Bone modeling and remodeling has been the subject of extensive experimental studies. There have been several mathematical models proposed to explain the observed behavior, as well. A different approach is taken here in which the bone is treated from a macroscopic view point. In this investigation, a one-dimensional analytical model is used to shed light on the factors which play the greatest role in modeling or growth of cortical bone at the periosteal surface. It is presumed that bone growth is promoted when increased amounts of bone nutrients, such as nitric oxide synthase (NOS) or messenger molecules, such as prostaglandin E2 (PGE2), seep out to the periosteal surface of cortical bone and are absorbed by osteoblasts. The transport of the bone nutrients is assumed to be a strain controlled process. Equations for the flux of these nutrients are written for a one-dimensional model of a long bone. The obtained partial differential equation is linearized and solved analytically. Based upon the seepage of nutrients out of the bone, the effect of loading frequency, number of cycles and strain level is examined for several experiments that were found in the literature. It is seen that bone nutrient seepage is greatest on the tensile side of the bone; this location coincides with the greatest amount of bone modeling
Strain-driven corrosion of branching cracks, initiated from a virtually plane surface, has been studied using a moving boundary technique. The material is assumed linear elastic and is subjected to fatigue loading under plain strain conditions. The surface of the material is covered by a protective film. During loading this film can be damaged if it is strained above a threshold value, thus revealing an unprotected surface. Corrosion advances by material dissolution, eventually evolving into cracks. The rate of surface evolution is a function of the degree of protective film damage. During unloading the protective film is assumed to develop and heal the surface. A low frequency cyclic loading is applied to ensure that total healing is assumed. The moving boundary technique, simulating corrosion, results in arc-shaped crack tips, rather than singular crack tip points, thus no crack growth criterion is needed in the analysis. For each load step, the strain distribution is found using the finite element method, followed by required the movement the boundary and then remeshing. The crack growth has been investigated for at least 2000 cycles. A more or less pronounced branching of the cracks is found to develop. The crack branches can be classified in three groups; main cracks that grows with maximum rate and branches further, branch cracks that initially retards and then find a steady state growth rate that is a fraction of maximum speed, and finally, arresting cracks that after a period of retardation stop growing. The crack patterns are realistic, showing a sort of self-similarity with tree-like structure, cf. the picture below that shows a typical finite element result. The width of a crack branch together with the shielding from the applied stresses, caused by the other branches and main cracks, seem to govern the evolution of the crack branch. A steady-state growth rate is achieved during parts of the evolution as the crack width and the strain field surrounding the crack tip is in balance, i.e. the crack widens while the crack grows longer. As the bluntness of the tip reaches an upper limit, branching results.
The kinking of a corrosion crack due to mixed-mode fatigue loading is studied using an adaptive finite element procedure. The rate of material dissolution is assumed to be proportional to the stretching of the corroding surface. The dissolution of material is governed by a corrosion law, where no criterion is needed for neither crack growth nor growth direction. The problem is treated as a general moving boundary problem. The kink angles are found to be in very good agreement with results for sharp cracks using criteria reported in the literature.
The growth of a crack subjected to corrosion fatigue is studied using adaptive finite elements. The crack is the image of a deep corrosion pit, and the growth is the result of a repeated cycle of dissolution of the material, formation of a protective oxide film and break-down of the oxide film. The break-down of the film is governed by the strain at the surface and the dissolution rate is assumed to be proportional to this stretching. A threshold strain is assumed to exist below which the oxide film remains intact. With this model, no criterion is needed, neither for crack growth, nor for prediction of the growth direction. The reason is that both are immediate results of the evolution of the body shape. The growth of a semi-infinite crack lying in an infinite strip subjected to different degrees of mixed-mode loading is studied and the results are compared to crack path criteria for sharp cracks. Additionally, the path of a corrosion fatigue crack starting at the surface of an elastic layer attached to a stiff substrate is simulated. The result showed some agreement with experimental results found in the literature.
Crack growth and crack path under a fracture mechanical experiment in polycarbonate is studied. The material, a polycarbonate, is photo elastic witch makes it possible to do an optical stress field analysis. The crack path and crack branching is also predicted by an adaptive finite element analysis. The crack growth is a result of stress induced, chemically activated material dissolution that completely determines the geometry of the specimen, including the shape and length of the crack. In this case, the mechanisms and processes leading to crack growth are assumed to be incorporated in a general material dissolution process. As a consequence of this assumption there is no need for a crack criterion or crack path hypothesis. The lack of crack path criterion imposes an arbitrary length scale where the crack width scales with the square of the applied load but the crack growth rate is independent of the load. Numerical calculations suggest that the crack is capable of branching and that the branches will continue to grow at the same rate as the original crack. The numerical results are verified by the photo elastic experiments and a reasonable agreement is established.
Sprickväg och sprickgrening studeras under brottmekaniska experiment i polykarbonat. Detta material är spänningsoptiskt och fotoelastiska analyser utförs. Sprickväg och sprickgrening beräknas också med hjälp av en adaptiv finit element-metod. Spricktillväxten är ett resultat av spänningsunderstödd, kemiskt eller biologiskt aktiverad, materialupplösning, som helt bestämmer provkroppens geometri inklusive sprickans form och längd. På så sätt är de processer som leder till sprickpropagering inkluderade i en allmän materialupplösningsprocess. Som en konsekvens av detta behövs varken brottkriterium eller sprickvägshypotes. Den fotoelastiska undersökningen jämförs med de numeriska resultaten. Avsaknaden av brottkriterium gör att sprickpropageringsprocessen saknar längdskala. Därmed följer att sprickans bredd skalar med kvadraten på den pålagda lasten medan spricktillväxthastigheten blir oberoende av pålagd last. Förhållandet gör vidare att sprickan förgrenas och att alla grenar fortsätter att växa utan att förlora fart.
Thin hard coatings on components and tools are used increasingly due to the rapid development in deposition techniques, tribological performance and application skills. The residual stresses in a coated surface are crucial for its tribological performance. Compressive residual stresses in PVD deposited TiN and DLC coatings were measured to be in the range of 0.03–4 GPa on steel substrate and 0.1–1.3 GPa on silicon. MoS2 coatings had tensional stresses in the range of 0.8–1.3 on steel and 0.16 GPa compressive stresses on silicon. The fracture pattern of coatings deposited on steel substrate were analysed both in bend testing and scratch testing. A micro-scale finite element method (FEM) modelling and stress simulation of a 2 μm TiN-coated steel surface was carried out and showed a reduction of the generated tensile buckling stresses in front of the sliding tip when compressive residual stresses of 1 GPa were included in the model. However, this reduction is not similarly observed in the scratch groove behind the tip, possibly due to sliding contact-induced stress relaxation. Scratch and bending tests allowed calculation of the fracture toughness of the three coated surfaces, based on both empirical crack pattern observations and FEM stress calculation, which resulted in highest values for TiN coating followed by MoS2 and DLC coatings, being KC = 4–11, about 2, and 1–2 MPa m1/2, respectively. Higher compressive residual stresses in the coating and higher elastic modulus of the coating correlated to increased fracture toughness of the coated surface.
A directional crack growth criterion in a compressed elastic perfectly plastic material is considered. The conditions at the crack-tip are evaluated for a straight stationary crack with a small incipient kink. Remote load is a combined hydrostatic pressure and pure shear applied via a boundary layer. Crack surfaces in contact are assumed to develop homogenous Coulomb friction. The crack opening displacement of an extended kink is examined in a finite element analysis to judge the risk of opening mode failure. It has been found that the direction that maximizes the crack opening displacement of an extended kink tip coincides very well with a prediction of the crack growth direction obtained by using a criterion for continued crack growth direction discussed by the authors elsewhere [Int. J. Fract. 108 (2001) 351]. Moreover, the by the model predicted incipient crack growth directions are qualitatively comparable with reported crack paths obtained in ductile materials in a limited number of experiments performed under a combined load of in-plane shear and compression.
Directional crack growth criteria in compressed elastic–plastic materials are considered. The conditions at the crack tip are evaluated for a straight stationary crack. Remote load is a combined hydrostatic stress and pure shear, applied via a boundary layer assuming small scale yielding. Strains and deformations are assumed to be small. Different candidates for crack path criteria are examined. Maximum non-negative hoop stress to judge the risk of mode I and maximum shear stress for mode II extension of the crack are examined in some detail. Crack surfaces in contact are assumed to develop Coulumb friction from the very beginning. Hence, a condition of slip occurs throughout the crack faces. The plane in which the crack extends is calculated using a finite element method. Slip-line solutions are derived for comparison with the numerically computed asymptotic field. An excellent agreement between numerical and analytical solutions is found. The agreement is good in the region from the crack tip to around halfway to the elastic–plastic boundary. The relation between friction stress and yield stress is varied. The crack is found to extend in a direction straight ahead in shear mode for sufficiently high compressive pressure. At a limit pressure a kink is formed at a finite angle to the crack plane. For lower pressures the crack extends via a kink forming an angle to the parent crack plane that increases with decreasing pressure.
A study of crack propagation paths in the situation where the crack is suppressed to open during growth due to high compressive forces has been performed. This problem was analyzed theoretically very recently by the authors and is here extended to involve a limited number of illustrative experimental results reported elsewhere in the literature. By analyzing the experimental crack growth patterns, the conclusion is that the model cannot describe the more realistic microscopic failure in detail. Since shear crack growth on the microscale strongly depend on inhomogenities in the material, like cavities, grains or inclusions; the closed crack growth patterns observed are not smooth or free of kinks. Nevertheless, the model show good agreement with the reported experimental observations of the paths of closed macroscopic mode II cracks on samples in brittle materials, induced under overall compression. Failure patterns experimentally observed supports the theory that the growth of mode II cracks under compression in brittle materials follow a propagation path described by a function y=k x^b, where b=3/2. This is strongly supported by the measured values obtained from various experiments. In all the studied experiments, the exponent b was found in the interval [1.43–1.58]. Further, an investigation of the curvature parameter k has been performed and the conclusion is that k does also agree with the simplified model, even though not as good as the exponent b. However, k differs in general <15% from the theoretical value predicted by the model. The process of crack growth is in the simplified model assumed to be controlled by the mode II stress intensity factor KII of the main crack and the difference between the compressive remote normal stress parallel with the crack plane (r11) and the compressive remote normal stress per- pendicular to the crack plane.
Abstract. A directional crack growth prediction in a compressed homogenous elastic isotropic material under plane strain conditions is considered. The conditions at the parent crack tip are evaluated for a straight stationary crack. Remote load is a combined biaxial compressive normal stress and pure shear. Crack surfaces are assumed to be frictionless and to remain closed during the kink formation wherefore the mode I stress intensity factor KI is vanishing. Hence the mode II stress intensity factor KII remains as the single stress intensity variable for the kinked crack. An expression for the local mode II stress intensity factor k2 at the tip of a straight kink has been calculated numerically with an integral equation using the solution scheme proposed by Lo (1978) and refined by He and Hutchinson (1989). The confidence of the solution is strengthened by verifications with a boundary element method and by particular analytical solutions. The expression has been found as a function of the mode II stress intensity factor KII of the parent crack, the direction and length of the kink, and the difference between the remote compressive normal stresses perpendicular to, and parallel with, the plane of the parent crack. Based on the expression, initial crack growth directions have been suggested. At a sufficiently high non-isotropic compressive normal stress, so that the crack remains closed, the crack is predicted to extend along a curved path that maximizes the mode II stress intensity factor k2. Only at an isotropic remote compressive normal stress the crack will continue straight ahead without change of the direction. Further, an analysis of the shape of the crack path has revealed that the propagation path is, according the model, required to be described by a function y = cxγ , where the exponent γ is equal to 3/2. In that case, when γ = 3/2, predicts the analytical model a propagation path that is selfsimilar (i.e. the curvature c is independent of any length of a crack extension), and which can be described by a function of only the mode II stress intensity factor KII at the parent crack tip and the difference between the remote compressive normal stress perpendicular to, and parallel with, the parent crack plane. Comparisons with curved shear cracks in brittle materials reported in literature provide limited support for the model discussed.
Stress intensity factors are calculated for long plane cracks with one tip interacting with a region of graded material characteristics. The material outside the region is considered to be homogeneous. The analysis is based on assumed small differences in stiffness in the entire body. The linear extent of the body is assumed to be large compared with that of the graded region. The crack tip, including the graded region, is assumed embedded in a square-root singular stress field. The stress intensity factor is given by a singular integral. Solutions are presented for rectangular regions with elastic gradient parallel to the crack plane. The limiting case of infinite strip is solved analytically, leading to a very simple expression. Further, a fundamental case is considered, allowing the solution for arbitrary variation of the material properties to be represented by Fourier's series expansion. The solution is compared with numerical results for finite changes of modulus of elasticity and is shown to have a surprisingly large range of validity. If an error of 5% is tolerated, modulus of elasticity may drop by around 40% or increase with around 60%.
Bi-materials composed of thin layers ideally bonded to large substrates are considered. Cracks emerging from an almost flat surface and propagating through the bi-materials are studied. The cracks acquire realistic geometrical shapes, where the tips are integral parts of the crack surfaces. Crack propagation is related to surface evolution resulting from material loss due to corrosion. Controlling mechanism for the evolution is the rupture of a brittle passive film, which is frequently building-up along the surface. The evolution rate is a function of the degree of film damage caused by the surface straining. The model leads to a moving boundary formulation, for which a numerical solution is used. The mismatch of the material plastic properties is being varied in the study. The results show how cracks pass the interface. The growth rate variation close to the interface is studied. Typical surface evolution for a crack passing through a soft-hard material interface is presented. The resulting crack morphology of the model resembles what has been observed in reality. It is shown how the results can be used in designing bi-material systems to inhibit corrosion crack growth.
This work proposes a model for corrosion driven crack growth. The model poses a moving boundary problem, where a chemical attack removes material from the body. The rate of the chemical attack is a function of the strain along the body surface. No crack growth criterion is needed for the analysis. A finite strain formulation is used and the material model is assumed hyperelastic. The problem is stated for a large body, containing a large crack. A low frequency cyclic loading is considered. Thus, corrosion is assumed to dissolve material with a rate approximately proportional to the strain rate. The problem is solved using finite element method based program, enhanced with a procedure handling the moving boundary. Parametric studies are performed for a series of different initial shapes of the near-tip region. Presented results show that the crack growth rate is largely dependent on the initial crack geometry. For a set of initial shapes and load levels steady-state conditions of growth are achieved, while for the others the cracks show tendency to branch.
A combined experimental/numerical method for determination of constitutive parameters in high strain-rate material models is presented. Impact loading, using moderate projectile velocities in combination with small specimens (sub mm) facilitate tensional strain rates in the order of 104–105 s−1. Loading force is measured from one-dimensional wave propagation in a rod using strain gauges and deformation is monitored with a high-speed camera equipped with a microscope lens. A sequence of digital photographs is taken during the impact loading and the plastic deformation history of the specimen is quantified from the photographic record. Estimation of material parameters is performed through so called inverse modelling in which results from repeated FE-simulations are compared with experimental results and a best choice of constitutive parameters is extracted through an iterative optimisation procedure using the simplex method. Results are presented from a preliminary tension test of a mild steel (A533B) at a strain rate well over 104 s−1. The sensitivity of the evaluated material parameters to errors in measured quantities is studied. The method, especially the optical technique for measurement of deformation will be further developed.
The crack tip driving force of a crack growing from a pre-crack that is perpendicular to and terminating at an interface between two materials is investigated using a linear fracture mechanics theory. The analysis is performed both for a crack penetrating the interface, growing straight ahead, and for a crack deflecting into the interface. The results from finite element calculations are compared with asymptotic solutions for infinitesimally small crack extensions. The solution is found to be accurate even for fairly large amounts of crack growth. Further, by comparing the crack tip driving force of the deflected crack with that of the penetrating crack, it is shown how to control the path of the crack by choosing the adhesion of the interface relative to the material toughness.
The fracture toughness of a polymer-metal laminate composite is obtained by mechanical testing of a specimen containing a pre-crack. The laminate is a material used for packaging. It consists of a thin aluminium foil and a polymer coating. A centre cracked panel test geometry is used. Each of the layers forming the laminate is also tested separately. The result is compared with the measured fracture strength of the individual layers. It is observed that the load carrying capacity increases dramatically for the laminate. At the strain when peak load is reached for the laminate only aluminium is expected to carry any substantial load because of the low stiffness of the LDPE. However, the strength of the laminate is almost twice the strength of the aluminium foil. The reason seems to be that the aluminium forces the polymer to absorb large quantities of energy at mmall nominal strain. The toughness compares well with the accumulated toughness of all involved layers. Possible fracture of the interface between the layers is discussed.
During fracture of membranes loading often produces buckles above and below the crack surface. This changes the stress state surrounding the crack-tip and stresses in the neighbourhood of the crack-tip posses a weaker singularity than r-1/2. As a result, fracture occurs when the crack-tip stress distribution is different as compared with that when buckling is artificially prevented. Therefore the conditions for transfer of lab results to real structures are changed. The weaker singularity is here utilised to formulate an adopted fracture mechanical theory. An approximate application is made based on an assumption that the buckled area of the paper is incapable of carrying load. This region is approximated with the region that is under compressive load at plane stress conditions. The result is compared with experiments performed on paper. The importance of the linear extent of the process region has on the energy available for fracture is discussed.
The stress driven growth of an expanding precipitate at a crack tip is studied. The material is assumed to be linearly elastic, and the expansion is considered to be isotropic or transversely isotropic. The extent of the precipitate is expected to be small as compared with the crack length and distance to boundaries. The problem has only a single length scale given by the squared ratio of the stress intensity factor and a critical hydrostatic stress that initiates the growth of the precipitate. Therefore, the growth occurs under self-similar conditions. The equations on non-dimensional form show that the free parameters are expansion strain, degree of anisotropy and Poisson's ratio. It is found that the precipitate, once initiated, grows without remote load for expansion strains above a critical value. The anisotropy of the expansion strongly affects the shape of the precipitate but does not have a large effect on the crack tip shielding.
A model is established that describes stress driven diffusion, resulting in formation and growth of an expanded precipitate at the tip of a crack. The new phase is transversely isotropic. A finite element method is used and the results are compared with a simplified analytical theory. A stress criterium for formation of the precipitate is derived by direct integration of the Einstein-Smoluchowski law for stress driven diffusion. Thus, the conventional critical concentration criterium for precipitate growth can be replaced with a critical hydrostatic stress. The problem has only one length scale and as a consequence the precipitate grows under self-similar conditions. The length scale is given by the stress intensity factor, the diffusion coefficient and critical stress versus remote ambient concentrations. The free parameters involved are the expansion strain, the degree of anisotropy and Poisson's ratio. Solutions are obtained for a variation of the first two. The key result is that there is a critical phase expansion strain below which the growth of the new phase is stable and controlled by the stress intensity factor. For supercritical expansion strains, the precipitate grows even without remote load. The anisotropy of the expansion strongly affects the shape of the precipitate, but does not have a large effect on the crack tip shielding. (C) 2018 The Authors. Published by Elsevier B.V.
The stress fields of expanding (precipitation) and contracting (dissolution) hydride plates were computed by finite element method using Zr–H solid solution and hydride properties at 25, 200 and 400 °C for fully and semi-constrained hydride plates. For the first time simultaneous hydride expansion and matrix contraction and vice-versa have been considered in a simulation of hydride precipitation and dissolution, respectively. It was observed that a fully constrained expanding hydride plate exerts a tensile stress field in the matrix close to the edge of the hydride plate while a partially contracting hydride plate exerts a tensile stress field in the hydride plate as well as a large compressive stress in the surrounding matrix close to the edge of the hydride plate. It is suggested that a compressive stress component in the matrix acting normal to a partially shrinking hydride plate could possibly explain an enhanced resistance to hydride embrittlement of Zr-alloy at elevated temperature.
Threshold stress, σth, for reorientation of hydrides in cold worked and stress-relieved (CWSR) Zr–2.5Nb pressure tube material was determined in the temperature range of 523–673 K. Using tapered gage tensile specimen, mean value of σth was experimentally determined by two methods, half thickness method and area compensation method. The difference between local values of σth measured across the thickness of the tube and the mean σth values yielded the residual stress variation across the tube thickness. It was observed that both the mean threshold stress and residual stress decrease with increase in reorientation temperature. Also, the maximum value of residual stresses was observed near the midsection of the tube.
Hydrogen migration under thermal stress gradient in zirconium alloys results in formation of hydride blisters [1]. An array of blisters makes Zirconium alloy components of nuclear reactors susceptible to fracture [2]. The whole process of hydride blister formation and fracture of these components is very complex and involves hydrogen migration under thermal gradient, hydride precipitation, straining of the matrix, setting up of hydrostatic stress gradient, enhanced hydrogen migration under the combined influence of thermal and stress gradient, stress-reorientation of hydrides [3], cracking of hydrides, crack growth by delayed hydride cracking mechanism [4], interlinking of blisters and spontaneous fracture of the component. In this work we estimate the stress components in hydride blisters and the surrounding matrix for certain assumed blister depths. The estimated stress predicts the hydride orientation in the matrix surrounding the blisters and will be subsequently used to model the hydrogen diffusion under hydrostatic stress and temperature gradients.
Hydrogen in excess of solid solubility precipitates as hydride phase of plate shaped morphology in hcp α-Zr with the broad face of the hydride plate coinciding with certain crystallographic plane of α-Zr crystal called habit plane. The objective of the present investigation is to predict the habit plane of δ-hydride precipitating in α-Zr at 298 K using strain energy minimization technique. The δ-hydride phase is modeled to undergo isotropic elasto-plastic deformation. The α-Zr phase was modeled to undergo transverse isotropic elastic deformation but isotropic plastic deformation. Accommodation strain energy of δ-hydride forming in α-Zr crystal was computed using initial strain method as a function of hydride nuclei orientation. Hydride was modeled as disk with round edge. Contrary to several habit planes reported in literature for δ- hydrides precipitating in α-Zr crystal, the total accommodation energy minima at 298 K suggests only basal plane i.e. (0001) as the habit plane.
Zirconium alloy tubes act as miniature pressure vessels in Pressurized Heavy Water Reactors (PHWR) and are subjected to aqueous corrosion resulting in hydrogen pick up. Hydrogen exceeding solid solubility precipitates out as brittle hydride phase and may cause embrittlement of the host matrix. Two forms of embrittlement have been recognized for dilute zirconium alloy pressure tubes – gross and localized. Gross embrittlement is caused due to uniformly distributed precipitate, whereas localized embrittlement is caused by hydrogen migration leading to damage accumulation within a small region over a period of time before catastrophic failure of the component may occur. The parameters like fracture toughness, threshold stress for reorientation of hydrides, threshold hydrogen concentration for blister formation, delayed hydride cracking (DHC) velocity, threshold stress intensity factor for DHC initiation are used as fitness criteria for service assessment of the pressure tubes. In this paper, some of the results generated using Indian pressure tube material will be discussed.
Delayed hydride cracking (DHC) velocity was determined at 203, 227, 250 and 283 °C using 17 mm width curved compact toughness specimens machined from an unirradiated Zr–2.5 wt.% Nb pressure tube spool, gaseously charged with 60 ppm of hydrogen by weight. Single CT specimen was used to determine DHC velocity at a constant temperature for a range of stress intensity factor (KI) obtained by load drop method. For a given temperature and KI > 15 MPa m1/2, DHC velocity was found to be practically independent of KI. For 15 > KI > 10 MPa m1/2, DHC velocity decreased significantly with decrease in stress intensity factor and extrapolation of the data suggested the threshold stress intensity factor to be about 9–11 MPa m1/2 in the aforementioned temperature range. The activation energy associated with DHC was observed to be 35.1 kJ/mol.
The misfit strains of both the isotopic forms of zirconium hydrides, viz., δ-ZrHx and δ-ZrDx, precipitated in zirconium were computed in the temperature range of 298–773 K and for 1.5 ≤ x ≤ 1.7, using the methodology proposed by Carpenter. Misfit strains along (0 0 0 1) plane and normal to it were observed to increase with increase in temperature for both the isotopic forms of hydrides. For a given isotopic atomic fraction, the misfit strains for deuteride were smaller than the corresponding values for hydride and with increase in atomic fraction of hydrogen isotopes, the misfit strains increased.
To honour the memory of Knut Bertram Broberg, and to promote continuing work on the many ideas which he shared so generously with students and colleagues throughout his lifetime, a symposium was arranged at University College of Dublin in 2007. The idea was to create an informal atmosphere, and to keep the num- ber of participants fairly low. This has now become a bi-annual symposium with the highlight being one or two lectures by outstanding world-class scientists. The second Broberg Memorial Symposium was arranged by the Lund Institute of Technology in Sweden (LTH) in May 2009. The venue of the conference was a renais- sance castle, Trolleholm, that was put to our disposal by the courtesy of the Rector of LTH. Trolleholm was once the home of Sophia Brahe whose careful obser- vations of planetary orbits later was used by Johannes Kepler to develop his laws of planetary motion. Kepler was at the time working as an assistant to Sophia’s brother Tycho Brahe.