This paper illustrates the formulation of viscoplastic-damage constitutive models using the framework of hyperplasticity.

The entire constitutive behaviour is derived from only two scalar potentials; a free energy potential and a dissipation

potential. This ensures that the model obeys the laws of thermodynamics

Birchall, T.J.

Coombs, W.M.

Koziara, T.

Osman, A.S.

This paper illustrates the formulation of viscoplastic-damage constitutive models using the framework of hyperplasticity. The entire constitutive behaviour is derived from only two scalar potentials; a free energy potential and a dissipation potential. This ensures that the model obeys the laws of thermodynamics

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International Conference on Computational Mechanics (CM13) 25-27 March 2013, Durham, UKOn the formulation of thermodynamically-consistent viscoplastic-damageconstitutive modelsT.J. Birchalla,∗, A.S. OsmanaaSchool of Engineering and Computing Sciences, Durham University, South Road, Durham, DH1 3LEAbstractThis paper illustrates the formulation of viscoplastic-damage constitutive models using the framework of hyperplasticity.The entire constitutive behaviour is derived from only two scalar potentials; a free energy potential and a dissipationpotential. This ensures that the model obeys the laws of thermodynamics.Keywords: viscoplasticity, damage, creep, constitutive model, hyperplasticity.1. IntroductionTime-dependent constitutive models are essential for modelling the deformation of underground cavities excavated increeping rock and are very useful in producing a successful design. Creep behaviour is prominent in soft rocks such as saltrock and potash, and has even been observed in hard rocks, highlighting the importance of designing for such behaviour.In this paper we present a new class of time-dependent constitutive models which are able to take into account boththe secondary and tertiary stages of creep deformation. Three stages of creep are typically observed in laboratory creeptests: a primary stage considered to be viscoelastic, a secondary stage considered to be viscoplastic and a tertiary stageconsidered to be caused by material damage, through the formation of cracks. Thus here we develop a viscoplastic-damageconstitutive model. These models are novel as they are derived using the hyperplastic framework of Houlsby and Puzrin[1] and therefore automatically obeys the laws of thermodynamics. In this approach the entire constitutive behaviourcan be derived from two scalar potentials; a free energy potential which provides the elasticity law, and a dissipationpotential which provides the yield function, the direction of plastic flow and the evolution of a damage variable. Noadditional assumptions are required. We assume that the viscoplasticity and damage are coupled, therefore they occursimultaneously. The classical sign convention in geomechanics is adopted; where compressive stresses are taken to bepositive. The constutive models are formulated in term of effective stresses.2. Formulation of a Von-Mises viscoplastic-damage constitutive modelThis section presents the derivation of the constitutive equations using the hyperplastic framework. The free energypotential can be written for a damaged material as followsψ =12{εe}(1 − αd)[De]{εe}, (1)where {εe} is the elastic strain vector, [De] is the elastic stiffness matrix and αd is a scalar damage parameter.Differentiating the free energy potential with respect to the elastic strain vector produces the elasticity law{σ} = ∂ψ∂{εe} = (1 − αd)[De]{εe}. (2)The free energy potential (equation (1)) can be expressed as follows in terms of deviatoric and mean stressesψ =q26G(1 − αd)+p22K(1 − αd), (3)∗T.J. BirchallEmail address: t.j.birchall@durham.ac.uk (T.J. Birchall)Draft extended abstract submitted to CM13 February 10, 2013where q is the von Mises equivalent stress defined as q =√3J2, where J2 = 12 tr([s][s]), where the deviatoric stress is[s] = [σ]− p[1] and [1] is the three by three identity matrix, p is the mean stress, G is the shear modulus and K is the bulkmodulus.The stresses associated with the damage can then be derived as followsχd =∂ψ∂αd=q26G(1 − αd)2+p22K(1 − αd)2. (4)For a coupled viscoplastic-damage material the dissipation potential Ḋ can be defined asḊ = σy√(rpα̇s1 + rpηn−1α̇ns2)2 + (rdρα̇d)2, (5)where α̇s1 and α̇s2are internal variables representing the equivalent deviatoric plastic strain rate α̇s(αs1 = αs2 = αs), η isthe viscous coefficient which controls the extent of plastic strain, n is a material constant, whilst rp and rd are constantsgoverning the ratio of viscoplasticity and damage. σy = σy0Π(αd), where σy0 is the uniaxial yield stress. ρ is defined asρ =q6G(1 − αd)2+p22K(1 − αd)2q, (6)It should be noted that for a rate-independent von Mises material, the above dissipation potential reduces to the form givenby Einav et al. [2].Differentiation of the dissipation potential with respect to the plastic strains gives the dissipative stresses.χq =∂Ḋ∗∂α̇s1+1n∂Ḋ∗∂α̇s2. (7)Noting that αs1 = αs2 = αs, we can show thatχq =σy(rpα̇s + rpηn−1α̇ns)rp(1 + (ηα̇s)n)√(rpα̇s + rpηn−1α̇ns)2 + (rdρα̇d)2. (8)Differentiating with respect to α̇d gives the dissipative stress due to damageχd =∂Ḋ∗∂α̇d=σyα̇d(rdρ)2√(rpα̇s + rpηn−1α̇ns)2 + (rdρα̇d)2. (9)Since there is no kinematic hardening, the true stresses and the dissipative stresses are identical. From equations (8)and(9): (qrpσy(1 + (ηα̇s)n−1))2+(qrdσy)2= 1. (10)Rearranging equation (10) we obtainα̇s =1η 1√(σyrpq)2 − ( rprd )2 − 11n−1(11)The plastic strain rate α̇s is a non negative quantity, therefore the expression inside the bracket in equation (11) mustbe greater than or equal to zero. Therefore the yield criterion is given byf =(qσy)2  1r2p + 1r2d − 1 (12)From this equation we can find that the constants rp and rd are related by the following expression1r2p+1r2d= 1, (13)2from which it follows that rp ≥ 1 and rd ≥ 1. In the limiting case when rp = 1, rd → ∞ and the constitutive modelsimplifies to the well-known Perzyna viscoplastic model [4].An expression for the damage internal variable can be found from equation (9), bearing in mind that ρ = χdqα̇d =α̇s(1 + (α̇sη)n−1)2ρ(rprd)2. (14)3. Formulation of a Drucker-Prager visco-plastic constitutive modelFor a frictional material, the dissipation function can be assumed to take the form:Ḋ =√(rp(c2 p + c1)α̇s1 + rpηn−1α̇ns2)2 + (rdρ(c2 p + c1)α̇d)2, (15)where c1 = dΠ(αd) and c2 = µΠ(αd), where d is a constant related to the cohesion, µ is a constant related to the frictionalangle and Π(αd) is the softening/hardening function. ρ is defined asρ =q26G(1 − αd)2(q − c3 p)+p22K(1 − αd)2(q − c3 p), (16)where c3 = βΠ(αd) and where β is a constant, related to the dilation angle. d, µ and β can be calculated from the cohesion,friction angle and dilation angle.As a side condition we will impose a linear relationship between the volumetric and shear strain ratesα̇v + c3α̇s = 0, (17)where αv = tr([εp]).Starting from the dissipation function of equation (15) together with the constrain given by equation (17) and followingthe procedure illustrated in the previous section, it can be shown thatα̇s =1η(c2 p + c1) 1√( rpck)2 − ( rprd )2 − 11n−1(18)whereck =(q − c3 p)(c2 p + c1). (19)And the damage internal variable is given byα̇d =α̇s(1 + (α̇sη)n−1(c2 p+c1))2ρ(rprd)2. (20)The yield criterion is given byf = ck2 1r2p + 1r2d − 1 (21)In order to use this constitutive model for practical application we must use a stress integration procedure. Here weuse the iterative implicit backward Euler (bE) stress integration scheme.34. Validation with triaxial dataThe Drucker-Prager viscoplastic-damage constitutive model, which is applicable to geomaterials, has been validatedusing data from triaxial compression testing of sandstone carried out by Yang and Jiang [3]. The experimental data wasobtained from short-term loading tests and long-term creep tests. Note that here we use a softening/hardening functionΠ(αd) equal to (1 − ααd)2, where α is the softening hardening parameter. This function fits well with the experimentaldata.The short-term loading tests were conducted by Yang and Jiang [3] under different confining pressures. We simulatedthese triaxial tests by using a single material point in MATLAB 7.9.0. We apply the initial confining pressure in the firsttime step and then apply an increment of strain over each subsequent time step, ensuring that the rate of applied straincorresponds to a rate of applied stress of 0.127 MPa/s, which is consistent with the rate adopted in the triaxial experiments.This rate of stress is applied over the full time period of the simulation (prior to and during material softening). Yang andJiang [3] conducted a creep test under multi levels of stress. The applied rate of stress (σ1 − σ3) was the same as thatused in the short-term tests (0.127 MPa/s), with a confinement σ3 of 5 MPa. When a required level of stress was obtainedit was held constant and the sandstone was allowed to creep for 48 hours. The final and highest level of stress used wasσ1 − σ3 = 160 MPa. Under this constant level of stress, the three typical stages of creep were observed. This stress wasbelow the peak strength of the material, but corresponded to the point where yield was initiated. In order to simulate acreep test, we modified the program used for the short-term tests by keeping σ1 − σ3 constant when yield begins. Herewe do not simulate a multi-stage creep test, but rather use a single level of stress equal to the final level of stress used byYang and Jiang [3] (σ1 − σ3 = 160 MPa). This is justified from the results of triaxial tests carried out by Yang and Jiang[3]. Their findings showed that very little creep occurred below the yield stress of the sandstone.Figure 1 compares the short-term stress-strain curves obtained by Yang and Jiang [3] and those obtained using thenew frictional constitutive model. The model parameters used in the simulations are shown in Table 1. We observe thatthe constitutive model provides a good prediction of the experimental data for all three confining pressures, at both peakstress and in the region of material softening. The experimental data obtained from the long-term tests is shown in Figure2, along with the response of the new constitutive model. Here we observe that the constitutive model underestimates thestrain shown in the experimental data. However the overall shape of the curves is very similar, therefore the constitutivemodel provides a useful approximation.(a)Figure 1: Validating constitutive model using short-term stress-strain triaxial datad[MPa] µ β η[MPa h] n α rp21.95 0 −2.38 1525 1.5 0.375 1.05Table 1: Values of constitutive model parameters used in the triaxial simulationIt should be noted that more experimental data is required in order to determine the constitutive model parametersmore accurately. Triaxial tests under different rates of loading are required, as the magnitude of the peaks of the short-term stress strain curves shown in Figure 1 is dependent on the rate of loading. A higher rate of loading could producehigher peaks, whilst a lower rate of loading could produce lower peaks. In order to obtain the strength parameters d, µand β more accurately it is necessary to conduct triaxial tests under low rates of loading.4Figure 2: Creep response of model under yield stress5. ConclusionsA novel viscoplastic-damage constitutive models have been derived within the framework of hyperplasticity. Thesemodels are able to describe both the secondary and tertiary stages of creep behaviour. The new frictional viscoplastic-damage model is validated with experimental data obtained by Yang and Jiang [3] for samples of sandstones. This modelcan be used for a variety of applications such as the creep analysis of underground tunnels and storage caverns as well asapplications outside the field of geomechanics. The model can be generalised by incorporating an appropriate lode angledependency function such as that suggested by Eekelen [5].References[1] G. T. Houlsby, A. M. Puzrin, Principles of hyperplasticity: an approach to plasticity theory based on thermodynamicprinciples, Springer-Verlag, 2006.[2] I. Einav, G. T. Houlsby, G. D. Nguyen, Coupled damage and plasticity models derived from energy and dissipationpotentials, International Journal of Solids and Structures 44 (2006) 2487–2508.[3] S. Yang, Y. Jiang, Triaxial mechanical creep behaviour of sandstone, Mining Science and Technology 20 (2010)339–349.[4] P. Perzyna, Fundamental problems in viscoplasticity, Advances in Applied Mechanics, 9 (1966) 243–377.[5] H. A. M. Eekelen, Isotropic yield surface in three dimensions for use in soil mechanics, International Journal forNumerical and Analytical Methods in Geomechanics 4 (1980) 89–101.5

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On the formulation of thermodynamically-consistent viscoplastic-damage constitutive models

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On the formulation of thermodynamically-consistent viscoplastic-damage constitutive models.

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