Constitutive equations of creep under changing multiaxial stresses
Jakob M. Klebanov
Department of Mechanical Engineering, Samara Federal Technical University, P.O. Box 4038, Samara 443110 Russia (Received 7 May 1996; revised and accepted 1 November 1998)
Abstract – Creep constitutive equations are derived here to describe creep behaviour of metals at proportional loading by generalisation of non-linear viscoelaticity equations equipped with temporal analogy of time-stress type. To obtain the new equations, integral dependences for the case of degenerative integral operators are first transformed into differential ones. It is shown that the latters imply the kinematic hardening law. In the proposed constitutive equations, mixed hardening is adopted as a composition of three hardening mechanisms connected with translation, size and shape of the potential surfaces respectively. Weight factors of these mechanisms are defined by two material constants which are obtained from the creep data of non-proportional loading. Two measures of isotropic hardening are introduced: the first is the equivalent strain and the second is a non-decreasing parameter. The constitutive equations introduced are evaluated on the base of experimental data on creep behaviour of D16T aluminium alloy and VT3-1 titanium alloy under non-proportional step loading. Comparison of the theoretical results with the experimental data indicates that the former give good predictions of the material responses. Noticeable differences between the two isotropic measures on predicting the isotropic hardening effect appear when the preceding stress vector rotation is not less than 90◦. The non-decreasing parameters are better in rotations of the kind.Elsevier, Paris creep / hardening / stress-rotation / model / anisotropy
1. Introduction
Various types of creep constitutive equations for metals at non-radial loading have been proposed by numerous authors: e.g. (Lagneborg, 1972; Hart, 1976; Gittus, 1976; Miller, 1976) within framework of physical and metallurgical points of view, (Sosnin, 1970; Malinin and Khadjinsky, 1972; Chaboche, 1977; Kadashevich and Novogilov, 1980; Murakami and Ohno, 1982; Ding and Findley, 1984; Mroz and Trampczynski, 1984; Gokhfeld and Sadakov, 1984; Ohno et al., 1985; Kawai, 1995; Rubin and Bodner, 1995; Yang, 1997) along the line of continuum mechanics approaches. Most constitutive equations are obtained by the way of generalisation of the simplest classical time- and strain-hardening laws.
The classical laws of creep are based on the assumption of isotropic hardening of materials. To describe anisotropic hardening, kinematic hardening and a combination of isotropic and kinematic hardening are introduced as extensions to the case of creep deformation of hardening theories in plasticity (Malinin and Khadjinsky, 1972; Miller, 1976; Chaboche, 1977; Krieg et al., 1978; Ohashi et al., 1982).
Physical and metallurgical analyses confirm that both isotropic and anisotropic effects due to the change of micro-structure in material appear in inelastic deformation of real materials (Murghabi, 1975; Miller, 1976; Krieg et al., 1978; Quesnel and Tsow, 1981; Ohashi et al., 1982; Kadashevich and Chernyakov, 1992). Anisotropic effects are connected with such mechanisms as dislocation pileup, bowing and recoil, or non-uniform distributions of inelastic micro-strain through adjacent grains. Isotropic effects are caused by dislocation tangles, sell-structure formation and so on.
In distinction from the classical time- and strain-hardening laws the theory of viscoelasticity reflects transient non-collinearity between the stress and the creep strain rate tensors under large stress-rotations. The theory can describe creep at essential deviations from radial loading (Bugakov, 1965). Equations of non-linear viscoelasticity equipped with temporal analogy of time-stress type are able to describe partial creep recovery at unloading and some other effects in solids (Shapery, 1969; Urgumtsev, 1982; Bugakov and Tchepovetskii, 1984). These properties allow to avoid the introduction of a superposition of recoverable and non-recoverable creep strains considered in (Findley and Lay, 1978; Ohno et al., 1985) as well as the decomposition of hardening mechanism into two parts associated with a creep loading process and a creep reorientation process suggested in (Mroz and Trampczynski, 1984). The equations of non-linear viscoelasticity equipped with temporal analogy lead to boundary value problems which are well-posed (Klebanov, 1996).
Nevertheless analysis of experimental data of non-proportional loading steps in multiaxial creep demon-strates that the integral dependences of viscoelasticity equations can predict the material behaviour under stress vector rotations limited by 50◦–60◦only.
2. Multiaxial creep constitutive equations
Equations of viscoelasticity are an appropriate base to elaborate constitutive relations of creep under large stress-rotations because of essential advantage of viscoelasticity at radial loadings and close to them. To obtain these relations integral dependences should be first transformed into differential ones that becomes possible in the case of degenerate integral operators.
The total strains are considered as the sum of elastic and creep components. To begin with, assume that the stress — creep strain law takes the form of non-linear viscoelasticity equipped with temporal analogy of time — stress type
pij(ξ )=
Z ξ
0
5(ξ−ζ )9
σeq(ζ )
8ij klσkl(ζ )dζ,
σeq=q8ij klσijσkl; i, j, k, l=1,2,3, (1)
where pij is the strain tensor; σij is the stress tensor; 5 is the kernel; σeq is the equivalent stress; 9 is a
non-negative monotone increasing function of σeq; 8ij kl denotes a fourth-order tensor of modules of initial
anisotropy. They possess the symmetries8ij kl=8klij =8j ikl.
The rate of change of the reduced timeξ is given by
dξ dt =g
σeq(t)
, (2)
wheret is the real time,gis the material scaling factor. Since ∂g/∂t=0, these constitutive equations satisfy the condition of invariance of material properties with respect to the zero time reference point. Atg≡1, they reduce to the equation of non-linear viscoelasticity without temporal analogy, which is the Boltzmann–Persoz equation. Constant loading creep curves are assumed to be similar on the reduced time scale.
The kernel is expressed in the form of exponential series
5(ξ )=
N
X
m=1
αmexp(−λmξ ), (3)
Now constitutive equations (1)–(3) can be changed by a system of
Let us introduce an equivalent strain of creep termpeq(m)as a strict convex first order homogeneous function of strain tensor components, such that atσij=const it is valid
peq(m)σeq=p(m)ij σij; m=1,2, . . . , N. (5)
It is easy to show that
peq(m)=
q
ij klpij(m)p (m)
kl ; ij kl8rskl=δirδj s; i, j, k, l, r, s=1,2,3,
whereδij is the Kronecker delta.
The equivalent total creep strain can be also introduced aspeq=p
ij klpijpkl. At constant loading we have
In the case ofλm=0, which corresponds to the secondary creep,
˙
The creep strain term obeys the kinematic hardening rule. Indeed, differential equations (4) can be rewritten as the following flow rule
where8mis the creep velocity potential;βij(m)is a tensor indicating the translation of the potential surface and
frequently referred to as a back stress.
The back stresses βij(m) have different evaluation rates since the coefficients λm many times differ among
Thus Eqs (4) imply the kinematic hardening law, introduced in plasticity by Prager. Many pioneering authors (Edelman and Drucker, 1951; Olszak and Urbanowski, 1957) assumed for initially anisotropic materials the quadratic Mises type plastic yield conditions based on the kinematic hardening law. It was then clarified by experimental results that, owing to the simplified assumption that the subsequent yield surface moves without changing its initial shape and size, kinematic hardening theories can not describe the complicated, actual inelastic behaviour well.
Assume that the actual hardening effect at creep can be represented as a combined one and introduce more general constitutive equations than Eqs (4) as follows:
pij=
It is easily convinced by expressing the creep velocity potential in the proper form that the three bracketed summands represent the kinematic, specific anisotropic and isotropic hardening respectively. The second is connected with changing in the potential surface shape. The ratio of the mixed invariant σijpij(m) toσeq is a
measure of the specific anisotropic hardening. The equivalent strain is a measure of the isotropic one. The constant multipliersω,γ and(1−γ −ω)are weight factors of the three hardenings. It is supposed that 06ω, γ 61.
It follows from (9) that at radial loading,σij(ξ )=σij◦ϕ(ξ ), all the termsp (m)
ij can be represented in the form
pij(m)(ξ )=ψm(ξ )8ij klσkl◦,
whereϕ,ψmare scalar functions of the reduced time,σij◦is a constant stress. This yields that, at radial loading,
Eqs (9) are reduced to Eqs (4) and that equalities (5) and (6) are valid not only at constant stress but at radial loading also.
According to (9), the effect of changing multiaxial loading can be removed by a subsequent radial loading. That corresponds to the experimentally observed inelastic behaviour of metals (Trampczynski and Mroz, 1992). For many metallic materials it is experimentally observed that inelastic deformation does not depend on hydrostatic pressure, thus (9) as well as (4) can be expressed in terms of the deviatoric stress componentssij,
where
sij=σij−
1 3δijσkk.
The restrictions imposed on the tensor8ij kl by the condition of incompressibility are given by the equalities 8iikl=0 and8iijj=0.
For inelastically incompressible and initially isotropic materials
8ij kl=
and Eqs (9) can be rewritten in the form
˙
eq are noninvariably convenient as measures of the hardening. Just imagine a case when after a reversal peq(m)=0 with material properties having been changed due to the previous loading. Let us consider also instead ofpeq(m) a non-decreasing measure of the isotropic hardening, the evolution of which is given as follows:
It follows from above notes that in the cases of constant and radial non-decreasing loadings the isotropic hardening parameterρm is equal topeq(m). However, after the direction or the sign of stress changed or sharp stress decreasing the value ofρm can not decrease so that the isotropic softening of materials is not possible.
The value ofρmincreases if it is less than the limit equivalent strainp∞(m)depending on current stress (8).
3. Evaluation of creep constitutive equations
All the functions and constants of constitutive equations (8)–(11) except the constantsγ andωare obtained from the experimental constant multiaxial creep curves and the succeeding creep recovery curves. The constantsγ andωare obtained from the creep experimental data of non-proportional loading.
The constant multiaxial creep curves and the succeeding creep recovery curves at different kinds of stress state should be first treated in order to obtain dependences of the reduced time and the coefficients of creep curve similarity on stresses. The ways of the treatment are represented in (Urgumtsev, 1982; Bugakov and Tchepovetskii, 1984). For many metals with relatively small recovery the dependence of the coefficients of creep curve similarity on stresses can be assumed to be linear in the scale of the reduced time. In that case the function9is changed by a constant.
Then, if the assumption of initial isotropy is rejected, the modules of initial anisotropy are identified with introduction of the restrictions imposed on the form of equivalent stress by orthotropy or another symmetry of material properties if any. Anisotropy modules for different kinds of symmetries are specified in (Olszak and Urbanowski, 1957). The modules can be determined by the method proposed in (Jakowluk and Mieleszko, 1982). For the plane stress state of thin-walled tubular specimens under tension and torsion assuming81111=1 we have
σeq2 =σ112 +481212σ122 +481112σ11σ12,
whereσ11,σ12are tensile and shear stresses for the specimen.
After that the constant multiaxial creep curves are represented bypeq-ξ curves at given values ofσeq. These curves should be fitted by Eqs (6)–(8) which include the material constantsαmand λm. It was pointed above
that the case of λm=0 corresponds with the secondary creep data. The serial method of the constants αm
Table I. Applied tension and torsion stressesσ11,σ12at the dwell stages and the duration of the stages1t in D16T aluminium alloy tests.
Program Stage I Stage II Stage III
σ11 σ12 1t σ11 σ12 1t σ11 σ12 1t
MPa MPa hr MPa MPa hr MPa MPa hr
1 58.83 90.21 5 124.5 68.84 5 171.6 31.38 4
2 58.83 90.21 5 171.6 31.38 5 171.6 −31.38 4
3 181.4 0 5 124.5 68.84 5 0 94.13 4
4 0 94.13 5 171.6 31.38 5 124.5 −68.64 4.5
5 171.6 31.38 5 58.83 −90.21 5 58.83 90.21 4
6 58.83 90.21 5 58.83 −90.21 5 58.83 90.21 5
Table II. Applied tension and torsion stressesσ11,σ12at the dwell stages and the duration of the stages1tin VT3-1 titanium alloy tests.
Stage Program 1 Program 2
σ11 σ12 1t σ11 σ12 1t
MPa MPa hr MPa MPa hr
I 588 0 9 0 339 4
II 667 0 24 0 368 21
III 686 0 21 0 396 30
IV 0 396 25 485 −279 25
V 686 0 25 0 396 25
VI 0 396 25 – – –
VII 686 0 25 – – –
VIII 0 396 25 – – –
among themselves in about ten or more times. One or two of the coefficients αm corresponding with large
coefficientsλmshould be then corrected in order to take into account the influence of initial loading on constant
load test data (Klebanov, 1980; Klebanov and Kokorev, 1985).
Finally, the constantsγ andωare appointed from a step test at non-proportional loading. To this end creep strain curves at a dwell stage after the direction of stress changes should be fitted by Eqs (9).
The capabilities of the constitutive equations developed are below evaluated by experimental comparison. Results of creep tests for type D16T aluminium alloy at the temperature of 200◦C are reported in (Sosnin, 1970). Programs of three-step non-proportional loading at constant equivalent stress are shown on table I.
The material employed in our experiments was VT3-1 titanium alloy. A description of thin-walled tubular specimens, the combined tension and torsion machine used for the experiments are given in (Klebanov, 1975; Sorokin and Klebanov, 1976). Outer and inner surfaces over the gauge length of 45±0.05 mm were finished with tolerances 17+0.02 mm and 18.2+0.02 mm in diameters. The tests were performed at the temperature of 250◦C. Programs of step loading are shown in table II. From 3 to 5 specimens were tested at each program.
All the material constants and functions were obtained as follows:
D16T aluminium alloy, 200◦C
σeq=
q
σ2
11+3.3σ122; p (m)
eq =
q
p(m)11 2
+ 2p(m)12 2
/3.3,
λ1=1.44·104; λ2=1.42; λ3=0.163; λ4=0,
g=(σeq/68.9)7.33; 9=1/68.9; ω=0.13; γ =0.6.
VT3-1 titanium alloy, 250◦C
σeq=σe; p(m)eq =p (m)
e ; N =4,
α1=5.2·10−5; α2=5.2·10−4; α3=1.65·10−5; α4=8.5·10−7,
λ1=20; λ2=2; λ3=0.05; λ4=0,
g=(σeq/677)31.62; 9=1/677; ω=0.003; γ =0.46,
ω=0.003; γ =0.78 at valuesρminstead ofρe(m),
where the values of the material constants are concerned with the units of MPa (σeq, σe) and hr−1(αm, λm).
Figure 1. Axial and torsional creep strain components of D16T aluminium alloy under complex loading at 200◦C.◦◦◦◦◦experimental axial component (Sosnin, 1970).• • • • •experimental torsional component (Sosnin, 1970). — components predicted by Eqs (9). - - - components predicted with usage
Figure 2. Axial and torsional creep strain components of VT3-1 titanium alloy under Program 1 loading at 250◦C. — average experimental components. — components predicted by Eqs (10). - - - components predicted with usage of valuesρm.
The values of the material constantsωand λwere determined from the results of creep test data for D16T alloy at the second stage of program three, table I and for VT3-1 alloy at the fourth stage of program one, table II.
Figures 1–3 show the variation of tension and torsion strains observed under large stress-rotations. Comparison of the theoretical results with the experimental data indicates that the formers give good predictions of the material responses. The noticeable differences between measures peq(m) and ρm on predicting of the
isotropic hardening effect appear when the preceding stress vector rotation is not less than 90◦. The non-decreasing parametersρmare better in rotations of the kind.
4. Conclusions
In this paper, the creep constitutive equations of non-linear viscoelasticity equipped with temporal analogy have been developed to describe creep behaviour of metals at non-proportional loading. The total creep strain is a sum of several terms which obeys the differential equations of the same kind. Unlike most other treatments of the problem, mixed hardening of each term is considered as the composition of the three hardening mechanisms connected with translation, size and shape of the potential surfaces respectively. Weight factors of these mechanisms are defined by two material constants.
Figure 3. Axial and torsional creep strain components of VT3-1 titanium alloy under Program 2 loading at 250◦C. — average experimental components. — components predicted by Eqs (10). - - - components predicted with usage of valuesρm.
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