Phase-field modeling of fracture and crack growth in friction stir processed pure copper


The present work aims to predict the fracture mechanism of a friction stir processed (FSPed) copper. In particular, a phase-field model integrated with finite element model is developed for ductile fracture modeling. The phase-field evolution governed by the nonlinear coupled system comprising the linear momentum equation and the diffusion-type equation is solved concurrently through a Newton–Raphson approach. The proposed fracture model is established by correlating the phase-field degradation function with a scalar measure of the plastic strain, and assuming that the fracture takes place once the accumulated plastic strain reaches a critical value. The numerical simulation results are validated with experimental investigations. The results show that the proposed model is capable of capturing the experimentally observed sequence of elastoplastic base material behavior, FSP deformation, and fracture phenomena in specimens.

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Y. Liao appreciates the financial support by startup funding from the Department of Mechanical Engineering at the University of Nevada, Reno.

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Correspondence to Yiliang Liao.

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Appendix A

Assembly algorithm for matrices

An algorithm of the assembly of the global stiffness matrix \( \mathbbm{K} \) from contributions of element stiffness matrices k can be expressed by the following pseudo-code:


n = number of degrees of freedom per elementN = total number of degrees of freedom in the domainE = number of elementsC[E, n] = connectivity arrayk[n, n] = element stiffness matrix\( \mathbbm{K}\left[N,N\right] \) = global stiffness matrix

Here for simplicity, element matrices are assembled fully in the full square global matrix. Since the global stiffness matrix is symmetric and sparse, these facts are used to economize space and time in actual finite element codes [42].

Appendix B

Detailed explanation of single element (*inp file)

This part demonstrates a practical single element example, which can be used to generate any model in ABAQUS/Standard with the implemented fracture model. The problem is a simple element subjected to uniaxial tension. The present *inp file demonstrates an element with eight nodes, in 2D, with material properties and twenty status variables. From this point, the displacement, boundaries and the analysis are defined usually as it is done in a normal input file. Moreover, putting a visualization command in the ASSEMBLY section pointing to the UMAT elements for post-processing purposes [31, 43].


Appendix C

Umat algorithmic paradigm


Appendix D

Thermodynamic and kinematic preliminaries

The constitutive equations that characterize a single-crystal undergoing small deformations consist of additive decomposition

$$ \nabla \mathbf{u}={\mathbf{H}}^e+{\mathbf{H}}^p $$

which He and Hp refers to elastic and plastic distortion respectively.

Based on the fundamental assumption used to establish the plasticity theory is that plastic deformation is isochoric or volume preserving. In the case of infinitesimal strain, this is expressed by:

$$ \mathrm{tr}\left({\mathbf{H}}^p\right)=0 $$

with postulate the elastic stress-strain relation

$$ {\mathbf{T}}^{\boldsymbol{e}}=\mathbb{C}{\mathbf{E}}^{\boldsymbol{e}} $$

in which

$$ {\mathbf{E}}^{\boldsymbol{e}}=\operatorname{sym}\nabla \mathbf{u} $$

is the elastic strain and T is the macroscopic stress.

Extension of balance equation to phase-field approach

Based on the following considerations [44], the internal power can be written as:

$$ \mathcal{L}\left(\mathrm{P}\right)={\int}_{\mathrm{P}}{\mathbf{T}}^{\boldsymbol{e}}:{\mathbf{E}}^{\boldsymbol{e}} dv+{\int}_{\mathrm{P}}\left(\pi \dot{\varphi}+\boldsymbol{\xi} .\nabla \dot{\varphi}\right) dv $$
  • an elastic-stress Te power conjugate to \( {\dot{\mathbf{H}}}^{\boldsymbol{e}} \)

  • a scalar internal microscopic force π power conjugate to \( \dot{\varphi} \)

a vector microscopic stress ξ α power-conjugate to the phase-field gradient \( \nabla \dot{\varphi} \)

with P an arbitrary subregion of the body and φ is the crack phase field.

In which

$$ {\int}_{\mathrm{P}}\left(\boldsymbol{\xi} .\nabla \dot{\varphi}\right) dv={\int}_{\mathrm{\partial P}}\left(\boldsymbol{\xi} .\mathbf{n}\right)\dot{\varphi} da-{\int}_{\mathrm{P}}\dot{\varphi}\mathrm{Div}\boldsymbol{\xi} dv $$

the power is extended externally by consideration of

a scalar microscopic traction Ξ(n)power-conjugate to \( \dot{\varphi} \)

$$ \mathcal{W}\left(\mathrm{P}\right)={\int}_{\mathrm{\partial P}}\mathbf{t}\left(\mathbf{n}\right).\dot{\mathbf{u}} da+{\int}_{\mathrm{P}}\left(\mathbf{b}.\dot{\mathbf{u}}\right) dv+{\int}_{\mathrm{\partial P}}\boldsymbol{\Xi} \left(\mathbf{n}\right).\dot{\varphi} da $$

to simplify the presentation, external microscopic forces do not include.

with considering (generalized) virtual velocities

$$ \mathcal{V}=\left(\overset{\sim }{\mathbf{u}},{\overset{\sim }{\mathbf{H}}}^e,{\overset{\sim }{\mathbf{H}}}^p,\overset{\sim }{\varphi}\right) $$

a consideration that leads to the following expressions

$$ \mathcal{L}\left(\mathrm{P},\mathcal{V}\right)={\int}_{\mathrm{P}}{\mathbf{T}}^{\boldsymbol{e}}:{\overset{\sim }{\mathbf{H}}}^e dv+{\int}_{\mathrm{P}}\left(\pi \dot{\overset{\sim }{\varphi }}+\boldsymbol{\xi} \nabla \dot{\overset{\sim }{\varphi }}\right) dv $$
$$ \mathcal{W}\left(\mathrm{P},\mathcal{V}\right)={\int}_{\mathrm{\partial P}}\mathbf{t}\left(\mathbf{n}\right).\dot{\overset{\sim }{\mathbf{u}}} da+{\int}_{\mathrm{P}}\left(\mathbf{b}.\dot{\overset{\sim }{\mathbf{u}}}\right) dv+{\int}_{\mathrm{\partial P}}\boldsymbol{\Xi} \left(\mathbf{n}\right).\dot{\overset{\sim }{\varphi }} da $$

given any subregion P of the body,


From the fundamental standpoint adopted in [45], the stress tensor governed by the symmetry condition Te = TeTderiving from objectivity.

and the local force balance is

$$ \mathrm{Div}\mathbf{T}+\mathbf{b}=\mathbf{0}. $$

By considering the traction condition and the mentioned macroscopic force balance

$$ \mathbf{t}\left(\mathbf{n}\right)=\mathbf{Tn} $$

on the other hand

$$ \overset{\mathcal{V}=\left(\overset{\sim }{\mathbf{u}},\nabla \overset{\sim }{\mathbf{u}},0,0\right)}{\Rightarrow}\mathcal{W}\left(\mathrm{P},\mathcal{V}\right)=\mathcal{L}\left(\mathrm{P},\mathcal{V}\right)\overset{\mathrm{leads}\ \mathrm{to}\ }{\Rightarrow}\left\{\begin{array}{c}\mathrm{Div}\mathbf{T}+\mathbf{b}=\mathbf{0}\\ {}\mathbf{t}\left(\mathbf{n}\right)=\boldsymbol{\upsigma} \mathbf{n}\end{array}\right. $$


$$ \mathcal{V}=\left(\mathbf{0},-{\overset{\sim }{\mathbf{H}}}^p,{\overset{\sim }{\mathbf{H}}}^p,0\right)\overset{\mathrm{delivers}}{\to }\ {\boldsymbol{\upsigma}}_{dev}={\boldsymbol{\upsigma}}^{\mathrm{p}} $$


$$ {\boldsymbol{\upsigma}}^{\boldsymbol{p}}:{\overset{\sim }{\mathbf{H}}}^p={\boldsymbol{\upsigma}}^{\boldsymbol{p}}:{\boldsymbol{\upvarepsilon}}^{\boldsymbol{p}}. $$


$$ \overset{\mathcal{V}=\left(0,0,0,\overset{\sim }{\varphi}\right)}{\Rightarrow}\mathcal{W}\left(\mathrm{P},\mathcal{V}\right)=\mathcal{L}\left(\mathrm{P},\mathcal{V}\right)\overset{\mathrm{yields}}{\to }\ {\int}_{\mathrm{P}}\left(\mathbf{b}.\dot{\overset{\sim }{\mathbf{u}}}\right) dv+{\int}_{\mathrm{\partial P}}\boldsymbol{\Xi} \left(\mathbf{n}\right).\dot{\overset{\sim }{\varphi }} da={\int}_{\mathrm{P}}\left(\pi .\dot{\overset{\sim }{\varphi }}+\boldsymbol{\xi} .\nabla \dot{\overset{\sim }{\varphi }}\right) dv. $$

Eq. (54) leads to phase-field macroscopic force balance;

$$ \mathit{\operatorname{div}}\ \boldsymbol{\xi} -\pi +\mathrm{b}=0 $$

with phase-field microscopic traction

$$ \boldsymbol{\Xi} \left(\mathbf{n}\right)=\boldsymbol{\xi} .\mathbf{n} $$

to extended to plasticity

$$ {\mathbf{n}}^{\mathrm{p}}=\frac{{\boldsymbol{\upsigma}}_{dev}}{\left\Vert {\boldsymbol{\upsigma}}_{dev}\right\Vert}\kern0.5em ,\kern1em {\mathbf{n}}^{\mathrm{p}}=\frac{{\dot{\upvarepsilon}}^{\mathrm{p}}}{\left\Vert {\dot{\upvarepsilon}}^{\mathrm{p}}\right\Vert}\kern0.5em ,\kern0.75em {\tau}^{\mathrm{p}}=\left\Vert {\boldsymbol{\upsigma}}_{dev}\right\Vert \kern0.5em ,\kern0.75em {\dot{e}}^{\mathrm{p}}=\left\Vert {\dot{\boldsymbol{\upvarepsilon}}}^{\mathrm{p}}\right\Vert \mathbf{\ge}\mathbf{0}\kern0.75em ,\kern0.75em {e}^{\mathrm{p}}(t)={\int}_0^t\left\Vert {\dot{\boldsymbol{\upvarepsilon}}}^{\mathrm{p}}\right\Vert d\tau . $$

As a result, the power energy takes form

$$ {\int}_{\mathrm{P}}\boldsymbol{\upsigma} :{\dot{\boldsymbol{\upvarepsilon}}}^{\boldsymbol{e}} dv+{\int}_{\mathrm{P}}{\tau}^{\mathrm{p}}:{\dot{e}}^{\mathrm{p}} dv+{\int}_{\mathrm{P}}\boldsymbol{\xi} .\nabla \dot{\varphi} dv+{\int}_{\mathrm{P}}\left(\mathrm{b}.\dot{\varphi}\right) dv={\int}_{\mathrm{\partial P}}\boldsymbol{t}\left(\boldsymbol{n}\right).\dot{\boldsymbol{u}} da+{\int}_{\mathrm{\partial P}}\boldsymbol{b}.\dot{\boldsymbol{u}} da+{\int}_{\mathrm{\partial P}}\boldsymbol{\Xi} \left(\mathbf{n}\right).\dot{\overset{\sim }{\varphi }} da+{\int}_{\mathrm{P}}\left(\pi \dot{\varphi}\right) dv. $$

Dissipation inequality and constitutive laws

by the agencies acting externally to the region, P energy imbalance is the postulate that for each region P: [16],

$$ {\int}_{\mathrm{P}}\dot{\overline{E_{\ell } dv}}-{\int}_{\mathrm{P}}{\mathbf{T}}^{\boldsymbol{e}}:{\mathbf{E}}^{\boldsymbol{e}} dv+{\int}_{\mathrm{P}}\left(\pi \dot{\varphi}+\boldsymbol{\xi} .\nabla \dot{\varphi}\right) dv=-{\int}_{\mathrm{P}}\delta dv\le 0, $$

using the transport and divergence theorems, we arrive at the local inequality:

$$ \dot{E_{\ell }}-{\mathbf{T}}^{\boldsymbol{e}}:{\mathbf{E}}^{\boldsymbol{e}}-\left(\pi \dot{\varphi}+\boldsymbol{\xi} .\nabla \dot{\varphi}\right)=-\delta \le 0 $$

inserting the J2plasticity relations Eq. (20) into the local energy imbalance Eq.(33) and writing

$$ \dot{E_{\ell }}-\boldsymbol{\upsigma} :{\dot{\boldsymbol{\upvarepsilon}}}^{\boldsymbol{e}}-\left\Vert {\boldsymbol{\upsigma}}_{\boldsymbol{dev}}\right\Vert :\left\Vert {\dot{\upvarepsilon}}^p\right\Vert -\left(\pi \dot{\varphi}+\boldsymbol{\xi} .\nabla \dot{\varphi}\right)=-\delta \le 0 $$

for brevity E = E(α1, α2, α3, α4, α5, …) specifically, can be written

$$ {E}_{\ell }={E}_{\ell}\left({\boldsymbol{\upvarepsilon}}^{\boldsymbol{e}},\left\Vert {\dot{\upvarepsilon}}^p\right\Vert, \alpha, \varphi, \nabla \varphi, \dots \right) $$

substitution of Eq. (20) in the free-energy imbalance, Eq. (25) leads to

$$ \left(\frac{\partial {E}_{\ell }}{\partial {\boldsymbol{\upvarepsilon}}^{\boldsymbol{e}}}-\boldsymbol{\upsigma} \right):{\dot{\boldsymbol{\upvarepsilon}}}^{\boldsymbol{e}}+\left(\frac{\partial {E}_{\ell }}{\partial {e}^{\mathrm{p}}}-{\tau}^{\mathrm{p}}\right){\dot{e}}^{\mathrm{p}}+\frac{\partial {E}_{\ell }}{\partial \alpha}\dot{\alpha}+\left(\frac{\partial {E}_{\ell }}{\partial \varphi }-\pi \right)\dot{\varphi}+\left(\frac{\partial {E}_{\ell }}{\partial \nabla \varphi }-\boldsymbol{\xi} \right)\nabla \dot{\varphi}=-\delta \le 0, $$

As a consequence

$$ \left\{\begin{array}{c}\left(\frac{\partial {E}_{\ell }}{\partial {\boldsymbol{\upvarepsilon}}^{\boldsymbol{e}}}-\boldsymbol{\upsigma} \right):{\dot{\boldsymbol{\upvarepsilon}}}^{\boldsymbol{e}}=0\Rightarrow \kern0.75em \frac{\partial {E}_{\ell }}{\partial {\boldsymbol{\upvarepsilon}}^{\boldsymbol{e}}}=\boldsymbol{\upsigma} \\ {}\left(\frac{\partial {E}_{\ell }}{\partial \varphi }-\pi \right)\dot{\varphi}+\left(\frac{\partial {E}_{\ell }}{\partial \nabla \varphi }-\boldsymbol{\xi} \right)\nabla \dot{\varphi}\le 0\Rightarrow \kern0.75em \frac{\partial {E}_{\ell }}{\partial \varphi }=\pi, \kern0.75em \frac{\partial {E}_{\ell }}{\partial \nabla \varphi }=\boldsymbol{\xi} \kern0.5em \end{array},\right. $$

phase field evolution equation:

$$ \overset{\mathrm{subs}\ \mathrm{Eq}.(28)\mathrm{intoEq}.(21)}{\Rightarrow}\mathit{\operatorname{div}}\left(\frac{\partial {E}_{\ell }}{\partial \varphi}\right)-\frac{\partial {E}_{\ell }}{\partial \nabla \varphi }=0 $$

as a result of Eq. (28) and introducing the thermodynamic force power-conjugate to \( \dot{\alpha} \)

$$ {t}_{\alpha }=-\frac{\partial {E}_{\ell }}{\partial \varphi }, $$

following reduced dissipation inequality derived from

$$ \delta ={\tau}^{\mathrm{p}}{\dot{e}}^{\mathrm{p}}+{t}_{\alpha}\dot{\alpha}-\frac{\partial {E}_{\ell }}{\partial {e}^{\mathrm{p}}}{\dot{e}}^{\mathrm{p}}\mathbf{\ge}0\overset{{\dot{e}}^{\mathrm{p}}\ge 0}{\Rightarrow }{\tau}^{\mathrm{p}}{\dot{e}}^{\mathrm{p}}+{t}_{\alpha}\dot{\alpha}\ge 0 $$

from the J2plasticity and flow rule

$$ \frac{\partial {E}_{\ell }}{\partial {e}^{\mathrm{p}}}\le 0\Longrightarrow \frac{\partial {E}_{\ell }}{\partial {e}^{\mathrm{p}}}=\frac{\mathrm{\partial g}}{\partial {e}^{\mathrm{p}}}{\psi_e}^{+}\left(\boldsymbol{\upvarepsilon} \right)\Longrightarrow \frac{\mathrm{\partial g}}{\partial {e}^{\mathrm{p}}}\le 0 $$

Finally, from the context of present works

$$ g\left(\varphi, {e}^{\mathrm{p}}\right)={\left(1- q\varphi \right)}^{2\frac{e^{\mathrm{p}}}{e_{cri}^p}}+\eta \Longrightarrow {e}_{cri}^p=\sqrt{\frac{3}{2}}{\upvarepsilon}_{eq, crit}^{\mathrm{p}}. $$

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Esmaeilzadeh, P., Behnagh, R.A., Pour, M.A.M. et al. Phase-field modeling of fracture and crack growth in friction stir processed pure copper. Int J Adv Manuf Technol 109, 2377–2392 (2020).

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  • Friction stir processing
  • Fracture
  • Phase-field
  • Finite element model