A numerical framework based on localizing gradient damage methodology for high cycle fatigue crack growth simulations

Abstract

Standard non-local gradient damage methodology for fatigue analysis has an intrinsic drawback of unusual widening of the damage zone. This causes a rapid growth of crack in the simulations which often violate experimental evidences. In order to tackle this undesirable behaviour, the localizing gradient damage methodology has been formulated for high cycle fatigue crack growth simulations. The framework comprises of coupling damage and elasticity through continuum mechanics, a fatigue damage law and an interaction function which reduces the influence of damaged regions on the surrounding locality. The present scheme prevents the spurious widening of the damage-band around the critically damaged area and therefore the non-physical growth of fatigue crack in the simulations is successfully countered. The developed framework is tested on various standard specimens under mode-I and mixed-mode high cycle fatigue loads. Nonlinear finite element analysis is used for this purpose. The discretized form of solver equations for the localizing framework is mathematically derived. Numerical examples show that the simulated crack-growth curves using proposed localizing framework agree closely with the experimental data and has a higher accuracy than the standard non-local framework.

Appendices

Appendix 1: Linearization of damage

At a particular load-step (n + 1), the damage at the end of iterations i and i + 1 are given below in Eqs. (48) and (49) respectively.

$$ D_{n + 1}^{(i + 1)} = D_{n} + \left( {(1 - \eta )f\left( {D_{n} ,e_{n} } \right) + \eta f\left( {D_{n + 1}^{(i + 1)} ,e_{n + 1}^{(i + 1)} } \right)} \right)\left( {e_{n + 1}^{(i + 1)} - e_{n} } \right) $$

(48)

$$ D_{n + 1}^{(i)} = D_{n} + \left( {(1 - \eta )f\left( {D_{n} ,e_{n} } \right) + \eta f\left( {D_{n + 1}^{(i)} ,e_{n + 1}^{(i)} } \right)} \right)\left( {e_{n + 1}^{(i)} - e_{n} } \right) $$

(49)

Subtracting, Eqs. (48) from Eq. (49),

$$ \begin{aligned} \delta D = & D_{n + 1}^{(i + 1)} - D_{n + 1}^{(i)} = (1 - \eta )f\left( {D_{n} ,e_{n} } \right)\left( {e_{n + 1}^{(i + 1)} - e_{n + 1}^{(i)} } \right) \\ & \quad + \eta f\left( {D_{n + 1}^{(i + 1)} ,e_{n + 1}^{(i + 1)} } \right)\left( {e_{n + 1}^{(i + 1)} - e_{n} } \right) \\ & \quad - \eta f\left( {D_{n + 1}^{(i)} ,e_{n + 1}^{(i)} } \right)\left( {e_{n + 1}^{(i)} - e_{n} } \right) \\ \end{aligned} $$

(50)

By applying first term of Taylor’s series expansion on f,

$$ f\left( {D_{n + 1}^{(i + 1)} ,e_{n + 1}^{(i + 1)} } \right) = f\left( {D_{n + 1}^{(i)} ,e_{n + 1}^{(i)} } \right) + \frac{\partial f}{{\partial D}}{\updelta }D + \frac{\partial f}{{\partial e}}{\updelta }e $$

(51)

Also, the linearization of non-local strain (e),

$$ e_{n + 1}^{(i + 1)} = e_{n + 1}^{(i)} + {\updelta }e $$

(52)

Substituting Eqs. (51) and (52) in (50) and neglecting the product of infinitesimal terms,

$$ \begin{aligned} \delta D = & (1 - \eta )f\left( {D_{n} ,e_{n} } \right)\delta e + \eta f\left( {D_{n + 1}^{(i)} ,e_{n + 1}^{(i)} } \right)\delta e \\ & \quad + \eta \frac{\partial f}{{\partial D}}\left( {e_{n + 1}^{(i + 1)} - e_{n} } \right)\delta D + \,\eta \frac{\partial f}{{\partial e}}\left( {e_{n + 1}^{(i + 1)} - e_{n} } \right)\delta e \\ \end{aligned} $$

(53)

At any gauss point, the incremental non-local strain (δe) can be written using linear shape functions S_e and the nodal vector of non-local strain (δe) as,

$$ {\updelta }e = {\mathbf{S}}_{e} \,{\updelta }{\mathbf{e}} $$

(54)

Substituting Eq. (54) into (53) and rearranging the terms, the linearized damage is obtained as shown below,

$$ {\updelta }D = p^{(i)} \,{\mathbf{S}}_{e} {\updelta }{\mathbf{e}} $$

(55)

where the expression of p⁽ⁱ⁾ is given as,

$$ p^{(i)} = \frac{{(1 - \eta )f\left( {D_{n} ,e_{n} } \right)\,\, + \,\,\eta f\left( {D_{n + 1}^{(i)} ,e_{n + 1}^{(i)} } \right)\, + \,\,\eta \frac{\partial f}{{\partial e}}\left( {e_{n + 1}^{(i + 1)} - e_{n} } \right)}}{{1 - \,\eta \frac{\partial f}{{\partial D}}\left( {e_{n + 1}^{(i + 1)} - e_{n} } \right)}} $$

(56)

Appendix 2: Linearization of moment stress

At a particular load-step (n + 1) and at a particular iteration (i), the moment stress is given as

$$ \left( {{{\varvec{\upvarepsilon}}}_{m} } \right)_{n + 1}^{(i)} = \varphi_{n + 1}^{(i)} \,h\,l_{s}^{2} \,\,\left( {\nabla e} \right)_{n + 1}^{(i)} $$

(57)

By applying first term of Taylor’s series expansion on ε_m,

$$ {\updelta }{{\varvec{\upvarepsilon}}}_{m} = \left( {\frac{{\partial {{\varvec{\upvarepsilon}}}_{m} }}{\partial \varphi }} \right)_{n + 1}^{(i)} \left( {\frac{\partial \varphi }{{\partial D}}} \right)_{n + 1}^{(i)} {\updelta }D\,\,\, + \,\,\,\left( {\frac{{\partial {{\varvec{\upvarepsilon}}}_{m} }}{{\partial \left( {\nabla e} \right)}}} \right)_{n + 1}^{(i)} {\updelta }\left( {\nabla e} \right) $$

(58)

Furthermore,

$$ \left( {\frac{{\partial {{\varvec{\upvarepsilon}}}_{m} }}{\partial \varphi }} \right)_{n + 1}^{(i)} = \,h\,l_{s}^{2} \,\,\left( {\nabla e} \right)_{n + 1}^{(i)} $$

(59)

$$ \left( {\frac{{\partial {{\varvec{\upvarepsilon}}}_{m} }}{{\partial \left( {\nabla e} \right)}}} \right)_{n + 1}^{(i)} = \,\varphi_{n + 1}^{(i)} \,h\,l_{s}^{2} \,\, $$

(60)

$$ \left( {\frac{\partial \varphi }{{\partial D}}} \right)_{n + 1}^{(i)} = \frac{ - \mu }{{1 - \exp ( - \mu )}}\exp ( - \mu D_{n + 1}^{(i)} ) $$

(61)

$$ \left( {\nabla e} \right)_{n + 1}^{(i)} = {\mathbf{B}}_{e} {\mathbf{e}}_{n + 1}^{(i)} $$

(62)

Substituting Eqs. (59, 60, 61, 62) and (55) into (58), the linearized moment stress is obtained as shown below,

$$ {\updelta }{{\varvec{\upvarepsilon}}}_{m} = h\,l_{s}^{2} \left( {{\mathbf{B}}_{e} {\mathbf{e}}_{n + 1}^{(i)} \,p^{(i)} q^{(i)} {\mathbf{S}}_{e} \,{\updelta }{\mathbf{e}}\,\, + \,\,\varphi_{(n + 1)}^{(i)} \,{\mathbf{B}}_{e} \,{\updelta }{\mathbf{e}}} \right) $$

(63)

where the expression of q⁽ⁱ⁾ is given as,

$$ q^{(i)} = \frac{ - \mu }{{1 - \exp ( - \mu )}}\exp ( - \mu D_{n + 1}^{(i)} ) $$

(64)

Appendix 3: Derivatives of equivalent strain

The equivalent Von-Mises strain is given as,

$$ \varepsilon_{eq} = \frac{1}{1 + \nu }\sqrt {3J_{2\varepsilon } } $$

(65)

where J_2ε is the second invariant of the deviatoric strain tensor which is given below for plane stress condition.

$$ J_{2\varepsilon } = \frac{1}{3}\left( {\varepsilon_{xx}^{2} + \varepsilon_{yy}^{2} + \varepsilon_{zz}^{2} - \varepsilon_{xx} \varepsilon_{yy} - \varepsilon_{yy} \varepsilon_{zz} - \varepsilon_{zz} \varepsilon_{xx} } \right) + \varepsilon_{xy}^{2} $$

(66)

The derivatives of equivalent strain are given as,

$$ \frac{{\partial \varepsilon_{eq} }}{{\partial \varepsilon_{xx} }} = \frac{{\partial \varepsilon_{eq} }}{{\partial J_{2\varepsilon } }}\frac{{\partial J_{2\varepsilon } }}{{\partial \varepsilon_{xx} }} = \frac{1}{2(1 + \nu )}\sqrt {\frac{3}{{J_{2\varepsilon } }}} \times \frac{1}{3}\left( {2\varepsilon_{xx} - \varepsilon_{yy} - \varepsilon_{zz} } \right) $$

(67)

$$ \frac{{\partial \varepsilon_{eq} }}{{\partial \varepsilon_{yy} }} = \frac{{\partial \varepsilon_{eq} }}{{\partial J_{2\varepsilon } }}\frac{{\partial J_{2\varepsilon } }}{{\partial \varepsilon_{yy} }} = \frac{1}{2(1 + \nu )}\sqrt {\frac{3}{{J_{2\varepsilon } }}} \times \frac{1}{3}\left( {2\varepsilon_{yy} - \varepsilon_{xx} - \varepsilon_{zz} } \right) $$

(68)

$$ \frac{{\partial \varepsilon_{eq} }}{{\partial \varepsilon_{xy} }} = \frac{{\partial \varepsilon_{eq} }}{{\partial J_{2\varepsilon } }}\frac{{\partial J_{2\varepsilon } }}{{\partial \varepsilon_{xy} }} = \frac{1}{2(1 + \nu )}\sqrt {\frac{3}{{J_{2\varepsilon } }}} \times 2\varepsilon_{xy} $$

(69)