Extended finite element modeling of fatigue crack growth microstructural mechanisms in alloys with secondary/reinforcing phases: model development and validation

Abstract

Light structural metals have been extensively applied throughout the transportation sector in recent years with greater impetus to reduce vehicle weight and enhance energy efficiency. However, their use is restricted by the engineering challenge of fatigue crack growth and the need to understand, simulate, and predict crack propagation mechanisms with respect to materials’ microstructure. To address this need, a comprehensive computational methodology has been developed using the extended finite element method to predict fatigue crack interaction mechanisms with characteristic microstructural features. Mathematical formulations and algorithmic development are rigorously addressed, with specific emphasis on the incorporation of relevant physical phenomena of plasticity and particle debonding/fracture. This approach is validated by comparison with analytical models and experiments on cast aluminum–silicon alloys using digital image correlation. The proposed methodology constitutes a framework for the successful development, application, and advancement of computational design of materials. Ultimately, this contributes to material/process selection and design for structural integrity by enabling rapid assessment of fatigue crack growth resistance without prior testing, thereby reducing of the extent of costly experimental investigations.

Appendices

Appendices

1.1 Appendix A: numerical integration of enriched elements

The assumptions of PU are violated by discontinuities and singularities cutting the support of enriched elements, and the procedure for numerical integration must be revised. Enriched elements are divided into non-overlapping “sub-triangle” elements that do not cross discontinuities by a Delaunay triangulation. The points used for the triangulation are those where an element’s support is cut (at the intersection of an edge with either crack flanks or particle interfaces), vertices of kinks in the crack flanks or particle interfaces, and the crack tip. An example of the creation of sub-triangles for crack tip and sign enriched elements is shown in Fig. 12 for a quadrilateral element. Gaussian quadrature is performed in a two-level procedure from sub-triangle Gauss coordinates, to quadrilateral Gauss coordinates, to global Cartesian coordinates (two coordinate transforms to calculate the total Jacobian). It should also be noted that this method does not increase the number of degrees of freedom.

Fig. 12

Numerical integration of the finite element mesh in a is conducted by dividing the parent quadrilateral elements into sub-triangles that do not cross the discontinuity. Examples of crack tip enriched elements b and sign enriched elements c identify the triangulation points, sub-triangles, and Gauss points

The conventional shape functions required for numerical integration of the governing equation are given for a four-node quadrilateral element by Eq. (A1), and require modification for enriched elements. The enriched shape functions for elements containing the crack flanks, interface, and/or crack tip are obtained by multiplying the standard shape functions by the shifted enrichment functions (evaluated relative to a particular node and its respective enriched degree(s) of freedom, as was done in Eq. (7)). The total shape function for an enriched node is then the concatenation of the standard and enriched shape functions, given by Eq. (A2).

$$ \begin{aligned} {\mathbf{N}}^{std} & = \frac{1}{4}\left[ \left( {1 - \xi } \right)\left( {1 - \eta } \right)\quad\left( {1 + \xi } \right)\left( {1 - \eta } \right)\quad\left( {1 + \xi } \right)\left( {1 + \eta } \right) \right. \quad\\ &\quad\quad \left. \left( {1 - \xi } \right)\left( {1 + \eta } \right) \right]^{T} \end{aligned} $$

(A1)

$$ \begin{aligned} & {\mathbf{N}} = \left[ {{\mathbf{N}}^{std} \, {\mathbf{N}}^{sign}\, {\mathbf{N}}^{ramp} \, {\mathbf{N}}^{tip} } \right] \\ & = \left[ {{\mathbf{N}}^{std}\, {\mathbf{N}}^{std} \bar{\psi }_{sign} \left( {\mathbf{x}} \right)\, {\mathbf{N}}^{std} \bar{\psi }_{ramp} \left( {\mathbf{x}} \right)\, {\mathbf{N}}^{std} \bar{\psi }_{tip} \left( {\mathbf{x}} \right)} \right] \end{aligned} $$

(A2)

The strain–displacement matrices are related to the element shape functions by the spatial derivative. They are obtained by element-wise pre-multiplication of each shape function by the matrix differential operator, L, given in Eq. (A3). This results in the standard and enriched strain–displacement matrices, Eqs. (A4) and (A5). The spatial derivatives of the standard and enriched shape functions can then be calculated by the chain rule, given in Eqs. (A6)–(A10) for rectangular elements. For the crack tip functions, which are defined in local polar coordinates at the crack tip, additional coordinate transformations to local Cartesian coordinates ($x_{1}^{'}$, $x_{2}^{'}$) and then to global Cartesian coordinates ($x_{1}$, $x_{2}$) must be conducted, Eq. (A11), resulting in the derivatives required for Eq. (A10). Then, similar to the shape functions, the total strain–displacement matrix B is the concatenation of the standard and enriched strain–displacement matrices, or ${\mathbf{B}} = \left[ {\begin{array}{*{20}c} {{\mathbf{B}}^{std} } & {{\mathbf{B}}^{enr} } \\ \end{array} } \right]$.

$$ {\mathbf{L}} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\partial /\partial x_{1} } \\ 0 \\ {\partial /\partial x_{2} } \\ \end{array} } & {\begin{array}{*{20}c} 0 \\ {\partial /\partial x_{2} } \\ {\partial /\partial x_{1} } \\ \end{array} } \\ \end{array} } \right] $$

(A3)

$$ {\mathbf{B}}^{std} = {\mathbf{L}} \circ \varvec{ }{\mathbf{N}}^{std} \left( {\mathbf{x}} \right) = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\partial {\mathbf{N}}^{std} /\partial x_{1} } \\ 0 \\ {\partial {\mathbf{N}}^{std} /\partial x_{2} } \\ \end{array} } & {\begin{array}{*{20}c} 0 \\ {\partial {\mathbf{N}}^{std} /\partial x_{2} } \\ {\partial {\mathbf{N}}^{std} /\partial x_{1} } \\ \end{array} } \\ \end{array} } \right] $$

(A4)

$$ {\mathbf{B}}_{p}^{enr} = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\partial \left( {{\mathbf{N}}^{std} \bar{\psi }_{p} \left( {\mathbf{x}} \right)} \right)/\partial x_{1} } \\ 0 \\ {\partial \left( {{\mathbf{N}}^{std} \bar{\psi }_{p} \left( {\mathbf{x}} \right)} \right)/\partial x_{2} } \\ \end{array} } & {\begin{array}{*{20}c} 0 \\ {\partial \left( {{\mathbf{N}}^{std} \bar{\psi }_{p} \left( {\mathbf{x}} \right)} \right)/\partial x_{2} } \\ {\partial \left( {{\mathbf{N}}^{std} \bar{\psi }_{p} \left( {\mathbf{x}} \right)} \right)/\partial x_{1} } \\ \end{array} } \\ \end{array} } \right] $$

(A5)

$$\begin{aligned}& \frac{\partial }{{\partial {\mathbf{x}}}}{\mathbf{N}}^{std}\\ & \quad = \left\{ {\begin{array}{*{20}c} {\frac{\partial }{{\partial x_{1} }}{\mathbf{N}}^{std} = \frac{\partial \xi }{{\partial x_{1} }}\frac{{\partial {\mathbf{N}}^{std} }}{\partial \xi } + \frac{\partial \eta }{{\partial x_{1} }}\frac{{\partial {\mathbf{N}}^{std} }}{\partial \eta } = \frac{2}{{l_{1} }}\frac{1}{4} } {\left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - \left( {1 - \eta } \right)} & {1 - \eta } \\ \end{array} } & {\begin{array}{*{20}c} {1 + \eta } & { - \left( {1 + \eta } \right)} \\ \end{array} } \\ \end{array} } \right]} \\ {\frac{\partial }{{\partial x_{2} }}{\mathbf{N}}^{std} = \frac{\partial \eta }{{\partial x_{2} }}\frac{{\partial {\mathbf{N}}^{std} }}{\partial \eta } + \frac{\partial \xi }{{\partial x_{2} }}\frac{{\partial {\mathbf{N}}^{std} }}{\partial \xi } = \frac{2}{{l_{2} }}\frac{1}{4} } { \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} { - \left( {1 - \xi } \right)} & { - \left( {1 + \xi } \right)} \\ \end{array} } & {\begin{array}{*{20}c} {1 + \xi } & {1 - \xi } \\ \end{array} } \\ \end{array} } \right]} \\ \end{array} } \right.\end{aligned} $$

(A6)

$$ \begin{aligned} & \frac{\partial }{{\partial {\mathbf{x}}}}{\mathbf{N}}^{std} \bar{\psi }_{p} \left( {\mathbf{x}} \right) = \frac{{\partial {\mathbf{N}}^{std} }}{{\partial {\mathbf{x}}}}\bar{\psi }_{p} \left( {\mathbf{x}} \right) \\ & + {\mathbf{N}}^{std} \frac{\partial }{{\partial {\mathbf{x}}}}\bar{\psi }_{p} \left( {\mathbf{x}} \right) \\ \end{aligned} $$

(A7)

$$ \frac{\partial }{{\partial {\mathbf{x}}}}\bar{\psi }_{sign} \left( {\mathbf{x}} \right) = 0 $$

(A8)

$$ \begin{aligned} \frac{\partial }{{\partial {\mathbf{x}}}}\bar{\psi }_{ramp} \left( {\mathbf{x}} \right) &= \frac{{\partial {\mathbf{N}}^{std} }}{{\partial {\mathbf{x}}}}\left| {\psi_{ramp} \left( {\mathbf{x}} \right)} \right| \\ &\quad - sign\left( {\psi_{ramp} \left( {\mathbf{x}} \right)} \right)\left( {\frac{{\partial {\mathbf{N}}^{std} }}{{\partial {\mathbf{x}}}}\psi_{ramp} \left( {\mathbf{x}} \right)} \right) \\ \end{aligned} $$

(A9)

$$ \begin{aligned} & \frac{{\partial \bar{\psi }_{tip}^{\alpha } }}{{\partial x_{1} }} = \frac{{\partial \bar{\psi }_{tip}^{\alpha } }}{{\partial x_{1}^{'} }}\cos \omega - \frac{{\partial \bar{\psi }_{tip}^{\alpha } }}{{\partial x_{2}^{'} }}\sin \omega , \\ & \frac{{\partial \bar{\psi }_{tip}^{\alpha } }}{{\partial x_{2} }} = \frac{{\partial \bar{\psi }_{tip}^{\alpha } }}{{\partial x_{1}^{'} }}\sin \omega + \frac{{\partial \bar{\psi }_{tip}^{\alpha } }}{{\partial x_{2}^{'} }}\cos \omega \\ \end{aligned} $$

(A10a,b)

$$ \begin{aligned} \frac{{\partial \bar{\psi }_{tip}^{1} }}{{\partial x_{1}^{'} }} &= - \frac{1}{2\sqrt r }\sin \frac{\theta }{2},\frac{{\partial \bar{\psi }_{tip}^{1} }}{{\partial x_{2}^{'} }} = \frac{1}{2\sqrt r }\cos \frac{\theta }{2},\\\frac{{\partial \bar{\psi }_{tip}^{2} }}{{\partial x_{1}^{'} }} & = \frac{1}{2\sqrt r }\cos \frac{\theta }{2},\frac{{\partial \bar{\psi }_{tip}^{2} }}{{\partial x_{2}^{'} }} = \frac{1}{2\sqrt r }\sin \frac{\theta }{2} \\ \end{aligned} $$

(A11a,b,c,d)

$$ \begin{aligned} & \frac{{\partial \bar{\psi }_{tip}^{3} }}{{\partial x_{1}^{'} }} = - \frac{1}{2\sqrt r }\sin \frac{3\theta }{2}\sin \theta , \\ & \frac{{\partial \bar{\psi }_{tip}^{3} }}{{\partial x_{2}^{'} }} = \frac{1}{2\sqrt r }\left( {\sin \frac{\theta }{2} + \sin \frac{3\theta }{2}\cos \theta } \right) \\ \end{aligned} $$

(A11e,f)

$$ \begin{aligned} & \frac{{\partial \bar{\psi }_{tip}^{4} }}{{\partial x_{1}^{'} }} = - \frac{1}{2\sqrt r }\cos \frac{3\theta }{2}\sin \theta , \\ & \frac{{\partial \bar{\psi }_{tip}^{4} }}{{\partial x_{2}^{'} }} = \frac{1}{2\sqrt r }\left( {\cos \frac{\theta }{2} + \cos \frac{3\theta }{2}\cos \theta } \right) \\ \end{aligned} $$

(A11g,h)

1.2 Appendix B: calculation of the interaction integral

The mode I and II stress intensity factors, Eqs. (A12) and (A13), for a crack tip in a homogeneous domain are calculated by means of the interaction integral which is determined with Eqs. (A14)–(A23). Much of the procedure is analogous to the construction of the global stiffness matrix and will not be discussed in depth here. It is calculated as a contour integral [39] around the crack tip(s), typically using the elements contained within a circle with a radius one to three element widths. The first two terms on the right hand side of Eq. (A14) must be solved by calculating the stress and strain for the present (1) and auxiliary (2) states, Eqs. (A15)–(A22). The third term, the interaction strain energy, W, is the sum of term-wise multiplication of Eqs. (A16) and (A22), given in Eq. (A23). The weight function q is given by Möes et al. [4] to take a value of 1 within the contour around the crack tip and zero elsewhere (i.e., only contributions from elements within the domain of the contour integral are considered).

$$ \begin{aligned} & K_{I,plane\, strain} = I^{\left( 1 \right)} \frac{E}{{2\left( {1 - \nu^{2} } \right)}}, \\ & K_{I,plane\, stress} = I^{\left( 1 \right)} \frac{E}{2} \\ \end{aligned} $$

(A12)

$$ \begin{aligned} & K_{II,plane \,strain} = I^{\left( 2 \right)} \frac{E}{{2\left( {1 - \nu^{2} } \right)}}, \\ & K_{II,plane\, stress} = I^{\left( 2 \right)} \frac{E}{2} \\ \end{aligned} $$

(A13)

$$ I^{{\left( {1,2} \right)}} = \mathop \smallint \limits_{A}^{ } \left[ {\sigma_{st}^{\left( 1 \right)} \frac{{\partial u_{s}^{\left( 2 \right)} }}{{\partial {\mathbf{x}}}} + \sigma_{st}^{\left( 2 \right)} \frac{{\partial u_{s}^{\left( 1 \right)} }}{{\partial {\mathbf{x}}}} - W^{{\left( {1,2} \right)}} \delta_{st} } \right]\frac{\partial q}{{\partial x_{t} }}dA $$

(A14)

$$ \left[ {\begin{array}{*{20}c} {\sigma_{11} } \\ {\sigma_{22} } \\ {\sigma_{12} } \\ \end{array} } \right] = {\mathbf{C}} {\mathbf{B}} \left[ {\begin{array}{*{20}c} {{\bar{\mathbf{u}}}} & {{\bar{\mathbf{a}}}} \\ \end{array} } \right]^{T} $$

(A15)

$$ \sigma_{st}^{\left( 1 \right)} = \left[ {\begin{array}{*{20}c} {\cos \omega } & {\sin \omega } \\ { - \sin \omega } & {\cos \omega } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\sigma_{11} } & {\sigma_{12} } \\ {\sigma_{12} } & {\sigma_{22} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \omega } & {\sin \omega } \\ { - \sin \omega } & {\cos \omega } \\ \end{array} } \right]^{T} $$

(A16)

$$ \sigma_{st}^{\left( 2 \right)} = \left[ {\begin{array}{*{20}c} {\frac{1}{{\sqrt {2\pi r} }}\cos \frac{\theta }{2}\left( {1 - \sin \frac{\theta }{2}\sin \frac{3\theta }{2}} \right)} & {\frac{1}{{\sqrt {2\pi r} }}\sin \frac{\theta }{2}\cos \frac{\theta }{2}\cos \frac{3\theta }{2}} \\ {\frac{1}{{\sqrt {2\pi r} }}\sin \frac{\theta }{2}\cos \frac{\theta }{2}\cos \frac{3\theta }{2}} & {\frac{1}{{\sqrt {2\pi r} }}\cos \frac{\theta }{2}\left( {1 + \sin \frac{\theta }{2}\sin \frac{3\theta }{2}} \right)} \\ \end{array} } \right] $$

(A17)

$$ \begin{aligned} \frac{{\partial u_{e} }}{{\partial x_{f} }} &= \mathop \sum \limits_{i = 1}^{{\mathcal{N}}} \frac{{\partial {\mathbf{N}}}}{{\partial x_{f} }}u_{e,i} \\ & \quad + \mathop \sum \limits_{p = 1}^{{\mathcal{P}}} \mathop \sum \limits_{j = 1}^{{\mathcal{J}}} \frac{{\partial {\mathbf{N}}}}{{\partial x_{f} }}\bar{\psi }_{p} \left( {\mathbf{x}} \right)a_{e,p,j} \left( {e,f = 1,2} \right) \\ \end{aligned} $$

(A18)

$$ \frac{{\partial u_{s}^{\left( 1 \right)} }}{{\partial {\mathbf{x}}}} = \left[ {\begin{array}{*{20}c} {\cos \omega } & {\sin \omega } \\ { - \sin \omega } & {\cos \omega } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\frac{{\partial u_{1} }}{{\partial x_{1} }}} & {\frac{{\partial u_{1} }}{{\partial x_{2} }}} \\ {\frac{{\partial u_{2} }}{{\partial x_{1} }}} & {\frac{{\partial u_{2} }}{{\partial x_{2} }}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\cos \omega } & {\sin \omega } \\ { - \sin \omega } & {\cos \omega } \\ \end{array} } \right]^{T} $$

(A19)

$$ \begin{aligned} & \frac{{\partial u_{1} }}{\partial r} = \frac{1}{{\sqrt {2\pi r} }}\frac{1 + \nu }{2E}\cos \frac{\theta }{2}\left( {\frac{3 - \nu }{1 + \nu } - \cos \theta } \right), \\ & \frac{{\partial u_{2} }}{\partial r} = \frac{1}{{\sqrt {2\pi r} }}\frac{1 + \nu }{2E}\sin \frac{\theta }{2}\left( {\frac{3 - \nu }{1 + \nu } - \cos \theta } \right) \\ \end{aligned} $$

(A20a,b)

$$ \frac{{\partial u_{1} }}{\partial \theta } = \sqrt {\frac{r}{2\pi }} \frac{1 + \nu }{E}\left\{ { - \frac{1}{2}\sin \frac{\theta }{2}\left( {\frac{3 - \nu }{1 + \nu } - \cos \theta } \right) + \cos \frac{\theta }{2}\sin \theta } \right\} $$

(A20c)

$$ \frac{{\partial u_{2} }}{\partial \theta } = \sqrt {\frac{r}{2\pi }} \frac{1 + \nu }{E}\left\{ {\frac{1}{2}\cos \frac{\theta }{2}\left( {\frac{3 - \nu }{1 + \nu } - \cos \theta } \right) + \sin \frac{\theta }{2}\sin \theta } \right\} $$

(A20d)

$$ \begin{aligned} & \frac{\partial r}{{\partial x_{1} }} = \cos \theta ,\frac{\partial r}{{\partial x_{2} }} = \sin \theta ,\,\frac{\partial \theta }{{\partial x_{1} }} \\ & = - \frac{\sin \theta }{r},\,\frac{\partial \theta }{{\partial x_{2} }} = \frac{\cos \theta }{r} \\ \end{aligned} $$

(A21a,b,c,d)

$$ \frac{{\partial u_{s}^{\left( 2 \right)} }}{{\partial {\mathbf{x}}}} = \left[ {\begin{array}{*{20}c} {\frac{{\partial u_{1} }}{\partial r}\frac{\partial r}{{\partial x_{1} }} + \frac{{\partial u_{1} }}{\partial \theta } \frac{\partial \theta }{{\partial x_{1} }}} & {\frac{{\partial u_{2} }}{\partial r}\frac{\partial r}{{\partial x_{1} }} + \frac{{\partial u_{2} }}{\partial \theta }\frac{\partial \theta }{{\partial x_{1} }}} \\ {\frac{{\partial u_{1} }}{\partial r}\frac{\partial r}{{\partial x_{2} }} + \frac{{\partial u_{1} }}{\partial \theta }\frac{\partial \theta }{{\partial x_{2} }}} & {\frac{{\partial u_{2} }}{\partial r}\frac{\partial r}{{\partial x_{2} }} + \frac{{\partial u_{2} }}{\partial \theta }\frac{\partial \theta }{{\partial x_{2} }}} \\ \end{array} } \right] $$

(A22)

$$ W^{{\left( {1,2} \right)}} = \mathop \sum \limits_{s = 1}^{2} \mathop \sum \limits_{t = 1}^{2} \sigma_{st}^{\left( 1 \right)} \frac{{\partial u_{st}^{\left( 2 \right)} }}{{\partial {\mathbf{x}}}} $$

(A23)