Belytschko T. - Introduction (779635), страница 50

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Integrating this relation from the initial length L0 to the currentlength L givesε true = ∫LL0dL= ln ( L L0 ) = ln λL(5.1.4)Taking the material time derivative of this expression givesε true =λ˙= D11λ(5.1.5)i.e., in the one-dimensional case, the time-derivative of the true strain is equal to the rate ofdeformation given by Eq. (3.3.19). This is not true in general unless the principal axes ofthe deformation are fixed.To plot true stress versus true strain, we need to know the cross-sectional area A asa function of the deformation and this can be measured during the test.

If the material isincompressible, then the volume remains constant and we have A0L0 = AL which can bewritten asA = A0 λ(5.1.6)and therefore the Cauchy stress is given byσ=TT=λ= λPAA0(5.1.7)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 ⊗ e1where P11 = P = T A0 . From Box 3.2, the Cauchy (or true) stress is given byσ = J −1F T ⋅P(5.1.8)where J = det F and it follows thatσ = σ11e1 ⊗ e1 = J −1 λP11e1 ⊗ e1(5.1.9)For the special case of an incompressible material J =1 and Eq.

(5.1.9) is equivalent to Eq.(5.1.7).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 asF = λ1 e1 ⊗ e1 + λ2e2 ⊗e 2 + λ3 e3 ⊗ 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 = λ 3 are the stretches in the lateraldirections. For an incompressible material J = det F = λ1λ 2λ3 =1 and thus λ2 = λ 3 = λ−1 2 .Now assume that we can represent the relationship between nominal stress andengineering strain in the form of a functionP11 = s0 (ε 11)(5.1.11)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 asσ11 = λs0 (ε11 ) = s( λ )(5.1.12)where the relation between the functions is s (λ ) = λs0 (λ −1) . This is an illustration of howwe 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 issaid to be rate-independent; otherwise it is rate-dependent. In Figures 5.4 a,b, the onedimensional response of a rate-independent and a rate-dependent material are shownrespectively for different nominal strain rates. The nominal strain rate is defined asε˙ = δ˙ L0 . Using the result δ˙ = L˙ and therefore δ˙ L0 = L˙ L0 = λ˙ it follows that thenominal 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.

Thisresults 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 ElasticityA 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 saidto be reversible. Furthermore, an elastic material is rate-independent (no dependence onstrain 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 measuresof 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 asσ x = s( ε x )(5.2.1)where σ x is the Cauchy stress and ε x = δ L0 is the linear strain, often known as theengineering strain.

Here s(ε x ) is assumed to be a monotonically increasing function. Theassumption that the function s(ε x ) is monotonically increasing is crucial to the stability ofthe material: if at any strain ε x , 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 σ xfor any strain ε x is uniquely given by (5.2.1).

It does not matter how the strain reaches thevalue ε x . 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 thatthere exists a potential function ρwint (ε x ) such thatρdwint (ε x )σ x = s( ε x ) =dε x(5.2.2)where ρwint (ε x ) is the strain energy density per unit volume. From Eq. (5.2.2) it followsthatρdwint (ε x ) = σ x dε x(5.2.3)which when integrated givesεxρwint = ∫ σ x dε x(5.2.4)0This can also be seen by noting that σ xdε x = σ x ˙ε x dt is the one-dimensional formof σ ij Dij 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 asσ x = Eε x(5.2.5)where the constant of proportionality is Young's modulus, E. This relation is oftenreferred to as Hooke's law.

From Eq. (5.2.4) the strain energy density is therefore givenbypw int =1 2Eε2 x(5.2.6)which is a qudratic function of strains. To avoid confusion of Young's modulus with theGreen 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 is( ( ) ( ))()assumed to be a convex function of strain, i.e., w int ε 1x − w ∫ ε 2x ε 1x − ε 2x ≥ 0 , equalityif ε 1x = ε 2x . If w int is non-convex function, this implies that energy is released by the bodyas 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 interrelatedpath− independence ⇔ reversible ⇔ nondissipativeThese properties can be embodied in a material model by modeling the material response byan 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 deformationprocess. We can choose the Green strain as a measure of strain Ex and writeSX =dΨdE X(5.2.7)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˙ =S E˙volume is given by ΨX 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=1EE 22 X(5.2.8)Then,S X = EEX(5.2.9)by equation (5.2.7), so the relation between these stress and strain measures is linear.

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