Skip to main content

Incrementally objective implicit integration of hypoelastic–viscoplastic constitutive equations based on the mechanical threshold strength model

Abstract

The present paper focuses on the development of a fully implicit, incrementally objective integration algorithm for a hypoelastic formulation of \(J_{2}\)-viscoplasticity, which employs the mechanical threshold strength model to compute the material’s flow stress, taking into account its dependence on strain rate and temperature. Heat generation due to high-rate viscoplastic deformation is accounted for, assuming adiabatic conditions. The implementation of the algorithm is discussed, and its performance is assessed in the contexts of implicit and explicit dynamic finite element analysis, with the aid of example problems involving a wide range of loading rates. Computational results are compared to experimental data, showing very good agreement.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

References

  1. 1.

    Rosakis P, Rosakis AJ, Ravichandran G, Hodowany J (2000) A thermodynamic internal variable model for the partition of plastic work into heat and stored energy in metals. J Mech Phys Solids 48:581–607

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Vogler TJ, Clayton JD (2008) Heterogeneous deformation and spall of an extruded tungsten alloy: plate impact experiments and crystal plasticity modeling. J Mech Phys Solids 56:297–335

    Article  Google Scholar 

  3. 3.

    Kocks UF, Argon AS, Ashby MF (1975) Thermodynamics and kinetics of slip, progress in materials science, vol 19. Pergamon, Oxford

    Google Scholar 

  4. 4.

    Hansen BL, Beyerlein IJ, Bronkhorst CA, Cerreta EK, Dennis-Koller D (2013) A dislocation-based multi-rate single crystal plasticity model. Int J Plast 44:129–146

    Google Scholar 

  5. 5.

    Preston DL, Tonks DL, Wallace DC (2003) Model of plastic deformation for extreme loading conditions. J Appl Phys 93: 211–220

    Google Scholar 

  6. 6.

    Weinberg K, Mota A, Ortiz M (2006) A variational constitutive model for porous metal plasticity. Comput Mech 37:142–152

    Article  MATH  Google Scholar 

  7. 7.

    Gurtin ME, Fried E, Anand L (2010) The mechanics and thermodynamics of continua. Cambridge University Press, Cambridge

    Book  Google Scholar 

  8. 8.

    Abu Al-Rub RK, Darabi MK (2012) A thermodynamic framework for constitutive modeling of time- and rate-dependent materials. Part I: theory. Int J Plast 34:61–92

    Article  Google Scholar 

  9. 9.

    Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: part I: yield criteria and flow rules for porous ductile media. J Eng Mater Technol 99:2–15

    Article  Google Scholar 

  10. 10.

    Brünig M, Gerke S (2011) Simulation of damage evolution in ductile metals undergoing dynamic loading conditions. Int J Plast 27:1598–1617

    Article  MATH  Google Scholar 

  11. 11.

    Gao X, Zhang T, Zhou J, Graham SM, Hayden M, Roe C (2011) On stress-state dependent plasticity modeling: significance of the hydrostatic stress, the third invariant of stress deviator and the non-associated flow rule. Int J Plast 27:217–231

    Article  Google Scholar 

  12. 12.

    Madou K, Leblond JB (2012) A Gurson-type criterion for porous ductile solids containing arbitrary ellipsoidal voids–II: determination of yield criterion parameters. J Mech Phys Solids 60:1037–1058

    Google Scholar 

  13. 13.

    Monchiet V, Bonnet G (2013) A Gurson-type model accounting for void size effects. Int J Solids Struct 50:320–327

    Article  Google Scholar 

  14. 14.

    Bodner SR, Partom Y (1975) Constitutive equations for elastic-viscoplastic strain-hardening materials. J Appl Mech 42: 385–389

    Google Scholar 

  15. 15.

    Bodner SR, Merzer A (1978) Viscoplastic constitutive equations for copper with strain rate history and temperature effects. J Eng Mater Technol 100:388–394

    Article  Google Scholar 

  16. 16.

    Steinberg DJ, Cochran SG, Guinan MW (1980) A constitutive model for metals applicable at high strain rate. J Appl Phys 51:1498–1504

    Article  Google Scholar 

  17. 17.

    Steinberg DJ, Lund CM (1989) A constitutive model for strain rates from \(10^{-4}\) to \(10^{6}\) \(\text{ s }^{-1}\). J Appl Phys 65:1528–1533

    Article  Google Scholar 

  18. 18.

    Johnson GR, Cook WH (1983) A constitutive model and data for metals subjected to large strains, high strain rates, and high temperatures. Proceeding of the 7th International Symposium on Ballistics, vol 21. The Hague, Netherlands, pp 541–547

  19. 19.

    Holmquist TJ, Johnson GR (1991) Determination of constants and comparison of results for various constitutive models. J Phys IV 01:C3–853–C3-860

    Google Scholar 

  20. 20.

    Johnson GR, Holmquist TJ, Anderson CE Jr, Nicholls AE (2006) Strain-rate effects for high-strain-rate computations. J Phys IV 134:391–396

    Google Scholar 

  21. 21.

    Zerilli FJ, Armstrong RW (1987) Dislocation-mechanics-based constitutive relations for material dynamics calculations. J Appl Phys 61:1816–1825

    Article  Google Scholar 

  22. 22.

    Zerilli FJ, Armstrong RW (1990) Description of tantalum deformation-behavior by dislocation mechanics based constitutive relations. J Appl Phys 68:1580–1591

    Article  Google Scholar 

  23. 23.

    Armstrong RW, Zerilli FJ (2010) High rate straining of tantalum and copper. J Phys D Appl Phys 43(492):002

    Google Scholar 

  24. 24.

    Abed FH, Voyiadjis GZ (2005) Plastic deformation modeling of AL-6XN stainless steel at low and high strain rates and temperatures using a combination of bcc and fcc mechanisms of metals. Int J Plast 21:1618–1639

    Article  MATH  Google Scholar 

  25. 25.

    Voyiadjis GZ, Abed FH (2005) Microstructural based models for bcc and fcc metals with temperature and strain rate dependency. Mech Mater 37:355–378

    Article  Google Scholar 

  26. 26.

    Khan AS, Liu H (2012) Variable strain rate sensitivity in an aluminum alloy: response and constitutive modeling. Int J Plast 36:1–14

    Article  Google Scholar 

  27. 27.

    Khan AS, Yu S, Liu H (2012) Deformation induced anisotropic responses of Ti–6Al–4V alloy. Part II: a strain rate and temperature dependent anisotropic yield criterion. Int J Plast 38: 14–26

    Google Scholar 

  28. 28.

    Kocks UF (1976) Laws for work-hardening and low-temperature creep. J Eng Mater Technol 98:76–85

    Article  Google Scholar 

  29. 29.

    Follansbee PS, Kocks UF (1988) A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable. Acta Metall 36:81–93

    Article  Google Scholar 

  30. 30.

    Follansbee PS, Gray GT (1989) An analysis of the low temperature, low and high strain-rate deformation of Ti6Al4V. Metall Trans A 20:863–874

    Article  Google Scholar 

  31. 31.

    Follansbee PS, Huang JC (1990) Low-temperature and high-strain-rate deformation of nickel and nickel–carbon alloys and analysis of the constitutive behavior according to an internal state variable model. Acta Metall Mater 38:1241–1254

    Article  Google Scholar 

  32. 32.

    Chen SR, Gray GT III (1996) Constitutive behavior of tantalum and tantalum–tungsten alloys. Metall Mater Trans A 27A:2994–3006

    Article  Google Scholar 

  33. 33.

    Addessio FL, Johnson JN (1993) Rate-dependent ductile failure model. J Appl Phys 74:1640–1648

    Article  Google Scholar 

  34. 34.

    Maudlin PJ, Mason TA, Zuo QH, Addessio FL (2003) Tepla-a: coupled anisotropic elastoplasticity and damage. Tech. Rep. LA-14015-PR, Los Alamos National Laboratory.

  35. 35.

    Simo JC, Hughes TJR (1998) Computational inelasticity. Springer, New York

    MATH  Google Scholar 

  36. 36.

    Schmidt I (2011) Numerical homogenization of an elasto-plastic model-material with large elastic strains: macroscopic yield surfaces and the Eulerian normality rule. Comput Mech 48:579–590

    Article  MATH  Google Scholar 

  37. 37.

    Gu Q, Conte JP, Yang Z, Elgamal A (2011) Consistent tangent moduli for multi-yield-surface J\(_{2}\) plasticity model. Comput Mech 48:97–120

    Article  MATH  Google Scholar 

  38. 38.

    Areias P, da Costa DD, Pires EB, Barbosa JI (2012) A new semi-implicit formulation for multiple-surface flow rules in multiplicative plasticity. Comput Mech 49:545–564

    Google Scholar 

  39. 39.

    Wang WM, Sluys LJ, De Borst R (1997) Viscoplasticity for instabilities due to strain softening and strain-rate softening. Int J Numer Methods Eng 40:3839–3864

    Article  MATH  Google Scholar 

  40. 40.

    Voyiadjis GZ, Abed FH (2006) Implicit algorithm for finite deformation hypoelastic-viscoplasticity in fcc metals. Int J Numer Methods Eng 67:933–959

    Article  MATH  Google Scholar 

  41. 41.

    Zaera R, Fernández-Sáez J (2006) An implicit consistent algorithm for the integration of thermoviscoplastic constitutive equations in adiabatic conditions and finite deformations. Int J Solids Struct 43:1594–1612

    Article  MATH  Google Scholar 

  42. 42.

    Becker R (2011) An alternative approach to integrating plasticity relations. Int J Plast 27:1224–1238

    Google Scholar 

  43. 43.

    Ben Bettaieb M, Lemoine X, Duchêne L, Habraken AM (2011) On the numerical integration of an advanced Gurson model. Int J Numer Methods Eng 85:1049–1072

    Article  MATH  Google Scholar 

  44. 44.

    Rempler HU, Wieners C, Ehlers W (2011) Efficiency comparison of an augmented finite element formulation with standard return mapping algorithms for elastic-inelastic materials. Comput Mech 48:551–562

    Article  MATH  MathSciNet  Google Scholar 

  45. 45.

    Tandaiya P, Ramamurty U, Narasimhan R (2011) On numerical implementation of an isotropic elastic–viscoplastic constutive model for bulk metallic glasses. Model Simul Mater Sci Eng 19(015):002

    Google Scholar 

  46. 46.

    Shutov AV, Ihlemann J (2012) A viscoplasticity model with an enhanced control of the yield surface distortion. Int J Plast 39:152–167

    Article  Google Scholar 

  47. 47.

    Varshni YP (1970) Temperature dependence of the elastic constants. Phys Rev B 2:3952–3958

    Article  Google Scholar 

  48. 48.

    Hughes TJR, Winget J (1980) Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis. Int J Numer Methods Eng 15:1862–1867

    Article  MATH  MathSciNet  Google Scholar 

  49. 49.

    ABAQUS 6.11 (2011) Abaqus Theory Manual. Dassault Systèmes Simulia Corp, Providence

  50. 50.

    Fish J, Shek K (1999) Computational aspects of incrementally objective algorithms for large deformation plasticity. Int J Numer Methods Eng 44:839–851

    Article  MATH  Google Scholar 

  51. 51.

    Seifert T, Maier G (2008) Consistent linearization and finite element implementation of an incrementally objective canonical form return mapping algorithm for large deformation inelasticity. Int J Numer Methods Eng 75:690–708

    Article  MATH  MathSciNet  Google Scholar 

  52. 52.

    Maudlin P, Bingert J (2003) Low-symmetry plastic deformation in BCC tantalum: experimental observations, modeling and simulations. Int J Plast 19:483–515

    Article  MATH  Google Scholar 

Download references

Acknowledgments

Helpful discussions over the course of this work with Drs. Rick Rauenzahn, Bradford Clements, and Jason Mayeur, of Los Alamos National Laboratory, are greatly appreciated. Funding for this work was provided by the Joint DoD/DOE Munitions Technology Development Program, the Advanced Simulation and Computing program (ASC), the NNSA Science Campaign 2—Dynamic Materials Properties, and the Laboratory Directed Research and Development program (LDRD) at Los Alamos National Laboratory. The authors gratefully acknowledge this support.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Hashem M. Mourad.

Appendix: Flow stress calculation and linearization

Appendix: Flow stress calculation and linearization

The purpose of the procedure described in this appendix is to calculate \({{}{\hat{\tau }}}^{(k)}\) and \(\frac{{\mathrm {d}}\,{{}{\hat{\tau }}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\), given \({{}\Delta \lambda }^{(k)}\). It is invoked at the end of the \(k\)th iteration of the Newton–Raphson scheme used to effect radial return mapping, cf. Eqs. (53)–(55). The procedure consists of six main, consecutive steps, listed below.

Step 1: Equivalent viscoplastic strain rate

We begin by using Eq. (46) to approximate the equivalent viscoplastic strain rate:

$$\begin{aligned} {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}=\min \left( \;\max \left( \sqrt{\tfrac{2}{3}}\;\frac{{{}\Delta \lambda }^{(k)}}{\Delta t},{\dot{\bar{\varepsilon }}}^{\text {vp}}_{\text {min}}\right) ,\;{\dot{\bar{\varepsilon }}}^{\text {vp}}_{\text {max}}\right) . \end{aligned}$$
(71)

Consequently, we have

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\left\{ \begin{array}{ll} \frac{\sqrt{2/3}}{\Delta t}\qquad \quad &{} \text {if }{\dot{\bar{\varepsilon }}}^{\text {vp}}_{\text {min}}\le \sqrt{\frac{2}{3}}\frac{{{}\Delta \lambda }^{(k)}}{\Delta t}\le {\dot{\bar{\varepsilon }}}^{\text {vp}}_{\text {max}},\\ 0 &{} \text {otherwise}. \end{array}\right. \end{aligned}$$
(72)

Step 2: Temperature and specific heat

The temperature and specific heat are determined with the aid of a one-point iteration based on Eqs. (47)–(48). In this iterative setup, we set

$$\begin{aligned}&{{}C_{p}}^{(k,l+1)}=C_{0}+C_{1}\,{{}\theta }^{(k,l)}+C_{2}\,\left( {{}\theta }^{(k,l)}\right) ^{-2}, \end{aligned}$$
(73)
$$\begin{aligned}&{{}\theta }^{(k,l+1)}={{}\theta }_{(n)}+\frac{\varPsi }{\rho \,{{}C_{p}}^{(k,l+1)}}\nonumber \\&\quad \quad \quad \quad \quad \times \left( \Vert {{}\mathbf{{S}}}^{(0)}\Vert -2G\,{{}\Delta \lambda }^{(k)}\right) \,{{}\Delta \lambda }^{(k)}, \end{aligned}$$
(74)

where \({{}(\cdot )}^{(k,l)}\) denotes the \(l\)th iterate (with \(l=0,1,\dots \)). The iterative procedure is initialized by setting \({{}\theta }^{(k,0)}={{}\theta }_{(n)}\), and continues until \([\,{{}\theta }^{(k,l+1)}-{{}\theta }^{(k,l)}\,]\) becomes smaller than a tolerance, indicating convergence. At that stage, we set \({{}\theta }^{(k)}=\min (\,{{}\theta }^{(k,l+1)},\theta _{\text {melt}}\,)\), preventing the temperature from exceeding its melting point, \(\theta _{\text {melt}}\), of the material. It can also be seen from Eqs. (47)–(48), or equivalently from (73)–(74), that

$$\begin{aligned} {\mathrm {d}}{{}\theta }^{(k)}=\frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}C_{p}}^{(k)}}{\mathrm {d}}{{}C_{p}}^{(k)}+\frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}\Delta \lambda }^{(k)}}{\mathrm {d}}{{}\Delta \lambda }^{(k)}, \end{aligned}$$
(75)

where \({\mathrm {d}}{{}C_{p}}^{(k)}=\left[ C_{1}-2C_{2}\left( {{}\theta }^{(k)}\right) ^{-3}\right] {\mathrm {d}}{{}\theta }^{(k)}\). Thus,

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}= \left\{ \begin{array}{ll} 0&{}\text {if }{{}\theta }^{(k)}=\theta _{\text {melt}},\\ \frac{\frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}\Delta \lambda }^{(k)}}}{1-\frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}C_{p}}^{(k)}}\left[ C_{1}-2C_{2}\left( {{}\theta }^{(k)}\right) ^{-3}\right] }\qquad &{}\text {otherwise}, \end{array}\right. \end{aligned}$$
(76)

where

$$\begin{aligned} \frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}C_{p}}^{(k)}}&= \frac{-\varPsi }{\rho \left( {{}C_{p}}^{(k)}\right) ^{2}}\big (\Vert {{}\mathbf{{S}}}^{(0)}\Vert -2G\,{{}\Delta \lambda }^{(k)}\big )\,{{}\Delta \lambda }^{(k)}, \end{aligned}$$
(77)
$$\begin{aligned} \frac{\partial \,{{}\theta }^{(k)}}{\partial \,{{}\Delta \lambda }^{(k)}}&= \frac{\varPsi }{\rho \,{{}C_{p}}^{(k)}}\big (\Vert {{}\mathbf{{S}}}^{(0)}\Vert -4G\,{{}\Delta \lambda }^{(k)}\big ). \end{aligned}$$
(78)

Step 3: Shear modulus

At this stage, the shear modulus is computed by substituting the known temperature, \({{}\theta }^{(k)}\), into Eq. (22):

$$\begin{aligned} {{}\mu }^{(k)}=\mu _{0}-\frac{D_{0}}{\exp \left( \frac{\theta _{0}}{{{}\theta }^{(k)}}\right) -1}. \end{aligned}$$
(79)

Accordingly,

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\theta }^{(k)}}\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$
(80)

where

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\theta }^{(k)}}=\frac{-D_{0}\theta _{0}\exp \left( \frac{\theta _{0}}{{{}\theta }^{(k)}}\right) }{\left( {{}\theta }^{(k)}\right) ^{2}\left[ \exp \left( \frac{\theta _{0}}{{{}\theta }^{(k)}}\right) -1\right] ^{2}}. \end{aligned}$$
(81)

Step 4: Rate- and temperature-dependent pre-multipliers

The pre-multiplying term, \({{}S_{i}}^{(k)}\), is calculated by substituting \({{}\theta }^{(k)}, {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}\), and \({{}\mu }^{(k)}\) into Eq. (23), i.e.

$$\begin{aligned} {{}S_{i}}^{(k)}=\left( 1-\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)} b^{3} g_{0i}}\log \left( \frac{{\dot{\varepsilon }}_{0i}}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\right) \right] ^{\frac{1}{q_{i}}}\right) ^{\frac{1}{p_{i}}}. \end{aligned}$$
(82)

Differentiation and application of the chain rule yield

$$\begin{aligned}&\frac{{\mathrm {d}}\,{{}S_{i}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}+\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}\theta }^{(k)}}\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&\qquad \qquad \qquad +\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}\mu }^{(k)}}\frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$
(83)

where the partial derivatives are given by

$$\begin{aligned}&\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}=\frac{{{}\fancyscript{D}_{i}}^{(k)}}{\log \left( \frac{{\dot{\varepsilon }}_{0i}}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\right) {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}, \end{aligned}$$
(84)
$$\begin{aligned}&\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}\theta }^{(k)}}=\frac{-{{}\fancyscript{D}_{i}}^{(k)}}{{{}\theta }^{(k)}}, \end{aligned}$$
(85)
$$\begin{aligned}&\frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}\mu }^{(k)}}=\frac{{{}\fancyscript{D}_{i}}^{(k)}}{{{}\mu }^{(k)}}, \end{aligned}$$
(86)

in terms of

$$\begin{aligned} {{}\fancyscript{D}_{i}}^{(k)}{{}:={}}\frac{{{}S_{i}}^{(k)}\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)}b^{3} g_{0i}}\log \left( \frac{{\dot{\varepsilon }}_{0i}}{{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}\right) \right] ^{\frac{1}{q_{i}}}}{p_{i}q_{i}\left( 1-\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)}b^{3} g_{0i}}\log \left( \frac{{\dot{\varepsilon }}_{0i}}{{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}\right) \right] ^{\frac{1}{q_{i}}}\right) }. \end{aligned}$$
(87)

Similarly for \({{}S_{\varepsilon }}^{(k)}\), using Eq. (24):

$$\begin{aligned} {{}S_{\varepsilon }}^{(k)}=\left( 1-\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)} b^{3} g_{0\varepsilon }}\log \left( \frac{{\dot{\varepsilon }}_{0\varepsilon }}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\right) \right] ^{\frac{1}{q_{\varepsilon }}}\right) ^{\frac{1}{p_{\varepsilon }}}, \end{aligned}$$
(88)

and we have

$$\begin{aligned}&\frac{{\mathrm {d}}\,{{}S_{\varepsilon }}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}+\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}\theta }^{(k)}}\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&\qquad \qquad \qquad +\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}\mu }^{(k)}}\frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$
(89)

where

$$\begin{aligned}&\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}=\frac{{{}\fancyscript{D}_{\varepsilon }}^{(k)}}{\log \left( \frac{{\dot{\varepsilon }}_{0\varepsilon }}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\right) {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}, \end{aligned}$$
(90)
$$\begin{aligned}&\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}\theta }^{(k)}}=\frac{-{{}\fancyscript{D}_{\varepsilon }}^{(k)}}{{{}\theta }^{(k)}}, \end{aligned}$$
(91)
$$\begin{aligned}&\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}\mu }^{(k)}}=\frac{{{}\fancyscript{D}_{\varepsilon }}^{(k)}}{{{}\mu }^{(k)}}, \end{aligned}$$
(92)

and

$$\begin{aligned} {{}\fancyscript{D}_{\varepsilon }}^{(k)}{{}:={}}\frac{{{}S_{\varepsilon }}^{(k)}\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)}b^{3} g_{0\varepsilon }}\log \left( \frac{{\dot{\varepsilon }}_{0\varepsilon }}{{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}\right) \right] ^{\frac{1}{q_{\varepsilon }}}}{p_{\varepsilon }q_{\varepsilon }\left( 1-\left[ \frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)}b^{3} g_{0\varepsilon }}\log \left( \frac{{\dot{\varepsilon }}_{0\varepsilon }}{{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}\right) \right] ^{\frac{1}{q_{\varepsilon }}}\right) }. \end{aligned}$$
(93)

Step 5: Structure-dependent MTS

In order to calculate the structure-dependent MTS, \({{}{\hat{\tau }}_{\varepsilon }}^{(k)}\), we begin by substituting \({{}\theta }^{(k)}, {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}\) into (51), which gives

$$\begin{aligned} {{}h_{0}}^{(k)}&= A_{0}+A_{1}\log \left( {{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}\right) +A_{2}\sqrt{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\nonumber \\&-A_{3}{{}\theta }^{(k)}+A_{4}\left( {{}\theta }^{(k)}\right) ^{-A_{5}}, \end{aligned}$$
(94)

and

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}h_{0}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\frac{\partial \,{{}h_{0}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\,\frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}+\frac{\partial \,{{}h_{0}}^{(k)}}{\partial \,{{}\theta }^{(k)}}\,\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$
(95)

where

$$\begin{aligned} \frac{\partial \,{{}h_{0}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}&= \frac{A_{1}}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}+\frac{1}{2}\frac{A_{2}}{\sqrt{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}},\end{aligned}$$
(96)
$$\begin{aligned} \frac{\partial \,{{}h_{0}}^{(k)}}{\partial \,{{}\theta }^{(k)}}&= -A_{3}-A_{4}A_{5}\left( {{}\theta }^{(k)}\right) ^{-(A_{5}+1)}. \end{aligned}$$
(97)

Similarly, using (52), we evaluate

$$\begin{aligned} {{}{\hat{\tau }}_{\varepsilon s}}^{(k)}={\hat{\tau }}_{\varepsilon s0}\left( \frac{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\dot{\varepsilon }}_{0\varepsilon s}}\right) ^{\frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)} b^{3}g_{0\varepsilon s}}}, \end{aligned}$$
(98)

and

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}&= \frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\,\frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}+\frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}\theta }^{(k)}}\,\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&+\frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}\mu }^{(k)}}\,\frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$
(99)

where

$$\begin{aligned} \frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}&= {{}{\hat{\tau }}_{\varepsilon s}}^{(k)}\frac{k{{}\theta }^{(k)}}{{{}\mu }^{(k)}b^{3}g_{0\varepsilon s}}\frac{1}{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}},\end{aligned}$$
(100)
$$\begin{aligned} \frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}\theta }^{(k)}}&= {{}{\hat{\tau }}_{\varepsilon s}}^{(k)}\frac{k}{{{}\mu }^{(k)}b^{3}g_{0\varepsilon s}}\log \left( \frac{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\dot{\varepsilon }}_{0\varepsilon s}}\right) ,\end{aligned}$$
(101)
$$\begin{aligned} \frac{\partial \,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{\partial \,{{}\mu }^{(k)}}&= {{}{\hat{\tau }}_{\varepsilon s}}^{(k)}\frac{-k{{}\theta }^{(k)}}{\left( {{}\mu }^{(k)}\right) ^{2}b^{3}g_{0\varepsilon s}}\log \left( \frac{{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\dot{\varepsilon }}_{0\varepsilon s}}\right) . \end{aligned}$$
(102)

Then, \({{}{\hat{\tau }}_{\varepsilon }}^{(k)}\) is determined iteratively using Newton’s method. In this context, the solution is updated via

$$\begin{aligned} {{}{\hat{\tau }}_{\varepsilon }}^{(k,l+1)}={{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}-\frac{{{}\fancyscript{R}_{\varepsilon }}^{(k,l)}}{{{}\fancyscript{R}_{\varepsilon }^{\prime }}^{(k,l)}}, \end{aligned}$$
(103)

where \({{}(\cdot )}^{(k,l)}\) denotes the \(l\)th iterate, \(l=0,1,\dots \), and in light of (49)–(50), the residual is defined as

$$\begin{aligned} {{}\fancyscript{R}_{\varepsilon }}^{(k,l)}{{}:={}}{{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}-\left[ {{}{\hat{\tau }}_{\varepsilon }}_{(n)}+{{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k,l)}\left( \sqrt{\tfrac{2}{3}}\,{{}\Delta \lambda }^{(k)}\right) \right] ,\nonumber \\ \end{aligned}$$
(104)

in terms of

$$\begin{aligned} {{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k,l)}={{}h_{0}}^{(k)}\left( 1-\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\right) ^{\kappa }. \end{aligned}$$
(105)

Thus, we also have

$$\begin{aligned}&{{}\fancyscript{R}_{\varepsilon }^{\prime }}^{(k,l)}{{}:={}}\,\frac{{\mathrm {d}}\,{{}\fancyscript{R}_{\varepsilon }}^{(k,l)}}{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}} =\,1+\bigg [\frac{{{}h_{0}}^{(k)}\,\kappa }{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\left( 1-\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\right) ^{(\kappa -1)}\nonumber \\&\qquad \qquad \qquad \times \left( \sqrt{\tfrac{2}{3}}\,{{}\Delta \lambda }^{(k)}\right) \bigg ]. \end{aligned}$$
(106)

Initialization is performed by setting \({{}{\hat{\tau }}_{\varepsilon }}^{(k,0)}={{}{\hat{\tau }}_{\varepsilon }}_{(n)}\), and the iterative process is terminated when \(|{{}\fancyscript{R}_{\varepsilon }}^{(k,l)}|\) becomes smaller than a tolerance. At that point, we set \({{}{\hat{\tau }}_{\varepsilon }}^{(k)}={{}{\hat{\tau }}_{\varepsilon }}^{(k,l)}\) and \({{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k)}={{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k,l)}\). It can also be seen from Eq. (49) that

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}=\frac{{\mathrm {d}}\,{{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\sqrt{\tfrac{2}{3}}\,{{}\Delta \lambda }^{(k)}+{{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k)}\sqrt{\tfrac{2}{3}}, \end{aligned}$$
(107)

and from (50),

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}&= \left( 1-\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\right) ^{\kappa }\frac{{\mathrm {d}}\,{{}h_{0}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&+{{}\fancyscript{D}}^{(k)}_{{\hat{\tau }}_{\varepsilon }}\left( \!\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}-\frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\!\right) , \end{aligned}$$
(108)

where

$$\begin{aligned} {{}\fancyscript{D}}^{(k)}_{{\hat{\tau }}_{\varepsilon }}=\frac{{{}h_{0}}^{(k)}\,\kappa }{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\left( 1-\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\right) ^{(\kappa -1)}. \end{aligned}$$
(109)

Thus,

$$\begin{aligned}&\frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&\quad =\left( \left[ \left( 1-\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\right) ^{\kappa }\frac{{\mathrm {d}}\,{{}h_{0}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}+{{}\fancyscript{D}}^{(k)}_{{\hat{\tau }}_{\varepsilon }}\frac{{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}\frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon s}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\right] \right. \nonumber \\&\qquad \left. \times \sqrt{\tfrac{2}{3}}{{}\Delta \lambda }^{(k)}\!+\!{{}\left[ \frac{{\mathrm {d}}\,{\hat{\tau }}_{\varepsilon }}{{\mathrm {d}}\,{\bar{\varepsilon }}^{\text {vp}}}\right] }^{(k)}\sqrt{\tfrac{2}{3}}\right) \!\bigg /\!\left( 1\!+\!{{}\fancyscript{D}}^{(k)}_{{\hat{\tau }}_{\varepsilon }}\sqrt{\tfrac{2}{3}}{{}\Delta \lambda }^{(k)}\!\right) .\nonumber \\ \end{aligned}$$
(110)

Step 6: Flow stress

Finally, the flow stress is evaluated using [cf. Eqs. (21), (45)]:

$$\begin{aligned} {{}{\hat{\tau }}}^{(k)}={\hat{\tau }}_{a}+\frac{{{}\mu }^{(k)}}{\mu _{0}}\left[ {{}S_{i}}^{(k)}{\hat{\tau }}_{i}+{{}S_{\varepsilon }}^{(k)}{{}{\hat{\tau }}_{\varepsilon }}^{(k)}\right] . \end{aligned}$$
(111)

Differentiation and use of the chain rule lead to:

$$\begin{aligned} \frac{{\mathrm {d}}\,{{}{\hat{\tau }}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}&= \frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\,\frac{{\mathrm {d}}\,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&+\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}\theta }^{(k)}}\,\frac{{\mathrm {d}}\,{{}\theta }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}\nonumber \\&+\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}\mu }^{(k)}}\,\frac{{\mathrm {d}}\,{{}\mu }^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}} +\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}\,\frac{{\mathrm {d}}\,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}{{\mathrm {d}}\,{{}\Delta \lambda }^{(k)}}, \end{aligned}$$
(112)

where

$$\begin{aligned}&\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}=\frac{{{}\mu }^{(k)}}{\mu _{0}}\left[ \frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}\,{\hat{\tau }}_{i}+\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}{\dot{\bar{\varepsilon }}^{\mathrm{vp}}}}^{(k)}}{{}{\hat{\tau }}_{\varepsilon }}^{(k)}\right] , \end{aligned}$$
(113)
$$\begin{aligned}&\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}\theta }^{(k)}}=\frac{{{}\mu }^{(k)}}{\mu _{0}}\left[ \frac{\partial \,{{}S_{i}}^{(k)}}{\partial \,{{}\theta }^{(k)}}\,{\hat{\tau }}_{i}+\frac{\partial \,{{}S_{\varepsilon }}^{(k)}}{\partial \,{{}\theta }^{(k)}}{{}{\hat{\tau }}_{\varepsilon }}^{(k)}\right] , \end{aligned}$$
(114)
$$\begin{aligned}&\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}\mu }^{(k)}}=\frac{1}{\mu _{0}}\left[ {{}S_{i}}^{(k)}\,{\hat{\tau }}_{i}+{{}S_{\varepsilon }}^{(k)}{{}{\hat{\tau }}_{\varepsilon }}^{(k)}\right] \end{aligned}$$
(115)
$$\begin{aligned}&\frac{\partial \,{{}{\hat{\tau }}}^{(k)}}{\partial \,{{}{\hat{\tau }}_{\varepsilon }}^{(k)}}=\frac{{{}\mu }^{(k)}}{\mu _{0}}\,{{}S_{\varepsilon }}^{(k)}. \end{aligned}$$
(116)

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Mourad, H.M., Bronkhorst, C.A., Addessio, F.L. et al. Incrementally objective implicit integration of hypoelastic–viscoplastic constitutive equations based on the mechanical threshold strength model. Comput Mech 53, 941–955 (2014). https://doi.org/10.1007/s00466-013-0941-9

Download citation

Keywords

  • Viscoplasticity
  • Mechanical threshold strength
  • MTS model
  • Finite deformation
  • Objective integration
  • Radial return mapping
  • Finite element method