Abstract The material point method can be regarded as a meshfree extension of the finite element method. This fact has two interesting consequences. On the one hand, this puts a vast literature at our disposal. On the other hand, many of this inheritance has been adopted without questioning it. A clear example of it is the use of the Voigt algebra, which introduces an artificial break point between the formulation of the continuum problem and its discretized counterpart. In the authors’ opinion, the use of the Voigt rules leads to a cumbersome formulation where the physical sense of the operations is obscured since the well-known algebra rules are lost. And with them, the intuition about how stresses and strains are related. To illustrate it, we will describe gently and meticulously the whole process to solve the nonlinear governing equations for isothermal finite strain elastodynamics, concluding with a compact set of expressions ready to be implemented effortless. In addition, the classic Newmark-β algorithm has been accommodated to the local maximum-entropy material point method framework by means of an incremental formulation.
Abstract The material point method can be regarded as a meshfree extension of the finite element method. This fact has two interesting consequences. On the one hand, this puts a vast literature at our disposal. On the other hand, many of this inheritance has been adopted without questioning it. A clear example of it is the use of the Voigt algebra, which introduces an artificial break point between the formulation of the continuum problem and its discretized counterpart. In the authors’ opinion, the use of the Voigt rules leads to a cumbersome formulation where the physical sense of the operations is obscured since the well-known algebra rules are lost. And with them, the intuition about how stresses and strains are related. To illustrate it, we will describe gently and meticulously the whole process to solve the nonlinear governing equations for isothermal finite strain elastodynamics, concluding with a compact set of expressions ready to be implemented effortless. In addition, the classic Newmark-β algorithm has been accommodated to the local maximum-entropy material point method framework by means of an incremental formulation. Read More
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