Elevated temperature creep of pearlitic steels: an experimental–numerical approach

Abstract

Pearlitic steels are well known for their high strength and hardness. This makes them the natural choice for applications in which structural integrity and minimum irreversible deformation over time are required. Although their room-temperature mechanical response has been intensively studied in the past, little information can be found in the literature regarding the effect of temperature on the mechanical response of pearlitic steels. In this paper, an experimental–numerical approach is used to study the mechanical response of pearlitic steels in the temperature range 20–500 °C. A finite-strain thermo-viscoplastic model is presented together with a set of elevated temperature tests (tensile and creep tests). The aim of the tests is twofold: first, to provide insight into the elevated-temperature mechanical response of the material; and second, to provide the data required to identify the corresponding material parameters. Furthermore, the model and the experimental data are instrumental in showing that the influence of temperature on the mechanical behavior of pearlitic steels becomes significant for temperatures above 350–400 °C.

Appendices

Appendix A: Tensor and vector operation notation

In this paper, Cartesian tensors and associated tensor products are used, using a Cartesian vector basis e ₁, e ₂, e ₃. The Einstein summation rule for repeated indices is used. Adopted notations are summarized below:

• Quantities:

  • Scalars: α,a,A,

  • Vectors: a=a _i e _i ,

  • Second-order tensors: A=A _ij e _i ⊗e _j ,

  • Second-order identity tensor: I=δ _ij e _i ⊗e _j ,

  • Fourth-order tensors: ${^{4}\mathbb{A} = A_{ijkl}\mathbf{e}_{i} \otimes\mathbf{e}_{j} \otimes\mathbf{e}_{k} \otimes\mathbf{e}_{l}}$,

  • Fourth-order identity tensor: ${^{4}\mathbb{I} = \delta_{il}\delta _{jk}\mathbf{e}_{i} \otimes\mathbf{e}_{j} \otimes\mathbf{e}_{k} \otimes\mathbf{e}_{l}}$,

  • Right transposed of the fourth-order identity tensor: ${^{4}\mathbb{I}^{RT} = \delta_{ik}\delta_{jl}\mathbf{e}_{i} \otimes\mathbf{e}_{j} \otimes\mathbf{e}_{k} \otimes\mathbf{e}_{l}}$.

• Operations:

  • Scalar multiplication: c=ab,c=a b,C=a B,

  • Dyadic product: C=a⊗b=a _i b _j e _i ⊗e _j ,

  • Inner product: c=a⋅b=a _i b _j ,C=A⋅B=A _ij B _jk e _i ⊗e _k ,

  • Double inner product: $c = \boldsymbol{{A}} : \boldsymbol{ {B}} = A_{ij}B_{ji}, \boldsymbol{{C}} = \mathbb{A} : \boldsymbol{ {B}} = A_{ijkl}B_{lk}\mathbf{e}_{i} \otimes\mathbf{e}_{j}$,

  • Transpose: A ^T=A _ji e _i ⊗e _j ,

  • Left transpose: $\mathbb{A}^{LT} = A_{jikl}\mathbf{e}_{i} \otimes \mathbf{e}_{j} \otimes\mathbf{e}_{k} \otimes\mathbf{e}_{l}$,

  • Right transpose: $\mathbb{A}^{RT} = A_{ijlk}\mathbf{e}_{i} \otimes \mathbf{e}_{j} \otimes\mathbf{e}_{k} \otimes\mathbf{e}_{l}$,

  • Gradient operator: $\boldsymbol{\nabla} = \mathbf{e}_{i} \frac{\partial }{\partial x_{i}}$.

Appendix B: Jacobian terms and derivatives

In the solution of (58a), (58b), the Jacobian terms given are required. These can be determined by taking the partial derivatives of the residuals r _τ and r _D with respect to the unknowns τ and D. This yields the following expressions:

$$\begin{aligned} \frac{\partial\boldsymbol{{r}}_{\boldsymbol{\tau}}}{\partial \boldsymbol{\tau}} \bigg|_{i} =& {^{4} \mathbb{I}} + \Delta t \theta(T){^{4}\mathbb{H}^{i}}: \biggl[ Z\bigl(\varphi^{i},D^{i}\bigr)\frac{\partial\boldsymbol{{N}}}{\partial \boldsymbol{\tau}} \bigg|_{i}+ \boldsymbol{{N}}^{i} \otimes\frac{\partial Z(\varphi,D)}{\partial \boldsymbol{\tau}} \bigg|_{i} \biggr], \end{aligned}$$

(59)

$$\begin{aligned} \frac{\partial\boldsymbol{{r}}_{\boldsymbol{\tau}}}{\partial D} \bigg|_{i} =& \Delta t \theta(T) \frac{\partial Z(\varphi,D)}{\partial D} \bigg|_{i}{^{4}\mathbb{H}^{i}}: \boldsymbol{{N}}^{i}, \end{aligned}$$

(60)

$$\begin{aligned} \frac{\partial r_{D}}{\partial\boldsymbol{\tau}} \bigg|_{i} =& -\Delta t \theta(T)h \bigl(D^{i}\bigr)\frac{\partial Z(\varphi,D)}{\partial \boldsymbol{\tau}} \bigg|_{i}, \end{aligned}$$

(61)

and

$$\begin{aligned} \frac{\partial r_{D}}{\partial D} \bigg|_{i} &= 1-\Delta t \theta(T) \biggl\{ h\bigl(D^{i}\bigr) \biggl[\frac{\partial Z(\varphi ,D)}{\partial D} \bigg|_{i}-\frac{d r(D)}{d D} \bigg|_{i} \biggr] \\ &\quad{}+\frac{d h(D)}{d D} \bigg|_{i} \bigl[ Z\bigl(\varphi^{i},D^{i} \bigr) - r\bigl(D^{i}\bigr) \bigr] \biggr\} . \end{aligned}$$

(62)

Furthermore, the required derivatives are:

$$\begin{aligned} \frac{\partial Z(\varphi,D)}{\partial\boldsymbol{\tau}} \bigg|_{i} =& \frac{ n Z(\varphi^{i},D^{i})}{ D^{i}} \coth \biggl[ \frac{\varphi^{i}}{ D^{i}} \biggr]\boldsymbol{{N}}^{i}, \end{aligned}$$

(63)

$$\begin{aligned} \frac{\partial Z(\varphi,D)}{\partial D} \bigg|_{i} =& -\frac{ n Z(\varphi^{i},D^{i})}{{D^{i}}^{2}} \coth \biggl[ \frac{\varphi^{i}}{ D^{i}} \biggr]\varphi^{i}, \end{aligned}$$

(64)

$$\begin{aligned} \frac{\partial\boldsymbol{{N}}}{\partial\boldsymbol{\tau}} \bigg|_{i} =& \frac{1}{\tau_{eq}} \biggl[ \frac {3}{2}{^{4}\mathbb{I}} -\frac{1}{2}\boldsymbol{{I}}\otimes \boldsymbol{{I}}- \boldsymbol{{N}}^{i}\otimes \boldsymbol{{N}}^{i} \biggr], \end{aligned}$$

(65)

$$\begin{aligned} \frac{d h(D)}{d D} \bigg|_{i} =& \frac{ m}{\delta C {\varphi_{D}}^{i}} \bigl[1 - {\varphi_{D}}^{i} \coth\bigl[ {\varphi_{D}}^{i} \bigr] \bigr]h\bigl(D^{i}\bigr), \end{aligned}$$

(66)

$$\begin{aligned} \frac{d r(D)}{d D} \bigg|_{i} =& \frac{ n}{\delta C} \coth \bigl[ {\varphi _{D}}^{i}\bigr] r\bigl(D^{i} \bigr). \end{aligned}$$

(67)