Researching | Theetical models of void nucleation growth f ductile metals under dynamic loading: A review

Journals >Matter and Radiation at Extremes >Volume 7 >Issue 1 >Page 018201 > Article

Matter and Radiation at Extremes
Vol. 7, Issue 1, 018201 (2022)

Theoretical models of void nucleation and growth for ductile metals under dynamic loading: A review

Haonan Sui^1、2, Long Yu¹, Wenbin Liu¹, Ying Liu¹, Yangyang Cheng¹, and Huiling Duan^1、2、a)

Author Affiliations

¹State Key Laboratory for Turbulence and Complex System, Department of Mechanics and Engineering Science, BIC-ESAT, College of Engineering, Peking University, Beijing 100871, People’s Republic of China

²CAPT, HEDPS and IFSA, Collaborative Innovation Center of MoE, Peking University, Beijing 100871, People’s Republic of China

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DOI: 10.1063/5.0064557 Cite this Article

Haonan Sui, Long Yu, Wenbin Liu, Ying Liu, Yangyang Cheng, Huiling Duan. Theoretical models of void nucleation and growth for ductile metals under dynamic loading: A review[J]. Matter and Radiation at Extremes, 2022, 7(1): 018201 Copy Citation Text

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Abstract

Void nucleation and growth under dynamic loading are essential for damage initiation and evolution in ductile metals. In the past few decades, the development of experimental techniques and simulation methods has helped to reveal a wealth of information about the nucleation and growth process from its microscopic aspects to macroscopic ones. Powerful and effective theoretical approaches have been developed based on this information and have helped in the analysis of the damage states of structures, thereby making an important contribution to the design of damage-resistant materials. This Review presents a brief overview of theoretical models related to the mechanisms of void nucleation and growth under dynamic loading. Classical work and recent research progress are summarized, together with discussion of some aspects deserving further study.

ε_{c} \geq \{\begin{cases} δ, & α < 1, \\ δ \sqrt{\frac{1}{α}}, & α \geq 1, \end{cases}

(1)

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δ = \sqrt{\frac{(7 - 5 ν) (1 + ν^{*}) + (1 + ν) (8 - 10 ν) α}{10 (7 - 5 ν)}} .

(2)

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ε_{c} \geq \{\begin{cases} β \sqrt{\frac{1}{2 r}}, & α < 1, \\ β \sqrt{\frac{1}{2 r α}}, & α \geq 1, \end{cases}

(3)

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β = \sqrt{l \frac{48 [(7 - 5 ν) (1 + ν^{*}) + (1 + ν) (8 - 10 ν) α] [(7 - 5 ν) (1 - ν^{*}) + 5 (1 - ν^{2}) α]}{{(7 - 5 ν)}^{2} [2 (1 - 2 ν^{*}) + (1 + ν) α]}},

(4)

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σ_{l} = κ μ b \sqrt{ρ_{l}},

(5)

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ρ_{l} = \frac{5 ε_{p}}{3 r b} .

(6)

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ε_{c} \geq \frac{1}{30} {(\frac{σ_{c}}{κ μ})}^{2} \frac{r}{b} .

(7)

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ε_{p}^{*} = \sqrt{\frac{b ε_{p}}{r}} .

(8)

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Δ E_{elas} + Δ E_{s} < 0,

(9)

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σ_{m} d v > γ d a,

(10)

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r_{c} = \frac{2 γ}{σ_{m}} .

(11)

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F (ξ) = \sqrt{2} σ b \frac{ξ}{{(ξ^{2} + \frac{1}{2})}^{2}} - \frac{μ b^{2}}{π a (1 - ν)} \frac{ξ (ξ^{4} + \frac{1}{4})}{{(ξ^{2} + \frac{1}{2})}^{2} (ξ^{4} - \frac{1}{4})},

(12)

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σ_{cr} = \frac{μ b / a}{\sqrt{2} π (1 - ν)} \frac{{(1 + \sqrt{2} w / a)}^{4} + 1}{{(1 + \sqrt{2} w / a)}^{4} - 1},

(13)

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\begin{align} F = σ b \frac{a^{2}}{r^{2}} \sin 2 (θ - φ) \\ + \frac{μ b^{2} \cos (θ - φ)}{2 π (1 - ν)} \frac{a^{2}}{r^{3}} (2 \frac{r^{2} - a^{2}}{r^{2}} \sin^{2} θ - \frac{r^{2}}{a^{2}} \frac{r^{2}}{r^{2} - a^{2}}) . \end{align}

(14)

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σ_{cr} = \frac{μ b / a}{4 π (1 - ν)} (\frac{r_{0}^{2}}{a^{2}} \frac{1}{f_{0}} - 2 f_{0}),

(15)

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r_{0}^{2} = a^{2} + w^{2} + 2 a w \cos θ, f_{0} = \frac{r_{0}^{2} - a^{2}}{r_{0}^{2}} \sin θ .

(16)

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{\frac{\partial σ_{c r}}{\partial θ}|}_{θ = θ_{cr}} = 0 .

(17)

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\begin{align} - \frac{1 + 2 η}{2} \cos 2 θ_{c r} + (1 - η) [\frac{3}{7 - 5 ν} (\sin^{4} θ_{c r} - 3 \sin^{2} θ_{c r} \cos^{2} θ_{c r}) \\ - 3 \cos 2 θ_{c r}] + \frac{3 γ}{a σ_{c r}} \cos 2 θ_{c r} = 0, \end{align}

(18a)

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\begin{align} τ_{g}^{i m a g e} (r_{0}, θ_{0}) + τ_{g}^{e x t e r n a l} (r_{0}, θ_{0}, σ_{cr}, f) + τ_{g}^{s u r f a c e} (r_{0}, θ_{0}) \\ + τ_{g}^{f r i c t i o n} = 0, \end{align}

(18b)

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\begin{align} \frac{d}{d t} (\frac{3}{2} ρ {\dot{a}}^{2} \frac{4 π a^{3}}{3}) + σ_{y} {\dot{ε}}_{0} {(\frac{2}{3 ε_{y}})}^{1 / n} \\ \times {(\frac{2 \dot{a}}{{\dot{ε}}_{0} a})}^{(m + 1) / m} f (\frac{a}{a_{0}}, m, n) \frac{4 π a^{3}}{3} = 3 p \frac{\dot{a}}{a} \frac{4 π a^{3}}{3}, \end{align}

(19a)

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\begin{align} \frac{3}{2} ρ {\dot{a}}^{2} \frac{4 π a^{3}}{3} + \frac{n σ_{y} ε_{y}}{n + 1} {(\frac{2}{3 ε_{y}})}^{(n + 1) / n} g (\frac{a}{a_{0}}, n) \frac{4 π a^{3}}{3} \\ = p \frac{a^{3} - a_{0}^{3}}{a^{3}} \frac{4 π a^{3}}{3}, \end{align}

(19b)

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\begin{align} f (\frac{a}{a_{0}}, m, n) & = \int_{1}^{\infty} {[\log (\frac{x}{x - 1 + a_{0}^{3} / a^{3}})]}^{1 / n} x^{- (m + 1) / m} d x, \end{align}

(20)

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\begin{align} g (\frac{a}{a_{0}}, n) & = \int_{1}^{\infty} {[\log (\frac{x}{x - 1 + a_{0}^{3} / a^{3}})]}^{(n + 1) / n} d x, \end{align}

(21)

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D = \frac{ρ a_{0}^{2}}{k} {\dot{ε}}_{r e f}^{2 - 1 / m},

(22)

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p (t) - \int_{0}^{2 \ln (a / a_{0})} \frac{σ_{e}}{\exp (3 ε / 2) - 1} d ε = ρ (a \overset{⃜}{a} + \frac{3 {\dot{a}}^{2}}{2}),

(23)

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I_{inertia} = \frac{ρ a_{0}^{2} {\dot{p}}^{2}}{σ_{y} p_{s}^{2}},

(24)

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\frac{σ_{e}}{σ_{y}} = f (ε) h (T^{*}),

(25)

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h (T^{*}) = \exp \{- \frac{c ω σ_{y}}{(n + 1) ρ c_{p}} [ε f (ε) - ε_{y}]\},

(26)

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p (t) - p_{c} = ρ (a \overset{⃜}{a} + \frac{3 {\dot{a}}^{2}}{2}),

(27)

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\begin{align} p_{c} & = \lim_{a / a_{0} \to \infty} \int_{0}^{2 \ln (a / a_{0})} \frac{σ_{e}}{\exp (3 ε / 2) - 1} d ε \\ = \frac{2}{3} σ_{y} + \int_{ε_{y}}^{\infty} \frac{σ_{e}}{\exp (3 ε / 2) - 1} d ε . \end{align}

(28)

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g (p_{c}) = \frac{β_{2}}{β_{1}} {(\frac{〈 p_{c} - p_{0 c} 〉}{β_{1}})}^{β_{2} - 1} \exp [- {(\frac{〈 p_{c} - p_{0 c} 〉}{β_{1}})}^{β_{2}}],

(29)

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a = \sqrt{\frac{8}{33}} \sqrt{\frac{\dot{p}}{ρ}} {〈t - \frac{p_{c}}{\dot{p}}〉}^{3 / 2} .

(30)

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\frac{d V_{void}}{d p_{c}} = \frac{4}{3} π a^{3} N g (p_{c}),

(31)

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f = \frac{V_{void}}{1 + V_{void}},

(32)

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V_{void} = \int_{p_{c}} \frac{4}{3} π a^{3} N g (p_{c}) d p_{c} .

(33)

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Φ = {(\frac{Σ_{e}}{σ_{0}})}^{2} + 2 f \cosh (\frac{3 Σ_{m}}{2 σ_{0}}) - (1 + f^{2}) = 0,

(34)

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Φ = {(\frac{Σ_{e}}{{\bar{σ}}_{e}})}^{2} + 2 q_{1} f^{*} \cosh (\frac{3 q_{2} Σ_{m}}{2 {\bar{σ}}_{e}}) - (1 + q_{3} f^{* 2}) = 0,

(35)

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Φ = {(\frac{Σ_{e} - Σ_{e}^{d}}{σ_{0}})}^{2} + 2 f \cosh (\frac{3}{2} \frac{Σ_{m} - Σ_{m}^{d}}{σ_{0}}) - (1 + f^{2}) = 0,

(36)

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Σ = \frac{1}{|Ω|} \int_{Ω} σ d V .

(37)

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Σ = \frac{1}{|Ω|} \int_{Ω} σ d V + \frac{1}{|Ω|} \int_{Ω} ρ \overset{⃜}{x} \otimes x d V,

(38)

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\begin{align} Σ - Σ^{static} = & ρ a^{2} \{\frac{1}{5} (f^{- 2 / 3} - f) [\dot{D^{'}} + D^{'} \cdot D^{'} - \frac{1}{3} tr (D^{'} \cdot D^{'}) I] \\ + (f^{- 2 / 3} - 1) [D_{m} D^{'} + \frac{1}{6} (D^{'} : D^{'}) I] \\ - (f^{- 2 / 3} - f^{- 1}) {\dot{D}}_{m} I \\ + (3 f^{- 1} - \frac{5}{2} f^{- 2 / 3} - \frac{1}{2} f^{- 2}) D_{m}^{2} I\}, \end{align}

(39)

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p (t) - p^{static} = ρ (ϕ_{1} a \overset{⃜}{a} + ϕ_{2} \frac{3 {\dot{a}}^{2}}{2}),

(40)

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p^{static} = \inf (p_{c}, p^{visco}),

(41)

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D^{p} = (\frac{N_{m} b \bar{v}}{\bar{σ}}) s,

(42)

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\bar{v} = \frac{\bar{L}}{t_{w} + t_{r}},

(43)

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B {\bar{v}}_{r} = τ_{eff} b,

(44)

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B = \frac{B_{0}}{1 - {({\bar{v}}_{r} / c_{s})}^{2}},

(45)

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τ_{μ} = κ μ b \sqrt{N_{im} + N_{m}},

(46)

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\begin{align} {\dot{N}}_{im} & = {\dot{N}}_{trap} - {\dot{N}}_{rec}, \end{align}

(47a)

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\begin{align} {\dot{N}}_{m} & = {\dot{N}}_{nuc} + {\dot{N}}_{mult} - {\dot{N}}_{ann} - {\dot{N}}_{trap}, \end{align}

(47b)

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\dot{ε} = \frac{a^{3}}{r^{3}} \frac{3 \dot{a}}{a} .

(48)

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{\bar{v}}_{r} = \frac{a^{3}}{r^{3}} \frac{3 \dot{a}}{a b N_{m}} .

(49)

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p (t) - \int_{a}^{r_{0}} \frac{4}{r} τ d r = ρ (ϕ_{1} a \overset{⃜}{a} + ϕ_{2} \frac{3 {\dot{a}}^{2}}{2}),

(50)

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\int_{a}^{r_{0}} \frac{4}{r} τ d r = R_{cr} + R_{dd},

(51)

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R_{dd} = \int_{a}^{r_{0}} \frac{4}{r} (τ - τ_{μ}) d r = \int_{a}^{r_{0}} \frac{4}{r} \frac{B_{0} {\bar{v}}_{r}}{b} \frac{1}{1 - {({\bar{v}}_{r} / c_{s})}^{2}} d r .

(52)

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Haonan Sui, Long Yu, Wenbin Liu, Ying Liu, Yangyang Cheng, Huiling Duan. Theoretical models of void nucleation and growth for ductile metals under dynamic loading: A review[J]. Matter and Radiation at Extremes, 2022, 7(1): 018201

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