EN222: Mechanics of Solids

 

 

 

 

 

  

 Division of Engineering

    Brown University

 

 

1.3 Nonlinear Elasticity

 

By definition, an elastic material model has the properties that:

(a)    The material is perfectly reversible: that is to say if you take a specimen of the material, and hold it at fixed temperature or prevent exchange of heat with its surroundings, and then subject it to a closed cycle of strain (i.e. start & end at the same point), the net work done on the solid is zero;

(b)   The stress at a `point’ in the solid depends only on some appropriate measure of strain at that point.

 

The linear elastic constitutive law discussed in 2.1 is one example of such a material model.  However, linear elasticity has two major limitations: (i) it can only be used to model small deformation, because it is based on a linearized deformation measure; (ii) even if the strains are small, it can only model linear stress-strain behavior.  For most practical purposes these restrictions are of no concern, since most materials only show elastic behavior for modest strains, and usually stress really is proportional to strain under these conditions.  However, there are some materials that require a more sophisticated model.  Some polymers may exhibit nonlinear stress-strain behavior even at modest strains (rare – I cant think of an example myself).   More importantly, there is a whole range of polymers (rubbers) that shows elastic behavior up to huge strains, and also shows complex nonlinear stress-strain behavior at finite strain.  Nonlinear elasticity theory is intended to account for these phenomena.

 

We will discuss three types of nonlinear elastic constitutive law

1.      Hypoelasticity – intended to model nonlinear stress-strain behavior but restricted to infinitesimal strains.  Its main application is an approximate theory of plasticity.

2.      The Hookean solid – intended to extend the linear elastic model to situations where stretches remain small, but rotations may be large.   

3.      General hyperelasticity – includes both nonlinear kinematics and nonlinear material behavior.  This has two main applications: (i) to model rubbers and (ii) to model elastic foams.

 

These constitutive laws are far simpler than plasticity theory.  The most interesting aspect of nonlinear elasticity  theory is to try to understand how the macroscopic response of the solid is governed by the molecular level mechanisms of deformation.

 

 

 

 

1.3.1 Hypoelasticity and `Deformation Theory’ plasticity

 

 

 

Hypoelasticity is used to model a material the exhibits nonlinear, but reversible, stress strain behavior even at small strains.  Its most common application is in the so-called `deformation theory of plasticity,’ which should really be called `junk plasticity,’ because like junk food, is attractive but not good for you.

 

Since we choose to restrict attention to small strains, we adopt the infinitesimal strain tensor

as our deformation measure.  We have no need to distinguish between the various stress measures and will use  to denote stress.

 

The stress strain relation is best defined through an elastic potential  such that

For an isotropic material, the potential can only depend on strain through the strain invariants

For a linear elastic solid, one can easily show that

For a hypoelastic solid, you can define any potential you like.  For example, one might choose

with  to simulate an incompressible material with a power-law type stress-strain curve.  The large value of K enforces , and for such deformations we find

where  are the components of deviatoric stress.

 

In ABAQUS, the constitutive law for a hypoelastic solid is specified rather differently.  The incremental stress strain relation is taken to have the form

but the values of  and  are assumed to be functions of the strain invariants.  The user must specify the variation of  and  with strain invariants by entering appropriate tables of values, or through a user subroutine.

 

Deformation Theory of Plasticity

 

The so-called `Deformation Theory’ of plasticity is intended to approximate the plastic response of a polycrystalline metal that is subjected to monotonically increasing proportional loading (i.e. the principal axes of stress do not rotate, and stresses steadily increase throughout loading).  If you were to subject a metal specimen to this kind of loading, you would notice that it had a nonlinear stress-strain curve, but you would never notice that its stress-strain behavior is irreversible.  So, you can’t distinguish it from a nonlinear elastic material.

 

 

 

There are certain advantages to replacing a plastic constitutive law with a nonlinear elastic one, as long as it makes no difference to the results.  For one thing, the constitutive law is simpler, because there is no irreversibility to fuss about.  For another, the existence of a strain energy potential for a solid allows one to develop energy theorems and variational principles that don’t apply for a true inelastic constitutive law.

 

Consequently, a special hypoelastic constitutive law exists for this purpose.  To mimic plasticity, the strain is decomposed into a linear part and a nonlinear part, as

The linear part is related to stress through the usual linear elastic constitutive relations

where  are the components of deviatoric stress.  It is most convenient to express the plastic strains in the form

where  is the secant modulus (the ratio of stress to strain in uniaxial tension) and E is the slope of the uniaxial stress-strain curve at , as shown in the figure.  A particularly convenient form for the uniaxial stress-strain curve is

 

whence the secant modulus can be calculated as , and for this stress-strain curve . This expression gives power-law hardening at large strain but has a finite slope at , which is convenient for numerical purposes.

 

For finite element implementation we need to invert this constitutive law to get an expression for stress in terms of strain.  To do so, first calculate the hydrostatic stress (by contracting i and j in the strain-stress equation) as

and substitute back into the stress-strain equation to see that

where  are the components of deviatoric strain (note that we are using total strain here – the sum of both elastic and plastic parts – because this in an FEM implementation we must calculate the stress corresponding to a given total strain).  Square both sides and rearrange to see that

where  is  the Von-Mises effective strain.  Finally  may be computed by inverting the uniaxial stress-strain curve, giving the following expression for Mises stress

wherupon the deviatoric stresses follow as

 

 

The finite element code ABAQUS has a similar, but rather more cumbersome, implementation of deformation plasticity.  In their model, the strain is decomposed into a linear part and a nonlinear part, as

The linear part is related to stress through the usual linear elastic constitutive relations

where  are the components of deviatoric stress, and  and  are the usual elastic constants. The nonlinear part of the strain is approximated as a power-law of the form

where  is the Von Mises effective stress, and  and n>1 are two material constants.  It is sensible to set , in which case  can be interpreted as the 2% yield stress (the stress at which the plastic strain reaches 2%).  Obviously, n controls the hardening rate – for large n there’s not much strain hardening, for small n there’s a lot.  Typically n is around 5-10.

 

One can readily verify that the uniaxial strain-stress relation becomes

The strain energy density for this material can be expressed in terms of stress as

An irritating feature of this model is that it is difficult to invert the stress-strain relation to obtain an expression for stress in terms of strain, because the uniaxial stress-strain curve can’t be inverted explicitly (ABAQUS solves it using a Newton-Raphson iteration).

 

 

 

 

1.3.2 Generalized Hooke’s law – small stretches, large rotations

 

It is easy to fix the linear elastic constitutive laws to deal with finite rotations – we merely need to adopt a sensible deformation measure.  It is convenient to use the Lagrange strain

As a stress measure, we must adopt the appropriate work-conjugate quantity, which you will recall is Material stress, denoted by  in our discussion of kinetics.

 

As long as stretches are small, we can adopt a linear-elasticity like constitutive law

where  is the usual elasticity tensor.  All the representations and symmetries discussed in Sect. 2.1 apply to this constitutive law.

 

 

1.3.3 Hyperelasticity

 

Finally we list a few common hyperelastic constitutive laws.  These theories are intended to be used to model materials that respond elastically up to very large strains, and account both for nonlinear material behavior and also for nonlinear kinematics.  The main applications of the theory are (i) to model the rubbery behavior of a polymeric material, and (ii) to model polymeric foams.  Although similar constitutive theories are used for both applications the actual behavior of a rubber is quite different to that of a foam.

 

In general, the response of a typical polymer is strongly dependent on temperature, strain history and loading rate.  The behavior will be described in more detail in the next section, where we present the theory of viscoelasticity.  For now, we note that polymers have various regimes of mechanical behavior, referred to as `glassy,’ `viscoelastic’ and `rubbery.’   The various regimes can be identified for a particular polymer by applying a sinusoidal variation of shear stress to the solid and measuring the resulting shear strain amplitude.  A typical result is shown below, by plotting the apparent shear modulus (ratio of stress amplitude to strain amplitude) as a function of temperature

 

At a critical temperature known as the glass transition temperature, a polymeric material undergoes a dramatic change in mechanical response.  Below this temperature, it behaves like a glass, with a stiff response – near the glass transition temperature the response is heavily rate dependent.  At the glass transition temperature, there is a dramatic drop in modulus.  Above this temperature, there is a regime where the polymer shows `rubbery’ behavior – the response is elastic without much rate or history dependence, and the modulus increases with temperature.  All polymers show these general trends, but the extent of each regime and the precise behavior within each regime depend on the solid’s molecular structure.  Heavily cross-linked polymers (elastomers) are the most likely to show ideal rubbery behavior.  

 

Hyperelastic constitutive laws are intended to approximate the `rubbery’ behavior of polymers.

The principal features of mechanical response in this regime are that

1.      The material is close to ideally elastic. i.e. (i) when deformed at constant temperature or adiabatically, stress is a function only of current strain and independent of history or rate of loading, (ii) the behavior is reversible: no net work is done on the solid when subjected to a closed cycle of strain under adiabatic or isothermal conditions.

2.      The material strongly resists volume changes.  The bulk modulus (the ratio of volume change to hydrostatic component of stress) is comparable to that of metals or covalently bonded solids;

3.      The material is very compliant in shear – shear modulus is of the order of  times that of most metals;

4.      Shear response is strongly temperature dependent: the material becomes stiffer as it is heated, in sharp contrast to metals;

5.      When stretched, the material gives off heat.

 

 

Polymeric foams (e.g. a sponge) share some of these properties – they are close to reversible, and show no rate or history dependence.  In contrast to rubbers, most foams are highly compressible – bulk and shear moduli are comparable.  Foams show complex true stress-true strain response, generally resembling the figure below.  The finite strain response of the foam in compression is quite different to that in tension, caused by buckling in the cell walls. 

 

 

Hyperelastic constitutive laws are used to model this type of behavior.  Most constitutive laws assume that

1.      One can define a specific Helmholtz free energy for the solid such that for any isothermal (constant temperature) motion the mechanical work done on the solid is equal to the rate of change of free energy, i.e.

where  is the Cauchy stress and  is the stretch rate tensor. (ABAQUS uses U to denote the free energy but this is usually used to denote internal energy.  Since many of the commonly used forms for the potential are in fact based on statistical mechanics computations of the Helmholtz free energy we will use  rather than U to denote the potential).  This assumption implies perfect reversibility under isothermal conditions; as well as history and rate independence.

2.      The solid is isotropic in its reference configuration. (The deformed configuration may be anisotropic, even if the reference configuration is isotropic)

 

In addition many rubbery materials are assumed to be incompressible.  This simplifies the constitutive relations, but it has the disadvantage that the stresses cannot be determined uniquely from the strains (the hydrostatic stress in indeterminate) and moreover it is painful to model fully incompressible materials using the finite element method.  Consequently we will outline a generalized version of the classical hyperelastic constitutive relations, which allows some volumetric distortion.  You can always take the incompressible limit if you need to do so.

 

To calculate the stress-strain relation, we can express the internal energy in terms of any convenient strain measure – assumption (1) then gives a way to calculate a stress-strain law in terms of the deformation measure of choice.  In addition, if the material is isotropic, it must be possible to express the constitutive law in terms of the principal values or invariants of your strain measure.  It then remains only to specify the precise form of the internal energy in terms of the strain invariants – this, of course, must be determined by experiment or by means of an appropriate physical model of the molecular mechanisms of deformation.

 

Usually the left Cauchy-Green deformation tensor is adopted as the deformation measure, because this leads to the simplest form of the stress strain relation.  To account for compressibility, the volumetric part of the deformation is handled separately.

 

Let  denote the components of the Lagrange strain tensor, and let  characterize volume changes.  Define

as a deformation measure that is independent of volume changes (defined in the same spirit as deviatoric infinitesimal strain in plasticity theory.  You can readily verify that  ).  Then define the deviatoric left C-G strain tensor in terms of

and introduce the two strain invariants

The free energy energy is then taken to be a function of J, the two strain invariants, and temperature .

 

We now need to find how to calculate the stress-strain relation given this form of .  Re-write the definition as

(you have to transform the integrals to the reference configuration to be able to take the time derivative inside the integral).  Now note

 

A straightforward (actually it’s not all that straightforward) but tedious (it’s DEFINITELY tedious) calculation shows that

where

Note also that we may write

where  are the components of deviatoric stress.  Substituting all this back into the expression for  shows that

 

 

and noting that this must hold for all  we obtain expressions for the hydrostatic stress as

and deviatoric stress

The last expression is written in this form because the hydrostatic component of  does no work through

 

This concludes the general development of stress-strain equations for hyperelastic solids.  It remains only to choose specific functions for .

 

Two classical examples are:

 

Neo-Hookean solid  (First used by Treloar Proc Phys Soc 60 135-44 1948)

where  is a temperature dependent constant (elementary statistical mechanics treatments predict that , where N is the number of polymer chains per unit volume and k is the Boltzmann constant).  This material is fully incompressible since  does not depend on J.  A modified version that allows compressibility is

Setting  (or very small) would make the material (almost) incompressible.

 

Mooney-Rivlin solid (First used by Mooney J Appl Phys 11 582 1940)

 

The original version was fully incompressible.  Here we give a modified version

where  and  are temperature dependent constants.

 

Ogden (First used by Ogden, Proc R Soc Lond A326, 565-84 (1972), ibid A328 567-83 (1972))

 

Instead of using strain invariants this model expresses the strain energy in terms of principal values of B, as

 

where , , , I=1…N are material constants.  For particular values of these material constants, this model will reduce to either the Neo-Hookean solid or the Mooney-Rivlin material. 

 

Arruda-Boyce

 

Also sometimes called the 8-chain model because it was derived by idealizing a polymer as 8 elastic chains inside a box, this constitutive law has a specific internal energy given by

 

where , L and D are material constants.

 

All these models are implemented in ABAQUS.

 

Calibrating nonlinear elasticity models

 

To use any of these constitutive relations, you will need to determine values for the material constants.  In some cases this is quite simple (the neo-Hookean material only has 1 constant!); for models like Ogden’s it is considerably more involved. 

 

Conceptually, however, the procedure is straightforward.  You can perform various types of test on a sample of the material, including

1.      Simple tension

2.      Pure shear

3.      Equibiaxial tension

4.      Volumetric compression (easier than tension!)

 

It is fairly straightforward to calculate the predicted stress-strain behavior for the specimen for each constitutive law.  The parameters can then be chosen to give the best fit to experimental behavior.  ABAQUS includes a capability to do the fitting for you, given appropriate experimental data.