Belytschko T. - Introduction (779635), страница 7
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The Cauchy stress is given byσ = TA(2.2.7)This measure of stress refers to the current area A. In the total Lagrangian formulation, wewill use the nominal stress. The nominal stress will be denoted by P and is given byTP= A0(2.2.8)It can be seen that it differs from the physical stress in that the net resultant force is dividedby the initial, or undeformed, area A0 . This is equivalent to the definition of engineeringstrain; however, in multi-dimensions, the nominal stress is not equivalent to theengineering stress, this is discussed further in Chapter 3.2-4T.
Belytschko, Chapter 2, December 16, 1998Comparing Eqs. (2.2.7) and (2.2.8), it can be seen that the physical and nominalstresses are related byAσ = A0 PP = AA σ0(2.2.9)Therefore, if one of the stresses is known, the other can always be computed if the currentand initial cross-sectional areas are known.Governing Equations. The nonlinear rod is governed by the following equations:1.2.3.4.5.conservation of mass;conservation of momentum;conservation of energy;a measure of deformation, often called a strain-displacement equation;a constitutive equation, which describes material behavior and relates stress to ameasure of deformation;In addition, we require the deformation to be continuous, which is often called acompatibility requirement.
The governing equations and initial and boundary conditionsare summarized in Box 1.Conservation of mass. The equation for conservation of mass for a Lagrangianformulation can be written as (see Appendix A for an engineering derivation):or ρ( X,t ) J( X, t ) = ρ0 ( X ) J0 ( X)ρJ = ρ0 J0(2.2.10)where the second expression is given to emphasize that the variables are treated asfunctions of the Lagrangian coordinates. Conservation of matter is an algebraic equationonly when expressed in terms of material coordinates.
Otherwise, it is a partial differentialequation. For the rod, we can use Eq. (2.2.4) to write Eq. (2.2.5) asρFA= ρ 0A 0(2.2.11)where we have used the fact that J0 = 1.Conservation of momentum. Conservation of momentum is written in terms of thenominal stress and the Lagrangian coordinates as (a derivation is given in Appendix A):( A0 P ), X + ρ0 A0 b = ρ0 A0˙u˙(2.2.12)where the superposed dots denote the material time derivative. The material time derivativeof the velocity, the acceleration, is written as D2u Dt 2 .
The subscript following a commadenotes partial differentiation with respect to that variable, i.e.P( X, t ), X ≡∂P( X, t)∂X(2.2.13)Equation (2.2.12) is called the momentum equation, since it represents conservationof momentum. If the initial cross-sectional area is constant in space, the momentumequation becomes2-5T. Belytschko, Chapter 2, December 16, 1998P, X + ρ 0 b = ρ 0˙u˙(2.2.14)Equilibrium Equation. When the inertial term ρ 0˙u˙ vanishes, i.e. when the problem isstatic, the momentum equation becomes the equilibrium equation( A0 P) , X + ρ0 A0 b = 0(2.2.15)Solutions of the equilibrium equations are called equilibrium solutions. Some authors callthe momentum equation an equilibrium equation regardless of whether the inertial term isnegligible; since equilibrium usually connotes a body at rest or moving with constantvelocity, this nomenclature is avoided here.Energy Conservation.
The energy conservation equation for a rod of constant area isgiven byρ 0 w˙ int = F˙ P− qx,X + ρ 0 s(2.2.16)where qx is the heat flux, s is the heat source per unit mass and w˙ int is the rate of change ofinternal energy per unit mass. In the absence of heat conduction or heat sources, theenergy equation gives˙ int = F˙ Pρ0w(2.2.17)which shows that the internal work is given by the product of the rate of the deformation Fand the nominal stress P.
The energy conservation equation is not needed for the treatmentof isothermal, adiabatic processes.ConstitutiveEquations. The constitutive equations reflect the stresses which aregenerated in the material as a response to deformation. The constitutive equations relate thestress to the measures of strain at a material point. The constitutive equation can be writteneither in total form, which relates the current stress to the current deformation(P( X,t ) = SPF F ( X, t ), F˙ ( X, t ), etc., t ≤ t)(2.2.18)or in rate form(P˙ ( X,t ) = StPF F˙ ( X, t ), F ( x,t ), P ( X, t ), etc., t ≤ t)(2.2.19)Here S PF and StPF are functions of the deformation which give the stress and stress rate,respectively.
The superscripts are here appended to the constitutive functions to indicatewhich measures of stress and strain they relate.˙ and on other stateAs indicated in Eq. (2.2.18), the stress can depend on both F and Fvariables, such as temperature, porosity; “etc.” refers to these additional variables whichcan influence the stress. The prior history of deformation can also affect the stress, as in anelastic-plastic material; this is explicitly indicated in Eqs. (2.2.18-2.2.19) by letting theconstitutive functions depend on deformations for all time prior to t.
The constitutiveequation of a solid is expressed in material coordinates because the stress in a solid usually2-6T. Belytschko, Chapter 2, December 16, 1998depends on the history of deformation at that material point. For example, in an elasticsolid, the stress depends on strain at the material point. If there are any residual stresses,these stresses are frozen into the material and move with the material point.
Therefore,constitutive equations with history dependence should track material points and are writtenin terms of the material coordinates. When a constitutive equation for a history dependentmaterial is written in terms of Eulerian coordinates, the motion of the point must beaccounted for in the evaluation of the stresses, which will be discussed in Chapter 7.The above functions should be continuos functions of the independent variables.Preferably they should be continuously differentiable, for otherwise the stress is lesssmooth than the displacements, which can cause difficulties.Examples of constitutive equations are:(a) linear elastic material:total form:P( X, t) = E PFε( X,t) = EPF ( F ( X,t )–1)˙ (X ,t ) = E PFε˙ ( X, t) = E PFF˙ ( X,t )rate form:P(2.2.20)(2.2.21)(b) linear viscoelasticP( X, t) = E PF ( F ( X,t )–1) +αF˙ ( X,t )[orP =E]PF( ε + αε˙ )(2.2.22)For small deformations the material parameter E PF corresponds to Young’s modulus; theconstant α determines the magnitude of damping.Momentum equation in terms of displacements.
A single governing equation forthe rod can be obtained by substituting the relevant constitutive equation, i.e. (2.2.18) or(2.2.19), into the momentum equation (2.2.12) and expressing the strain measure in termsof the displacement by (2.2.6). For the total form of the constitutive equation (2.2.18), theresulting equation can be written as( A0 P(u, X , u˙ , X ,..)), X + ρ0 A0b = ρ0 A0u˙˙(2.2.23)which is a nonlinear partial differential equation (PDE) in the displacement u(X,t).
Thecharacter of this partial differential equation is not readily apparent from the above anddepends on the details of the constitutive equation. To illustrate one form of this PDE, weconsider a linear elastic material. For a linear elastic material, Eq. (2.2.20), the constitutiveequation and (2.2.23) yield( A0 E PF u, X ), X + ρ 0 A0b = ρ0 A0u˙˙(2.2.24)It can be seen that in this PDE, the highest derivatives with respect to the materialcoordinate X is second order, and the highest derivative with respect to time is also secondorder, so the PDE is second order in X and time t. If the stress in the constitutive equationonly depends on the first derivatives of the displacements with respect to X and t asindicated in (2.2.18) and (2.2.19), then the momentum equation will similarly be a secondorder PDE in space and time.2-7T.
Belytschko, Chapter 2, December 16, 1998For a rod of constant cross-section and modulus, if the body force vanishes, i.e. whenb = 0, the momentum equation for a linear material becomes the well known linear waveequationu, XX = c12 ˙u˙(2.2.25)where c is the wave speed relative to the undeformed configuration and given byPFc 2 = Eρ0(2.2.26)Boundary Conditions. The independent variables of the momentum equation are thecoordinate X and the time t.
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