Belytschko T. - Introduction (779635), страница 27

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(E3.9.3-E3.9.5) with Eq. (3.4.14), whichgivesS11 =l 0  Aσ x l  Ao (E3.9.8)where the quantity in the parenthesis can be recognized as the nominal stress. Itcan be seen from the above that it is difficult to ascribe a physical meaning to thePK2 stress. This, as will be seen in Chapter 5, influences the selection of stressmeasures for plasticity theories, since yield functions must be described in termsof physical stresses. Because of the nonphysical nature of the nominal and PK2stresses, it is awkward to formulate plasticity in terms of these stresses.3.5 CONSERVATION EQUATIONS3.5.1ConservationLaws.

One group of the fundamental equations ofcontinuum mechanics arises from the conservation laws. These equations mustalways be satisfied by physical systems. Four conservation laws relevant tothermomechanical systems are considered here:1. conservation of mass2. conservation of linear momentum, often called conservation ofmomentum3. conservation of energy4. conservation of angular momentumThe conservation laws are also known as balance laws, e.g. the conservation ofenergy is often called the balance of energy.The conservation laws are usually expressed as partial differentialequations (PDEs).

These PDEs are derived by applying the conservation laws toa domain of the body, which leads to an integral equation. The followingrelationship is used to extract the PDEs from the integral equation:if f ( x,t) is C −1 and∫ f ( x,t )dΩ= 0 for any subdomain Ω of ΩΩand time t ∈[0, t ] , thenf ( x,t) = 0 in Ω for t ∈[0, t ](3.5.1)In the following, Ω is an arbitrary subdomain of the body under consideration.Prior to deriving the balance equations, several theorems useful for this purposeare derived.3.5.2 Gauss’s Theorem.In the derivation of the governing equations,Gauss's theorem is frequently used.

This theorem relates integrals of differentdimensions: it can be used to relate a contour integral to an area integral or asurface integral to a volume integral.The one dimensional form of Gauss’stheorem is the fundamental theorem of calculus, which we used in Chapter 2.3-39T. Belytschko, Continuum Mechanics, December 16, 199840Gauss’s theorem states that when f(x) is a piecewise continuouslydiffrentiable, i.e. C1 function, then∂f (x )dΩ=∂xiΩ∫∫Ω0∫ f ( x)ni dΓ∫ ∇f (x )dΩ= ∫ f (x )ndΓorΓ∂f (X)dΩ0 =∂XiΩ∫ f (X )ni dΓ00Γ∫orΩ0Γ0(3.5.2a)∇X f (X)dΩ 0 =∫ f ( X)n 0 dΓ0 (3.5.2b)Γ0As seen in the above, Gauss's theorem applies to integrals in both the current andreference configurations.The above theorem holds for a tensor of any order; for example if f(x) isreplaced by a tensor of first order, then∂gi (x )dΩ= ∫ gi ( x)nidΓ∂xiΩΓ∫∫ ∇⋅ g(x )dΩ = ∫ n⋅g(x )dΓorΩ(3.5.3)Γwhich is often known as the divergence theorem. The theorem also holds forgradients of the vector field:∂gi ( x)dΩ= ∫ gi (x)n j dΓ∂xjΩΓ∫∫ ∇g( x)dΩ = ∫ n⊗ g(x )dΓorΩ(3.5.3b)Γand to tensors of arbitrary order.If the function f (x) is not continuously differentiable, i.e.

if its derivativesare discontinuous along a finite number of lines in two dimensions or on surfacesin three dimensions, then Ω must be subdivided into subdomains so that thefunction is C1 within each subdomain. Discontinuities in the derivatives of thefunction will then occur only on the interfaces between the subdomains. Gauss’stheorem is applied to each of the subdomains, and summing the results yields thefollowing counterparts of (3.5.2) and (3.5.3):∂f∫ ∂xi dΩ = ∫ fni dΓ+ ∫ΩΓ∂g∫ ∂xii dΩ = ∫ gi nidΓ + ∫fni dΓΓintΩΓgi ni dΓΓintwhere Γ int is the set of all interfaces between these subdomains andn⋅g are the jumps defined byf =fA(3.5.4)− fBfand(3.5.5a)()()n⋅g = gi ni = giAniA + giB niB = giA − giB niA = giB − giA niB3-40(3.5.5b)T. Belytschko, Continuum Mechanics, December 16, 199841where A and B are a pair of subdomains which border on the interface Γ int , n Aand n B are the outward normals for the two subdomains and fA and fB are thefunction values at the points adjacent to the interface in subdomains A and B,respectively.

All the forms in (3.5.5b) are equivalent and make use of the fact thaton the interface, n A =-n B. The first of the formulas is the easiest to rememberbecause of its symmetry with respect to A and B.3.5.3Material Time Derivative of an Integral and Reynold’sTransportTheorem. The material time derivative of an integral is the rate ofchange of an integral on a material domain.

A material domain moves with thematerial, so that the material points on the boundary remain on the boundary andno flux occurs across the boundaries. A material domain is analogous to aLagrangian mesh; a Lagrangian element or group of Lagrangian elements is a niceexample of a material domain. The various forms for material time derivatives ofintegrals are called Reynold;s transport theorem, which is employed in thedevelopment of conservation laws.The material time derivative of an integral is defined byD1f dΩ= lim ∆t ( ∫ f ( x, τ + ∆t)dΩ x − ∫ f ( x, τ )dΩx )∫∆t →0Dt ΩΩΩτ +∆t(3.5.6)τwhere Ωτ is the spatial domain at time τ and Ωτ +∆t the spatial domain occupiedby the same material points at time τ +∆ t .

The notation on the left hand side is alittle confusing because it appears to refer to a single spatial domain. However, inthis notation, which is standard, the material derivative on the integral implies thatthe domain refers to a material domain. We now transform both integrals on theright hand side to the reference domain using (3.2.18) and change the independentvariables to the material coordinates, which givesD1f dΩ= lim ∆t ( ∫ f (X,τ +∆ t )J (X,τ +∆t )dΩ0 − ∫ f ( X,τ ) J ( X, τ ) dΩ0 ) (3.5.7)∫∆t →0Dt ΩΩ0Ω0The function is now f (φ(X,t ),t ) ≡ f o φ , but we adhere to our convention that thesymbol represents the field and leave the symbol unchanged.Since the domain of integration is now independent of time, we can pullthe limit operation inside the integral and take the limit, which yieldsf dΩ= ∫ ∂ ( f ( X,t ) J ( X,t ))dΩ 0∫∂tDtDΩ(3.5.9)Ω0The partial derivative with respect to time in the integrand is a material timederivative since the independent space variables are the material coordinates.

Wenext use the product rule for derivatives on the above:D∂ ∂f∂J f dΩ= ∫ ( f ( X,t )J ( X,t ))dΩ0 = ∫  J + f  dΩ 0∫ ∂tDt Ω∂t∂t Ω0Ω03-41T. Belytschko, Continuum Mechanics, December 16, 199842Bearing in mind that the partial time derivatives are material time derivativesbecause the independent variables are the material coordinates and time, we canuse (3.2.19) to obtain ∂f ( X,t )D∂v f dΩ = ∫ J + fJ i  dΩ0∫Dt Ω∂t∂xi Ω0 (3.5.12)We can now transform the RHS integral to the current domain using (3.2.18) andchange the independent variables to an Eulerian description, which gives Df ( x,t )D∂vi f(x,t)dΩ=+f dΩ∫∫ DtDt Ω∂xi Ω(3.5.11)where the partial time derivative has been changed to a material time derivativebecause of the change of independent variables; the material time derivativesymbol has been changed with the change of independent variables, sinceDf (x,t ) Dt ≡∂ f (X,t ) ∂t as indicated in (3.2.8).An alternate form of Reynold’s transport theorem can be obtained byusing the definition of the material time derivative, Eq.

(3.2.12) in (3.5.11). ThisgivesDDt∂f∂f∂v∫ f dΩ =∫( ∂t + vi ∂x i + ∂xiiΩ∫f ) dΩ = (ΩΩ∂f ∂( vi f )+) dΩ∂t∂x i(3.5.13)which can be written in tensor form asDDt∫∫f dΩ = (ΩΩ∂f+ div( vf )) dΩ∂t(3.5.14)Equation (3.5.14) can be put into another form by using Gauss’s theorem on thesecond term of the RHS, which givesDDt∂f∫ f dΩ= ∫ ∂t dΩ + ∫ fvini dΓΩΩΓorDDt∂f∫ f dΩ= ∫ ∂t dΩ + ∫ fv⋅ndΓΩΩ(3.5.15)Γwhere the product fv is assumed to be C1 in Ω . Reynold’s transport theorem,which in the above has been given for a scalar, applies to a tensor of any order.Thus to apply it to a first order tensor (vector) gk , replace f by gk in Eq. (3.5.14),which gives ∂g ∂ (vi gk ) Dgk dΩ = ∫  k + dΩ∫Dt Ω∂t∂xiΩ3-42(3.5.16)T. Belytschko, Continuum Mechanics, December 16, 19983.5.5 Massby43Conservation.

The mass m(Ω ) of a material domain Ω is given∫m(Ω ) = ρ( X, t) dΩ(3.5.17)Ωwhere ρ( X,t ) is the density. Mass conservation requires that the mass of amaterial subdomain be constant, since no material flows through the boundariesof a material subdomain and we are not considering mass to energy conversion.Therefore, according to the principle of mass conservation, the material timederivative of m(Ω ) vanishes, i.e.Dm D=ρ dΩ = 0Dt Dt∫(3.5.18)ΩApplying Reynold’s theorem, Eq. (3.5.11), to the above yields Dρ∫  Dt + ρ div(v ) dΩ= 0(3.5.19)ΩSince the above holds for any subdomain Ω , it follows from Eq.(3.5.1) thatDρ+ρ div(v) = 0 orDtDρ+ ρ vi ,i = 0 or ˙ρ + ρvi, i = 0Dt(3.5.20)The above is the equation of mass conservation, often called the continuityequation. It is a first order partial differential equation.Several special forms of the mass conservation equation are of interest.When a material is incompressible, the material time derivative of the densityvanishes, and it can be seen from equation (3.5.20) that the mass conservationequation becomes:div(v ) = 0vi, i = 0(3.5.21)In other words, mass conservation requires the divergence of the velocity field ofan incompressible material to vanish.If the definition of a material time derivative, (3.2.12) is invoked in(3.5.20), then the continuity equation can be written in the form∂ρ∂t∂ρ+ ρ ,i vi + ρvi,i = ∂ t + ( ρvi ) ,i = 0(3.5.22)This is called the conservative form of the mass conservation equation.

It is oftenpreferred in computational fluid dynamics because discretizations of the aboveform are thouught to more accurately enforce mass conservation.3-43T. Belytschko, Continuum Mechanics, December 16, 199844For Lagrangian descriptions, the rate form of the mass conservationequation, Eq. (3.5.18), can be integrated in time to obtain an algebraic equationfor the density. Integrating Eq. (3.5.18) in time gives∫ ρ dΩ= constant = ∫ ρ 0dΩ 0Ω(3.5.23)Ω0Transforming the left hand integral in the above to the reference domain by(3.2.18) gives∫ ( ρJ − ρ0 ) dΩο = 0(3.5.24)Ω0Then invoking the smoothness of the integrand and Eq. (3.5.1) gives the followingequation for mass conservationρ (X, t ) J ( X, t ) = ρ 0 (X)ρJ = ρ 0or(3.5.25)We have explicitly indicated the independent variables on the left to emphasizethat this equation only holds for material points; the fact that the independentvariables must be the material coordinates in these equations follows from the factthat the integrand and domain of integration in (3.5.24) must be expressed for amaterial coordinate and material subdomain, respectively.As a consequence of the integrability of the mass conservation equation inLagrangian descriptions, the algebraic equation (3.5.25) are used to enforce massconservation in Lagrangian meshes.

In Eulerian meshes, the algebraic form ofmass conservation, Eq. (3.5.25), cannot be used, and mass conservation isimposed by the partial differential equation, (3.5.20) or (3.5.22), i.e. thecontinuity equation.3.5.5 Conservation of Linear Momentum.The equation emanatingfrom the principle of momentum conservation is a key equation in nonlinear finiteelement procedures. Momentum conservation is a statement of Newton’s secondlaw of motion, which relates the forces acting on a body to its acceleration. Weconsider an arbitrary subdomain of the body Ω with boundary Γ . The body issubjected to body forces ρb and to surface tractions t, where b is a force per unitmass and t is a force per unit area.

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