Fracture analysis of a metal specimen

I analyzed a specimen of ductile material with explicit dynamic method about one year ago. Recently, I want to review the model and summarize the process of analysis. Stress and strain of material in ABAQUS Stress and strain in the definition of material in abauqs should be converted to true stress \(\sigma\) and true plastic strain \({\varepsilon _p}\) from the data obtained in experiments. The stress and strain achieved in experiments are engineering stress \({\sigma _{{\rm{eng}}}}\) and engineering strain \({{\varepsilon

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Probabilistic system analysis – Part III

Limit theorem Chebyshev’s inequality \begin{align} {\sigma ^2} &= \int {{{\left( {x – \mu } \right)}^2}{f_X}\left( x \right)dx} \\ &\ge \int_{ – \infty }^{u – c} {{{\left( {x – \mu } \right)}^2}{f_X}\left( x \right)dx} + \int_{u + c}^\infty {{{\left( {x – \mu } \right)}^2}{f_X}\left( x \right)dx} \\ &\ge {c^2}\int_{ – \infty }^{u – c} {{f_X}\left( x \right)dx} {\rm{ + }}{c^2}\int_{u + c}^\infty {{f_X}\left( x \right)dx} \\ &={c^2}{\rm{P}}\left( {\left| {X – \mu } \right| \ge c} \right) \end{align} \[{\rm{P}}\left( {\left| {X –

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Probabilistic system analysis – Part II

Bernoulli process \begin{align} {\rm{P}}\left( {{\rm{sucess}}} \right) &= {\rm{P}}\left( {{X_i} = 1} \right) = p\\ {\rm{P}}\left( {{\rm{failure}}} \right) &= {\rm{P}}\left( {{X_i} = 0} \right) = 1 – p\\ {\rm{E}}\left[ {{X_t}} \right] &= p,{\mathop{\rm var}} \left( {{X_t}} \right) = p\left( {1 – p} \right) \end{align} PMF of # of arrivals (number of success \(S\) in \(n\) time slots): binomial pmf \begin{align} {\rm{P}}\left( {S = k} \right) &= \left( \begin{array}{l} n\\ k \end{array} \right){p^k}{\left( {1 – p} \right)^{n – k}}, \ k =

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Probabilistic system analysis – Part I

Independent and disjoint Two events are independent or disjoint are two different concept. If two events are independent, there is no relation between but they can happen together. It is noted that independence can be affected by conditioning. If two events are disjoint, they has the relation that when one happens the other cannot happen and they are not independent. Independent: \({\rm{P}}\left( {B\left| A \right.} \right) = {\rm{P}}\left( B \right) \Leftrightarrow {\rm{P}}\left( {A\left| B \right.} \right) = {\rm{P}}\left( A \right)\) Disjoint: \({\rm{P}}\left(

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Differential equation – Part IV ODE system

ODE system \[\left\{ \begin{array}{l} {x^{\prime}} = f\left( {x,y,t} \right)\\ {y^{\prime}} = g\left( {x,y,t} \right) \end{array} \right.,x\left( {{t_0}} \right) = {x_0},y\left( {{t_0}} \right) = {y_0}\] Linear system: \[\left\{ \begin{array}{l} {x^{\prime}} = ax + by + {r_1}\left( t \right)\\ {y^{\prime}} = cx + dy + {r_2}\left( t \right) \end{array} \right.\] \(a\), \(b\), \(c\), \(d\) can be functions of \(t\). If they are constant, then the system is called constant coefficient system. When \(r_1\left( t \right) = {r_2}\left( t \right) = 0\), the system is linearly homogeneous. Example:

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