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Computational homogenisation for thermoviscoplasticity: application to thermally sprayed coatings

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Abstract

Metal forming processes require wear-resistant tool surfaces in order to ensure a long life cycle of the expensive tools together with a constant high quality of the produced components. Thermal spraying is a relatively widely applied coating technique for the deposit of wear protection coatings. During these coating processes, heterogeneous coatings are deployed at high temperatures followed by quenching where residual stresses occur which strongly influence the performance of the coated tools. The objective of this article is to discuss and apply a thermo-mechanically coupled simulation framework which captures the heterogeneity of the deposited coating material. Therefore, a two-scale finite element framework for the solution of nonlinear thermo-mechanically coupled problems is elaborated and applied to the simulation of thermoviscoplastic material behaviour including nonlinear thermal softening in a geometrically linearised setting. The finite element framework and material model is demonstrated by means of numerical examples.

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Notes

  1. http://www.openmp.org.

  2. http://eigen.tuxfamily.org.

  3. http://faculty.cse.tamu.edu/davis/suitesparse.html.

  4. http://www.openblas.net.

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Correspondence to Rolf Berthelsen.

A Consistent linearisation of the thermoviscoplastic model

A Consistent linearisation of the thermoviscoplastic model

In order to obtain the consistently linearised Jacobian of the finite element framework, (25), the total derivative of the stress \({\varvec{{\sigma }}}\), (52), and the mechanical heat production \(\hat{r}\), (58), with respect to strain \({\varvec{{\varepsilon }}}\) at constant temperature \(\theta \) must be determined

(A.1)
(A.2)

where \(\bullet \) represents a scalar-valued contraction. Analogously, the total derivatives of the stress and the mechanical heat production with respect to the temperature at constant strain are defined as

(A.3)
(A.4)
figure j

In order to find a relation between \(\text {d}\tilde{\mathbf {z}}\) and \(\text {d}{\varvec{{\varepsilon }}}\) and, respectively, between \(\text {d}\tilde{\mathbf {z}}\) and \(\text {d}\theta \), the linearisation of the local residual equation (68) at constant temperature as well as at constant strain must be carried out

(A.5)
(A.6)

In the above equations, the expression (69) is inserted. With this at hand, Eqs. (A.1)–(A.4) can be rewritten as

$$\begin{aligned} {{\varvec{{\mathsf {E}}}}}_\mathrm {alg}&= \dfrac{\partial {{\varvec{{\sigma }}}}}{\partial {\varvec{{\varepsilon }}}} - \dfrac{\partial {{\varvec{{\sigma }}}}}{\partial \tilde{\mathbf {z}}} \cdot {\mathbf {J}}_\mathrm {loc}^{-1} \cdot \dfrac{\partial {\mathbf {r}}_\mathrm {loc}}{\partial {\varvec{{\varepsilon }}}} \; , \end{aligned}$$
(A.7)
$$\begin{aligned} {{\varvec{{\gamma }}}}_\mathrm {alg}&= \dfrac{\partial \hat{r}}{\partial {\varvec{{\varepsilon }}}} - \dfrac{\partial \hat{r}}{\partial \tilde{\mathbf {z}}} \cdot {\mathbf {J}}_\mathrm {loc}^{-1} \cdot \dfrac{\partial {\mathbf {r}}_\mathrm {loc}}{\partial {\varvec{{\varepsilon }}}} \;, \end{aligned}$$
(A.8)
$$\begin{aligned} {{\varvec{{\beta }}}}_\mathrm {alg}&= \dfrac{\partial {{\varvec{{\sigma }}}}}{\partial \theta } - \dfrac{\partial {{\varvec{{\sigma }}}}}{\partial \tilde{\mathbf {z}}} \cdot {\mathbf {J}}_\mathrm {loc}^{-1} \cdot \dfrac{\partial {\mathbf {r}}_\mathrm {loc}}{\partial \theta } \;, \end{aligned}$$
(A.9)
$$\begin{aligned} {\zeta }_\mathrm {alg}&= \dfrac{\partial \hat{r}}{\partial \theta } - \dfrac{\partial \hat{r}}{\partial \tilde{\mathbf {z}}} \cdot {\mathbf {J}}_\mathrm {loc}^{-1} \cdot \dfrac{\partial {\mathbf {r}}_\mathrm {loc}}{\partial \theta } \;, \end{aligned}$$
(A.10)

such that, subsequently, merely the arising partial derivatives are required to evaluate the given expressions. The partial derivative of Eq. (52) with respect to the strain is

$$\begin{aligned} \begin{aligned} \dfrac{\partial {{\varvec{{\sigma }}}}}{\partial {{\varvec{{\varepsilon }}}}}&= {K\varvec{I}} \otimes \dfrac{\partial {\hbox {tr}}({{\varvec{{\varepsilon }}}})}{{{\varvec{{\varepsilon }}}}} + 2 \, G \, \dfrac{\partial {{\varvec{{\varepsilon }}}}_{\mathrm {dev}}}{\partial {{\varvec{{\varepsilon }}}}} - 2\,G\,{\varDelta }\lambda \,\dfrac{\partial {\varvec{{\nu }}}}{\partial {\varvec{{\varepsilon }}}}\\ \end{aligned} \end{aligned}$$
(A.11)

with

$$\begin{aligned} \dfrac{\partial {{\varvec{{\nu }}}}}{\partial {{\varvec{{\varepsilon }}}}} = \dfrac{2\,G}{\Vert {{\varvec{{\sigma }}}}_{\mathrm {dev}}^{\mathrm {tr}} \Vert } \, \bigg [\, {{\varvec{{\mathsf {I}}}}}^{\mathrm {dev}} - {{\varvec{{\nu }}}} \otimes {{\varvec{{\nu }}}} \,\bigg ] \;, \end{aligned}$$
(A.12)

wherein \({\varvec{{\mathsf {I}}}}^\mathrm{dev}\) denotes the deviatoric and symmetric fourth order identity tensor, it follows

$$\begin{aligned} \dfrac{\partial {{\varvec{{\sigma }}}}}{\partial {{\varvec{{\varepsilon }}}}} = {K\varvec{I}} \otimes \varvec{I} + \bigg [ \, 2 \, G - \dfrac{4\,G^2\,{\varDelta }\lambda }{\Vert {{\varvec{{\sigma }}}}_{\mathrm {dev}}^{\mathrm {tr}} \Vert } \, \bigg ] \, {{\varvec{{\mathsf {I}}}}}^{\mathrm {dev}} + \dfrac{4\,G^2\,{\varDelta }\lambda }{\Vert {{\varvec{{\sigma }}}}_{\mathrm {dev}}^{\mathrm {tr}} \Vert } \, {{\varvec{{\nu }}}} \otimes {{\varvec{{\nu }}}} \; . \end{aligned}$$
(A.13)

Furthermore, the partial derivative of Eq. (52) with respect to the temperature is given by

$$\begin{aligned} \dfrac{\partial {{\varvec{{\sigma }}}}}{\partial \theta } = -3 \, \alpha \,{K\varvec{I}}. \end{aligned}$$
(A.14)

Finally, the partial derivative of Eq. (52) with respect to \(\tilde{\mathbf {z}}\) is

$$\begin{aligned} \dfrac{\partial {{\varvec{{\sigma }}}}}{\partial \tilde{\mathbf {z}}} = \begin{bmatrix} -2\,G\,{{\varvec{{\nu }}}} \\ \mathbf 0 \end{bmatrix} \;. \end{aligned}$$
(A.15)

The partial derivative of the mechanical heat production, which is given by Eq. (58), with respect to the strain is

$$\begin{aligned} \dfrac{\partial \hat{r}}{\partial {\varvec{{\varepsilon }}}} = -\dfrac{3\,\alpha \,K\,\theta }{{\varDelta }t} \, {\varvec{{I}}} + \dfrac{2\,G\,{\varDelta }\lambda \,{\varvec{{\nu }}}}{{\varDelta }t} \end{aligned}$$
(A.16)

The partial derivative of the mechanical heat production with respect to the temperature is

$$\begin{aligned} \dfrac{\partial \hat{r}}{\partial \theta } = - 3 \, \alpha \, K \, {\hbox {tr}}(\dot{{\varvec{{\varepsilon }}}}) + \sqrt{\dfrac{2}{3}}\,\dfrac{\partial {\tilde{\kappa }}}{\partial \theta }\,\dfrac{{\varDelta }\lambda }{{\varDelta }t} \; , \end{aligned}$$
(A.17)

wherein the abbreviation

$$\begin{aligned} \tilde{\kappa } = \kappa - \theta \, \dfrac{\partial \kappa }{\partial \theta } \; , \end{aligned}$$
(A.18)

is introduced. The partial derivative of the plastic mechanical heat production with respect to \(\tilde{\mathbf {z}}\) is

$$\begin{aligned} \dfrac{\partial \hat{r}}{\partial \tilde{\mathbf {z}}} = \begin{bmatrix} \bigg [\,\Vert {{\varvec{{\sigma }}}}_\mathrm {dev}^\mathrm {tr} \Vert - 4\,G\,{\varDelta }\lambda +\sqrt{\dfrac{2}{3}}\,\tilde{\kappa } \,\bigg ] \, \dfrac{1}{{\varDelta }t} \\ \sqrt{\dfrac{2}{3}}\,\dfrac{\partial \tilde{\kappa }}{\partial k}\,\dfrac{{\varDelta }\lambda }{{\varDelta }t} \end{bmatrix} \; . \end{aligned}$$
(A.19)

The partial derivatives of the local residual (68) with respect to the strain and as well to the temperature are

$$\begin{aligned} \dfrac{\partial {\mathbf {r}}_\mathrm {loc}}{\partial {\varvec{{\varepsilon }}}}&= \begin{bmatrix} 2\,G\,{\varvec{{\nu }}}\\ \mathbf 0 \end{bmatrix} \; , \end{aligned}$$
(A.20)
$$\begin{aligned} \dfrac{\partial {\mathbf {r}}_\mathrm {loc}}{\partial \theta }&= \begin{bmatrix} -\sqrt{\dfrac{2}{3}}\,\bigg [\, \dfrac{\partial y_0}{\partial \theta } - \dfrac{\partial \kappa }{\partial \theta } \,\bigg ] \\ 0 \end{bmatrix} \; . \end{aligned}$$
(A.21)

With the above partial derivatives, the tangent moduli (A.7)-(A.10) finally read

$$\begin{aligned}&\begin{aligned} {{\varvec{{\mathsf {E}}}}}_\mathrm {alg} =&K \, {\varvec{{I}}} \otimes {\varvec{{I}}} + \bigg [\, 2\,G - \dfrac{4\,G^2\,{\varDelta }\lambda }{\Vert {{\varvec{{\sigma }}}}_\mathrm {dev}^\mathrm {tr} \Vert } \,\bigg ] \, {{\varvec{{\mathsf {I}}}}}^{\mathrm {dev}}\\&+ \bigg [\, \dfrac{4\,G^2\,{\varDelta }\lambda }{\Vert {{\varvec{{\sigma }}}}_\mathrm {dev}^\mathrm {tr} \Vert } - \dfrac{4\,G^2}{\det ({\mathbf {J}}_\mathrm {loc})} \,\bigg ] \, {\varvec{{\nu }}}\otimes {\varvec{{\nu }}}\; , \end{aligned} \end{aligned}$$
(A.22)
$$\begin{aligned}&\begin{aligned}&{{\varvec{{\gamma }}}}_\mathrm {alg} = - \dfrac{3 \, \alpha \, K\,\theta }{{\varDelta }t} \, {\varvec{{I}}} + \bigg [\, \dfrac{2\,G\,{\varDelta }\lambda }{{\varDelta }t}\\&\phantom {+ \qquad \bigg [}- \dfrac{2\,\mathrm {J}_{\mathrm {loc}\,22}\,G\,\big [\, \Vert {{\varvec{{\sigma }}}}_\mathrm {dev}^\mathrm {tr} \Vert - 4\,G\,{\varDelta }\lambda + \sqrt{2/3}\,\tilde{\kappa } \,\big ]}{{\varDelta }t \, \det ({\mathbf {J}}_\mathrm {loc})}\\&\phantom {+ \qquad \bigg [}+ \dfrac{2\,\sqrt{6}\, \mathrm {J}_{\mathrm {loc}\,21}\,G\,{\varDelta }\lambda }{3\,{\varDelta }t\,\det ({\mathbf {J}}_\mathrm {loc})} \, \dfrac{\partial \tilde{\kappa }}{\partial k}\,\bigg ] \, {\varvec{{\nu }}}\;, \end{aligned} \end{aligned}$$
(A.23)
$$\begin{aligned}&\begin{aligned}&{{\varvec{{\beta }}}}_\mathrm {alg} = - 3\, \alpha \,K \, {\varvec{{I}}} + \dfrac{2\,G}{\det ({\mathbf {J}}_\mathrm {loc})} \, \bigg [\, \mathrm {J}_{\mathrm {loc}\,12}\,\dfrac{\partial k}{\partial \theta } \\&- \mathrm {J}_{\mathrm {loc}\,22} \, \sqrt{\dfrac{2}{3}} \, \Big [\, \dfrac{\partial y_0}{\partial \theta } - \dfrac{\partial \kappa }{\partial \theta } \,\Big ] \,\bigg ] \, {\varvec{{\nu }}}\;, \end{aligned} \end{aligned}$$
(A.24)
$$\begin{aligned}&\begin{aligned} {\zeta }_\mathrm {alg} =&- \dfrac{3\,\alpha \,K\,\theta \,[{\hbox {tr}}({\varvec{{\varepsilon }}})-{\hbox {tr}}({\varvec{{\varepsilon }}}_n)]}{{\varDelta }t} + \sqrt{\dfrac{2}{3}}\,\dfrac{\partial \tilde{\kappa }}{\partial \theta }\,\dfrac{{\varDelta }\lambda }{{\varDelta }t}\\&\quad - \dfrac{1}{{\varDelta }t \, \det ({\mathbf {J}}_\mathrm {loc})} \, \Bigg [\, \bigg [\, \Vert {{\varvec{{\sigma }}}}_\mathrm {dev}^\mathrm {tr} \Vert - 4\,G\,{\varDelta }\lambda + \sqrt{\dfrac{2}{3}}\,\tilde{\kappa } \,\bigg ]\\&\phantom {-} \bigg [\, \mathrm {J}_{\mathrm {loc}\,12}\,\dfrac{\partial k}{\partial \theta } - \sqrt{\dfrac{2}{3}} \, \mathrm {J}_{\mathrm {loc}\,22}\,\Big [\, \dfrac{\partial y_0}{\partial \theta } - \dfrac{\partial \kappa }{\partial \theta } \,\Big ] \,\bigg ] \,\Bigg ]\\&\quad - \dfrac{1}{{\varDelta }t \, \det ({\mathbf {J}}_\mathrm {loc})} \, \Bigg [\, \bigg [\, \sqrt{\dfrac{2}{3}}\,\dfrac{\partial \tilde{\kappa }}{\partial k} \, \dfrac{{\varDelta }\lambda }{{\varDelta }t} \,\bigg ]\\&\phantom {-} \bigg [\,-\mathrm {J}_{\mathrm {loc}\,11}\,\dfrac{\partial k}{\partial \theta } + \sqrt{\dfrac{2}{3}}\,\mathrm {J}_{\mathrm {loc}\,21}\,\Big [\, \dfrac{\partial y_0}{\partial \theta } - \dfrac{\partial \kappa }{\partial \theta } \,\Big ] \,\bigg ] \,\Bigg ] \;. \end{aligned} \end{aligned}$$
(A.25)

Furthermore, the total derivative of the heat flux (59) with respect to the temperature gradient \(\nabla _{{\varvec{{x}}}}\theta \) reads

$$\begin{aligned} {\varvec{{L}}}_\mathrm{alg} = \dfrac{\text {d}{\varvec{{q}}}}{\text {d}\nabla _{{\varvec{{x}}}}\theta } = -\varLambda \, {\varvec{{I}}} \; . \end{aligned}$$
(A.26)

Details regarding the total derivative of the surface heat flux contribution, cf. Eq. (20)\(_4\), can be found in [1].

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Berthelsen, R., Denzer, R., Oppermann, P. et al. Computational homogenisation for thermoviscoplasticity: application to thermally sprayed coatings. Comput Mech 60, 739–766 (2017). https://doi.org/10.1007/s00466-017-1436-x

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