CN102819633A - Method for establishing constitutive relation between welding thermal cycle temperature and thermal deformation history material and senior nonlinear finite element analysis software (MSC. MARC) secondary development - Google Patents

Abstract

The invention relates to a method for establishing a constitutive relation between a welding thermal cycle temperature and a thermal deformation history material and a senior nonlinear finite element analysis software (MSC. MARC) secondary development, belonging to the simulation field of a finite element numerical value of elastic-plastic mechanics, aiming at solving a problem that a real material constitutive relation related to a structural change and the like cannot be flexibly applied by a conventional model. The method comprises the following steps of: calculating a stress increment depending on an elastic relation; depending on the Von Mises yield condition, calculating a trial stress; implementing an integral operation to a differential of a plastic strain tensor through a backward Euler algorithm, to obtain an equation which takes an accumulated plastic strain increment as an independent variable; updating a deviatoric stress and a stress, the constitutive relation of material isotropic hardening, the constitutive relation of material mixing hardening, thereby obtaining a material constitutive relation after the secondary development. The method is applicable to a finite element vertical simulation field in elasticity mechanics.

Description

Historical material constitutive relation establishing method of Thermal Cycle temperature and thermal deformation and MSC.MARC secondary development

Technical field

The present invention relates to historical material constitutive relation establishing method of Thermal Cycle temperature and thermal deformation and MSC.MARC secondary development, belong to the finite element numerical simulation field of plasto-elasticity.

Background technology

The constitutive model of material is one of key influence factor of welding process-mechanical couplings simulation, is determining the welding stress and distortion precision of numerical simulation.In the welding process of heat treatment reinforcement metal and flow harden metal, sweating heat history causes changes in material properties, and promptly the material constitutive of welding process relation is different because of the material classification.The mechanics of materials constitutive relation that relates in the welding process; Comprise the deformation of creep common in the flow of metal, strain hardening character; And change directly related mechanical property with the material internal microstructure and develop, as the second phase precipitation separate out, annealing, transformation plasticity that the martensite phase transformation causes, volumetric expansion that diffusion transformation causes or contraction etc.Changing noticeably of material mechanical performance influences welding process stress, the distribution of the differentiation of strain and last unrelieved stress and distortion.Therefore, need to investigate the temperature and the deformation history dependence of the mechanical property of materials, to improve simulation accuracy.Yet in most of numerical simulation of welding processes, the material therefor model does not consider that welding is historical to effect of material performance.

At present, MSC.MARC becomes hot-working field use face one of the most limited software.Adopt finite element software MSC.MARC definition material model that two aspect problems are arranged, the one, experimental data is numerous and jumbled, adds relatively difficulty like the stress-strain curve under a large amount of different temperatures through the three-dimensional table mode; In addition, the strain hardening type that this software can apply has only etc. to sclerosis, and the definition of kinematic hardening and mixed hardening is invalid command, therefore, can't apply neatly and organizes relevant real material constitutive relation such as variation through existing model.

Summary of the invention

The present invention can't apply the problem that concerns with relevant real material constitutives such as organizing variation through existing model neatly for solving at present, and then proposes a kind of Thermal Cycle temperature and historical material constitutive relation establishing method of thermal deformation and MSC.MARC secondary development.

The present invention addresses the above problem the technical scheme of taking to be: the concrete steps of the method for building up of material constitutive relation according to the invention are following:

Step 1, mixed hardening such as are at the weighted sums to sclerosis and kinematic hardening, set the weighting coefficient that waits to sclerosis and kinematic hardening and are 0.5, wait to the sclerosis characteristic and adopt the power exponent rule of hardening to be expressed as:

σ=Eε 0<σ<σ _S ①

σ=σ _S+m(ε ^pl) ⁿ σ>σ _S ?②

The 1. 2. middle σ of formula is a stress, and ε is strain, σ _sBe initial yield strength, E is a Young modulus, ε ^PlBe equivalent plastic strain, m is a material constant, and n is the parameter of material forming;

Step 2, employing pula lattice bilinearity kinematic hardening model tormulation formula:

db _ij=Cdε _ij ^pl ③

The 3. middle b of formula _IjBe back stress, ε _Ij ^PlBe the plastic strain tensor, C pula lattice hardening model constant;

Step 3, can know relation below existing between pula lattice hardening model constant C and the plastic modulus by uniaxial tension stress-strain relation and Feng Misaisi yield condition:

$C=\frac{2}{3}{\mathrm{<figure-callout\; id="4"\; label="E\; p"\; filenames="HDA00001941407100031.png,HDA00001941407100032.png"\; state="\{\{state\}\}">E</figure-callout>}}_p$ ④

The 4. middle E of formula _pBe plastic modulus;

The Feng Misaisi yield condition is:

$f=J_2-\frac{{{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00001941407100032.png,HDA00001941407100033.png"\; state="\{\{state\}\}">y</figure-callout>}}}^2}{3}$

Wherein f is a loading function,

Be second invariant of deviatoric stress, S _IjBe deviatoric stress tensor, σ _yFollow-up yield strength for material; If elastic-plastic deformation or plastic yield take place in f>=0, if elastic deformation takes place in f＜0;

Step 4, employing Ziegler model, internal variable back stress and S _Ij-b _IjProportional:

${\mathrm{db}}_{\mathrm{ij}}=a\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{pl}}}(S_{\mathrm{ij}}-b_{\mathrm{ij}})$ ⑤

The 5. middle S of formula _IjBe deviatoric stress,

Be the accumulated plastic strain differential, a is positive scale factor;

Step 5, for the kinematic hardening material, Feng Misaisi yield condition and plasticity associated flow rule combine and can obtain:

dε _ij ^pl=dλ(S _ij-b _ij)?⑥

To 6. both sides self dot product of formula, obtain:

$\mathrm{d\&lambda;}=\frac{3}{2}\frac{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{pl}}}}{{\mathrm{\&sigma;}}^y}$

Promptly $d{{\mathrm{\&epsiv;}}_{\mathrm{Ij}}}^{\mathrm{Pl}}=\frac{3}{2}\frac{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{Pl}}}}{{\mathrm{\&sigma;}}^y}(S_{\mathrm{Ij}}-b_{\mathrm{Ij}})$ 7.

${\mathrm{db}}_{\mathrm{ij}}=C\frac{3}{2}\frac{\stackrel{\&OverBar;}{{\mathrm{d\&epsiv;}}^{\mathrm{pl}}}}{{\mathrm{\&sigma;}}^y}(S_{\mathrm{ij}}-b_{\mathrm{ij}})$ ⑧

$a=\frac{E_p}{{\mathrm{\&sigma;}}^y}$ ⑨

The 6. middle d λ of formula is a scale factor;

Step 6, employing power law formal equivalence effect creep strain rate are expressed, and the constitutive relation between steady state creep speed

creep temperature T and the creep stress σ can be expressed as:

${\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^c=A{\mathrm{\&sigma;}}^n{\mathrm{\&epsiv;}}^{c^m}{T}^p\left({\mathrm{qt}}^{q-1}\right)$

ε wherein ^cBe equivalent creep strain, A is a pre-exponential factor, and s is a stress exponent, and r is equivalent creep strain index, and p is a humidity index, and q is a time index, and T is a thermodynamic temperature, and t is the time;

Step 7, have the stress-strain relation of elastoplasticity and creeping property at last, promptly the material constitutive model is:

${{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^e=\frac{1+\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{ij}}-\frac{\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{kk}}{\mathrm{\&delta;}}_{\mathrm{ij}}$

ε _ij ^pl=λS _ij

ε _ij ^c=λ ^cS _ij

${\mathrm{\&epsiv;}}_{\mathrm{ij}}={{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^e+{{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^{\mathrm{pl}}+{{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^c=\frac{1+\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{ij}}-\frac{\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{kk}}{\mathrm{\&delta;}}_{\mathrm{ij}}+\mathrm{\&lambda;}{S}_{\mathrm{ij}}+{\mathrm{\&lambda;}}^c{S}_{\mathrm{ij}}$

Wherein, σ _Ij, ε _IjBe respectively stress tensor and strain tensor, ε _Ij ^e, ε _Ij ^cBe respectively elastic strain tensor sum creep strain tensor, λ ^cBe and the historical relevant constant of the deformation of creep; Above stress-strain relation is expressed as with stiffness matrix:

σ _ij=D _ijkl(ε _kl-ε _kl ^c)=D _ijkl[ε _kl-λ ^cS _ij]

Wherein, when material has only elastic deformation, deformation of creep series connection, D _IjklBe elastic matrix, when elastic deformation, plastic yield and deformation of creep series connection, D _IjklBe the plasticity matrix.When FEM calculation, the creep item shows stress relaxation phenomenon when faking load Processing.

The concrete steps of the MSC.MARC secondary development of the historical material constitutive relation of Thermal Cycle temperature according to the invention and thermal deformation are following:

Step 1, according to resilient relationship calculated stress increment:

Δσ _ij=λΔε _kkδ _ij+2GΔe _ij

Wherein, Δ σ _IjBe the increment of stress tensor, Δ ε _KkBe the increment of hydrostatic strain, Δ e _IjBe current inclined to one side strain increment, δ _IjBe Crow Nellie symbol, G is a modulus of shearing, and λ is a Lame's constant, λ=E υ/(1+ υ) (1-2 υ), and υ is a Poisson ratio;

Step 2, according to the Feng Misaisi yield condition, calculate tentative calculation stress

${\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}=\sqrt{\frac{3}{2}{S}_{\mathrm{ij}}^{\mathrm{pr}}{S}_{\mathrm{ij}}^{\mathrm{pr}}}$

Wherein,

is the tentative calculation deviatoric stress;

is the deviatoric stress tensor that i calculates incremental step; When the yield strength of tentative calculation stress greater than present material; When promptly satisfying yield condition; Get into the plastic yield stage, if do not satisfy, according to elastic stiffness matrix update stress state;

Step 3, according to Prandtl-Roy's Si quadrature flow rule, the differential of plastic strain tensor is:

$d{{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^{\mathrm{pl}}=\frac{3}{2}\frac{({S_{\mathrm{ij}}}^i-{b_{\mathrm{ij}}}^i)}{{\mathrm{\&sigma;}}_y}\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{pl}}}$

Wherein, d ε ^PlBe the differential of equivalent plastic strain, b _Ij ⁱIt is the back stress tensor of current calculating incremental step;

Step 4, the laggard euler algorithm of employing carry out integral operation to the differential of plastic strain tensor, obtain with accumulation plastic strain increment Δ ε ^PlEquation for independent variable:

${\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}-3\mathrm{G\&Delta;}{\mathrm{\&epsiv;}}^{\mathrm{pl}}={\mathrm{\&sigma;}}_y$

In view of σ _yBe ε ^PlFunction, right

The equation solution of form adopts Newton iteration method, to the power exponent hardening model, finds the solution V ε ^PlThe Newton iterative calculation formula be:

$\mathrm{\&Delta;}{{\mathrm{\&epsiv;}}_{j+1}}^{\mathrm{pl}}=\mathrm{\&Delta;}{{\mathrm{\&epsiv;}}_j}^{\mathrm{pl}}+\frac{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}-{\mathrm{\&sigma;}}_y-3\mathrm{G\&Delta;}{{\mathrm{\&epsiv;}}_j}^{\mathrm{pl}}}{\mathrm{mn}\&times;{({\mathrm{\&epsiv;}}^{\mathrm{pl}}+\mathrm{\&Delta;}{{\mathrm{\&epsiv;}}_j}^{\mathrm{pl}})}^{n-1}+3G}$

Δ ε wherein _J+1 ^Pl, Δ ε _j ^PlBe respectively j+1 and the j iterative computation accumulation plastic strain increment in step; The above iterative computation that circulates, until convergence error | Δ ε _J+1 ^Pl-Δ ε _j ^Pl|<10 ^-8Stop, so far trying to achieve the V ε that i calculates incremental step ^PlThe accumulation plastic strain increment of current calculating incremental step

$\mathrm{\&Delta;}{\mathrm{\&epsiv;}}_{\mathrm{ij}}^{\mathrm{pl}}=\frac{3}{2}(S_{\mathrm{ij}}^{\mathrm{pr}}-{b_{\mathrm{ij}}}^i)\mathrm{\&Delta;}{\mathrm{\&epsiv;}}^{\mathrm{pl}}/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}$

Step 5, renewal deviatoric stress and stress:

$\left\{\begin{array}{c}{S_{\mathrm{ij}}}^i+\mathrm{\&Delta;}{S}_{\mathrm{ij}}-{b_{\mathrm{ij}}}^i={\mathrm{\&sigma;}}_y\&times;(S_{\mathrm{ij}}^{\mathrm{pr}}-{b_{\mathrm{ij}}}^i)/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}\\ {\mathrm{\&sigma;}}_{\mathrm{ij}}=\frac{1}{3}{\mathrm{\&delta;}}_{\mathrm{ij}}{\mathrm{\&sigma;}}_{\mathrm{kk}}^{\mathrm{pr}}+{S_{\mathrm{ij}}}^i+\mathrm{\&Delta;}{S}_{\mathrm{ij}}\end{array}\right.$

Wherein, Δ S _IjBe the deviatoric stress increment of current calculating incremental step,

For next calculates the hydrostatic force of incremental step, ε _Kk ⁱHydrostatic strain for current calculating incremental step;

Step 6, material etc. to the constitutive relation of sclerosis are:

Δσ _ij=λ ^*δ _ijΔε _kk+2G ^*Δε _ij+h'×η _ijη _klΔε _kl

Wherein $G^{*}=G\frac{{\mathrm{\&sigma;}}_y}{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}},$ ${\mathrm{\&lambda;}}^{*}=K-\frac{2}{3}{G}^{*},$ $K=\frac{E}{3(1-2\mathrm{\&upsi;})},$ $h^{\&prime;}=\frac{h}{1+h/3G}-3G^{*},$ $h=\frac{d{\mathrm{\&sigma;}}_y}{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{Pl}}}},$ ${\mathrm{\&eta;}}_{\mathrm{Ij}}=S_{\mathrm{Ij}}^{\mathrm{Pr}}/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}};$

The constitutive relation of material kinematic hardening is:

Δσ _ij=λ ^*δ _ijΔε _kk+2G ^*Δε _ij+h′η _ijη _klΔε _kl

Wherein $G^{*}=G\frac{({\mathrm{\&sigma;}}_s+\mathrm{H\&Delta;}{\mathrm{\&epsiv;}}^{\mathrm{Pl}})}{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}},$ ${\mathrm{\&lambda;}}^{*}=K-\frac{2}{3}{G}^{*},$ $K=\frac{E}{3(1-2\mathrm{\&upsi;})},$ $h^{\&prime;}=\frac{h}{1+h/3G}-3G^{*},$ H=a σ _S, ${\mathrm{\&eta;}}_{\mathrm{Ij}}=(S_{\mathrm{Ij}}^{\mathrm{Pr}}-{b_{\mathrm{Ij}}}^i)/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}},$ b _Ij ⁱIt is the back stress that i calculates incremental step;

The constitutive relation of step 7, material mixing sclerosis, promptly the relation of the material constitutive after the secondary development is:

Δσ _ij=λ ^*δ _ijΔε _kk+2G ^*Δε _ij+h'×η _ijη _klΔε _kl

Wherein $G^{*}=G\frac{{\mathrm{\&sigma;}}_y+h{\mathrm{\&Delta;\; \&epsiv;}}^{\mathrm{Pl}}}{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}},$ ${\mathrm{\&lambda;}}^{*}=K-\frac{2}{3}{G}^{*},$ $K=\frac{E}{3(1-2\mathrm{\&upsi;})},$ $h^{\&prime;}=\frac{h}{1+h/3G}-3G^{*},$ $h=a{\mathrm{\&sigma;}}_s+\frac{d{\mathrm{\&sigma;}}_y}{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{Pl}}}},$ ${\mathrm{\&eta;}}_{\mathrm{Ij}}=(S_{\mathrm{Ij}}^{\mathrm{Pr}}-{b_{\mathrm{Ij}}}^i)/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}.$

The invention has the beneficial effects as follows: the present invention has set up the elastoplasticity (mixed hardening) and creeping property (strain softening) constitutive relation of the material that is applicable to Thermal Cycle; And utilize MSC.MARC second development interface and Fortran language development user subroutine; Two aspect problems of present employing finite element software MSC.MARC definition material model have been solved; One of which is that experimental data is numerous and jumbled; Add difficulty like the stress-strain curve under a large amount of different temperatures through the three-dimensional table mode; The strain hardening type that its two this software can apply has only etc. to sclerosis, and the definition of kinematic hardening and mixed hardening is invalid command, can't add neatly through existing model to change relevant material constitutive with tissue and concern; Consider sweating heat history to effect of material performance, can improve the precision of welding residual stress and distortion finite element analogy.The present invention has successfully realized the FEM calculation of plasticity incremental constitutive; And having realized the data transfer and the storage of outer bevel program and MSC.MARC software, it is good that subroutine has convergence, but advantages such as result data aftertreatment; And for elastic-plastic constitutive relation; Obtained the general expression formula of stiffness matrix, correlation model and algorithm can be extrapolated and be applicable to the description to other metal material mechanics constitutive relation in the hot procedure, greatly reduce experiment measuring cost and post analysis cost.Because based on plasto-elasticity, metal metallurgy smelting etc., theoretical solid, the person's that is easy to obtain the numerical simulation study approval is with a wide range of applications.

Description of drawings

Fig. 1 is the process flow diagram of the MSC.MARC secondary development of the historical material constitutive relation of Thermal Cycle temperature and thermal deformation; Fig. 2 is the uniaxial loading strain-stress relation of different sclerosis types of material; Fig. 3 is the temperature-uniaxial tension plastic strain and the yield strength relation curve of A7N01-T4 aluminum alloy materials; Fig. 4 adopts the single shaft displacement of the constitutive relation of three-dimensional table temperature-equivalent plastic strain-yield strength relation and the present invention's proposition to load when being 20 ℃ stress-time relationship compares; Fig. 5 adopts the single shaft displacement of the constitutive relation of three-dimensional table temperature-equivalent plastic strain-yield strength relation and the present invention's proposition to load when being 58 ℃ stress-time relationship compares; Fig. 6 adopts stress-time relationship that the single shaft displacement of the constitutive relation that three-dimensional table temperature-equivalent plastic strain-yield strength relation and the present invention propose loads relatively when being 400 ℃, Fig. 7 adopts stress-time relationship of the single shaft displacement loading of the constitutive relation that three-dimensional table temperature-equivalent plastic strain-yield strength relation and the present invention propose to compare when being 550 ℃.

Embodiment

Embodiment one: combine Fig. 2 that this embodiment is described, the concrete steps of the material constitutive relation establishing method that said Thermal Cycle temperature of this embodiment and thermal deformation are historical are following:

Step 1, mixed hardening such as are at the weighted sums to sclerosis and kinematic hardening, set the weighting coefficient that waits to sclerosis and kinematic hardening and are 0.5, wait to the sclerosis characteristic and adopt the power exponent rule of hardening to be expressed as:

σ=Eε 0<σ<σ _S ①

σ=σ _S+m(ε ^pl) ⁿ σ>σ _S ②

The 1. 2. middle σ of formula is a stress, and ε is strain, σ _sBe initial yield strength, E is a Young modulus, ε ^PlBe equivalent plastic strain, m is a material constant, and n is the parameter of material forming;

Step 2, employing pula lattice bilinearity kinematic hardening model tormulation formula:

db _ij=Cdε _ij ^pl?③

The 3. middle b of formula _IjBe back stress, ε _Ij ^PlBe the plastic strain tensor, C pula lattice hardening model constant;

Step 3, can know relation below existing between pula lattice hardening model constant C and the plastic modulus by uniaxial tension stress-strain relation and Feng Misaisi yield condition:

$C=\frac{2}{3}{\mathrm{<figure-callout\; id="4"\; label="E\; p"\; filenames="HDA00001941407100031.png,HDA00001941407100032.png"\; state="\{\{state\}\}">E</figure-callout>}}_p$ ④

The 4. middle E of formula _pBe plastic modulus;

The Feng Misaisi yield condition is:

$f=J_2-\frac{{{\mathrm{\&sigma;}}_{\mathrm{<figure-callout\; id="2"\; label="y"\; filenames="HDA00001941407100032.png,HDA00001941407100033.png"\; state="\{\{state\}\}">y</figure-callout>}}}^2}{3}$

Wherein f is a loading function,

Be second invariant of deviatoric stress, S _IjBe deviatoric stress tensor, σ _yFollow-up yield strength for material; If elastic-plastic deformation or plastic yield take place in f>=0, if elastic deformation takes place in f＜0;

Step 4, employing Ziegler model, internal variable back stress and S _Ij-b _IjProportional:

${\mathrm{db}}_{\mathrm{ij}}=a\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{pl}}}(S_{\mathrm{ij}}-b_{\mathrm{ij}})$ ⑤

The 5. middle S of formula _IjBe deviatoric stress,

Be the accumulated plastic strain differential, the scale factor that a is positive;

Step 5, for the kinematic hardening material, Feng Misaisi yield condition and plasticity associated flow rule combine and can obtain:

dε _ij ^pl=dλ(S _ij-b _ij)⑥

To 6. both sides self dot product of formula, obtain:

$\mathrm{d\&lambda;}=\frac{3}{2}\frac{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{pl}}}}{{\mathrm{\&sigma;}}^y}$

Promptly $d{{\mathrm{\&epsiv;}}_{\mathrm{Ij}}}^{\mathrm{Pl}}=\frac{3}{2}\frac{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{Pl}}}}{{\mathrm{\&sigma;}}^y}(S_{\mathrm{Ij}}-b_{\mathrm{Ij}})$ 7.

${\mathrm{db}}_{\mathrm{ij}}=C\frac{3}{2}\frac{\stackrel{\&OverBar;}{{\mathrm{d\&epsiv;}}^{\mathrm{pl}}}}{{\mathrm{\&sigma;}}^y}(S_{\mathrm{ij}}-b_{\mathrm{ij}})$ ⑧

$a=\frac{E_p}{{\mathrm{\&sigma;}}^y}$ ⑨

The 6. middle d λ of formula is a scale factor;

Step 6, employing power law formal equivalence effect creep strain rate are expressed, and the constitutive relation between steady state creep speed creep temperature T and the creep stress σ can be expressed as:

${\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^c=A{\mathrm{\&sigma;}}^n{\mathrm{\&epsiv;}}^{c^m}{T}^p\left({\mathrm{qt}}^{q-1}\right)$

ε wherein ^cBe equivalent creep strain, A is a pre-exponential factor, and s is a stress exponent, and r is equivalent creep strain index, and p is a humidity index, and q is a time index, and T is a thermodynamic temperature, and t is the time;

Step 7, have the stress-strain relation of elastoplasticity and creeping property at last, promptly the material constitutive model is:

${{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^e=\frac{1+\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{ij}}-\frac{\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{kk}}{\mathrm{\&delta;}}_{\mathrm{ij}}$

ε _ij ^pl=λS _ij

ε _ij ^c=λ ^cS _ij

${\mathrm{\&epsiv;}}_{\mathrm{ij}}={{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^e+{{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^{\mathrm{pl}}+{{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^c=\frac{1+\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{ij}}-\frac{\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{kk}}{\mathrm{\&delta;}}_{\mathrm{ij}}+\mathrm{\&lambda;}{S}_{\mathrm{ij}}+{\mathrm{\&lambda;}}^c{S}_{\mathrm{ij}}$

Wherein, σ _Ij, ε _IjBe respectively stress tensor and strain tensor, ε _Ij ^e, ε _Ij ^cBe respectively elastic strain tensor sum creep strain tensor, λ ^cBe and the historical relevant constant of the deformation of creep; Above stress-strain relation is expressed as with stiffness matrix:

σ _ij=D _ijkl(ε _kl-ε _kl ^c)=D _ijkl[ε _kl-λ ^cS _ij]

Wherein, when material has only elastic deformation, deformation of creep series connection, D _IjklBe elastic matrix, when elastic deformation, plastic yield and deformation of creep series connection, D _IjklBe the plasticity matrix; When FEM calculation, the creep item shows stress relaxation phenomenon when faking load Processing.

By the simulation of cyclic shift loading uniaxial tension process, verify the correctness of material strain sclerosis constitutive relation secondary development in the present embodiment.Load forward yield strength and the oppositely relation of maximum stress down based on cyclic shift, can confirm the descriptor compound Li Yiyi of subprogram, i.e. strain hardening under the low temperature and the strain softening characteristic under the high temperature for the sclerosis constitutive relation.

Embodiment two: combine Fig. 1 that this embodiment is described, the concrete steps of the MSC.MARC secondary development of the historical material constitutive relation of said Thermal Cycle temperature of this embodiment and thermal deformation are following:

Step 1, according to resilient relationship calculated stress increment:

Δσ _ij=λΔε _kkδ _ij+2GΔe _ij

Wherein, Δ σ _IjBe the increment of stress tensor, Δ ε _KkBe the increment of hydrostatic strain, Δ e _IjBe current inclined to one side strain increment, δ _IjBe Crow Nellie symbol, G is a modulus of shearing, and λ is a Lame's constant, λ=E υ/(1+ υ) (1-2 υ), and υ is a Poisson ratio;

Step 2, according to the Feng Misaisi yield condition, calculate tentative calculation stress

${\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}=\sqrt{\frac{3}{2}{S}_{\mathrm{ij}}^{\mathrm{pr}}{S}_{\mathrm{ij}}^{\mathrm{pr}}}$

Wherein,

is the tentative calculation deviatoric stress;

is the deviatoric stress tensor that i calculates incremental step; When the yield strength of tentative calculation stress greater than present material; When promptly satisfying yield condition; Get into the plastic yield stage, if do not satisfy, according to elastic stiffness matrix update stress state;

Step 3, according to Prandtl-Roy's Si quadrature flow rule, the differential of plastic strain tensor is:

$d{{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^{\mathrm{pl}}=\frac{3}{2}\frac{({S_{\mathrm{ij}}}^i-{b_{\mathrm{ij}}}^i)}{{\mathrm{\&sigma;}}_y}\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{pl}}}$

Wherein, d ε ^PlBe the differential of equivalent plastic strain, b _Ij ⁱIt is the back stress tensor of current calculating incremental step;

Step 4, the laggard euler algorithm of employing carry out integral operation to the differential of plastic strain tensor, obtain with accumulation plastic strain increment Δ ε ^PlEquation for independent variable:

${\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}-3\mathrm{G\&Delta;}{\mathrm{\&epsiv;}}^{\mathrm{pl}}={\mathrm{\&sigma;}}_y$

In view of σ _yBe ε ^PlFunction, right

The equation solution of form adopts Newton iteration method, to the power exponent hardening model, finds the solution V ε ^PlThe Newton iterative calculation formula be:

$\mathrm{\&Delta;}{{\mathrm{\&epsiv;}}_{j+1}}^{\mathrm{pl}}=\mathrm{\&Delta;}{{\mathrm{\&epsiv;}}_j}^{\mathrm{pl}}+\frac{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}-{\mathrm{\&sigma;}}_y-3\mathrm{G\&Delta;}{{\mathrm{\&epsiv;}}_j}^{\mathrm{pl}}}{\mathrm{mn}\&times;{({\mathrm{\&epsiv;}}^{\mathrm{pl}}+\mathrm{\&Delta;}{{\mathrm{\&epsiv;}}_j}^{\mathrm{pl}})}^{n-1}+3G}$

Δ ε wherein _J+1 ^Pl, Δ ε _j ^PlBe respectively j+1 and the j iterative computation accumulated plastic strain increment in step; The above iterative computation that circulates, until convergence error | Δ ε _J+1 ^Pl-Δ ε _j ^Pl<10 ^-8Stop, so far trying to achieve the V ε that i calculates incremental step ^PlThe accumulated plastic strain increment of current calculating incremental step

$\mathrm{\&Delta;}{\mathrm{\&epsiv;}}_{\mathrm{ij}}^{\mathrm{pl}}=\frac{3}{2}(S_{\mathrm{ij}}^{\mathrm{pr}}-{b_{\mathrm{ij}}}^i)\mathrm{\&Delta;}{\mathrm{\&epsiv;}}^{\mathrm{pl}}/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}$

Step 5, renewal deviatoric stress and stress:

$\left\{\begin{array}{c}{S_{\mathrm{ij}}}^i+\mathrm{\&Delta;}{S}_{\mathrm{ij}}-{b_{\mathrm{ij}}}^i={\mathrm{\&sigma;}}_y\&times;(S_{\mathrm{ij}}^{\mathrm{pr}}-{b_{\mathrm{ij}}}^i)/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}\\ {\mathrm{\&sigma;}}_{\mathrm{ij}}=\frac{1}{3}{\mathrm{\&delta;}}_{\mathrm{ij}}{\mathrm{\&sigma;}}_{\mathrm{kk}}^{\mathrm{pr}}+{S_{\mathrm{ij}}}^i+\mathrm{\&Delta;}{S}_{\mathrm{ij}}\end{array}\right.$

Wherein, Δ S _IjBe the deviatoric stress increment of current calculating incremental step,

For next calculates the hydrostatic force of incremental step, ε _Kk ⁱHydrostatic strain for current calculating incremental step;

Step 6, material etc. to the constitutive relation of sclerosis are:

Δσ _ij=λ ^*δ _ijΔε _kk+2G ^*Δε _ij+h'×η _ijη _klΔε _kl

Wherein $G^{*}=G\frac{{\mathrm{\&sigma;}}_y}{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}},$ ${\mathrm{\&lambda;}}^{*}=K-\frac{2}{3}{G}^{*},$ $K=\frac{E}{3(1-2\mathrm{\&upsi;})},$ $h^{\&prime;}=\frac{h}{1+h/3G}-3G^{*},$ $h=\frac{d{\mathrm{\&sigma;}}_y}{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{Pl}}}},$ ${\mathrm{\&eta;}}_{\mathrm{Ij}}=S_{\mathrm{Ij}}^{\mathrm{Pr}}/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}};$

The constitutive relation of material kinematic hardening is:

Δσ _ij=λ ^*δ _ijΔε _kk+2G ^*Δε _ij+h′η _ijη _klΔε _kl

Wherein $G^{*}=G\frac{({\mathrm{\&sigma;}}_s+\mathrm{H\&Delta;}{\mathrm{\&epsiv;}}^{\mathrm{Pl}})}{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}},$ ${\mathrm{\&lambda;}}^{*}=K-\frac{2}{3}{G}^{*},$ $K=\frac{E}{3(1-2\mathrm{\&upsi;})},$ $h^{\&prime;}=\frac{h}{1+h/3G}-3G^{*},$ H=a σ _S, ${\mathrm{\&eta;}}_{\mathrm{Ij}}=(S_{\mathrm{Ij}}^{\mathrm{Pr}}-{b_{\mathrm{Ij}}}^i)/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}},$ b _Ij ⁱIt is the back stress that i calculates incremental step;

The constitutive relation of step 7, material mixing sclerosis, promptly the relation of the material constitutive after the secondary development is:

Δσ _ij=λ ^*δ _ijΔε _kk+2G ^*Δε _ij+h'×η _ijη _klΔε _kl

Wherein $G^{*}=G\frac{{\mathrm{\&sigma;}}_y+h{\mathrm{\&Delta;\; \&epsiv;}}^{\mathrm{Pl}}}{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}},$ ${\mathrm{\&lambda;}}^{*}=K-\frac{2}{3}{G}^{*},$ $K=\frac{E}{3(1-2\mathrm{\&upsi;})},$ $h^{\&prime;}=\frac{h}{1+h/3G}-3G^{*},$ $h=a{\mathrm{\&sigma;}}_s+\frac{d{\mathrm{\&sigma;}}_y}{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{Pl}}}},$ ${\mathrm{\&eta;}}_{\mathrm{Ij}}=(S_{\mathrm{Ij}}^{\mathrm{Pr}}-{b_{\mathrm{Ij}}}^i)/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}.$

Embodiment three: the concrete steps that this embodiment said employing MSC.MARC second development interface and Fortran language carry out the MSC.MARC secondary development to the mixed hardening constitutive relation are following:

Step 1, write " hypela2.f " subroutine, in " hypela2.f " program, preserve current elastic strain, plastic strain and hardening parameter are to state variable space t (5)-t (23);

Step 2,19 state variable t (5)-t (23) are composed give initial value 0, promptly set up 19 starting condition;

Step 3, in JOB, apply defined starting condition, comprise icond1-icond19 and temperature starting condition;

Step 4, in JOB RESULTS the aftertreatment that will extract of definition user defined var# (user sub plotv) as a result, adopt plotv subroutine and state variable t (5)-t (23) to extract the variable that users such as accumulated plastic strain, equivalent plastic strain need;

The viscoelastroplasticity pattern of step 5, separation promptly belongs to Mike Si Weier type with the deformation of creep and the plastic yield series connection of material, and the deformation of creep defines according to the power exponent equation.

Through the simulation of positive-displacement loading uniaxial tension process, verify the correctness of material strain hardening elastoplasticity (mixed hardening) and the secondary development of creeping property (strain softening) constitutive relation in this embodiment.Stress-time relationship that Fig. 4 loads for the single shaft displacement of the constitutive relation that to adopt three-dimensional table respectively be temperature-equivalent plastic strain-yield strength relation proposes with the present invention under the different temperatures relatively, both are identical better.

Embodiment

Embodiment one, combination Fig. 3 explain present embodiment to scheming Fig. 7, set up A7N01-T4 aluminum alloy materials constitutive relation according to embodiment one and embodiment two described methods:

At first, measure the true stress-strain curve of A7N01-T4 aluminum alloy materials under different temperatures, under 20 ℃, 58 ℃, 102 ℃, 150 ℃, 200 ℃, along with the accumulation of plastic yield, yield strength raises gradually; Under 252 ℃, 302 ℃, 350 ℃, 400 ℃, 470 ℃, 550 ℃, 600 ℃, along with the accumulation of plastic yield, yield strength reduces gradually.The present invention weighs the strain softening characteristic that is caused by tissue variations such as dynamic recovery and crystallizations again with the high-temerature creep model; Therefore, at low temperatures, the A7N01-T4 aluminum alloy materials is the elastic-plastic material with strain hardening characteristic; At high temperature; Except elastic-plastic deformation, also, cause stress relaxation with the place creep distortion with strain hardening characteristic.

Weight coefficient Deng to sclerosis and kinematic hardening is 0.5, based on the uniaxial tension curve of material, obtains true stress and logarithmic strain relation, and then obtains m and n.According to the uniaxial tension stress-strain diagram of material, calculate the kinematic hardening model parameter of material.For high-temp strain sclerosis character, the hardening curve extrapolation match during according to low temperature obtains, and the result is as shown in table 1.

The correlation parameter of table 1 strain hardening model

At hot stage; Be that temperature is during greater than 252 ℃; Adopt creep model to describe the strain softening character of A7N01-T4 aluminum alloy materials; Adopt the power law formal equivalence to imitate creep strain rate and express, the stress-time relationship of the single shaft displacement loading through the constitutive relation that to adopt three-dimensional table respectively be temperature-equivalent plastic strain-yield strength relation proposes with the present invention under the contrast different temperatures is come match creep equation expression formula.Finally, for the A7N01-T4 aluminum alloy materials, the constitutive relation between steady state creep speed, thermodynamic temperature and the stress can be expressed as:

${\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^c=7.6\&times;{10}^{-12}{\mathrm{\&sigma;}}^{3.6}{\mathrm{\&epsiv;}}^0{T}^{0.2}\&times;{\mathrm{<figure-callout\; id="0"\; label="t"\; filenames="HDA00001941407100031.png,HDA00001941407100032.png"\; state="\{\{state\}\}">t</figure-callout>}}^0$

Because the viscoelastroplasticity pattern of separating promptly belongs to Mike Si Weier (Maxwell) type with the deformation of creep and the plastic yield series connection of material, the deformation of creep defines according to power exponent equation (power law).

Load the simulation of uniaxial tension process through positive-displacement; The correctness of checking material strain hardening elastoplasticity (mixed hardening) and the secondary development of creeping property (strain softening) constitutive relation; Adopting three-dimensional table under the different temperatures respectively is that temperature-equivalent plastic strain-yield strength concerns when the constitutive relation of (as shown in Figure 3) and the present invention's proposition is simulated; Stress-time relationship that the single shaft displacement loads is coincide good; Like Fig. 4,5,6, shown in 7, the constitutive relation that this explanation the present invention is directed to the proposition of heat treatment precipitation reinforced aluminium alloy welding process is rational.

Claims (2)

1. the historical material constitutive relation establishing method of Thermal Cycle temperature and thermal deformation; It is characterized in that: the material constitutive model adopts elastoplasticity and creeping property to express, and the concrete steps of the material constitutive relation establishing method that said Thermal Cycle temperature and thermal deformation are historical are following:

Step 1, mixed hardening such as are at the weighted sums to sclerosis and kinematic hardening, set the weighting coefficient that waits to sclerosis and kinematic hardening and are 0.5, wait to the sclerosis characteristic and adopt the power exponent rule of hardening to be expressed as:

σ=Eε 0<σ<σ _S ①

σ=σ _S+m(ε ^pl)n σ>σ _S ②

The 1. 2. middle σ of formula is a stress, and ε is strain, σ _sBe initial yield strength, E is a Young modulus, ε ^PlBe equivalent plastic strain, m is a material constant, and n is the parameter of material forming;

Step 2, employing pula lattice bilinearity kinematic hardening model tormulation formula:

db _ij=Cdε _ij ^pl?③

The 3. middle b of formula _IjBe back stress, ε _Ij ^PlBe the plastic strain tensor, C pula lattice hardening model constant;

Step 3, can know relation below existing between pula lattice hardening model constant C and the plastic modulus by uniaxial tension stress-strain relation and Feng Misaisi yield condition:

$C=\frac{2}{3}{E}_p$ ④

The 4. middle E of formula _pBe plastic modulus;

The Feng Misaisi yield condition is:

$f=J_2-\frac{{{\mathrm{\&sigma;}}_y}^2}{3}$

Wherein f is a loading function, Be second invariant of deviatoric stress, S _IjBe deviatoric stress tensor, σ _yFollow-up yield strength for material; If elastic-plastic deformation or plastic yield take place in f>=0, if elastic deformation takes place in f＜0;

Step 4, employing Ziegler model, internal variable back stress and S _Ij-b _IjProportional:

${\mathrm{db}}_{\mathrm{ij}}=a\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{pl}}}(S_{\mathrm{ij}}-b_{\mathrm{ij}})$ ⑤

The 5. middle S of formula _IjBe deviatoric stress,

Be the accumulated plastic strain differential, the scale factor that a is positive;

Step 5, for the kinematic hardening material, Feng Misaisi yield condition and plasticity associated flow rule combine and can obtain:

dε _ij ^pl=dλ(S _ij-b _ij)?⑥

To 6. both sides self dot product of formula, obtain:

$\mathrm{d\&lambda;}=\frac{3}{2}\frac{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{pl}}}}{{\mathrm{\&sigma;}}^y}$

Promptly $d{{\mathrm{\&epsiv;}}_{\mathrm{Ij}}}^{\mathrm{Pl}}=\frac{3}{2}\frac{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{Pl}}}}{{\mathrm{\&sigma;}}^y}(S_{\mathrm{Ij}}-b_{\mathrm{Ij}})$ 7.

${\mathrm{db}}_{\mathrm{ij}}=C\frac{3}{2}\frac{\stackrel{\&OverBar;}{{\mathrm{d\&epsiv;}}^{\mathrm{pl}}}}{{\mathrm{\&sigma;}}^y}(S_{\mathrm{ij}}-b_{\mathrm{ij}})$ ⑧

$a=\frac{E_p}{{\mathrm{\&sigma;}}^y}$ ⑨

The 6. middle d λ of formula is a scale factor;

Step 6, employing power law formal equivalence effect creep strain rate are expressed, and the constitutive relation between steady state creep speed

creep temperature T and the creep stress σ can be expressed as:

${\stackrel{\&CenterDot;}{\mathrm{\&epsiv;}}}^c=A{\mathrm{\&sigma;}}^n{\mathrm{\&epsiv;}}^{c^m}{T}^p\left({\mathrm{qt}}^{q-1}\right)$

ε wherein ^cBe equivalent creep strain, A is a pre-exponential factor, and s is a stress exponent, and r is equivalent creep strain index, and p is a humidity index, and q is a time index, and T is a thermodynamic temperature, and t is the time;

Step 7, have the stress-strain relation of elastoplasticity and creeping property at last, promptly the material constitutive model is:

${{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^e=\frac{1+\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{ij}}-\frac{\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{kk}}{\mathrm{\&delta;}}_{\mathrm{ij}}$

ε _ij ^pl=λS _ij

ε _ij ^c=λ ^cS _ij

${\mathrm{\&epsiv;}}_{\mathrm{ij}}={{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^e+{{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^{\mathrm{pl}}+{{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^c=\frac{1+\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{ij}}-\frac{\mathrm{\&upsi;}}{E}{\mathrm{\&sigma;}}_{\mathrm{kk}}{\mathrm{\&delta;}}_{\mathrm{ij}}+\mathrm{\&lambda;}{S}_{\mathrm{ij}}+{\mathrm{\&lambda;}}^c{S}_{\mathrm{ij}}$

Wherein, σ _Ij, ε _IjBe respectively stress tensor and strain tensor, ε _Ij ^e, ε _Ij ^cBe respectively elastic strain tensor sum creep strain tensor, λ ^cBe and the historical relevant constant of the deformation of creep; Above stress-strain relation is expressed as with stiffness matrix:

σ _ij=D _ijkl(ε _kl-ε _kl ^c)=D _ijkl[ε _kl-λ ^cS _ij]

Wherein, when material has only elastic deformation, deformation of creep series connection, D _IjklBe elastic matrix, when elastic deformation, plastic yield and deformation of creep series connection, D _IjklBe the plasticity matrix; When FEM calculation, the creep item shows stress relaxation phenomenon when faking load Processing.

2. the MSC.MARC secondary development of the historical material constitutive relation of Thermal Cycle temperature and thermal deformation; It is characterized in that: the MSC.MARC secondary development of the historical material constitutive relation of Thermal Cycle temperature and thermal deformation is based on Feng's Mises yield criterion, associated flow and hardening model, and the concrete steps of the MSC.MARC secondary development of the historical material constitutive relation of said Thermal Cycle temperature and thermal deformation are following:

Step 1, according to resilient relationship calculated stress increment:

Δσ _ij=λΔε _kkδ _ij+2GΔe _ij

Wherein, Δ σ _IjBe the increment of stress tensor, Δ ε _KkBe the increment of hydrostatic strain, Δ e _IjBe current inclined to one side strain increment, σ _IjBe Crow Nellie symbol, G is a modulus of shearing, and λ is a Lame's constant, λ=E υ/(1+ υ) (1-2 υ), and υ is a Poisson ratio;

Step 2, according to the Feng Misaisi yield condition, calculate tentative calculation stress

${\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}=\sqrt{\frac{3}{2}{S}_{\mathrm{ij}}^{\mathrm{pr}}{S}_{\mathrm{ij}}^{\mathrm{pr}}}$

Wherein, is the tentative calculation deviatoric stress; is the deviatoric stress tensor that i calculates incremental step; When the yield strength of tentative calculation stress greater than present material; When promptly satisfying yield condition; Get into the plastic yield stage, if do not satisfy, according to elastic stiffness matrix update stress state;

Step 3, according to Prandtl-Roy's Si quadrature flow rule, the differential of plastic strain tensor is:

$d{{\mathrm{\&epsiv;}}_{\mathrm{ij}}}^{\mathrm{pl}}=\frac{3}{2}\frac{({S_{\mathrm{ij}}}^i-{b_{\mathrm{ij}}}^i)}{{\mathrm{\&sigma;}}_y}\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{pl}}}$

Wherein, d ε ^PlBe the differential of equivalent plastic strain, b _Ij ⁱIt is the back stress tensor of current calculating incremental step;

Step 4, the laggard euler algorithm of employing carry out integral operation to the differential of plastic strain tensor, obtain with accumulation plastic strain increment Δ ε ^PlEquation for independent variable:

${\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}-3\mathrm{G\&Delta;}{\mathrm{\&epsiv;}}^{\mathrm{pl}}={\mathrm{\&sigma;}}_y$

In view of σ _yBe ε ^PlFunction, right The equation solution of form adopts Newton iteration method, to the power exponent hardening model, finds the solution V ε ^PlThe Newton iterative calculation formula be:

$\mathrm{\&Delta;}{{\mathrm{\&epsiv;}}_{j+1}}^{\mathrm{pl}}=\mathrm{\&Delta;}{{\mathrm{\&epsiv;}}_j}^{\mathrm{pl}}+\frac{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}-{\mathrm{\&sigma;}}_y-3\mathrm{G\&Delta;}{{\mathrm{\&epsiv;}}_j}^{\mathrm{pl}}}{\mathrm{mn}\&times;{({\mathrm{\&epsiv;}}^{\mathrm{pl}}+\mathrm{\&Delta;}{{\mathrm{\&epsiv;}}_j}^{\mathrm{pl}})}^{n-1}+3G}$

Δ ε wherein _J+1 ^Pl, Δ ε _j ^PlBe respectively j+1 and the j iterative computation accumulation plastic strain increment in step; The above iterative computation that circulates, until convergence error | Δ ε _J+1 ^Pl-Δ ε _j ^Pl|<10 ^-8Stop, so far trying to achieve the V ε that i calculates incremental step ^PlThe accumulation plastic strain increment of current calculating incremental step

$\mathrm{\&Delta;}{\mathrm{\&epsiv;}}_{\mathrm{ij}}^{\mathrm{pl}}=\frac{3}{2}(S_{\mathrm{ij}}^{\mathrm{pr}}-{b_{\mathrm{ij}}}^i)\mathrm{\&Delta;}{\mathrm{\&epsiv;}}^{\mathrm{pl}}/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}$

Step 5, renewal deviatoric stress and stress:

$\left\{\begin{array}{c}{S_{\mathrm{ij}}}^i+\mathrm{\&Delta;}{S}_{\mathrm{ij}}-{b_{\mathrm{ij}}}^i={\mathrm{\&sigma;}}_y\&times;(S_{\mathrm{ij}}^{\mathrm{pr}}-{b_{\mathrm{ij}}}^i)/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{pr}}\\ {\mathrm{\&sigma;}}_{\mathrm{ij}}=\frac{1}{3}{\mathrm{\&delta;}}_{\mathrm{ij}}{\mathrm{\&sigma;}}_{\mathrm{kk}}^{\mathrm{pr}}+{S_{\mathrm{ij}}}^i+\mathrm{\&Delta;}{S}_{\mathrm{ij}}\end{array}\right.$

Wherein, Δ S _IjBe the deviatoric stress increment of current calculating incremental step,

For next calculates the hydrostatic force of incremental step, ε _Kk ⁱHydrostatic strain for current calculating incremental step;

Step 6, material etc. to the constitutive relation of sclerosis are:

Δσ _ij=λ ^*σ _ijΔε _kk+2 _G ^*Δε _ij+h'×η _ijη _klΔε _kl

Wherein $G^{*}=G\frac{{\mathrm{\&sigma;}}_y}{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}},$ ${\mathrm{\&lambda;}}^{*}=K-\frac{2}{3}{G}^{*},$ $K=\frac{E}{3(1-2\mathrm{\&upsi;})},$ $h^{\&prime;}=\frac{h}{1+h/3G}-3G^{*},$ $h=\frac{d{\mathrm{\&sigma;}}_y}{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{Pl}}}},$ ${\mathrm{\&eta;}}_{\mathrm{Ij}}=S_{\mathrm{Ij}}^{\mathrm{Pr}}/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}};$

The constitutive relation of material kinematic hardening is:

Δσ _ij=λ ^*δ _ijΔε _kk+2G ^*Δε _ij+h′η _ijη _klΔε _kl

Wherein $G^{*}=G\frac{({\mathrm{\&sigma;}}_s+\mathrm{H\&Delta;}{\mathrm{\&epsiv;}}^{\mathrm{Pl}})}{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}},$ ${\mathrm{\&lambda;}}^{*}=K-\frac{2}{3}{G}^{*},$ $K=\frac{E}{3(1-2\mathrm{\&upsi;})},$ $h^{\&prime;}=\frac{h}{1+h/3G}-3G^{*},$ H=a σ _S, ${\mathrm{\&eta;}}_{\mathrm{Ij}}=(S_{\mathrm{Ij}}^{\mathrm{Pr}}-{b_{\mathrm{Ij}}}^i)/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}},$ b _Ij ⁱIt is the back stress that i calculates incremental step;

The constitutive relation of step 7, material mixing sclerosis, promptly the relation of the material constitutive after the secondary development is:

Δσ _ij=λ ^*δ _ijΔε _kk+2G ^*Δε _ij+h'×η _ijη _klΔε _kl

Wherein $G^{*}=G\frac{{\mathrm{\&sigma;}}_y+h{\mathrm{\&Delta;\; \&epsiv;}}^{\mathrm{Pl}}}{{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}},$ ${\mathrm{\&lambda;}}^{*}=K-\frac{2}{3}{G}^{*},$ $K=\frac{E}{3(1-2\mathrm{\&upsi;})},$ $h^{\&prime;}=\frac{h}{1+h/3G}-3G^{*},$ $h=a{\mathrm{\&sigma;}}_s+\frac{d{\mathrm{\&sigma;}}_y}{\stackrel{\&OverBar;}{d{\mathrm{\&epsiv;}}^{\mathrm{Pl}}}},$ ${\mathrm{\&eta;}}_{\mathrm{Ij}}=(S_{\mathrm{Ij}}^{\mathrm{Pr}}-{b_{\mathrm{Ij}}}^i)/{\stackrel{\&OverBar;}{\mathrm{\&sigma;}}}^{\mathrm{Pr}}.$

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