Nonlinear Finite Elements for Continua and Structures Part 7 pdf

In the present discussion, emphasis is on the mechanical response, althoughcoupling to energy equations and thermal effects are considered.The implementation of the constitutive relation

T Belytschko, Lagrangian Meshes, December 16, 1998

where l0 is the initial length of the element In this example, the coordinates X, Y are used in a

somewhat different sense than before: it is no longer true that x t( =0)=X However, the

definition used here corresponds to a rotation and translation of x t( =0) Since neither rotation

nor translation effects E or any strain measure, this choice of an X, Y coordinate system is

perfectly acceptable We could have used the element coordinates ξ as material coordinates, butthis complicates the definition of physical strain components

The spatial coordinates are given in terms of the element coordinates by

x=x1(1− ξ)+x2ξ

y=y1(1−ξ)+y2ξ or

x y

The deformation gradient F is not a square matrix for the rod since there are two space dimensions

but only one independent variable describes the motion, (E4.8.2)

The only nonzero stress is along the axis of the rod To take advantage of this, we use the

nodal force formula in terms of the PK2 stress, since S11 is the only nonzero component of this

stress For the nominal stress, P11 is not the only nonzero component The X axis as defined here

is corotational with the axis of the rod, so S11 is always the stress component along the axis of therod Substituting (E4.8.5) and (E4.8.6) into Eq (4.9.19) then gives the following expression forthe internal nodal forces:

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T Belytschko, Lagrangian Meshes, December 16, 1998

Since the deformation is constant in the element, we can assume the integrand is constant, so

multiplying the integrand by the volume A0l0 we have

l0[−cosθ −sinθ cosθ sinθ]

The expression for the nodal forces, (4.5.19) then becomes

Example 4.9 Triangular Element Develop expressions for the deformation gradient,

nodal internal forces and nodal external forces for the 3-node, linear displacement triangle Theelement was developed in the updated Lagrangian formulation in Example 4.1; the element isshown in Fig 4.2

The motion of the element is given by the same linear map as in Example 4.1, Eq (E4.1.2)

in terms of the triangular coordinates ξI The B0 matrix is given by (4.9.7):

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T Belytschko, Lagrangian Meshes, December 16, 1998

Voigt Notation The expression for the internal nodal forces in Voigt notation requires the B0

matrix Using Eq (4.9.24) and the derivatives of the shape functions in Eq (E4.9.1) gives

T Belytschko, Lagrangian Meshes, December 16, 1998

Example 4.10 Two-Dimensional Isoparametric Element Construct the discrete

equations for two- and three-dimensional isoparametric elements in indicial matrix notation andVoigt notation The element is shown in Fig 4.4; the same element in the updated Lagrangianform was considered in Example 4.2

The motion of the element is given in Eq (E4.2.1), followed by the shape functions andtheir derivatives with respect to the spatial coordinates The key difference in the formulation ofthe isoparametric element in the total Lagrangian formulation is that the matrix of derivatives of theshape functions with respect to the material coordinates must be found By implicit differentiation

T Belytschko, Lagrangian Meshes, December 16, 1998

ξ =det X( ),ξ =det F( )ξ0 (E4.10.8)

If the Voigt form is used, the internal forces are computed by Eq (4.9.22) in terms of S

The external nodal forces, particularly those due to pressure, are usually best computed in theupdated form The mass matrix was computed in the total Lagrangian form in Example 4.2

Example 4.12 Three-Dimensional Element Develop the strain and nodal force equations

for a general three-dimensional element in the total Lagrangian format The element is shown inFigure 4.5 The parent element coordinates are

ξ = ξ( 1, ξ2, ξ3)≡ ξ( ,η,ζ) for an isoparametricelement, ξ = ξ( 1, ξ2, ξ3) for a tetrahedral element, where for the latter ξi are the volume(barycentric) coordinates

Matrix Form The standard expressions for the motion, Eqs (4.9.1-5) are used The

deformation gradient is given by Eq (4.9.6) The Jacobian matrix relating the referenceconfiguration to the parent is

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T Belytschko, Lagrangian Meshes, December 16, 1998

where X,ξ-1 is evaluated numerically from Eq (E4.12.1) The Green-strain tensor can be

computed directly from F, but to avoid round-off errors, it is better to compute

The Green-strain tensor is then given by Eq (???)

If the constitutive law relates the PK2 stress S to E, the nominal stress is then computed by

P=SFT, using F from Eq (??.2) The nodal internal forces are then given by

T Belytschko, Lagrangian Meshes, December 16, 1998

4.9.3 Variational Principle For static problems, weak forms for nonlinear analysis with

path-independent materials can be obtained from variational principles For many nonlinearproblems, variational principles can not be formulated However, when constitutive equations andloads are path-independent and nondissipative, a variational priniciple can be written because thestress and load can be obtained from potentials The materials for which stress is derivable from apotential are called hyperelastic materials, see Section 5.4 In a hyperelastic material, the nominalstress is given in terms of a potential by Eq (5.4.113) which is rewritten here

Note the order of the subscripts on the stress, which follows from the definition

For the existence of a variational principle, the loads must also be conservative, i.e theymust be independent of the deformation path Such loads are also derivable from a potential, i.e.the loads must be related to a potential so that

Theoem of Stationary Potential Energy When the loads and constitutive equations posses

potentials, then the stationary points of

The theorem is proven by showing the equivalence of the stationary principle to the weakform for equilibrium, traction boundary conditions and the interior continuity conditions We firstwrite the stationary condition of (4.9.30), which gives

T Belytschko, Lagrangian Meshes, December 16, 1998

which is the weak form given in Eq (4.8.7) for the case when the accelerations vanish The samesteps given in Section 4.8 can then be used to establish the equivalence of Eq (4.8.7) to the strongform of the equilibrium equation

Stationary principles are thus in a sense more restrictive weak forms: they apply only toconservative, static problems However they can improve our understanding of stability problemsand are used in the study of the existence and uniqueness of solutions

The discrete equations are obtained from the stationary principle by using the usual finiteelement approximation to motion with a Lagrangian mesh, Eqs (4.12) to (4.9.5), which we write

in the form

The potential energy can then be expressed in terms of the nodal displacements, giving

W d( )=W int( )d −W ext( )d (4.9.34)The solutions to the above correspond to the stationary points of this function, so the discreteeqautions are

Example 4.11 Rod Element by Stationary Principle Consider a structural model

consisting of two-node rod elements in three dimensions Let the internal potential energy be givenby

For the two-node element, the displacement field is linear and the Green strain is constant, so Eq

(E4.11.3) can be simplified by multiplying the integrand by the initial volume of the element A0l0:

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T Belytschko, Lagrangian Meshes, December 16, 1998

If we make the finite element

approximation z=z I N I , where N I are the shape functions given in Eq (E4.8.4) then

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T Belytschko, Lagrangian Meshes, December 16, 1998

T Belytschko and B.J Hsieh, "Nonlinear Transient Finite Element Analysis with Convected

Coordinates," International Journal for Numerical Methods in Eng., 7, pp 255-271, 1973.

T.J.R Hughes (1997), The Finite Element Method, Prentice-Hall, Englewood Cliffs, New

Jersey

L.E Malvern (1969), Introduction to the Mechanics of a Continuous Medium, Prentice-Hall,

Englewood Cliffs, New Jersey

J.T Oden and J.N Reddy (1976), An Introduction to the Mathematical Theory of Finite

Elements, John Wiley & Sons, New York.

G Strang and G.J Fix (1973), An Analysis of the Finite Element Method, Prentice Hall, New

York

G.A Wempner (1969), "Finite elements, finite rotations and small strains," Int J Solids and

Structures, 5, 117-153.

Figure 4.1 Initial and Current Configurations of an Element and Their

Relationships to the Parent Element (p 18)

Figure 4.2 Triangular Element Showing Node Numbers and the Mappings of the

Initial and Current Configurations to the Parent Element (p 29)

Figure 4.3 Triangular Element Showing the Nodal Force and Velocity Compenents

T Belytschko, Lagrangian Meshes, December 16, 1998

Figure 4.8 Exploded view of a tieline; when joined together, the velocites of nodes 3 and 5 equal

the nodal velocities of nodes 1 and 2 and the velocity of node 4 is given in terms ofnodes 1 and 2 by a linear constraint (p 45)

Figure 4.9 Two-node rod element showing initial configuration and current configuration and the

corotational coordinate (p 50)

Figure 4.10 Initial, current, and parent elements for a three-node rod; the corotational base vector

e ˆ x is tangent to the current configuration (p 52)

Figure 4.11 Triangular three-node element treated by corotational coordinate system (p 54)Figure 4.12 Rod element in rwo dimensions in total Lagrangian formulation (p 68)

Box 4.1 Governing Equations for Updated Lagrangian Formulation (p 3)

Box 4.2 Weak Form in Updated Lagrangian Formulation: Principle of Virtual

Power (p 9)

Box 4.3 untitled

Box 4.5 Governing Equations for Total Lagrangian Formulation (p 48)

Box 4.6 untitled (p 53)

Box 4.7 Internal Force Computation in Total Lagrangian Fomulation (p 57)

Box 4.8 Discrete Equations for the Updated Lagrangian Formulation and Internal Nodal Force

Algorithm (p 75)

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to multiaxial stress states Special emphasis is placed on the elastic-plastic constitutiveequations for both small and large strains Some fundamental properties such asreversibility, stability and smoothness are also dsicussed An extensive body of theoryexists on the thermodynamic foundations of constituive equations at finite strains and theinterested reader is referred to Noll (1973), Truesdell and Noll (1965) and Truesdell(1969) In the present discussion, emphasis is on the mechanical response, althoughcoupling to energy equations and thermal effects are considered.

The implementation of the constitutive relation in a finite element code requires aprocedure for the evaluation of the stress given the deformation (or an increment ofdeformation from a previous state) This may be a straightforward function evaluation as

in hyperelasticity or it may require the integration of the rate or incremental form of theconstitutive equations The algorithm for the integration of the rate form of the constitutive

relation is called a stress update algorithm Several stress update algorithms are presented

and discussed along with their numerical accuracy and stability The concept of stress ratesarises naturally in the specification of the incremental or rate forms of constitutive equationsand this lays the framework for the discussion of linearization of the governing equations inChapter 6

In the following Section, the tensile test is introduced and discussed and used tomotivate different classes of material behavior One-dimensional constitutive relations forelastic materials are then discussed in detail in Section 5.2 The special and practicallyimportant case of linear elasticity is considered in Section 5.3 In this section, theconstitutive relation for general anisotropic linear elasticity is developed The case of linearisotropic elasticity is obtained by taking account of material symmetry It is also shownhow the isotropic linear elastic constitutive relation may be developed by a generalization ofthe one-dimensional behavior observed in a tensile test

Multixial constitutive equations for large deformation elasticity are given in Section5.4 The special cases of hypoelasticity (which often plays an important role in largedeformation elastic-plastic constitutive relations) and hyperelasticity are considered Well-known constitutive models such as Neo-Hookean, Saint Venant Kirchhoff and Mooney-Rivlin material models are given as examples of hyperleastic constitutive relations

In Section 5.5, constitutive relations for elastic-plastic material behavior formultiaxial stress states for both rate-independent and rate-dependent materials are presented

for the case of small deformations The commonly used von Mises J2-flow theoryplasticity models (representative of the behavior of metals) for rate-independent and rate-dependent plastic deformation and the Mohr-Coulomb relation (for the deformation of soilsand rock) are presented The constitutive behavior of elastic-plastic materials undergoinglarge deformations is presented in Section 5.6

Well-established extensions of J2-flow theory constituve equations to finite strainresulting in hypoelastic-plastic constitutive relations are discussed in detail in Section 5.7.The Gurson constitutive model which accounts for void-growth and coalescence is given as

an illustration of a constitutive relation for modeling material deformation together withdamage and failure The constitutive modeling of single crystals (metal) is presented as anillustration of a set of micromechanically motivated constitutive equation which has provenvery successful in capturing the essential features of the mechanical response of metalsingle crystals Single crystal plasticity models have also provided a basis for largedeformation constitutive models for polycrstalline metals and for other classes of materialundergoing large deformation Hyperelastic-plastic constitutive equations are alsoconsidered In these models, the elastic response is modeled as hyperelastic (rather thanhypoelastic) as a means of circumventing some of the difficulties associated with rotations

in problems involving geometric nonlinearity

Constitutive models for the viscoelastic response of polymeric materials aredescribed in section 5.8 Straightforward generalizations of one-dimensional viscoelasticmodels to multixial stress states are presented for the cases of small and large deformations

Stress update algorithms for the integration of constitutive relations are presented insection 5.9 The radial return and associated so-called return-mappng algorithms for rate-independent materials are presented first Stress-update schemes for rate dependent materialare then presented and the concept of algorithmic tangent modulus is introduced Issues ofaccuracy and stability of the various schemes are introduced and discussed

5.1 The Stress-Strain Curve

The relationship between stress and deformation is represented by a constitutveequation In a displacement based finite element formulation, the constitutive relation isused to represent stress or stress increments in terms of displacment or displacementincrements respectively Consequently, a constitutve equation for general states of stressand stress and deformation histories is required for the material The purpose of thischapter is to present the theory and development of constitutive equations for the mostcommonly observed classes of material behavior To the product designer or analyst, thechoice of material model is very important and may not always be obvious Often the onlyinformation available is general knowledge and experience about the material behavioralong with perhaps a few stress strain curves It is the analyst's task to choose theappropriate constitutive model from available libraries in the finite element code or todevelop a user supplied constitutive routine if no suitable constitutive equation is available

It is important for the engineer to understand what the key features of the constitutive modelfor the material are, what assumptions have gone into the development of the model, howsuitable the model is for the material in question, how appropriate the model is for theexpected load and deformation regime and what numerical issues are involved in theimplentation of the model to assure accuracy and stability of the numerical procedure Aswill be seen below, the analyst needs to have a broad understanding of relevant areas ofmechanics of materials, continuum mechanics and numerical methods

Many of the essential features of the stress-strain behavior of a material can beobtained from a set of stress-strain curves for the material response in a state of one-dimensional stress Both the physical and mathematical descriptions of the materialbehavior are often easier to describe for one-dimensional stress states than for any other.Also, as mentioned above, often the only quantitative information the analyst has about thematerial is a set of stress strain curves It is essential for the analyst to know how tocharacterize the material behavior on the basis of such stress-strain curves and to knowwhat additional tests, if any, are required so that a judicious choice of constitutive equationcan be made For these reasons, we begin our treatment of constitutive models and theirimplementation in finite element codes with a discussion of the tensile test As will beseen, constitutive equations for multixial states are often based on simple generalizations ofthe one-dimensional behavior observed in tensile tests.

5.1.1 The Tensile Test

The stress strain behavior of a material in a state of uniaxial (one-dimensional)stress can be obtained by performing a tensile test (Figure 5.1) In the tensile test, aspecimen is gripped at each end in a testing machine and elongated at a prescribe rate Theelongation δ of the gage section and the force T required to produce the elongation are

measured A plot of load versus elongation (for a typical metal) is shown in Figure 5.1.This plot represents the response of the specimen as a structure In order to extractmeaningful information about the material behavior from this plot, the contributions of thespecimen geometry must be removed To do this, we plot load per unit area (or stress) ofthe gage cross-section versus elongation per unit length (or strain) Even at this stage,decisions need to be made: Do we use the the original area and length or the instantaneousones? Another way of stating this question is what stress and strain measures should weuse? If the deformations are sufficently small that distinctions between original and currentgeometries are negligible for the purposes of computing stress and strain, a small straintheory is used and a small strain constitutive relation developed Otherwise, full nonlinearkinematics are used and a large strain (or finite deformation) constitutive relation isdeveloped From Chapter 3 (Box 3.2), it can be seen that we can always transform fromone stress or strain measure to another but it is important to know precisely how theoriginal stress-strain relation is specified A typical procedure is as follows:

Define the stretch λ = L L0 where L= L0+δ is the length of the gage sectionassociated with elongation δ Note that λ = F11 where F is the deformation gradient The

nominal (or engineering stress) is given by

σ =T

where A is the current (instantaneous) area of the cross-section A measure of true strain is

derived by considering an increment of true strain as change in length per unit current

length, i.e., dεtrue=d L L Integrating this relation from the initial length L0 to the current

To plot true stress versus true strain, we need to know the cross-sectional area A as

a function of the deformation and this can be measured during the test If the material is

incompressible, then the volume remains constant and we have A0L0 =AL which can be

A plot of true stress versus true strain is given in Figure 5.3

The nominal or engineering stress is written in tensorial form as P=P11e1⊗e1

where P11=P=T A0 From Box 3.2, the Cauchy (or true) stress is given by

Prior to the development of instabilities (such as the well known phenomenon ofnecking) the deformation in the gage section of the bar can be taken to be homogeneous.The deformation gradient, Eq (3.2.14), is written as

F= λ1e1⊗e1+ λ2e2⊗e2 + λ3e3⊗e13 (5.1.10)where λ1= λ is the stretch in the axial direction (taken to be aligned with the x1-axis of arectangular Cartesian coordinate system) and λ2 = λ3 are the stretches in the lateral

directions For an incompressible material J =det F= λ1λ2λ3=1 and thus λ2= λ3= λ−1 2

.Now assume that we can represent the relationship between nominal stress andengineering strain in the form of a function

where ε11= λ −1 is the engineering strain We can regard (5.1.11) as a stress-strainequation for the material undergoing uniaxial stressing at a given rate of deformation Atthis stage we have not introduced unloading or made any assumptions about the materialresponse From equation (5.1.9), the true stress (for an incompressible material) can bewritten as

where the relation between the functions is s( )λ = λs0( )λ −1 This is an illustration of how

we obtain different functional representations of the constitutive relation for the samematerial depending on what measures of stress and deformation are used It is especiallyimportant to keep this in mind when dealing with multiaxial constitutive relations at largestrains

A material for which the stress-strain response is independent of the rate of deformation is

said to be rate-independent; otherwise it is rate-dependent In Figures 5.4 a,b, the

one-dimensional response of a rate-independent and a rate-dependent material are shownrespectively for different nominal strain rates The nominal strain rate is defined as

ε =˙ δ ˙ L0 Using the result δ =˙ L and therefore ˙ δ ˙ L0=L L ˙

0=λ ˙ it follows that the

nominal strain rate is equivalent to the rate of stretching, i.e., ˙ ε =λ =˙ F ˙ 11 As can be seen,the stress-strain curve for the rate-independent material is independent of the strain ratewhile for the rate-dependent material the stress strain curve is elevated at higher rates Theelevation of stress at the higher strain rate is the typical behavior observed in most materials(such as metals and polymers) A material for which an increase in strain rate gives rise to adecrease in the stress strain curve is said to exhibit anomolous rate-dependent behavior

In the description of the tensile test given above no unloading was considered InFigure 5.5 unloading behaviors for different types of material are illustrated For elasticmaterials, the unloading stress strain curve simply retraces the loading one Upon completeunloading, the material returns to its inital unstretched state For elastic-plastic materials,however, the unloading curve is different from the loading curve The slope of theunloading curve is typically that of the elastic (initial) portion of the stress strain curve This

results in permanent strains upon unloading as shown in Figure 5.5b Other materialsexhibit behaviors between these two extremes For example, the unloading behavior for abrittle material which develops damage (in the form of microcracks) upon loading exhibitsthe unloading behavior shown in Figure 5.5c In this case the elastic strains are recoveredwhen the microcracks close upon removal of the load The initial slope of the unloadingcurve gives information about the extent of damage due to microcracking.

In the following section, constitutive relations for one-dimensional linear andnonlinear elasticity are introduced Multixial consitutive relations for elastic materials arediscussed in section 9.3 and for elastic-plastic and viscoelastic materials in the remainingsections of the chapter

5.2 One-Dimensional Elasticity

A fundamental property of elasticity is that the stress depends only on the currentlevel of the strain This implies that the loading and unloading stress strain curves areidentical and that the strains are recovered upon unloading In this case the strains are said

to be reversible Furthermore, an elastic material is rate-independent (no dependence on

strain rate) It follows that, for an elastic material, there is a one-to-one correspondencebetween stress and strain (We do not consider a class of nonlinearly elastic materialswhich exhibit phase transformations and for which the stress strain curve is not one-to-one.For a detailed discussion of the treatment of phase transformations within the framework ofnonlinear elasticity see (Knowles, ).)

We focus initially on elastic behavior in the small strain regime When strains androtations are small, a small strain theory (kinematics, equations of motion and constitutiveequation) is often used In this case we make no distinction between the various measures

of stress and strain We also confine our attention to a purely mechanical theory in whichthermodyanamics effects (such as heat conduction) are not considered.For a nonlinearelastic material (small strains) the relation between stress and strain can be written as

where σx is the Cauchy stress and εx = δ L0 is the linear strain, often known as the

engineering strain Here s( )εx is assumed to be a monotonically increasing function The

assumption that the function s( )εx is monotonically increasing is crucial to the stability ofthe material: if at any strain εx, the slope of the stress strain curve is negative, i.e.,

ds dε x <0 then the material response is unstable Such behavior can occur in constitutivemodels for materials which exhibit phase transformations (Knowles) Note thatreversibility and path-independence are implied by the structure of (5.2.1): the stress σx

for any strain εx is uniquely given by (5.2.1) It does not matter how the strain reaches thevalue εx The generalization of (5.2.1) to multixial large strains is a formidablemathematical problem which has been addressed by some of the keenest minds in the 20thcentury and still enocmpasses open questions (see Ogden, 1984, and references therein).The extension of (5.2.1) to large strain uniaxial behaior is presented later in this Section.Some of the most common multiaxial generalizations to large strain are discussed in Section5.3

In a purely mechanical theory, reversibility and path-independence also imply theabsence of energy dissipation in deformation In other words, in an elastic material,deformation is not accompanied by any dissipation of energy and all energy expended indeformation is stored in the body and can be recovered upon unloading This implies that

there exists a potential function ρwint( )εx such that

This can also be seen by noting that σx dε x =σx ˙ ε x dt is the one-dimensional form

of σij D ij dt for small strains.

One of the most obvious characteristics of a stress-strain curve is the degree ofnonlinearity it exhibits For many materials, the stress strain curve consists of an initiallinear portion followed by a nonlinear regime Also typical is that the material behaveselastically in the initial linear portion The material behvior in this regime is then said to belinearly elastic The regime of linear elastic behavior is typically confined to strains of nomore than a few percent and consequently, small strain theory is used to describe linearelastic materials or other materials in the linear elastic regime

For a linear elastic material, the stress strain curve is linear and can be written as

where the constant of proportionality is Young's modulus, E This relation is often

referred to as Hooke's law From Eq (5.2.4) the strain energy density is therefore givenby

pw

int =1

2Eε x2

(5.2.6)

which is a qudratic function of strains To avoid confusion of Young's modulus with the

Green strain, note that the Green (Lagrange) strain is always subscripted or in boldface.

Because energy is expended in deforming the body, the strain energy w int isassumed to be a convex function of strain, i.e.,

w

int( )ε1x −w∫( )εx2

( )(ε1x − εx2)≥0 , equality

if εx1= εx2

If w int is non-convex function, this implies that energy is released by the body

as it deforms, which can only occur if a source of energy other than mechanical is presentand is converted to mechanical energy This is the case for materials which exhibit phasetransformations Schematics of convex and non-convex energy functions along with thecorresponding stress strain curves given by (5.2.2) are shown in Figure 5.6

In summary, the one-dimensional behavior of an elastic material is characterized bythree properties which are all interrelated

path−independence⇔reversible⇔nondissipative

These properties can be embodied in a material model by modeling the material response by

an elastic potential

The extension of elasticity to large strains in one dimension is ratherstraightforward: it is only necessary to choose a measure of strain and define an elasticpotential for the (work conjugate) stress Keep in mind that the existence of a potentialimplies reversibility, path-independence and absence of dissipation in the deformation

process We can choose the Green strain as a measure of strain E x and write

S X = dΨ

The fact that the corresponding stress is the second Piola-Kirchhoff stress follows from thework (power) conjugacy of the second Piola-Kirchhoff stress and the Green strain, i.e.,recalling Box 3.4 and, specializing to one dimension, the stress power per unit reference

volume is given by Ψ =˙ S X E ˙

X.The potential Ψ in (5.2.7) reduces to the potential (5.2.2) as the strains becomesmall Elastic stress-strain relationships in which the stress can be obtained from apotential function of the strains are called hyperelastic

The simplest hyperelastic relation (for large deformation problems in onedimension) results from a potential which is quadratic in the Green strain:

Ψ = 1

2EE X

2

(5.2.8)Then,

U =U−I is a valid strain measure (called the Biot strain), and that in one-dimension the

conjugate stress is the nominal stress P X,so

P X= dΨ

dU X = dΨ

We can write the second form in (5.2.10) because the unit tensor I is constant and hence

dU X =dU X It is interesting to observe that linearity in the relationship between a certainpair of stress and strain measures does not imply linearity in other conjugate pairs For

example if S X =EE X it follows that P X= E U( X2+2U X) 2

A material for which the rate of Cauchy stress is related to the rate of deformation is

said to be hypoelastic The relation is generally nonlinear and is given by

which is a hyperelastic relation and thus path-independent However, for multiaxial

problems, hypoelastic relations can not in general be transformed to hyperelastic Multixialconstitutive models for hypoelastic, elastic and hyperelastic materials are described inSections 5.3 and 5.4 below

A hypoelastic material is, in general, strictly path-independent only in the dimensional case (• check) However, if the elastic strains are small, the behavior is closeenough to path-independent to model elastic behavior Because of the simplicity ofhypoelastic laws, a muti-axial generalization of (5.2.11) is often used in finite elementsoftware to model the elastic response of materials in large strain elastic-plastic problems(see Section 5.7 below)

one-For the case of small strains, equation (9.2.12) above can be written as

which is the rate form (material time derivative) of Hooke's law (5.2.5)