A constitutive equation for ceramic materials used in lightweight armors

A constitutive model to simulate the behavior of ceramic materials under impact loading is proposed in order to achieve a better representation of the damage process due to the material fragmentation. To integrate the proposed constitutive equations, a semi implicit algorithm (implicit for the stresses and explicit for the damage variable) has been developed, leading to the generalized expressions of the clas sical return mapping algorithm. The model was implemented in a commercial finite element code and its performance was demonstrated by comparing its predictions with experimental results obtained by other authors.The authors are indebted to the Spanish Ministry of Education (Project DPI2008 06408/DPI), and to the Region of Madrid (Project CCG08 UC3M/MAT 4464) for the financial support.Publicad

An implicit consistent algorithm for the integration of thermoviscoplastic constitutive equations in adiabatic conditions and finite deformations

19 pages, 4 figures.The so-called viscoplastic consistency model, proposed by Wang, Sluys and de Borst, is extended here to the integration of a thermoviscoplastic constitutive equation for J2 plasticity and adiabatic conditions. The consistency condition in this case includes not only strain rate but also the effect of temperature on the yield function. Using the backward Euler integration scheme to integrate the constitutive equations, an implicit algorithm is proposed, leading to generalized expressions of the classical return mapping algorithm for J2 plasticity, both for the iterative calculation of the plastic multiplier increment and for the consistent tangent operator when strain rate and temperature are considered also as state variables of the hardening equation. The model was implemented in a commercial finite element code and its performance is demonstrated with the numerical simulation of four Taylor impact tests.This research was done with the financial support of the Comunidad Autónoma de Madrid under Project 07N/0040/2002.Publicad

An analysis of Gurson model with parameters dependent on triaxiality based on unitary cells

The Gurson model is widely used in the continuum-mechanics framework to analyse the ductile fracture process promoted by the nucleation, growth, and coalescence of voids. Further works improved the original Gurson model by introducing two parameters, q1 and q2, to adjust model predictions to the numerical results of a periodic array of cylindrical and spherical voids in hardening materials. This modified model is known as the Gurson–Tvergaard (GT) model. Commonly, these parameters are considered constants or dependent only on the material-hardening properties. However, there is evidence that these parameters also depend on the triaxiality of the stress field, as well as on initial porosity. In this work, a consistent fully implicit integration of the constitutive equations of the GT model, considering the q-parameters dependent on the triaxiality and the initial porosity of the stress field, is presented, and the corresponding consistent tangent operator is proposed. The model is validated by comparing the stress–strain behaviour, as well as the evolution of void volume fraction, of a voided cell and the equivalent cell of GT material with dependent parameters. The cases considered correspond to variable triaxiality stress fields, present in non-proportional loading conditions.This research was performed with the financial support of the Spanish Ministry of Education under Project reference DPI2005- 06769, and of the Region of Madrid, under Project reference CCG06-UC3M/DPI-0796.Publicad

Dynamic fracture-initation toughness determination of AI 7075-T651 aluminum alloy

We present values of the static and dynamic fracture-initiation toughness at different loading rates of the aluminum alloy Al 7075-T651 obtained on three-point bend specimens with different thicknesses and initial crack lengths. In static conditions, the critical stress intensity factor was measured according to well known procedures. The methodology of dynamic fracture initiation toughness is not yet standardized, and the dynamic tests were done with a modified Hopkinson pressure bar coupled to high-speed cameras to measure the crack mouth opening displacement CMOD of the specimens. The critical stress intensity value was obtained from the CMOD at the time that crack starts to grow. This instant was detected by means of crack gauges at the crack tip.Publicad

Three-dimensional effects on the dynamic fracture determination of Al 7075-T651 using TPB specimens

Here we present a numerical analysis of three dimensional effects on the dynamic three point bending fracture tests on Al 7075 T651 alloy specimens performed in a modified Split Hopkinson Pressure Bar. Using the Finite Element Method implemented in a commercial code the whole experimental device has been modeled. Different thicknesses and initial crack to width ratio specimens are simulated at different impact velocities to study the possible effect on the Crack Mouth Open ing Displacement and, using a local stress fracture criterion, in the critical Stress Intensity Factor. The numerical results are compared with experimental results found by the authors.Publicad

On the use of variable-separation method for the analysis of vibration problems with time-dependent boundary conditions

In this article, the axial vibrations of a rod with a clamped end and the transversal vibrations of a cantilever beam, both with a time-dependent and non-harmonic force applied on their free ends, are analysed. These are problems in which the traction and the shear, for the rod and the beam, respectively, prescribed in the boundaries of the bodies vary with time. The problems can be solved by the method proposed by Mindlin and Goodman. However, it is usual to solve this problem by the classic variable-separation method (which does not properly fulfil the time-dependent boundary conditions). The displacements and the forces along the systems are derived from both cited methods, and the results are compared. These results highlight the importance of using the proper solution method for the vibration problems with time-dependent boundary conditions.Comisión Interministerial de Ciencia y Tecnología of the Spanish Government and to the Comunidad Autónoma de Madrid for partial support of this work through the research projects DPI2011-23191 and CCG10-UC3M-DPI-5596, respectively.Publicad

On the interplay between strain rate and strain rate sensitivity on flow localization in the dynamic expansion of ductile rings

In this work a stability analysis on flow localization in the dynamic expansion of ductile rings is con ducted. Within a 1 D theoretical framework, the boundary value problem of a radially expanding thin ring is posed. Based on a previous work, the equations governing the stretching process of the expanding ring are derived and solved using a linear perturbation method. Then, three different perfectly plastic material constitutive behaviours are analysed: the rate independent material, the rate dependent mate rial showing constant logarithmic rate sensitivity and the rate dependent material showing non constant and non monotonic logarithmic rate sensitivity. The latter allows to investigate the interaction between inertia and strain rate sensitivity on necking formation. The main feature of this work is rationally dem onstrate that under certain loading conditions and material behaviours: (1) decreasing rate sensitivity may not lead to more unstable material, (2) increasing loading rate may not lead to more stable material. This finding reveals that the relation between rate sensitivity and loading rate controls the unstable flow growth. Additionally a finite element model of the ring expansion problem is built in ABAQUS/Explicit. The stability analysis properly reflects the results obtained from the numerical simulations. Both proce dures, perturbation analysis and numerical simulations, allow for emphasizing the interplay between rate sensitivity and inertia on strain localization.The financial support of the Comunidad Autónoma de Madrid (Project CCG10 UC3M/DPI 5596) and of the Ministerio de Ciencia e Innovación de España (Project DPI/2008 06408) is kindly acknowledged.Publicad

Axisymmetric free vibration of closed thin spherical nano-shell

This work investigates the free axisymmetric vibrations of a closed spherical nano-shell using the Eringen nonlocal elasticity theory. The motion equations are properly formulated considering the hypotheses of thin shells and the solution is obtained using the classical separation of variables method. The effect of the nonlocal parameter on the natural frequencies and modal shapes are discussed in comparison to their local counterparts. This study could be useful in biomedical and bioengineering applications as well as in other fields related with the nanotechnology.Ministerio de Ciencia e Innovación de España (Projects DPI/2011-24068 and DPI/2011-23191)Publicad

Análisis del comportamiento frente a impacto de elementos de estructuras aeronáuticas y aeroespaciales

El grupo de “Dinámica y Fractura de Elementos Estructurales” ofrece su experiencia en el análisis de problemas de mecánica de sólidos para estudiar el comportamiento de elementos estructurales sometidos a cargas de impacto. Para ello el grupo dispone de herramientas de simulación y equipamiento experimental específico para sus actividades de investigación. Sus actividades pueden resultar atractivas a empresas aeronáuticas que precisen de técnicas de análisis no convencionales

General expressions for the stress intensity factor of a one-point bend beam

The aim of this work is to find general expressions to determine the stress intensity factor of a one point bend beam like specimen, whether from the measurement of the applied load or the crack mouth opening displacement. The expressions, obtained by applying the superposition principle, involve the decomposition of the general case into three auxiliary problems. The solution of two of them (pure bending and three point bending) is well known, while the solution of the third (one point bending) is developed in the present work. The proposed expressions are compared to numerical results obtained by the finite element method and their accuracy is equal to or better than available expressions published elsewhere.Publicad