CN104036150B - A kind of technology that one-dimensional ideal elastic-plastic solid is calculated under eulerian coordinate system - Google Patents

Abstract

The present invention proposes a kind of technology that one-dimensional ideal elastic-plastic solid is calculated under eulerian coordinate system, and the main contents of its invention are the technology for calculating its related physical quantity using the model of one-dimensional ideal elastic-plastic solid under eulerian coordinate system of complete set.Innovative point of the invention is mainly reflected in the calculation of individual derivative in one-dimensional Hooke laws under eulerian coordinate system.Proposition of the invention, can be used directly to calculate one-dimensional ideal elastic-plastic solid, and significant in one-dimensional ideal elastic-plastic solid is by the practical engineering application such as external force effect and the effect of other medium couples.

Description

A kind of technology that one-dimensional ideal elastic-plastic solid is calculated under eulerian coordinate system

Technical field

The present invention relates to a kind of technology for calculating one-dimensional ideal elastic-plastic solid, and in particular to one kind is under eulerian coordinate system Calculate the technology of one-dimensional ideal elastic-plastic solid.

Background technology

One-dimensional ideal elastic-plastic solid model can be compared with accurate description solid (such as metals such as aluminium, steel) by common intensity Each physical quantity variation situation under external force effect.Therefore, the computing technique of ideal elastic-plastic solid is studied, is had in Practical Project There is important application value and be widely applied prospect.

At present, although existed some calculate ideal elastic-plastic solids technologies, but with technology proposed by the present invention It is different.Such as, M.L.Wilkins 1964 propose ideal elastic-plastic solid model after, using finite difference calculus pair The model is solved, and which uses the approximate shceme form of complexity.For another example, B.P.Howell used Free in 2000 Lagrange methods are calculated ideal elastic-plastic solid.The method is calculated under Largrangian coordinates, although Be can be simplified on some variables (such as deviatoric stress) are calculated, but become sufficiently complex when being generalized to higher-dimension.In order that preferable The calculating of elastic-plastic solid had both simply had accurately, and the present invention is directly calculated under eulerian coordinate system, only need to be by Hooke laws In derivative processed as individual derivative.It is noted that the present invention is in the work of M.B.Tyndall in 1993 Inspired and proposed.However, but there are some mistakes in the computational methods of M.B.Tyndall.First, Euler's seat is being calculated Lower each mesh point of mark is at the position of a upper time step, and it is average that he takes the time on fixed mesh point to speed.This It is correct under Largrangian coordinates to plant computational methods, is but Eulerian coordinates that are wrong, being set up with him under Eulerian coordinates Under governing equation contradict.Secondly, he when related physical quantity value at a time step on each mesh point is calculated, Employ the Quadratic interpolation (quadratic function interpolation) of this both sides mesh point.But, the governing equation of ideal elastic-plastic solid is Hyperbolic equations, propagating has directionality, can cause the inaccurate of calculating using Quadratic interpolation, or even cause unstable and produce mistake By mistake.Actual numerical computations also demonstrate his method and are implicitly present in some mistakes.It is of the invention then directly adopt for this problem Take linear interpolation windward.In a word, the computing technique of one-dimensional ideal elastic-plastic solid proposed by the present invention has taken into account the simple of method Property and correctness.

The content of the invention

The technology of the one-dimensional ideal elastic-plastic solid of calculating proposed by the present invention, its content of the invention is mainly reflected in Euler The technology of the one-dimensional ideal elastic-plastic solid of calculating of the complete set under coordinate system, its innovative point is mainly reflected in one-dimensional Hooke The calculation of individual derivative in law under eulerian coordinate system.

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

Herein, ρ is density, and u is speed, and p is pressure, and E is total energy, σ_xIt is the total stress in x directions.Additionally, for ideal Elastic-plastic solid, its total stress and pressure also meet following relation：

σ_x=-p+s_x

Wherein, s_xIt is the deviatoric stress in x directions.When ideal elastic-plastic solid is in elastic stage, have

With

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

With

Wherein c₀, ρ₀, γ_sIt is the constant relevant with specific solid, Y₀It is yield strength；For deviatoric stress s_x, positive sign table Show that solid is in extended state, symbol represents that solid is in compressive state.It is when ideal elastic-plastic solid meets following equation Elastic stage

When above-mentioned inequality is invalid, solid is in mecystasis.

The specific content of the invention of the invention can be attributed to the technology of being calculated as below.Assuming that known one-dimensional ideal elastic-plastic solid In n-th each variate-value of time stepNeed for these variate-values to be advanced to (n+1)th time step, obtain ArriveIts computing technique is realized by following six step：

1. governing equation (1) is solved, by n-th each variate-value of time step in governing equationIt is updated to (n+1)th time step, obtains

2. each mesh point under Eulerian coordinates is calculatedN-th position of time step, x is designated as_old, have

3., using linear interpolation windward, ρ, p, s are calculated_xIn x_oldThe value at place, is denoted as ρ_old, p_old, s_xold, such as

p_oldAnd s_xoldAlso can be calculated by similar fashion.

4. Hooke laws and windward linear interpolation are utilized, obtains preliminary

5. judged in each Eulerian mesh point by von Mises yield conditionsThe elastic-plastic behavior at place, and update pressure Force value is extremelyIf meeting von Mises yield conditions at a certain mesh point, i.e.,Then solid is in elasticity State, and pressureCalculated by Hooke laws

If being unsatisfactory for von Mises yield conditions in the mesh point, i.e.,Then solid is in moulding state, And pressureCalculated by state equation

Meanwhile, make deviatoric stressMeet ideal plasticity condition

6. return to step 1 is required until reaching the time iteration of setting.

Brief description of the drawings

Fig. 1 is the flow chart that the present invention calculates one-dimensional ideal elastic-plastic solid under eulerian coordinate system；

Fig. 2 to Fig. 4 is the numerical results that the present invention calculates one-dimensional ideal elastic-plastic solid under eulerian coordinate system.

Specific embodiment

In order to illustrate specific embodiment of the invention, an example will be demonstrated below.Consider in aluminium Dimension Riemannian problem, wherein the dimensionless initial value on the left of the Riemannian problem are u_L=20.0, p_L=1.0, ρ_L=2.7, s_L=0.0, it is right The dimensionless initial value of side is u_R=-20.0, p_R=1.0, ρ_R=2.7, s_R=0.0.It is equidistant in nondimensional solution interval [0,1] 2000 Eulerian mesh points are dispersed with, and the initial interface of Riemannian problem is 0.0.Meanwhile, the ideal elastoplastic model of aluminium Related dimensionless group is respectively ρ₀=2.71, c₀=538.0, γ_s=2.71, K=740000.0, μ=265000.0, Y₀= 3000.0。

The problem will simultaneously produce elastic wave and plastic wave in the interface left and right sides.Take time step Δ t= 0.0000015, calculated using Lax-Friedrich forms, obtain negative total stress, speed in aluminium in time t=0.001 Degree, density are as shown in Figures 2 to 4.

Claims (2)

1. a kind of method that one-dimensional ideal elastic-plastic solid is calculated under eulerian coordinate system, it is characterised in that the method is a set of Complete calculates one-dimensional ideal elastic-plastic solid related physical quantity density p, speed u, pressure p, deviatoric stress under eulerian coordinate system s_xMethod, the one-dimensional ideal elastic-plastic solid be aluminium；

Derivative in Hooke laws needs to be converted into individual derivative calculating under eulerian coordinate system, and specific discrete form is expressed as

$p_i^{n+1}=p_{old}+Kln\left(\frac{{\mathrm{\ρ}}_i^{n+1}}{{\mathrm{\ρ}}_{old}}\right)$

And

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

Wherein, ρ_old、p_old、s_xoldThe mesh point under eulerian coordinate system is represented respectivelyIn upper time step positionThe density value at place, pressure value, deviatoric stress value, they are by the line windward of similar following form Property interpolation is obtained

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

p_old、s_xoldSimilarly.

2. the method for one-dimensional ideal elastic-plastic solid being calculated under eulerian coordinate system as claimed in claim 1, it is characterised in that When it is in elastic stage, meet

Hooke laws

With

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

With

Wherein c₀, ρ₀, γ_sIt is the constant relevant with specific solid, Y₀It is yield strength；For deviatoric stress s_x, positive sign is represented to be consolidated Body is in extended state, and negative sign represents that solid is in compressive state, and the yield condition of the solid is vonMises surrender bars Part, i.e., when its deviatoric stress meets

${s_x}^2\&le;{\left(\frac{2}{3}{Y}_0\right)}^2$

When, solid is in elastic stage, and when above-mentioned inequality is invalid, solid is in ideal plasticity state.