Belytschko T. - Introduction (779635), страница 55
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The penalty parameter $\lambda$ must be large enough so thatthecompressibility error is negligible (i.e., $I_3$ is approximately equalto $1$), yet not so large that numerical ill-conditioning occurs.Numerical experiments reveal that $\lambda = 10^3\times {\rm max}(C_1,C_2)$ to $\lambda = 10^7\times {\rm max}(C_1,C_2)$ is adequate for floating-pointword length of 64 bits. The constant $p_o$ is chosen so that the componentsof ${\bf S}$ are all zero in the initial configuration, i.e,\begin{equation}po = -(C_1 + 2 C_2)\end{equation}$\bullet$ Exercises\setcounter{equation}{0}\subsection {Plasticity in One Dimension}Materials for which permanent strains are developed upon unloading arecalled plastic materials.
Many materials (such as metals) exhibit elastic(often linear) behavior up tp a well defined stress levlel called theyeild strength. Onec loaded beyond the initial yield strength, plasticstrains are developed. Elastic plastic materials are further subdividedinto rate-independent materials, where the stress is independent of thestrain rate, i.e., the rate of loading has no effect on the stresses andrate-dependent materials , in which the stress depends on the strainrate; such materials are often called strain rate-sensitive.The major ingredients of the theory ofplasticity are\begin{enumerate}\item A decomposition of each increment of strain into an elastic,reversible component $d\varepsilon^e$ and an irreversible plastic part$d\varepsilon^p$.\item A yield function $f$ which governs the onset and continuance ofplastic deformation.\item A flow rule which governs the plastic flow, i.e., determines theplastic strain increments.\item A hardening relation which governs the evolution of the yield function.\end{enumerate}There are two classes of elastic-plastic laws:\begin{itemize}\item Associative models, where the yield function and the potentialfunction are identical\item Nonassociative models where the yield function and flow potential aredifferent.\end{itemize}Elastic-plastic laws are path-dependent and dissipative.
A large part ofthe work expended in deforming the solid is irreversibly converted toother forms of energy, particularly heat, which can not be converted tomechanical work. The stress depends on the entire history of thedeformation, and can notbe written as a single valued function of the strain as in ( ) and ( ).The stress is path-dependent and dependes on the history of thedeformation. We cannot therefore write an explicit relation for the stressin termsof strain, but only as a relation between rates of stress and strainThe constitutive relations for rate-independent and rate-dependentplasticity in one-dimension are given in the following sections.\subsubsection{\bf Rate-Independent Plasticity in One-Dimension}A typical stress-strain curve for a metal under uniaxial stress is shownin Figure~\ref{fig:stress-strain}.
Upon initial loading, the materialbehaves elastically (usually assumed linear) until the initial yield stressis attained. The elastic regime is followed by an elastic-plasticregime where permanent irreversible plastic strains are induced upon furtherloading.Reversal of the stress is called unlaoding. In unloading, thestress-strain response is typicallyassumed to be governed by the elastic modulus and the strainswhich remain after complete unloading are calledthe plastic strains. The increments in strain areassumed to be additively decomposed into elastic and plastic parts. Thuswe write\begin{equation}d\varepsilon = d\varepsilon^e +d\varepsilon^p\end{equation}Dividing both sides of this equation by a differential time increment$dt$ gives the rate form\begin{equation}\dot{\varepsilon} = \dot{\varepsilon}^e + \dot{\varepsilon}^p\end{equation}The stress increment (rate) is related to the increment (rate) ofelastic strain.
Thus\begin{equation}d\sigma = Ed\varepsilon^e, \quad \dot\sigma = E\dot\varepsilon^e\end{equation}relates the increment in stress to the increment in elastic strain.In the nonlinearelastic-plastic regime, the stress-strain relation is given by( see Figure ( ))\begin{equation}d\sigma = Ed\varepsilon^e = E^{\rm tan} d\varepsilon\end{equation}where the elastic-plastic tangent modulus, $E^{\rm tan}$, is the slope of thestress-strain curve. In rate form, the relation is written as\begin{equation}\dot{\sigma} = E\dot{\varepsilon^e} = E^{\rm tan}\dot{\varepsilon}\end{equation}The above relations are homogeneous in the rates of stress and strain whichmeans that if timeis scaled by an arbitrary factor, the constitutive relation remainsunchanged and therefore the material response is {\em rate-independent}even though it is expressed in terms of a strain rate.
In the sequel,the rate form of the constitutive relations will be used as the notationbecause the incremental form can get cumbersome especially for large strainformulations.$\bullet$ kinematic hardeningThe increase of stress after initial yield is called work or strainhardening. The hardening behavior of the material is generally afunction of the prior history of plastic deformation.In metal plasticity, the history of plastic deformation is oftencharcterized by asingle quantitiy $\bar{\varepsilon}$ called the accumulated plastic strainwhich is given by\begin{equation}\bar{\varepsilon} = \int\dot{{\bar\varepsilon}}dt\end{equation}where\begin{equation}\dot{\bar{\varepsilon}} = \sqrt{\dot{\varepsilon}^p\dot{\varepsilon}^p}\end{equation}is the effective plastic strain rate. The plastic strain rate is given by\begin{equation}\dot{\varepsilon}^p = \dot{\lambda}{\rm sgn}(\sigma)\end{equation}where\begin{equation}{\rm sign}(\sigma) = \left\{\begin{array}{cc} 1 & {\mbox{if $\sigma>0$}} \\-1 & {\mbox{if $\sigma <0$}}\end{array}\right.\end{equation}>From ( ) it follows that\begin{equation}\dot{\lambda} = \dot{\bar{\varepsilon}}\end{equation}The accumulatedplastic strain $\bar{\varepsilon}$, isan example of an internal variable used to characterize the inelasticresponse of the material.
An alternative, internal variable used in therepresentation of hardening is the plastic work which is given by (Hill,1958)\begin{equation}W^P = \int \sigma\dot{\varepsilon}^p dt\end{equation}The hardening behavior is often expressed through thedependence of the yield stress, $Y$, on the accumulated plastic strain, i.e.,$Y = Y(\bar{\varepsilon})$.
More general constitutive relations useadditional internal variables.A typical hardening curve is shown in Figure ( ). The slope of thiscurve is the plastic modulus, $H$, i.e.,\begin{equation}H = {dY(\bar{\varepsilon})\over d\bar{\varepsilon}}\end{equation}The effective stress is defined as\begin{equation}\bar{\sigma} = \sqrt{\sigma^2}\equiv |\sigma | = \sigma {\rm sgn}(\sigma)\end{equation}The yield condition is written as\begin{equation}f = \bar{\sigma} - Y(\bar{\varepsilon}) = 0\end{equation}which is regarded as the equation for the yield point (or surface whenmultiaxial stress states are considered). Note that the plastic strainrate can be written as\begin{equation}\dot{\varepsilon}^p = \dot{\bar{\varepsilon}}{\rm sign}(\sigma) =\dot{\bar{\varepsilon}}{\partial f\over \partial \sigma}\end{equation}where the result $\partial \bar{\sigma}/\partial \sigma = {\rmsign}(\sigma)$ has been used.
For plasticity in one-dimension(uniaxial stress),the distinction between associated and non-associated plasticity is nostpossible. Also, the lateral strain which accompanies the axialstrain has both elastic and plastic parts. This point will be addressedfurther in Section X on multiaxial plasticity.Plastic deformation occurs only when the yield condition is met.
Uponplastic loading, the stress must remain at yield, which is called the{\em consistency} condition, and is given by\begin{equation}\dot{f} = \dot{\bar{\sigma}} - \dot{Y}(\bar{\varepsilon}) = 0.\end{equation}>From ( ) it follows that, during plastic loading,\begin{equation}\dot{\bar{\sigma}} = {dY(\bar{\varepsilon})\overd\bar{\varepsilon}}\dot{\bar{\varepsilon}} = H\dot{\bar{\varepsilon}}\end{equation}Using ( ), ( ) and ( ) in ( ) gives\begin{equation}{1\over E^{\rm tan}} = {1\over E} + {1\over H}\end{equation}or\begin{equation}E^{\rm tan} = {EH\over E + H} = E - {E^2\over E + H}\end{equation}The plastic switch parameter $\alpha$ is introduced with $\alpha=1$corresponding to plasticloading and $\alpha=0$ corresponding to purely elastic response (loading orunlaoding).
Thus the tangent modulus is written\begin{equation}E^{\rm tan} = E - \alpha{E^2\over E + H}\end{equation}An alternative way of writing the laoding-unlaoding conditions withoutusing the switch parameter $\alpha$ is through the use of the Kuhn-Tuckerconditions, which play an important role in mathematical programmingtheory [Ref?]For plasticity, the conditions are:\begin{equation}\dot{\lambda}\dot{f}=0,\quad \dot{\lambda}\ge 0, \quad \dot{f}\le 0\end{equation}Thus for plastic loading, $\dot{\lambda}\ge 0$ and the consistencycondition $\dot{f}=0$ is satisfied. For purely elastic loading orunloading, $\dot{f}\ne 0$ and it follows that $\dot{\lambda}=0$.The constitutive relations for rate-independent plasticity in 1D aresummarized in Box 9.1.\subsubsection{Rate-Dependent Plasticity in One Dimension}In rate dependent plasticity, the plastic response of the materialdepends on the rate of loading.
The elastic response is given as before(in rate form) as\begin{equation}\dot{\sigma} = E\dot{\varepsilon}^e\end{equation}which may be written using the elastic-plastic decomposition of the totalstrainrate (Equation ) as\begin{equation}\dot{\sigma}=E(\dot{\varepsilon}-\dot{\varepsilon}^p).\end{equation}For plastic deformation to occur the yield condition must be met orexceeded. This differs from the rate-independent case in that inrate-dependent plasticity the stress can exceed the yield stress.The plastic strain rate is given by\begin{equation}\dot\varepsilon^p = \dot{\bar\varepsilon} {\rm sgn}\sigma\end{equation}Formany rate-dependent materials, the plastic response is characterized by anoverstress model of the form\begin{equation}\dot{\bar{\varepsilon}} = {\phi (\sigma, \bar{\varepsilon})\over \eta}\end{equation}where, $\phi$ is the overstress and $\eta$ is the viscosity.For example, the overstress model introduced by Perzyna (19xx) is given by\begin{equation}\phi = Y\bigl({\bar{\sigma}\over Y}-1\bigr)^n\end{equation}where $n$ is called the rate-sensivityexponent.
Using ( ) and ( ) the expression for the stress rate is given by\begin{equation}\dot{\sigma}=E\biggl(\dot{\varepsilon} - {\phi(\sigma, \bar{\varepsilon})\over \eta}{\rm sgn}(\sigma)\biggr).\end{equation}which is a differential equation for the evolution of the stress.Comparing this expression to ( ), it can be seen that ( ) isinhomogeneous in the rates and therefore the material response is{\em rate-dependent}. Models of this type are often used to model thestrain-rate dependence observed in materials. More elaborate models withadditional internal variables and perhaps different response in differentstrain-rate regimes have been developed (see for examplethe unified creepplasticity model [Ref]).
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