On a Constitutive Material Model to Capture Time Dependent Behaviour of Cortical Bone

It is commonly known that cortical bone exhibits viscoelastic-viscoplastic behavior which affects the biomechanical response when an implant is subjected to an external load. In addition, long term effects such as creep, relaxation and remodeling affect the success of the implant over time. Constitutive material models are commonly derived from data obtained in\ua0in vitro\ua0experiments. However during function, remodeling of bone greatly affects the bone material over time. Hence it is essential to include long term\ua0in vivo\ua0effects in a constitutive model of bone. This paper proposes a constitutive material model for cortical bone incorporating viscoelasticity, viscoplasticity, creep and remodeling to predict stress-strain at various strain rates as well as the behavior of bone over time\ua0in vivo. The rheological model and its parameters explain the behavior of bone subjected to longitudinal loading. By a proper set of model parameters, for a specific cortical bone, the present model can be used for prediction of the behavior of this bone under specific loading conditions. In addition simulation with the proposed model demonstrates excellent agreement to\ua0in vitro\ua0and\ua0in vivo\ua0experimental results in the literature

The numerical simulation of standard concrete tests and steel reinforcement using force flux peridynamics

Peridynamics is a numerical particle-based solid mechanics method that enables the simulation of brittle and quasi-brittle materials, as well as ductile materials. It allows cracking to appear spontaneously in the arms joining the particles and can therefore be used to simulate progressive fracture. In this article, we apply our version of peridynamics, which we call force flux peridynamics, to the simulation of concrete where the appearance of cracks plays an important role in the global mechanical properties. It is not difficult to modify the material parameters in peridynamics to achieve a given tensile strength or a given compressive strength. However, it is much more difficult to choose parameters which will model all the strength parameters of a material within the same model. When concrete fails in compression it may split or spall showing a complex relationship between compressive and tensile failure. We therefore set ourselves the simple task of producing a single peridynamics model which can predict the stiffness and strength behavior of concrete in standard compression and tension tests for which we chose the American Society for Testing and Materials standards for the cylinder compression test, the split cylinder test, and the modulus of rupture test. A parameter sensitivity study was performed based on the cylinder compression test to tune the key peridynamics parameters that determine the global material behavior. The compressive and tensile strengths were then determined from the combined simulation data. While the fracture modes, crack branching pattern and also the stress–strain curve show promising results, the maximum tensile strength was found to be significantly larger than physical experiments suggest. This is probably due to imperfections within real concrete at the interface between aggregate particles and cement paste and it shows that the detailed numerical modeling of the failure of concrete is highly complex with a large number of unknown material parameters

Architectural design methods used in engineering Master\u27s thesis projects

By letting structural engineering thesis students explore questions using architectural design\ua0methods, they creatively and systematically addressed\ua0 holistic questions while maintaining\ua0a technical depth. The approach may serve as a model to increase engineering students\u27\ua0ability to insightfully contribute to solutions for complex societal problems

The architectural application of shells whose boundaries subtend a constant solid angle

Surface geometry plays a central role in the design of bridges, vaults and shells, using various techniques for generating a geometry which aims to balance structural, spatial, aesthetic and construction requirements. In this paper we propose the use of surfaces defined such that given closed curves subtend a constant solid angle at all points on the surface and form its boundary. Constant solid angle surfaces enable one to control the boundary slope and hence achieve an approximately constant span-to-height ratio as the span varies, making them structurally viable for shell structures. In addition, when the entire surface boundary is in the same plane, the slope of the surface around the boundary is constant and thus follows a principal curvature direction. Such surfaces are suitable for surface grids where planar quadrilaterals meet the surface boundaries. They can also be used as the Airy stress function in the form finding of shells having forces concentrated at the corners. Our technique employs the Gauss-Bonnet theorem to calculate the solid angle of a point in space and Newton's method to move the point onto the constant solid angle surface. We use the Biot-Savart law to find the gradient of the solid angle. The technique can be applied in parallel to each surface point without an initial mesh, opening up for future studies and other applications when boundary curves are known but the initial topology is unknown. We show the geometrical properties, possibilities and limitations of surfaces of constant solid angle using examples in three dimensions

Adaptive bone re-modelling for optimization of porous structural components

This paper presents a speculative application of adaptive bone-remodelling to generate porous structures for building components using a numerical meshless method. We hypothesize that such porous structures could then be 3d printed to achieve light weight and material efficientbuilding components. The meshless model is built up of particles that are connected by arms to their neighbours within a distance called a horizon. The re-modelling adaption is then based on the ratio of arms strain over average arm strain which is mapped to a third-order polynomial function and used to scale the arm stiffness in a way that mimics the resorption and densification of bone tissue. The method is shown to work rather well in the recreation of the structural patterns found in cross section of a femur bone. The translation to a geometry which can be manufactured with additive techniques is not tackled specifically and suggest a direction for further work

The Use of Peridynamic Virtual Fibres to Simulate Yielding and Brittle Fracture

The forces in the ‘arms’ joining the particles in a peridynamic analysis depend upon the state of stress in the equivalent continuum and the orientation, length and density of the arms. Short and long arms carry less force than medium length arms as controlled by the weighting kernel. We introduce an intermediate step of imagining a mat of long fibres in which the fibre forces only depend upon the stress, the fibre orientation and the length of fibres per unit volume without the added complexity of the arm lengths. The effect of the arm lengths can then be considered as a separate exercise, which does not involve the continuum properties. The arm length is proportional to size of the particles and the separation of length from the state of stress allows for modelling of variable particle density in the discretisation of a problem domain, which enables computationally efficient accurate analysis. We then introduce the concept of arm elongation to fracture in order to model surface energy in fracture mechanics. This means that shorter arms have a larger strain to fracture than longer arms. The numerical implementation demonstrates that this produces a fracture stress that is inversely proportional to the square root of the crack length as predicted by the Griffith theory

Implant stability and bone remodeling up to 84 days of implantation with an initial static strain. An in vivo and theoretical investigation

ObjectivesWhen implants are inserted, the initial implant stability is dependent on the mechanical stability. To increase the initial stability, it was hypothesized that bone condensation implants will enhance the mechanical stability initially and that the moderately rough surface will further contribute to the secondary stability by enhanced osseointegration. It was further hypothesized that as the healing progresses the difference in removal torque will diminish. In addition, a 3D model was developed to simulate the interfacial shear strength. This was converted to a theoretical removal torque that was compared to the removal torque obtained invivo. Material and methodsCondensation implants, inducing bone strains of 0.015, were installed into the left tibia of 24 rabbits. Non-condensation implants were installed into the right tibia. All implants had a moderately rough surface. The implants had an implantation time of 7, 28, or 84days before the removal torque was measured. The interfacial shear strength at different healing time was estimated by the means of finite element method. ResultsAt 7days of healing, the condensation implant had an increased removal torque compared to the non-bone-condensation implant. At 28 and 84days of healing, there was no difference in removal torque. The simulated interfacial shear strength ratios of bone condensation implants at different implantation time were in line with the invivo data. ConclusionsModerately rough implants that initially induce bone strain during installation have increased stability during the early healing period. In addition, the finite element method may be used to evaluate differences in interlocking capacity

Prestressed Gridshell Structures

This paper describes a method for the form finding of shell structures composed of both compression and tension members which may lie in one layer or two layers. The length of some of the members can be constrained to a fixed length yielding some control of the resulting form found shape. The form finding is accomplished by adjusting the nodal positions until an equilibrium state is reached using dynamic relaxation. If part of a structure is unstable due to compression forces, then a negative mass must be used in the dynamic relaxation. The length constraint is met by adjusting the force density during form finding, again using dynamic relaxation. Finally, case studies are presented where the applied load and the prestress is used to govern the form found shape

Form Finding Nodal Connections in Grid Structures

Nodes for grid structures are often manufactured in a rather material intensive and inefficient way, increasing the weight of the structure and thus the load. Recent development of additive manufacturing techniques, have resulted in a rising interest in large-scale metal 3D printing. Topology optimization has become the obvious companion in the design of structural parts for 3D printing, and rightfully so. The technique is demonstrably able to provide material efficient solutions and is well suited for a manufacturing technique with few formal restrictions. However, from a designer’s perspective one could argue that topology optimization have some limitations. Like other “automated processes”, it tends to take over and does not leave much room for other form drivers. This paper presents an alternative method for designing material efficient nodes in grid structures that builds on the conventional form-finding techniques, usually applied to create minimal surface tensile structures or gravity shell like structures. The technique works by modelling the node as a hollow shell with a mesh, applying a set of tensile forces derived from the structural action from elements adjacent to the node (where compression is converted to tension) and running a form finding simulation. After the simulation, the shell is then thickened and analysed for the real load case (which consider both tension and compression) using FE-analysis. The benefit of such technique is that the designer has control over the topology of the design which enables more creative control and free exploration of a range of design variations. The form finding is done using dynamic relaxation and introduces spline elements with bending capability to control deviation from the pure spring network solution

Simulation of the mechanical interlocking capacity of a rough bone implant surface during healing

Background: When an implant is inserted in the bone the healing process starts to osseointegrate the implant by creating new bone that interlocks with the implant. Biomechanical interlocking capacity is commonly evaluated in in vivo experiments. It would be beneficial to find a numerical method to evaluate the interlocking capacity of different surface structures with bone. In the present study, the theoretical interlocking capacity of three different surfaces after different healing times was evaluated by the means of explicit finite element analysis. Methods: The surface topographies of the three surfaces were measured with interferometry and were used to construct a 3D bone-implant model. The implant was subjected to a displacement until failure of the bone-to-implant interface and the maximum force represents the interlocking capacity. Results: The simulated ratios (test/control) seem to agree with the in vivo ratios of Halldin et al. for longer healing times. However the absolute removal torque values are underestimated and do not reach the biomechanical performance found in the study by Halldin et al. which might be a result of unknown mechanical properties of the interface. Conclusion: Finite element analysis is a promising method that might be used prior to an in vivo study to compare the load bearing capacity of the bone-to-implant interface of two surface topographies at longer healing times