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5 本构关系
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⎪⎪⎬⎫====)(ˆ)(ˆ)(ˆ)(ˆL L L L ηηεεq
q T T 在“纯力学”的研究中，本构关系常成为“应力－应变关系”
(1) 各向同性和各向异性
(3) 弹塑性和粘弹性
蠕变松弛
Newtonian fluid
Non-Newtonian fluid
Newtonian fluid
Viscoelastic fluid
5.2 本构关系的一般原理
确定性原理：物体在时刻t 的状态和行为由物体在该时刻以前的全部运动历史和温度历史所确定。
局部作用原理：物体中某一点在时刻t 的行为只由该点任意小邻域的运动历史所确定。
减退记忆原理：决定材料当前力学行为的各种变量的历史中，距今越远的历史对当前的力学行为影响越小。
客观性原理：物体的力学和热学的性质
不随观察者的变化而变化。
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目录1简介2基本假设3研究对象4古典连续介质力学5近代连续介质力学6主要分支学科简介研究连续介质宏观力学性状的分支学科。
宏观力学性状是指在三维欧氏空间和均匀流逝时间下受牛顿力学支配的物质性状。
连续介质力学对物质的结构不作任何假设。
它与物质结构理论并不矛盾，而是相辅相成的。
物质结构理论研究特殊结构的物质性状，而连续介质力学则研究具有不同结构的许多物质的共同性状。
连续介质力学的主要目的在于建立各种物质的力学模型和把各种物质的本构关系用数学形式确定下来，并在给定的初始条件和边界条件下求出问题的解答。
它通常包括下述基本内容：①变形几何学，研究连续介质变形的几何性质，确定变形所引起物体各部分空间位置和方向的变化以及各邻近点相互距离的变化，这里包括诸如运动，构形、变形梯度、应变张量、变形的基本定理、极分解定理等重要概念。
②运动学，主要研究连续介质力学中各种量的时间率，这里包括诸如速度梯度，变形速率和旋转速率，里夫林－埃里克森张量等重要概念。
③基本方程，根据适用于所有物质的守恒定律建立的方程，例如，热力连续介质力学中包括连续性方程、运动方程、能量方程、熵不等式等。
④本构关系。
⑤特殊理论，例如弹性理论、粘性流体理论、塑性理论、粘弹性理论、热弹性固体理论、热粘性流体理论等。
⑥问题的求解。
根据发展过程和研究内容，客观上连续介质力学已分为古典连续介质力学和近代连续介质力学。
基本假设连续介质力学的最基本假设是“连续介质假设”：即认为真实的流体和固体可以近似看作连续的，充满全空间的介质组成，物质的宏观性质依然受牛顿力学的支配。
这一假设忽略物质的具体微观结构（对固体和液体微观结构研究属于凝聚态物理学的范畴），而用一组偏微分方程来表达宏观物理量（如质量，数度，压力等）。
这些方程包括描述介质性质的方程（constitutive equations）和基本的物理定律，如质量守恒定律，动量守恒定律等。
研究对象固体：固体不受外力时，具有确定的形状。
R, ,V b ,a )b A ()a A ()b a (A ~~~~~~~~~∈βα∈∀⋅β+⋅α=β+α⋅~j~i ij ~i~i ij ~A,g g A g g A A ~⊗∂Π∂=⊗∂Π∂=∂Π∂=Π~i i mng g g g T TT ⊗⊗⊗∂=∂=Chapter 4Stress4.1 The Stress Vector and the Stress Principle of Euler and Cauchy1 External loads1) Body force: a distributed force per unit mass.in ΩThe body force is a long-range action.mFf m ΔΔ=→Δ~~lim2) Surface force: a force per unit area on surfaceon ∂ΩThe surface force is a short-range action, which manifests itself in the form of the contact forceSTp S ΔΔ=→Δ~~lim2 Interaction in the interior of continuum1) The Euler-Cauchy postulate(1) The interaction by a part applied on another part in the interior of a continuum is characterized by a stress vector field, which is defined upon an imagined interfacial surface S .(2) The stress vector field applied on the material occupying the space interior to S is equipollent to theaction of the exterior material upon it.p nS(3) P n remains unchanged for all surfaces having at X the same normal vector n . This means that P n is independent of the surfaces chosen as long as they all have at X the same normal vector.XP nnComments:The Euler-Cauchy postulate implies, (1) no long-range interaction exists in the deformable continuum;(2) Interactions within the continuum are momentless.2) Stress vector (Traction)ST p nS n ΔΔ=→Δ~~limNote:The stress vector refers to adeformed surface element whose unit normal vector is n and represents a contact force per unit area of the deformed surface.Discuss:(1) What differences are there between the surface force and the stress vector? (2) The force state of a mass particle is determined by magnitude and direction of all forces applied on it. How should we describe the force state of a point within continuum?4.2 The Cauchy Stress FormulaCauchy tetrahedronwith inclined face having an arbitrary orientation n ,constructed about some material point and to be shrunk onto that point in the limit to be taken.The Cauchy stress tensorThe Cauchy stress formulaii g p ~~~⊗=σ~~~np n ⋅=σThe components of the Cauchy stress tensor and their meaningsThe component of σare denoted by σij , whose first index indicates the direction of the force of action and the second index indicates the direction of the normal to the surface on which σij acts.x 1x 2x 3σ11σ21σ31σ22σ12σ33σ13σ23Note:(1) The Cauchy stress tensor is an invariant independent of the choice of basis vectors.(2) The Cauchy stress tensor isdefined on the current configuration. (3) For non-polar materials, the Cauchy stress tensor is symmetric.Normal stress and shear stress~~~~~nn p n n n ⋅⋅=⋅=σσ2~~~~22~nnn n n n p σσσστ−⋅⋅⋅=−=n I σσ≤Chapter 5Balance Equations In this chapter we examine the local forms of the various conservations laws as expressed in the initial and current configuration we have introduced. To expedite our development, we first discuss how integral representations of equilibrium can be converted to pointwise conservation principles, a process known as localization. The principle of energy conjugate and physical component are also given here.5.1 LocalizationSuppose we consider an arbitrary volume of material, V⊂Ω, in the reference configuration of a solid body, as depicted in Figure below: Suppose further that wecan establish the followinggeneral integral relationover this volume:The localized hypothesis asserts that (5-1) holds true for each and everysubvolume V of Ω,i.e.,)()(~~=∫ΩXdVXϕ)()(~~=∫V XdVXϕ(5-1)(5-2)According to the localization theorem, from Eq.(5-2) it yieldsϕ(X)= 0pointwise in Ω(5-3)Note: The procedure as same as the above can also be applied on a spatial domain.5.2 Balance of massConservation law of mass: The total mass of a closed continuum remains constant in motion.Integral representation:)(~=∫ΩxdvDtDρDifferential representation:)()(~~=∇⋅+⋅∇+∂∂v v tρρρ)(~=∇⋅+∂∂v tρρρρ=J 5.3 Conservation of momentumFor a given set of particles, the time rate ofchange of the total linear momentumequates to the vector sum of all the external forces acting on the particles of the set. This is expressed mathematically in the Equation below∫∫∫ΩΩΩ∂=+dv v DtDdv f ds p n ~~~ρρSubstituting the Cauchy formula into theEquation above yieldsApplying the divergence theorem and thetransport theorem gives∫∫∫ΩΩΩ∂=+⋅dv v DtDdv f ds n ~~~~ρρσ0)(~~~=−+∇⋅∫Ωdv DtvD f ρρσ By the localization theorem, it deduces toNote:1 The equilibrium equation of momentum isdefined in the current configuration.2 A nonlinearity is implicitly included in the equilibrium equation of momentum.DtvD f ~~~ρρσ=+∇⋅ Discuss: Can the equilibrium equation ofmomentum be represented by the second Piola-Kirchhoff stress?~0~0~~)(af F ρρ=+∇⋅Π⋅ 5.4 Balance of moment of momentumThe conservation of moment of momentumequates the time rate of change of the total moment of momentum for a set of particles to the vector sum of the moments of the external forces acting on the system. In the current configuration, we can write its balance of angular momentum via∫∫∫ΩΩ∂ΩΩ×=×+Ω×d v r Dt Dds p r d f r ~~~~~~ρρBy a series of operations and the localizedhypothesis, the equation above deduces to~~~~0:σσσ=⇔=∈TTThe conservation law of the moment ofmomentum guarantees that the stresstensor is symmetric under the condition that there does not exist the moment of body force and surface force. Therefore, there are only six independent components of stress—three normal components and three shear components.5.5 Physical ComponentsIn the polar coordinates, write out thebalance equation of two dimensional staticproblem in terms of the tensorial laws.Take the balance equation of momentum as example. If the body force is not concerned, the it is represented as;=βαβσ In order to calculate the Cristtoffelsymbol,we firstly give the transformation between the polar and Cartesian coordinates as follows:x =r cos θ, y =r sin θ.,=Γ+Γ+αγβγβγβαγββαβσσσ Therefore, we get g r =cos θe 1+sin θe 2g θ=–r sin θe 1+r cos θe 2g rr =1, g θθ=r 2, g r θ= g θr =0;g rr =1, g θθ=1/r 2, g r θ= g θr =0;Γθr θ=Γr θθ=–Γθθr =g θθ,r /2=r ;Γθr θ=Γr θθ=1/r, Γθθr =–r.Note: The Base vector has dimensions.As thus,However, in the standard textbook ofelasticity, the balance equation ofmomentum has the form below:030=+∂∂+∂∂=−+∂∂+∂∂rr r r r r r rr r θθθθθσθσσσσθσσ02101=+∂∂+∂∂=−+∂∂+∂∂rr r r r r r r r r r θθθθθσθσσσσθσσWhy ?This is because the physical dimensions ofbase vectors.Let us define the base vectors without dimensions as follows:.ˆ,ˆ~~~~><><==ααααααααgggg g gSo we haveβααββαββαααββααββααββαββαααββααβσσσσσσσ~~~~~~~~~~~~~ˆˆˆˆˆˆˆˆˆˆg g g gggg g g g g gg g g g ⊗=⊗=⊗=⊗=⊗=⊗=><><><>< Cleverly,These are the so-called physicalcomponents of tensor. By this concept, wecan obtain the standard balance equation of momentum..ˆ ,ˆ.ˆ ,ˆ><><><><><><><><====ββαααβαβββαααβαβββαααβαβββαααβαβσσσσσσσσggg g g g g g 5.6 Balance of energy1 Balance equation of energyThe global form of the equation of energyconservation may be written as∫∫∫∫∫ΩΩ∂ΩΩΩ∂Ω+⋅=⋅−Ω+Ω⋅+⋅d e v v Dt D sd h rd d f v ds p v )21(~~~~~~~~ρρρρDue to the fact that∫∫∫∫∫∫ΩΩΩΩΩ∂Ω∂Ω∇⋅σ⋅+σ=Ω∇⋅σ⋅+σ=Ω∇⋅σ⋅+σ∇⊗=Ω∇⋅σ⋅=⋅σ⋅=⋅d )](v d :[d )](v :L [d )](v :)v [(d )v (ds n v ds p v ~~~~~~~~~~~~~~~~~~~∫∫ΩΩ∂Ω∇⋅=⋅d h s d h ~~~∫∫∫ΩΩΩΩ+⋅=Ω+⋅=Ω+⋅d ea v d e v v Dt D d e v v Dt D][])(21[)21(~~~~~~&&ρρρρρρthe equation of energy conservation reducesto∫∫ΩΩΩρ=Ωσ+ρ+∇⋅−d ed )d :r h (~~~& By the localized hypothesis, one hased :r h ~~~&ρ=σ+ρ+∇⋅−2 Energy conjugateProblem:1 When a strain measure is given, is thechoice of stress arbitrary?Select a pairs of strain and stress according to what criterion?~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:1)(:1)(1)(1:)(1:1)(1)(1)(1:)(1::E JF d F J F L F tr J F F tr J F F J FJ F tr J F L tr J L F tr J L F J L d eTT T T T TT &&&&&&Σ=⋅⋅Σ=⋅⋅⋅Σ=⋅⋅Σ=Σ⋅==⋅=⋅⋅=⋅⋅=⋅===πππππσσρ Different strain measure needs differentstress to be associated with it in order toenable to uniquely determine the internal energy of body. This is the principle of energy conjugate.。
