements :${\ displaystyle {\ boldsymbol {\ varepsilon}}}$ ${\ displaystyle \ int _ {a} {\ vec {t}} \ cdot {\ vec {u}} \, \ mathrm {d} a + \ int _ {v} {\ vec {b}} \ cdot {\ vec {u}} \, \ mathrm {d} v = \ int _ {v} {\ boldsymbol {\ sigma}}: {\ boldsymbol {\ varepsilon}} \, \ mathrm {d} v \ ,.}$ This theorem from Clapeyron assumes sufficient smoothness and continuity of the fields.

In a linear elastic body, the product of the stresses and the strains is half the work of deformation.

If the external forces are conservative, then that follows

#### Principle of the minimum of potential energy and supplementary energy

The principle of the minimum of potential energy says that of all displacement fields that meet certain boundary conditions in an elastic solid that is loaded by conservative external forces, those displacements that meet the equilibrium conditions minimize the potential energy. (The potential energy is the sum of the work of the conservative, external forces and the deformation energy.)

The principle of the minimum of the supplementary energy states that in an elastic solid, of all stress states that meet the boundary conditions, the state that meets the equilibrium condition minimizes the supplementary energy.

The specific supplementary energy U c and the specific deformation energy U are related

${\ displaystyle U ^ {c} = {\ boldsymbol {\ sigma}}: {\ boldsymbol {\ varepsilon}} - U \ ,.}$ #### Theorem of Betti

If a linear hyperelastic body is exposed to external forces, this results in a deformation which minimizes the deformation energy. The system of tension, elongation and displacement is an elastic state of the body that belongs to the attacking force system .

If there is a second system of forces that causes a second elastic state, then Betti's theorem applies:

The work of the first system of forces on the displacements of the second elastic state is equal to the work of the second system of forces on the displacements of the first elastic state.

These reciprocal work of external forces correspond to reciprocal work of deformation:

The work of the stresses of the first elastic state on the extensions of the second elastic state is equal to the work of the stresses of the second elastic state on the extensions of the first elastic state.

Betti's theorem is a basis of the boundary element method .

#### Compatibility Conditions

When a body moves through space, deformations occur in the cases that are interesting for continuum mechanics, which can be quantified by the distortions. There are six components of the distortions in the general three-dimensional case. If the three components of the movement (i.e. the displacements) in the three spatial directions are to be reconstructed from them, the distortions cannot be independent of one another; instead, they must comply with the compatibility conditions formulated for them. By expressing the distortions in the linear elastic material with the stresses, corresponding compatibility conditions are created for the stresses.

In the case of linear elasticity, the compatibility conditions can be met with reasonable effort and thus open up the possibility of solving a boundary value problem with stress functions .

#### Stress functions

In equilibrium, only tensions and gravity appear in the local momentum balance. Here the stresses can be selected as primary unknowns and expressed with stress functions that automatically maintain the equilibrium conditions. The solution of a boundary value problem is reduced to finding stress functions that meet the given boundary conditions and the compatibility conditions for the stresses.

The plane case with Airy's stress function , with the help of which analytical solutions of many boundary value problems in the plane are available today , has been particularly well investigated .

## Mathematical theory

A real material deforms under the action of force in such a way that the deformation energy is minimized. The mathematical theory of elasticity examines u. a. the question under which conditions a deformation exists in the mathematical model that minimizes the deformation energy.

In this context, an important and plausible requirement of the deformation energy is that it tends towards infinity in the case of infinitely large deformation, so that the deformation energy is a coercive function of the deformation. Namely, if the strain energy is a coercive and convex function of the deformation, then a strain which minimizes the strain energy certainly exists.

If the deformation energy also grows beyond all limits when the body is compressed to zero volume, which is plausible, then it cannot be convex. Hence, convexity is an untenable demand on the strain energy.

In contrast, the polyconvexity according to John M. Ball and the coercivity of the deformation energy guarantee the existence of a deformation that minimizes the deformation energy:

• for isotropic hyperelasticity there are a number of such deformation energy functions that are polyconvex and coercive.
• in the case of anisotropic hyperelasticity, JM Ball asked the question: “Are there ways of verifying polyconvexity […] for a useful class of anisotropic stored-energy functions?” (in German: “Are there ways that polyconvexity […] for a useful Class of anisotropic deformation energy functions? ”) The search for the answer to this question is still the subject of lively research activity in the twenty-first century.