Analysis and Numerical Approximation of an Integro-differential Equation Modeling Non-local Effects in Linear Elasticity

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.Long-range interactions for linearly elastic media resulting in nonlinear dispersion relations are modeled by an initial-value problem for an integro-differential equation (IDE) that incorporates non-local effects. Interpreting this IDE as an evolutionary equation of second order, well-posedness in L ∞(ℝ) as well as jump relations are proved. Moreover, the construction of the micromodulus function from the dispersion relation is studied. A numerical approximation based upon quadrature is suggested and carried out for two examples, one involving jump discontinuities in the initial data corresponding to a Riemann-like problem

The Cauchy Problem for a One Dimensional Nonlinear Peridynamic Model

This paper studies the Cauchy problem for a one-dimensional nonlinear peridynamic model describing the dynamic response of an infinitely long elastic bar. The issues of local well-posedness and smoothness of the solutions are discussed. The existence of a global solution is proved first in the sublinear case and then for nonlinearities of degree at most three. The conditions for finite-time blow-up of solutions are established.Comment: To appear in Journal of Differential Equations (added references, corrected typos, minor revision in Section 2, 18 pages

Determination of Ballistic Limit for IM7/8552 Using Peridynamics

Significant testing is required to design and certify primary aircraft structure subject to High Energy Dynamic Impact (HEDI) events; current work under the NASA Advanced Composites Consortium (ACC) HEDI Project seeks to determine the state-of-the-art of dynamic fracture simulations for composite structures in these events. This paper discusses one of four Progressive Damage Analysis (PDA) methods selected for this project: peridynamics, through EMU implementation. A brief discussion of peridynamic theory is provided, followed by an outline of ballistic impact testing performed for model development and assessment. Detailed modeling approach and test-analysis correlation for a single open test case are presented, followed by the results of a series of blind predictions made prior to testing and test-analysis correlation performed with measured NASA test results. Specifically, we present simulation results for the ballistic limit (V50) of IM7/8552 composite panels ballistically tested with an impactor representative of a high-velocity fan-blade-out condition. In particular, force and displacement history and the damage state determined analytically are compared to measured results. Ultimately, peridynamics has the ability to predict damage patterns, impact force and deflections during a high energy dynamic impact event on composite panels of different layups using two different types of impactors. Blind predictions were promising and increased confidence in the model for impact simulation. There are open questions regarding the fidelity of the test fixture idealization in regards to stiffness and damping which will need to be addressed in future work

Numerical Dispersion Error in Finite Methods, Exemplified by the Perfectly Straight Beam Undergoing Bending Oscillations

In chapter (1) we present numerical simulations of the eigenstates of a MBS-model (Multi-Body-System) for a continuous ring structure. Both eigenfrequencies and eigenforms show a systematic error for high mode numbers. In order to understand the source of this error, in chapter (2) we calculate analytical solutions for the eigenfrequencies of a MBS- and a FEM-model (Finite-Element-Method) for the straight beam undergoing bending oscillations. We summarize our results in chapter (3)

Produção de diferentes genótipos de mangabeira (harconia speciosa gomes) sob a aplicação de diversos tipos de biofertilizantes no município de Humaitá, AM

The culture of mangaba can be cited as one of the fruit species cultures with wide growth and great acceptance in the market, presenting aromatic fruits, with excellent flavors, rich in nutrients, used both for fresh consumption and for industry, not to mention in the extraction of its latex. The main objective of the present work was to study and evaluate which biofertilizer composition promoted the greatest growth and development of the mangabeira genotypes (Hancornia speciosa GOMES). The experiment was carried out in a greenhouse at the Institute of Education, Agriculture and Environment of the Federal University of Amazonas Campus – Humaitá. The test was conducted with samples collected from 0 to 20 cm from the soil called Cambissolo Haplico. For promoting satisfactory results and differing significantly between the other treatments for the variables stem diameter, shoot dry mass, root fresh mass and root dry mass, promoting the highest averages for each variable (DC: 2.18mm), (MSPA: 1.83g), (MFR: 4.01), (MSR: 1.30g), treatment 02 (100% bovine manure) becomes viable to be used on mangaba seedlings, being the best among the other treatments tested.A cultura da mangaba pode ser citada como uma das culturas de espécies frutíferas com amplo crescimento e grande aceitação no mercado, apresentando frutos aromáticos, com excelentes sabores, ricos em nutrientes, utilizados tanto para o consumo in natura, quanto para a indústria, sem contar no extrativismo de seu látex. O presente trabalho teve como principal objetivo, estudar e avaliar qual composição de biofertilizante promoveu o maior crescimento e desenvolvimento aos genótipos da mangabeira (Hancornia speciosa GOMES). O experimento foi realizado em casa de vegetação do Instituto de Educação, Agricultura e Ambiente da Universidade Federal do Amazonas Campus – Humaitá. O ensaio foi conduzido com amostras coletadas de 0 a 20 cm do solo denominado Cambissolo Háplico. Por promover resultados satisfatórios e diferindo-se significativamente entre os demais tratamentos para às variáveis diâmetro do caule, massa seca da parte aérea, massa fresca da raiz e massa seca da raiz, promovendo as maiores médias para cada variável (DC: 2,18mm), (MSPA: 1,83g), (MFR: 4,01), (MSR: 1,30g), o tratamento 02 (100% de esterco bovino) torna-se viável para ser usado nas mudas de mangaba, sendo o melhor entre os demais tratamentos testados.2Nã

Crack nucleation in a peridynamic solid

A condition for the emergence of a discontinuity in an elastic peridynamic body is proposed, resulting in a material stability condition for crack nucleation. The condition is derived by determining whether a small discontinuity in displacement, superposed on a possibly large deformation, grows over time. Stability is shown to be determined by the sign of the eigenvalues of a tensor field that depends only on the linearized material properties. This condition for nucleation of a discontinuity in displacement can be interpreted in terms of the dynamic stability of plane waves with very short wavelength. A numerical example illustrates that cracks in a peridynamic body form spontaneously as the body is loaded

Analysis and numerical approximation of an integro-differential equation modelling non-local effects in linear elasticity

Long-range interactions for linearly elastic media resulting in nonlinear dispersion relations are modelled by an initial-value problem for an integro-differential equation (IDE) that incorporates non-local effects. Interpreting this IDE as an evolutionary equation of second order, well-posedness in L^{\infty}(\rz) as well as jump relations are proved. A numerical approximation based upon quadrature is suggested and carried out for two examples, one involving jump discontinuities in the initial data corresponding to a Riemann-like problem

Avaliação do crescimento e desenvolvimento das mudas do mamão havaí (Carica Papaya L.) sob o efeito da aplicação de diferentes composições de biofertilizantes

Apesar da existência de muitos estudos que relatam sobre a cultura do mamão havaí (Carica Papaya L.), poucos são os trabalhos que tem mostrado estudar o comportamento da cultura em relação a aplicação de biofertilizantes. Dessa forma o objetivo deste trabalho é avaliar o crescimento e o desenvolvimento de mudas de mamão havaí (Carica Papaya L.) sob o efeito da aplicação de diferentes composições de biofertilizantes. O experimento será instalado em casa de vegetação do Instituto de Educação, Agricultura e Ambiente da Universidade Federal do Amazonas, Campus Humaitá. O delineamento estatístico será em blocos casualizados com 5 tratamentos: testemunha (sem aplicação de biofertilizante); biofertilizante puro (composto com 100% de esterco bovino fresco); biofertilizante 1 (composto com 75% de esterco bovino fresco + 25% de caroço de açaí fresco triturado); biofertilizante 2 (composto com 50% de esterco bovino fresco + 50% de caroço de açaí fresco triturado) e biofertilizante 3 (composto com 25% de esterco bovino fresco + 75% de caroço de açaí fresco triturado), com 4 repetições em 4 blocos totalizando 80 unidades experimentais. As sementes serão coletadas de frutos de mamoeiro sadios e em seguida será seca a sombra e semeada em vasos com capacidade para dez litros de solo a uma profundidade de 1,5 cm, sendo colocadas três sementes por vaso. Após a etapa de germinação e crescimento será realizado o desbate das mudas aos 4 cm, deixando uma muda por vaso. As mudas serão avaliadas nos seguintes parâmetros: índice de velocidade de emergência, número de folhas, diâmetro do caule (mm), altura das plantas, biomassa fresca da parte aérea (g), massa fresca e seca das raízes e contagem dos ramos secundários. Os dados coletados serão submetidos à análise de variância (teste F), teste tukey ao nível de 5% de probabilidade e analise de regressão para o estudo das diferentes composições de biofertilizantes.CNP