Belytschko T. - Introduction (779635), страница 49
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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 asin 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 constitutiverelation is called a stress update algorithm. Several stress update algorithms are presentedand 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.
Wellknown constitutive models such as Neo-Hookean, Saint Venant Kirchhoff and MooneyRivlin 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 presentedfor the case of small deformations. The commonly used von Mises J2 -flow theoryplasticity models (representative of the behavior of metals) for rate-independent and ratedependent 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 asan 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 rotationsin 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 rateindependent 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 CurveThe 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 onedimensional 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 TestThe 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 aremeasured. 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. Thenominal (or engineering stress) is given byP=TA0(5.1.1)where A0 is the original cross-sectional area. The engineering strain is given byε=δ= λ −1L0(5.1.2)A plot of engineering stress versus engineering strain for a typical metal is given in Figure5.2.Alternatively, the stress strain response may be given in terms of true stress. TheCauchy (or true) stress is given byσ=TA(5.1.3)where A is the current (instantaneous) area of the cross-section. A measure of true strain isderived by considering an increment of true strain as change in length per unit currentlength, i.e., dε true = d L L .
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