CN104036150A - Technology for calculating one-dimensional ideal elastic-plastic solid under Eulerian coordinate system - Google Patents

Abstract

The present invention proposes a technique for calculating one-dimensional ideal elastoplastic solids in the Euler coordinate system. technology of physical quantities. The innovation of the present invention is mainly reflected in the calculation method of the material derivative in the one-dimensional Hooke's law in the Euler coordinate system. The proposal of the present invention can be directly used to calculate one-dimensional ideal elastic-plastic solids, and is of great significance in practical engineering applications such as the one-dimensional ideal elastic-plastic solids are subjected to external forces and coupled with other media.

Description

A kind of technology of calculating one dimension ideal elastic-plastic solid under Eulerian coordinates system

Technical field

The present invention relates to a kind of technology of calculating one dimension ideal elastic-plastic solid, be specifically related to a kind of technology of calculating one dimension ideal elastic-plastic solid under Eulerian coordinates system.

Background technology

One dimension ideal elastic-plastic solid model can be compared with accurate description solid (as the metal such as aluminium, steel) the each physical quantity situation of change under the External Force Acting that is subject to common intensity.Therefore, the computing technique of research ideal elastic-plastic solid has important using value and application prospect widely in Practical Project.

At present, although the technology that has existed some to calculate ideal elastic-plastic solid, the technology all proposing from the present invention is different.Such as, M.L.Wilkins after the model of proposition ideal elastic-plastic solid, adopted method of finite difference to solve this model in 1964, had wherein used complicated full discrete form.For another example, B.P.Howell adopted Free Lagrange method to calculate ideal elastic-plastic solid in 2000.The method is calculated under Largrangian coordinates, although calculating on some variablees (as deviatoric stress) and can being simplified, become very complicated while being generalized to higher-dimension.For the calculating of ideal elastic-plastic solid had both simply been had accurately, the present invention directly calculates under Eulerian coordinates system, only the derivative in Hooke law need be processed as individual derivative.It is worth mentioning that, the present invention is inspired and is proposed in the work of M.B.Tyndall in 1993.But, in the computing method of M.B.Tyndall, but there are some mistakes.First,, in the time calculating position at a upper time step of each net point under Eulerian coordinates, he has taked time average to speed on fixing net point.This computing method are correct under Largrangian coordinates, are but wrong under Eulerian coordinates, and the governing equation under the Eulerian coordinates of setting up with him contradicts.Secondly, he has adopted the parabolic interpolation (quadratic function interpolation) of this both sides net point in calculating the value at related physical quantity time step place on each net point., the governing equation of ideal elastic-plastic solid is hyperbolic equations, propagates and has a directivity, adopts parabolic interpolation can cause the inaccurate of calculating, even causes unstable and produces mistake.Actual numerical evaluation has also verified that his method exists some mistakes really.For this problem, the present invention directly takes linear interpolation windward.The computing technique of the one dimension ideal elastic-plastic solid that in a word, the present invention proposes has been taken into account simplicity and the correctness of method.

Summary of the invention

The present invention propose the technology of calculating one dimension ideal elastic-plastic solid, its summary of the invention is mainly reflected in the technology of a set of complete calculating one dimension ideal elastic-plastic solid under Eulerian coordinates systems, and its innovative point is mainly reflected in individual derivative in the one dimension Hooke law account form under Eulerian coordinates system.

For one-dimensional case, the governing equation of ideal elastic-plastic solid under Eulerian coordinates system is

$\frac{\∂U}{\∂t}+\frac{\∂F\left(U\right)}{\∂x}=0---\left(1\right)$

$U=\left(\begin{array}{c}\mathrm{\ρ}\\ \mathrm{\ρu}\\ E\end{array}\right),F\left(U\right)=\left(\begin{array}{c}\mathrm{\ρu}\\ {\mathrm{\ρu}}^2-{\mathrm{\σ}}_x\\ (E-{\mathrm{\σ}}_x)u\end{array}\right)$

Herein, ρ is density, and u is speed, and p is pressure, and E is total energy, σ _xit is the total stress of x direction.In addition,, for ideal elastic-plastic solid, its total stress and pressure also meet relation below:

σ _x＝-p+s _x

Wherein, s _xit is the deviatoric stress of x direction.When ideal elastic-plastic solid is in elastic stage, have

$\stackrel{\·}{p}=K\frac{\stackrel{\·}{\mathrm{\ρ}}}{\mathrm{\ρ}}$ With ${\stackrel{\·}{s}}_x=-\frac{4}{3}\mathrm{\μ}\frac{\stackrel{\·}{\mathrm{\ρ}}}{\mathrm{\ρ}}$

Wherein K is bulk modulus, and μ is modulus of shearing.When ideal elastic-plastic solid is in mecystasis, have

$p=c_0^2(\mathrm{\ρ}-{\mathrm{\ρ}}_0)+({\mathrm{\γ}}_s-1)\mathrm{\ρe}$ With $s_x=\±\frac{2}{3}{Y}_0$

Wherein c ₀, ρ ₀, γ _sbe the constant relevant with concrete solid, Y ₀it is yield strength; For deviatoric stress s _x, positive sign represents that solid is in extended state, symbol represents that solid is in compressive state.Below ideal elastic-plastic solid meets, be elastic stage when equation

${s_x}^2,,{\left(\frac{2}{3}{Y}_0\right)}^2$

In the time that above-mentioned inequality is false, solid is in mecystasis.

Concrete summary of the invention of the present invention can be summed up as following computing technique.Suppose the each variate-value of known one dimension ideal elastic-plastic solid at n time step these variate-values need to be advanced to n+1 time step, obtain its computing technique realizes by following six steps:

1. solve governing equation (1), by each variate-value of n time step in governing equation be updated to n+1 time step, obtain ${\mathrm{\ρ}}_i^{n+1},u_i^{n+1},E_i^{n+1}.$

2. calculate each net point under Eulerian coordinates in the position of n time step, be designated as x _old, have

3. adopt linear interpolation windward, calculate ρ, p, s _xat x _oldthe value at place, is denoted as ρ _old, p _old, s _xold, as

$\left\{\begin{array}{c}{\mathrm{\&rho;}}_{\mathrm{old}}={\mathrm{\&rho;}}_i^n-\frac{{\mathrm{\&rho;}}_i^n-{\mathrm{\&rho;}}_{i-1}^n}{\mathrm{\&Delta;x}}(x_i^{n+1}-x_{\mathrm{old}}),u_i^{n+1}\&GreaterEqual;0\\ {\mathrm{\&rho;}}_{\mathrm{old}}={\mathrm{\&rho;}}_i^n+\frac{{\mathrm{\&rho;}}_{i+1}^n-{\mathrm{\&rho;}}_i^n}{\mathrm{\&Delta;x}}(x_{\mathrm{old}}-x_i^{n+1}),u_i^{n+1}<0\end{array}\right.$

P _oldand s _xoldalso can calculate by similar fashion.

4. utilize Hooke law and linear interpolation windward, obtain preliminary

$\left\{\begin{array}{c}{s}_{\mathrm{xi}}^{n+1}=s_{\mathrm{xold}}+2\mathrm{\&mu;}[\frac{u_i^{n+1}-u_{i-1}^{n+1}}{\mathrm{\&Delta;x}}\mathrm{\&Delta;t}+\frac{1}{3}\ln \left(\frac{{\mathrm{\&rho;}}_i^{n+1}}{{\mathrm{\&rho;}}_{\mathrm{old}}}\right)],u_i^{n+1}\&GreaterEqual;0\\ s_{\mathrm{xi}}^{n+1}=s_{\mathrm{xold}}+2\mathrm{\&mu;}[\frac{u_{i+1}^{n+1}-u_i^{n+1}}{\mathrm{\&Delta;x}}\mathrm{\&Delta;t}+\frac{1}{3}\ln \left(\frac{{\mathrm{\&rho;}}_i^{n+1}}{{\mathrm{\&rho;}}_{\mathrm{old}}}\right)],u_i^{n+1}<0\end{array}\right.$

5. judge at each Eulerian mesh point by von Mises yield condition the elastic-plastic behavior at place, and upgrade force value extremely if a certain net point place meets von Mises yield condition, solid is in elastic stage, and pressure calculate by Hooke law

$p_i^{n+1}=p_{\mathrm{old}}+K\ln \left(\frac{{\mathrm{\&rho;}}_i^{n+1}}{{\mathrm{\&rho;}}_{\mathrm{old}}}\right)$

If do not meet von Mises yield condition at this net point, solid is in moulding state, and pressure calculate by state equation

$p_i^{n+1}=c_0^2({\mathrm{\&rho;}}_i^{n+1}-{\mathrm{\&rho;}}_0)+({\mathrm{\&gamma;}}_s-1)(E_i^{n+1}-\frac{1}{2}{\mathrm{\&rho;}}_i^{n+1}{u}_i^{n+1}{u}_i^{n+1})$

Meanwhile, make deviatoric stress meet ideal plasticity condition

$s_{\mathrm{xi}}^{n+1}=\frac{2}{3}{Y}_0\frac{s_{\mathrm{xi}}^{n+1}}{\left|s_{\mathrm{xi}}^{n+1}\right|}$

6. return to step 1 until reach the time iteration requirement of setting.

Brief description of the drawings

Fig. 1 is the process flow diagram that the present invention calculates one dimension ideal elastic-plastic solid under Eulerian coordinates system;

Fig. 2 to Fig. 4 is the numerical results that the present invention calculates one dimension ideal elastic-plastic solid under Eulerian coordinates system.

Embodiment

For the specific embodiment of the present invention is described, will demonstrate an example below.Consider the one dimension Riemannian problem in aluminium, wherein the dimensionless initial value in this Riemannian problem left side is u _l=20.0, p _l=1.0, ρ _l=2.7, s _l=0.0, the dimensionless initial value on right side is u _r=-20.0, p _r=1.0, ρ _r=2.7, s _r=0.0.2000 the Eulerian mesh points that are being equally spaced in nondimensional solution interval [0,1], and the initial interface of Riemannian problem is 0.0.Meanwhile, the relevant dimensionless group of the ideal elastoplastic model of aluminium is respectively ρ ₀=2.71, c ₀=538.0, γ _s=2.71, K=740000.0, μ=265000.0, Y ₀=3000.0.

This problem will produce elastic wave and plastic wave in the left and right sides, interface simultaneously.Get time step Δ t=0.0000015, adopt Lax-Friedrich form to calculate, obtain in the time of time t=0.001 the negative total stress in aluminium, speed, density as shown in Figures 2 to 4.

Claims (3)

1. a technology of calculating one dimension ideal elastic-plastic solid under Eulerian coordinates systems, is characterized in that, this technology is a set ofly complete under Eulerian coordinates system, to calculate one dimension ideal elastic-plastic solid related physical quantity (density p, speed u, pressure p, deviatoric stress s _x) technology.

2. one dimension ideal elastic-plastic solid as claimed in claim 1, is characterized in that, when it is in elastic stage, meets Hooke law

$\stackrel{\&CenterDot;}{p}=K\frac{\stackrel{\&CenterDot;}{\mathrm{\&rho;}}}{\mathrm{\&rho;}}$ With ${\stackrel{\&CenterDot;}{s}}_x=-\frac{4}{3}\mathrm{\&mu;}\frac{\stackrel{\&CenterDot;}{\mathrm{\&rho;}}}{\mathrm{\&rho;}}$

Wherein K is bulk modulus, and μ is modulus of shearing; When it is in mecystasis, meet relation

$p=c_0^2(\mathrm{\&rho;}-{\mathrm{\&rho;}}_0)+({\mathrm{\&gamma;}}_s-1)\mathrm{\&rho;e}$ With $s_x=\&PlusMinus;\frac{2}{3}{Y}_0$

Wherein c ₀, ρ ₀, γ _sbe the constant relevant with concrete solid, Y ₀it is yield strength; For deviatoric stress s _x, positive sign represents that solid is in extended state, negative sign represents that solid is in compressive state, and the yield condition of this solid is von Mises yield condition, when its deviatoric stress meets

${s_x}^2,,{\left(\frac{2}{3}{Y}_0\right)}^2$

Time, solid is in elastic stage, and in the time that above-mentioned inequality is false, solid is in ideal plasticity state.

3. the computing technique of one dimension ideal elastic-plastic solid as claimed in claim 1, is characterized in that, the derivative in Hooke law needs to convert to individual derivative and calculates under Eulerian coordinates system, and concrete discrete form is expressed as

$p_i^{n+1}=p_{\mathrm{old}}+K\ln \left(\frac{{\mathrm{\&rho;}}_i^{n+1}}{{\mathrm{\&rho;}}_{\mathrm{old}}}\right)$

And

$\left\{\begin{array}{c}{s}_{\mathrm{xi}}^{n+1}=s_{\mathrm{xold}}+2\mathrm{\&mu;}[\frac{u_i^{n+1}-u_{i-1}^{n+1}}{\mathrm{\&Delta;x}}\mathrm{\&Delta;t}+\frac{1}{3}\ln \left(\frac{{\mathrm{\&rho;}}_i^{n+1}}{{\mathrm{\&rho;}}_{\mathrm{old}}}\right)],u_i^{n+1}\&GreaterEqual;0\\ s_{\mathrm{xi}}^{n+1}=s_{\mathrm{xold}}+2\mathrm{\&mu;}[\frac{u_{i+1}^{n+1}-u_i^{n+1}}{\mathrm{\&Delta;x}}\mathrm{\&Delta;t}+\frac{1}{3}\ln \left(\frac{{\mathrm{\&rho;}}_i^{n+1}}{{\mathrm{\&rho;}}_{\mathrm{old}}}\right)],u_i^{n+1}<0\end{array}\right.$

Wherein, ρ _old, p _old, s _xoldrepresent respectively the net point under Eulerian coordinates system in a upper time step position density value, force value, the deviatoric stress value at place, they all obtain by the linear interpolation windward of similar following form

$\left\{\begin{array}{c}{\mathrm{\&rho;}}_{\mathrm{old}}={\mathrm{\&rho;}}_i^n-\frac{{\mathrm{\&rho;}}_i^n-{\mathrm{\&rho;}}_{i-1}^n}{\mathrm{\&Delta;x}}(x_i^{n+1}-x_{\mathrm{old}}),u_i^{n+1}\&GreaterEqual;0\\ {\mathrm{\&rho;}}_{\mathrm{old}}={\mathrm{\&rho;}}_i^n+\frac{{\mathrm{\&rho;}}_{i+1}^n-{\mathrm{\&rho;}}_i^n}{\mathrm{\&Delta;x}}(x_{\mathrm{old}}-x_i^{n+1}),u_i^{n+1}<0\end{array}\right.$

P _old, s _xoldin like manner.