Finite Element Analysis - Thermomechanics of Solids Part 19 doc

243 Inelastic and Thermoinelastic Materials 19.1 PLASTICITY Plasticity and thermoplasticity are topics central to the analysis of important appli- cations, such as metal forming, ballistics, and welding. The main goal of this section is to present a model of plasticity and thermoplasticity, along with variational and finite-element statements, accommodating the challenging problems of finite strain and kinematic hardening. 19.1.1 K INEMATICS Elastic and plastic deformation satisfies the additive decomposition (19.1) from which we can formally introduce strains: (19.2) The Lagrangian strain E satisfies the decomposition (19.3) Typically, plastic strain is viewed as permanent strain. As illustrated in Figure 19.1, in a uniaxial tensile specimen, the stress, S 11 , can be increased to the point A, and then unloaded along the path AB. The slope of the unloading portion is E, the same as that of the initial elastic portion. When the stress becomes equal to zero, there still is a residual strain, E i , which is identified as the inelastic strain. However, if instead the stress was increased to point C, it would encounter reversed loading at point D, which reflects the fact that the elastic region need not include the zero- stress value. 19.1.2 P LASTICITY We will present a constitutive equation for plasticity to illustrate how the tangent modulus is stated. The ideas leading to the equation will be presented subsequently in the section on thermoplasticity. With χχ χχ e = ITEN 22( D e ) and D e denoting the 19 DD D=+ ri , ∃= ∃ = ∃ = ∫∫∫ DD Ddt dt dt rr ii . ˙ .EE E== = ∫∫ FDF FDF FDF TT r T i ri dt dt 0749_Frame_C19 Page 243 Wednesday, February 19, 2003 5:34 PM © 2003 by CRC CRC Press LLC 244 Finite Element Analysis: Thermomechanics of Solids tangent-modulus tensor relating the elastic-strain rate to the stress rate (assuming a linear relation), the constitutive equation of interest is (19.4) In Equation 19.4, ΨΨ ΨΨ i is the yield function. ΨΨ ΨΨ i = 0 determines a closed convex surface in stress space called the yield surface. (We will see later that ΨΨ ΨΨ i also serves as a [complementary] dissipation potential.) The stress point remains on the yield surface during plastic flow, and is moving toward its exterior. The plastic strain rate, expressed as a vector, is typically assumed to be normal to the yield surface at the stress point. If the stress point is interior to, or moving tangentially on, the yield surface, only elastic deformation occurs. On all interior paths, for example, due to unloading, the response is only elastic. Plastic deformation induces “hardening,” corresponding to a nonvanishing value of C i . Finally, k is a history-parameter vector, introduced to represent dependence on the history of plastic strain, for example, through the amount of plastic work. The yield surface is distorted and moved by plastic strain. In Figure 19.2(a), the conventional model of isotropic hardening is illustrated in which the yield surface expands as a result of plastic deformation. This model is unrealistic in predicting a growing elastic region. Reversed plastic loading is encountered at much higher stresses than isotropic hardening predicts. An alternative is kinematic hardening (see Figure 19.2[b]), in which the yield surface moves with the stress point. Within a few percentage points of plastic strain, the yield surface may cease to encircle the origin. FIGURE 19.1 Illustration of inelastic strain. S 11 E 11 A C E E E F E i B D ˙˙˙ ˙ , ˙ ˙ . eCseCsC K C eii e ee i i ii i i ii HH == == = ∂ ∂       ∂ ∂ =− ∂ ∂ + ∂ ∂       ∂ ∂               − , , , χχ ΨΨΨΨΨΨΨΨΨΨ 1 k ss ek K s TT e 0749_Frame_C19 Page 244 Wednesday, February 19, 2003 5:34 PM Inelastic and Thermoinelastic Materials 245 A reference point interior to the yield surface, sometimes called the back stress, must be identified to serve as the point at which the elastic strain vanishes. Com- bined isotropic and kinematic hardening are shown in Figure 19.2(c). However, the yield surface contracts, which is closer to actual observations (e.g., Ellyin [1997]). FIGURE 19.2(a) Illustration of yield-surface expansion under isotropic hardening. FIGURE 19.2(b) Illustration of yield-surface motion under kinematic hardening. S ll S l S lll principal stresses S l S ll S lll path of stress point S lll S ll S l path of stress point 0749_Frame_C19 Page 245 Wednesday, February 19, 2003 5:34 PM 246 Finite Element Analysis: Thermomechanics of Solids The rate of movement must exceed the rate of contraction for the material to remain stable with a positive tangent modulus. Combining the elastic and inelastic portions furnishes the tangent-modulus tensor: (19.5) Suppose that in uniaxial tension, the elastic modulus is E e and the inelastic modulus-relating stress and inelastic strain increments are E i , and E i << E e . The total uniaxial modulus is then 19.2 THERMOPLASTICITY As in Chapter 18, two potential functions are introduced to provide a systematic way to describe irreversible and dissipative effects. The first is interpreted as the Helmholtz free-energy density, and the second is for dissipative effects. To accom- modate kinematic hardening, we also assume an extension of the Green and Naghdi (G-N) (1965) formulation, in which the Helmholtz free energy decomposes into reversible and irreversible parts, with the irreversible part depending on the “plastic strain.” Here, it also depends on the temperature and a workless internal state variable . 19.2.1 B ALANCE OF E NERGY The conventional equation for energy balance is augmented using a vector-valued, work- less internal variable, αα αα 0 , regarded as representing “microstructural rearrangements”: (19.6) FIGURE 19.2(c) Illustration of combined kinematic and isotropic hardening. S lll S ll S l path of stress point χχχχχχχχ=+ [] =+ − − − ei eie 1 1 1 CC[].I E EE i ie (/) . 1 + ρρ ori o h ˙ ˙˙ ˙ , χ 00000 =+−∇++se e TT T T qs ββαα 0749_Frame_C19 Page 246 Wednesday, February 19, 2003 5:34 PM Inelastic and Thermoinelastic Materials 247 where χ o is the internal energy per unit mass in the undeformed configuration and s = VEC ( S ), e = VEC ( E ), and ββ ββ 0 is the flux per unit mass associated with αα αα 0 . However, note that ββ ββ = 0 , thus ββ ββ i = −ββ ββ r . Also, c is the internal energy per unit mass, q 0 is the heat-flux vector referred to undeformed coordinates, and h is the heat input per unit mass, for simplicity’s sake, assumed independent of temperature. The state variables are E r , E i , T, and αα αα 0 . The next few paragraphs will go over some of the same ground as for damped elastomers in Chapter 18, except for two major points. In that chapter, the stress was assumed to decompose into reversible and irreversible portions in the spirit of elementary Voigt models. In the current context, the strain shows the decomposition in the spirit of the classical Maxwell model. In addition, as seen in the following, it proves beneficial to introduce a workless internal variable to give the model the flexibility to accommodate phenomena such as kinematic hardening. The Helmholtz free energy, φ , per unit mass and the entropy, η , per unit mass are introduced using (19.7) Now, (19.8) 19.2.2 E NTROPY -P RODUCTION I NEQUALITY The entropy now satisfies (19.9) Viewing φ r as a differentiable function of e r , T, and αα αα 0 , we conclude that (19.10) Extending the G-N formulation, let s * T = ρ 0 ∂ φ i / ∂ e i and assume that η i = −∂ φ i / ∂ T and ρ 0 ∂ φ i / ∂αα αα 0 = . Now, (19.11) The entropy-production inequality (see Equation 19.9) is now restated as (19.12) φ =− χη T. ∇ 00 0 0 0 TT q −= +− − −+ ρρηρηρφ 000 TTh T r T i ss ˙˙ ˙ ˙ ˙ ˙ .ee ββαα ρη ρ ρφ ρ η ρη 00000 000000 TT/T TTT/T ˙ ˙ ˙˙ ˙ ˙ ˙ . ≥− + + ≥−−+ + + − ∇∇ ∇ TT T ri hqq q TT se se T ββαα ∂∂ ∂∂ ∂∂ φφρηφ rr T rrrr T // /.es==− = T 000 ααββ ββ 0i T se* T ii i i i i T ==− = ρφ η φ ρφ 0000 ∂∂ ∂∂ ∂∂///. T ααββ () ˙ .sse TT i *−− ≥q T 00 0∇ T/T 0749_Frame_C19 Page 247 Wednesday, February 19, 2003 5:34 PM 248 Finite Element Analysis: Thermomechanics of Solids The inequality shown in Equation 19.12 can be satisfied if (19.13) The first inequality involves a quantity, s* = VEC(S*), with dimensions of stress. In the subsequent sections, s* will be viewed as a reference stress, often called the back stress, which is interior to a yield surface and can be used to characterize the motion of the yield surface in stress space. In classical kinematic hardening in which the hyperspherical yield surface does not change size or shape but just moves, the reference stress is simply the geometric center. If kinematic hardening occurs, as stated before, the yield surface need not include the origin even with small amounts of plastic deformation. Thus, there is no reason to regard as vanishing at the origin. Instead, = 0 is now associated with a moving-reference stress interior to the yield surface, identified here as the back stress s*. 19.2.3 DISSIPATION POTENTIAL As in Chapter 18, we introduce a specific dissipation potential, Ψ i , for which (19.14a) from which, with Λ i > 0 and Λ t > 0, (19.14b) On the expectation that properties governing heat transfer are not affected by strain, we introduce the decomposition into inelastic and thermal portions: (19.15a) where Ψ i represents mechanical effects and is identified in the subsequent sections. The thermal constitutive relation derived from the dissipation potential implies Fourier’s law: (19.15b) The inelastic portion is discussed in the following section. () ˙ sse TT i *−≥ −∇≥00 00 (a) T/T (b).q T ˙ e r ˙ e r ˙ / /ess ii i T it *=∋−∇=∂ ∋=− ρρ 0000 ΛΛ (i) T / T (ii) (iii),∂∂ ∂ΨΨ T q ρρ 000 0ΛΛ it (/ (/ ) .∂∂ ∂ >Ψ∋)∋+ ∂Ψ 0 qq ΨΨΨ Ψ=+ = it t t ρ 000 2 Λ qq T , −∇ = 00 T/T q /.Λ t 0749_Frame_C19 Page 248 Wednesday, February 19, 2003 5:34 PM Inelastic and Thermoinelastic Materials 249 19.3 THERMOINELASTIC TANGENT-MODULUS TENSOR The elastic strain rate satisfies a thermohypoelastic constitutive relation: (19.16) C r is a 9 × 9 second-order, elastic compliance tensor, and a r is the 9 × 1 thermoelastic expansion vector, with both presumed to be known from measure- ments. Analogously, for rate-independent thermoplasticity, we seek tensors C i and a i , depending on , e i , and T such that (19.17a) (19.17b) During thermoplastic deformation, the stress and temperature satisfy a thermo- plastic yield condition of the form (19.18) and Π i is called the yield function. Here, the vector k is introduced to represent the effect of the history of inelastic strain, , such as work hardening. It is assumed to be given by a relation of the form (19.19) The “consistency condition” requires that = 0 during thermoplastic flow, from which (19.20) We introduce a thermoplastic extension of the conventional associated flow rule, whereby the inelastic strain-rate vector is normal to the yield surface at the current stress point, (19.21) ˙ () ˙ .ess rr r *=−+ • CTa ∋ ˙ () ˙ eCss ii i *=−+ • a T ˙ []()() ˙ ess=+ − ++ • CC ri ri * aaT. Π ii i (, , , , ) ,∋=e k T η 2 0 ˙ e i ˙ (,,) ˙ .kK k= ee ii T ˙ Π i d d d d d d d d d d ii i i ii i i ΠΠ ΠΠΠ ∋ ∋ ˙ ˙ ˙ ˙ ˙ .++++ = e e k k T T i η η 0 ˙ ˙ e ii i T i i d d d d =       =Λ Π Λ Π ∋ (a) (b). i i η η 0749_Frame_C19 Page 249 Wednesday, February 19, 2003 5:34 PM 250 Finite Element Analysis: Thermomechanics of Solids Equation 19.14a suggests that the yield function may be identified as the dissi- pation potential: Π i = ρ 0 Ψ i . Standard manipulation furnishes (19.22) and H must be positive for Λ i to be positive. Note that, in the current formulation, the dependence of the yield function on temperature accounts for c i . The thermody- namic inequality shown in Equation 19.13a is now satisfied if H > 0. Next, note that s* depends on e i , T, and αα αα 0 since s* T = ρ 0 ∂ φ i /∂e i . For simplicity’s sake, we neglect dependence on αα αα 0 and assume that a relation of the following form can be measured for s*: (19.23) From Equations 19.16 and 19.17, the thermoinelastic tangent-modulus tensor and thermal thermomechanical vector are obtained as (19.24a) (19.24b) If appropriate, the foregoing formulation can be augmented to accommodate plastic incompressibility. 19.3.1 EXAMPLE We now provide a simple example using the Helmholtz free-energy density function and the dissipation-potential function to derive constitutive relations. The following expression involves a Von Mises yield function, linear kinematic hardening, linear work hardening, and linear thermal softening. ˙ ˙ ˙ ˙ ˙ ˙ e ii i i i T i i ii ii i T i i ii i i HH T HH =∋+ = ∋+ = ∂ ∂∋       ∂ ∂∋ == ∂ ∂∋       ∂ ∂ = ∂ ∂       =− ∂ ∂ + ∂ ∂       ∂ C T c T T c ab ab ek K T η 2 2 C ΨΨ ΨΨ ΨΨΨ ΨΨΨΨ i T i i i ∂∋       + ∂ ∂ ∂ ∂         η 2 T ˙˙ ˙ /se ee e* i ii T T ii =+ =       =∂ ∂ ∂ΓΓΓΓ ϑϑ T T. I ∂ ∂ ∂ ∂ ΨΨ 2 ˙˙ ˙ es=+C aT CCCII C C CC CCI C =+ −+ =++ + − + + − − ()[()] [()()()] ri i i ri r ii r i i ΓΓΓΓ ΓΓΓΓ 1 1 aaa a ϑ 0749_Frame_C19 Page 250 Wednesday, February 19, 2003 5:34 PM Inelastic and Thermoinelastic Materials 251 i. Helmholtz free-energy density: (19.25) in which is a known constant. From Equation 19.28, (19.26) Finally, (19.27) ii. Dissipation potential: (19.28) (19.29) Straightforward manipulations serve to derive (19.30) (19.31) Consider a two-stage thermomechanical loading, as illustrated schematically in Figure 19.3. Let S I , S II , S III denote the principal values of the 2 nd Piola-Kirchhoff stress, and suppose that S III = 0. In the first stage, with the temperature held fixed at T 0 , the stresses are applied proportionally well into the plastic range. The center of the yield surface moves along a line in the (S 1 , S 2 ) plane, and the yield surface expands as it moves. In the second stage, suppose that the stresses S 1 and S 2 are fixed, but that the temperature increases to T 1 and then to T 2 and T 3 . The plastic strain must increase, thus, the center of the yield surface moves. In addition, strain hardening tends to cause the yield surface to expand, while the increased temperature tends to make it contract. However, in this case, thermal softening must dominate strain hardening, and contraction must occur since the center of the yield surface must move further along the path shown even as the yield surface continues to “kiss” the fixed stresses S I and S II . φφφ φ φ =+ = =−−+ ′ − −− ri i ii rrrr r r r k ln T T c T (T / T ) ρ ρρ 03 0 11 00 0 21 ee CC T TT aee e/() ( ) ′ c r ∋= ∂ ∂ = − − − ρφ 0 1 0 ( / ) [ ( )]. rr rrr ee T C a TT cT T c r r =− ∂ ∂ = ′ 2 2 φ ΨΨΨ Ψ=+ = it t t T ρ 000 2 Λ qq Ψ i T i kkkk k=∋∋− + − − = =∋ T [ ( )] ˙ ˙ 01 2 0 0TT e Hk kk kkk=∋∋= +− − 110120 T [()]TT Cab T T T iiii HkH k H= ∋∋ ∋∋ === ∋ ∋∋ c 2 2 2 0749_Frame_C19 Page 251 Wednesday, February 19, 2003 5:34 PM 252 Finite Element Analysis: Thermomechanics of Solids Unfortunately, accurate finite-element computations in plasticity and thermo- plasticity often require close attention to the location of the front of the yielded zone. This front will occur within elements, essentially reducing the continuity order of the fields (discontinuity in strain gradients). Special procedures have been devel- oped in some codes to address this difficulty. The shrinkage of the yield surface with temperature provides an element of the explanation of the phenomenon of adiabatic shear banding, which is commonly encountered in some materials during impact or metal forming. In rapid processes, plastic work is mostly converted into heat and on into high temperatures. There is not enough time for the heat to flow away from the spot experiencing high deformation. However, the process is unstable while the stress level is maintained. Namely, as the material gets hotter, the rate of plastic work accelerates, thanks to the softening evident in Figure 19.3. The instability is manifested in small, peri- odically spaced bands, in the center of which the material is melted and resolidified, usually in a much more brittle form than before. These bands can nucleate brittle failure. 19.4 TANGENT-MODULUS TENSOR IN VISCOPLASTICITY The thermodynamic discussion in the previous section applies to thermoinelastic deformation, for which the first example given concerned quasi-static plasticity and thermoplasticity. However, it is equally applicable when rate sensitivity is present, in which case viscoplasticity and thermoviscoplasticity are attractive models. An example FIGURE 19.3 Effect of load and temperature on yield surface. S lll S ll S l T 0 T 0 T 0 T 0 T 0 T 0 T 1 T 2 T 3 T 4 0749_Frame_C19 Page 252 Wednesday, February 19, 2003 5:34 PM [...]... flow occurs at a rate dependent on the distance to the exterior of the reference surface This situation is illustrated in Figure 19. 4 Sll S ∗ stress point Sl Slll FIGURE 19. 4 Illustration of reference surface in viscoplasticity 0749_Frame_C19 Page 254 Wednesday, February 19, 2003 5:34 PM 254 Finite Element Analysis: Thermomechanics of Solids It should be evident that viscoplasticity and thermoviscoplasticity... Bonora (199 7) The contribution to the ˙ irreversible entropy production can be introduced in the form D D ≥ 0, in which D ˙ is the “force” associated with flux D Positive dissipation is assured if we assume D = ∂Ψd , ˙ ∂D Ψd = 1 ˙ Λ (e , T, k ) D2 , 2 d i Λ d (ei , T, k ) > 0 (19. 38) 0749_Frame_C19 Page 256 Wednesday, February 19, 2003 5:34 PM 256 Finite Element Analysis: Thermomechanics of Solids ˙... f = Ξ( f , ei , T ), (19. 37) 0749_Frame_C19 Page 255 Wednesday, February 19, 2003 5:34 PM Inelastic and Thermoinelastic Materials 255 S11 Ep1 Sy1 Ep3 Ep2 Sy3 Sy2 E2 E3 E1 E11 FIGURE 19. 5 Illustration of effect of damage on elastic-plastic properties for which several specific forms have been proposed To this point, a nominal stress is used in the sense that the reduced ability of material to support... critical value, the element is considered to have failed It is then removed from the mesh (considered to be no longer supporting the load) The displacement and temperature fields are recalculated to reflect the element deletion There are two different schools of thought on the suitable notion of a damage parameter One, associated with Gurson (197 7), Tvergaard (198 1), and Thomasson (199 0), considers damage... ∂Ψi     ∂∋  T dVo (19. 36) 19. 5 CONTINUUM DAMAGE MECHANICS Ductile fracture occurs by processes associated with the notion of damage An internal-damage variable is introduced that accumulates with plastic deformation It also manifests itself in reductions in properties, such as the experimental values of the elastic modulus and yield stress When the damage level in a given element reaches a known... and thermal shrinkage of the reference-yield surface The tangent-modulus matrix now reduces to elastic relations, and viscoplastic effects can be treated as an initial force (after canceling the variation) since χ  k ˙  ˙ ˙ ⋅s = χ r e + χ r α r T −  Γ − r  < 1 − > Ψi µv    ∂Ψi     ∂∋  ∂Ψi ∂∋ T  ∂Ψi     ∂∋  T (19. 35) In particular, the Incremental Principle of Virtual Work is now... Element Analysis: Thermomechanics of Solids ˙ An example of a satisfactory function is Λd (ei , T, k) = Λdo ∫(s − s*) ei dt, Λdo a positive constant, showing damage to depend on plastic work Specific examples of constitutive relations for damage are given, for example, in Bonora (199 7) At the current values of the damage parameter, the finite -element equations are solved for the nodal displacements,... inelastic strains can be computed This information can then be used to update the damage-parameter values Upon doing so, the damage-parameter values are compared to critical values As stated previously, if the critical value is obtained, the element is deleted The path of deleted elements can be viewed as a crack The code LS-DYNA Ver 9.5, (2000) incorporates a material model that includes viscoplasticity...0749_Frame_C19 Page 253 Wednesday, February 19, 2003 5:34 PM Inelastic and Thermoinelastic Materials 253 of a constitutive model, for example, following Perzyna (197 1), is given in undeformed coordinates as ˙ µω ei =< 1 − k > Ψi  ∂Ψi     ∂∋  ∂Ψi ∂∋ T  ∂Ψi     ∂∋  T , =  Ψi Ψi  0,  k ≥0 Ψi otherwise, if 1− (19. 32) and Ψi(∋, ei, k, T, ηi) is a... process: nucleation of voids, their subsequent growth, and their coalescence to form a macroscopic defect The coalescence event is used as a criterion for element failure The parameter used to measure damage is the void-volume fraction f Models and criteria for the three processes have been formulated For both nucleation and growth, evolution of f is governed by a constitutive equation of the form ˙ f . S lll path of stress point S lll S ll S l path of stress point 0749_Frame_C19 Page 245 Wednesday, February 19, 2003 5:34 PM 246 Finite Element Analysis: Thermomechanics of Solids The rate of. 0749_Frame_C19 Page 251 Wednesday, February 19, 2003 5:34 PM 252 Finite Element Analysis: Thermomechanics of Solids Unfortunately, accurate finite -element computations in plasticity and thermo- plasticity. (b). i i η η 0749_Frame_C19 Page 249 Wednesday, February 19, 2003 5:34 PM 250 Finite Element Analysis: Thermomechanics of Solids Equation 19. 14a suggests that the yield function may be identified as the dissi- pation