A note on the Mooney-Rivlin material moel I-Shih Liu Instituto e Matemática Universiae Feeral o Rio e Janeiro 2945-97, Rio e Janeiro, Brasil Abstract In finite elasticity, the Mooney-Rivlin material moel for the Cauchy stress tensor T in terms of the left Cauchy-Green strain tensor B is given by T = pi + s B + s 2B, where p is the pressure an s, s 2 are two material constants. It is usually assume that s > an s 2, known as E-inequalities, base on the assumption that the free energy function be positive efinite for any eformation. In this note, we shall relax this assumption an with a thermoynamic stability analysis, prove that s 2 nee not be negative so that some typical behavior of materials uner contraction can also be moele. Keywors: Mooney-Rivlin material, E-inequalities, uniaxial extension, stability analysis, stress-strain curve. Deicate to Ingo Müller on the occasion of his 75 th birthay Introuction In finite elasticity, the Mooney-Rivlin material moel for incompressible isotropic elastic solis is given by the Cauchy stress tensor T in terms of the left Cauchy-Green strain tensor B, T = pi + s B + s 2 B, where p is the pressure an s, s 2 are two material constants. Base on the assumption that the free energy function be positive efinite for any eformation, it is usually assume that s > an s 2, known as E-inequalities, which is strongly supporte from experimental evience for rubber-like materials. However, the assumption s 2 gives rise to a very restrictive behavior in boies uner contraction, that the compressive stress-strain curves are increasing an concave upwar, which means that it is getting harer an harer to contract as the compressive strain increases. This behavior may seem to be reasonable for rubber-like materials an it is also known in soil mechanics as the locking behavior observe in ry san uner confine compression. Nevertheless, the locking behavior is not typical for most materials, such as soil an rock, that sustain essentially compressive stresses. In this note, we shall relax the assumption that lea to the E-inequalities an with a thermoynamic stability analysis for an isothermal uniaxial experiment, prove that s 2 nee
not be negative so that some typical behavior of materials uner contraction can also be moele for moerate compressive strains as Mooney-Rivlin materials. 2 Mooney-Rivlin thermoelastic materials We shall briefly re-erive a constitutive equation for moerate strains of an incompressible isotropic thermoelastic soli, starting start with the free energy function ψ, an the Cauchy stress T given by ψ = ψθ, I B, II B, T = p I + 2ρ ψ B, B where θ is the temperature, ρ the mass ensity an p the ineterminate pressure for incompressibility, while {I B, II B, III B = } are the three principal invariants of the left Cauchy- Green tensor B an I is the ientity tensor. By Taylor series expansion, we can write ψθ, I B, II B = ψ + ψ I B 3 + ψ 2 II B 3 + oδ 2, δ 2 2, δ δ 2, 2 where ψ k = ψ k θ for k =,, 2, an oδ 2, δ 2 2, δ δ 2 stans for the secon orer terms in I B 3 = oδ an II B 3 = oδ 2, say, respectively of the orer of δ < an δ 2 <. Since the eformation graient F = I + H, we have B = F F T = I + A, where A = H + H T + HH T. Therefore, if we assume that the isplacement graient H is of the orer of δ <, write, H = oδ, it follows that A is of the same orer, A = oδ. Note that for B = I + A, one can show that I B = 3 + I A, In particular, for et B = III B =, we have II B = 3 + 2 I A + II A, III B = + I A + II A + III A. I A = II A III A. Therefore, since II A = oδ 2, of the orer δ 2, so is I A = oδ 2, an we have I B 3 = I A = oδ = oδ 2, II B 3 = 2 I A + II A = oδ 2 = oδ 2, which implies oδ 2, δ 2 2, δ δ 2 = oδ 4. Consequently, from 2 we have ψθ, I B, II B = ψ + ψ I B 3 + ψ 2 II B 3 + oδ 4. In other wors, the free energy function, ψθ, I B, II B = ψ + ψ I B 3 + ψ 2 II B 3, 3 is a general representation with an error in the fourth orer of the isplacement graient. By the use of the relations I B B = I, II B B = I BI B, 2
from, we obtain where with T + p I = s B + s 2 B 2 I B B, 4 s θ = 2ρψ θ, s 2 θ = 2ρψ 2 θ. 5 Note that the constitutive equation 4 can be written as T = p I + t B + t 2 B 2, t = s s 2 I B, t 2 = s 2. Consequently, t is not a function of θ only. However, by the use of the Cayley-Hamilton theorem, with III B =, B 2 I B B = B II B I, it gives T = p I + s B + s 2 B, in which the term II B I is absorbe into the ineterminate pressure p I. In this case, the material parameters s an s 2 are functions of the temperature only. Therefore we conclue that: Mooney-Rivlin material. The constitutive equation T = p I + s B + s 2 B, 6 with s θ an s 2 θ, is general enough to account for incompressible thermoelastic isotropic solis at moerate strain with an error in the fourth orer of the isplacement graient in free energy. We claim no originality in the erivation of this result, since similar erivation can be foun elsewhere for example, see []. However, we have inclue it here to facilitate later iscussions on the moel. 3 Isothermal uniaxial experiment In orer to examine some possible restrictions on the values of the material parameters s an s 2, we shall consier a simple isothermal uniaxial experiment: For the uniaxial eformation λ >, x = λ X, y = Y, z = Z, λ λ of a cyliner V, with cross-section area A an length X L, in the reference state κ, we have B = λ 2 e x e x + λ e y e y + e z e z, I B = λ 2 + 2 λ, II B = λ 2 + 2λ, where {e x, e y, e z } are the base vectors in the Cartesian coorinate system. The free energy function 3 in this case takes the following form, ψλ = ψ + ψ λ 2 + 2 λ 3 + ψ 2 λ 2 + 2λ 3. 7 3
One can easily show that both λ 2 + 2 λ 3 an λ 2 +2λ 3 are non-negative for any λ >. Therefore, the free energy function ψλ is positive efinite if ψ >, ψ, ψ 2, which, from 5, justifies the well-known empirical E-inequalities, s >, s 2, 8 strongly supporte by experimental evience for rubber-like materials see [2] an Sect. 55 of [3]. 3. Thermoynamic stability analysis To analyze thermoynamic stability of an isothermal uniaxial experiment, we start with the energy equation an the entropy inequality, ρε + t V 2ẋ ẋv + q T ẋ n a =, V ρη v + q n a. t θ V Elimination of the heat flux q between these relations leas to ρψ + t 2ẋ ẋv ẋ T n a, V V where the free energy function ψ = ε θη. Assuming the experiment is quasi-static by neglecting the kinetic energy, we can rewrite the above inequality in the reference state as t V ρψ v κ ẋ T κ n κ a κ, V κ V κ where T κ is the Piola Kirchhoff stress on the reference bounary V κ. In this case, we have the following bounary conitions: T κ = on the lateral bounary of the cyliner V κ an is constant on the top an the bottom, where the velocities are given by ẋ X= =, ẋ X=L = λl e x. Therefore, it follows after integration that ρψ T κ xx λ AL. t If we call Ψλ = ρψλ T κ xx λ the available energy, as the sum of the free energy an the potential energy, then we have Ψ t. This allows us to establish a stability criterion. If we assume the equilibrium state at the eformation λ is a stable state, then any small perturbation from this state will eventually return to this state as time tens to infinity. Suppose that such a perturbation is represente by a ynamic eformation process λt uner the same bounary conitions, it follows that lim λt = λ, t an since Ψλt is a ecreasing function of time, it implies that Ψλt Ψλ t >. In other wors, the available energy Ψλt attains its minimum at λ. Therefore, we arrive at the following criterion: 4
A thermoynamic stability criterion. A state at the uniaxial eformation λ is stable as t if the value of the available energy function Ψλ attains its minimum at this state. The minimization of Ψλ at given λ requires that Ψλ λ =, an From the expression 7, the first conition implies from 5 that Ψλ λ 2 Ψλ λ 2. = 2ρψ λ λ 2 2ρψ 2 λ 3 T κ xx =, 9 T κ xx = s λ λ 2 + s 2 λ 3. This is the axial stress per unit original area of the cyliner in the experiment, which can also be obtaine irectly from the constitutive equation 6 for the uniaxial eformation λ. The secon conition, on the other han, is more interesting, we have 2 Ψλ λ 2 = 2ρψ + 2 λ 3 Since ψ 2 /ψ = s 2 /s by assuming s >, it follows that 3 + 2ρψ 2. λ4 s 2 s 3 2λ + λ4. Obviously, this is a restriction for possible values of the material parameters s an s 2. In particular, for λ =, it gives s s 2, which, in turns, implies that the requirement is always satisfie for uniaxial extension, λ >. On the other han, for uniaxial contraction as λ, it requires s 2. Therefore, the E-inequalities satisfies the conition for any eformation λ >. However, for most materials, it is almost impossible to reach the limit at λ in uniaxial contraction without structure failure, an since the Mooney-Rivlin material is a goo material moel for thermoelastic soli with moerate strain, say, λ λ L, λ U, for some λ U > an > λ L >, the conition requires that s 2 γ s, for γ = 3 2λ L + λ 4 L <. 2 In other wors, for a material boy which has a finite lower limit in uniaxial contraction, it is not necessary to require that the material parameter s 2 be negative as propose by the E-inequalities, but rather only s 2 < s. 3.2 Stress-strain curves In the uniaxial experiment, let the axial stress σ = T κ xx be the force per unit area an the axial strain ɛ = λ be the elongation per unit length in the reference state κ. Then from the first conition of thermoynamic stability 9, we have the stress-strain relation, σɛ = s + ɛ + ɛ 2 + s 2 + ɛ 3, 5
5 4 h = -.5 -.5 Tensile stress 3 2.3.5.5.5 2 2.5 3 Tensile strain Figure : Stress-strain curve uner tension, σ/s vs ɛ, for h =.5 up to h =.5 an the secon conition becomes σɛ ɛ Consequently, for thermoynamic stability, the stress-strain curve must be non-ecreasing in uniaxial experiments. From 2, let h = s 2 s <, an express the stress-strain curve as + ɛ σ s =. + ɛ 2 + h + ɛ 3. 3 In Fig., for tensile test, we plot the stress-strain curve, σ/s vs ɛ, for h =.5,.5,, an for some positive values h =.3,.4,.5. Note that all these curves are increasing, therefore, thermoynamically stable, incluing the positive values of s 2. For compressive test, it is customary to plot the axial compressive stress σ versus the axial compressive strain ɛ. Therefore, let σ = σ, ɛ = ɛ, an the compressive stress-strain relation can be written as σ ɛ = σɛ. In Fig. 2, the compressive stress-strain curve is plotte as σ/s vs ɛ for several values of h. It is clear that for h, so that E-inequalities hol, the curves are increasing, therefore, thermoynamically stable. However, they are concave upwar, or mathematically, we have 2 σ ɛ ɛ 2 = 2 σɛ ɛ 2 = 2 σɛ ɛ 2 > for h, ɛ <. 6
2 -.5.5 -.5 Compressive stress.5.3.4 h =.5..2.3.4.5.6.7.8 Compressive strain Figure 2: Stress-strain curve uner compression, σ/s vs ɛ, for h =.5 own to h =.5 Physically, a concave upwar compressive stress-strain curve means that the material is getting harer to contract as the compressive strain increases. This kin of behavior may actually be reasonable for rubber-like materials but may not be appropriate for some other kin of materials, like soil an rock, which might exhibit concave-ownwar compressive stress-strain curves for example, see Fig. 2 of [4]. On the other han, Fig. 2 also shows that for h.3, the curves are inee concave ownwar for moerate compressive strains, which assures that by allowing s 2 to be positive, it is possible to moel some kin of materials exhibiting such behavior. To analyze the qualitative behavior of such materials [4], we consier the compressive stress-strain curve for h =.7 shown in Fig. 3. It is a concave-ownwar curve with a maximum compressive stress marke at the compressive strain ɛ =.78. Note that thermoynamic stability requires that σ ɛ ɛ = σɛ ɛ = σɛ ɛ Therefore, the otte part of the curve with negative slope is thermoynamically unstable. Likewise, the part of the curves with negative slope for h.3 in Fig. 2 are unstable. Moreover, at the maximum compressive stress, the compressive strain ɛ =.78 correspons to λ = ɛ =.829. One can easily check from the conition 2 that, for γ = h =.7, the lower contraction limit λ L gives. 3 2λ L + λ 4 L =.7 λ L =.829. Therefore, ɛ =.78 is the maximum compressive strain for thermoynamic stability, an contraction beyon the lower limit woul possibly mean a certain structural failure of the experimental specimen, such as crack, shear ben etc. From the above observation, there is a simple way to etermine the values of the material constants s an s 2 from a concave-ownwar compressive stress-strain curve of an uniaxial 7
.2. x.8 Compressive stress.6.4.2.5..5.2.25.3 Compressive strain Figure 3: Stress-strain curve uner compression, σ/s vs ɛ, for h =.7. The point marke at the maximum stress correspons to ɛ =.78. The otte part of the curve is thermoynamically unstable experiment. From the experimental curve, one can etermine the maximum compressive strain ɛ max at the maximum compressive stress σ max, an hence, obtain λ L = ɛ max, h = s 2 s = 3 2λ L + λ 4 L. After that, from 3, it is easy to etermine the value of s with the calculate value of h, σ max = ɛ max s ɛ max 2 + h ɛ max 3. One can then ecie whether the Mooney-Rivlin moel is an acceptable material moel by comparison of the fitte curve with the experimental one. References [] Haupt, P.: Continuum Mechanics an Theory of Materials, Secon Eition, Springer 22. [2] Müller, I.: Rubber an Rubber Balloons, Springer 24. [3] Truesell, C.; Noll, W.: The Non-Linear Fiel Theories of Mechanics, Thir Eition, Springer 24. [4] Liang, W.; Yang, C.; Zhao, Y.; Dusseault, M.B.; Liu, J.: Experimental investigation of mechanical properties of bee salt rock. Int. J. Rock Mech. Min. Sci. 44, 4-4. 27, oi:.6/j.ijrmms.26.9.7. 8