1 Introduction
2 Governing equations
2.1 Kinematics and thermodynamics
2.2 Constitutive model
2.3 Balance of linear momentum
3 Numerical solution procedure
3.1 Diagonally implicit Runge–Kutta method
(a) | (b) | (c) | |
---|---|---|---|
\(\begin{array}{c|cccc} c_1 &{} a_{11} &{} 0 &{} \cdots &{} 0 \\ c_2 &{} a_{21} &{} a_{22} &{} \cdots &{} 0 \\ \vdots &{}\vdots &{}\vdots &{}\ddots &{}\vdots \\ c_s &{} b_{1} &{} b_{2} &{} \cdots &{} b_{s} \\ \hline &{} b_{1} &{} b_{2} &{} \cdots &{} b_{s} \\ &{} \hat{b}_{1} &{} \hat{b}_{2} &{} \cdots &{} \hat{b}_{s} \end{array}\)
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\(\begin{array}{c|cc} \alpha &{} \alpha &{} 0 \\ 1 &{} 1-\alpha &{} \alpha \\ \hline &{} 1-\alpha &{} \alpha \\ &{} 1-\hat{\alpha } &{}\hat{\alpha } \end{array}\)
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\(\begin{array}{l} \alpha = 1-\frac{1}{2} \sqrt{2} \\ \hat{\alpha } = 2-\frac{5}{4} \sqrt{2} \\ \end{array}\)
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\(\begin{array}{c|c} 1 &{} 1 \\ \hline &{} 1 \\ \end{array}\)
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3.2 Implicit backward Euler method
3.3 Exponential update
3.4 Plastic incompressibility
4 Numerical examples
Parameter |
G
|
K
|
Q
|
\(G_0\)
|
q
|
\(\dot{\gamma }_0\)
|
m
|
B
|
\(g^{\alpha }_0\)
|
---|---|---|---|---|---|---|---|---|---|
Value | 80 MPa | 164 MPa | 200 MPa | 55 MPa | 1.4 | \(10^{-3}\) 1/s | 50 | 8 |
\( 7\times 10^{-3}\)
|
Parameter |
\(\epsilon _r\)
|
\(\epsilon _u\)
|
\(\epsilon _{F^p}\)
|
\(\epsilon _g\)
|
\(f_{max}\)
|
\(f_{min}\)
|
\(f_s\)
|
---|---|---|---|---|---|---|---|
Value |
\(10^{-9}\)
|
\(2 \times 10^{-3}\)
|
\(2.7 \times 10^{-3}\)
|
\(1.8 \times 10^{-3}\)
| 2 | 0.5 | 0.9 |