A theoretical investigation into the effect of growth stress variation in tree stems on population measurments
Growth stress intro
A substantial problem which is not well studied or understood with regard to growth stress is the characterisation of the stress field existing within the stem. There is currently no known technology which has the ability to directly or indirectly measure either the surface or volume stress field with a degree of accuracy which would provide insight into the scale of local inhomogeneity. It is suspected, once a reliable technology is developed to investigate the field our understanding and way of thinking about growth stress from both a theoretical and applied view will change significantly. -- add comments from intro about boyeds early work etc where they cut the tree open, jacobs too.
Currently rudimentary testing technologies such as strain gauges are limited to measuring surface strains with an unknown level of accuracy. There are no current testing procedures which are non-destructive, and hence repeated testing on a unique (all wooden samples are unique) samples is imposable. Most techniques us multiple measurements of surface strain around the stem which are then averaged (--refs--) to provide a single quantification of 'growth strain' however, the accuracy of any one of these given testing procedures can not be tested as measurement error and variation on the stem surface are completely confounded. The same problem exists for the splitting test --ref-- the only testing procedure fast enough to be used for tree breeding.
A more fundamental problem also exists; the idea that growth strain is quantifiable as a mean surface strain, whether obtained through multiple surface tests or through some geometric averaging as is implicit in the splitting tests. This assertion is particularly problematic for wood scientists who are interested in identifying pieces of timber which are unlikely to bend during sawing whether that be developing in-line screening technology for mills or to assist breeders identifying favorable genetics in breeding programs. --- comment on jacobs images with respect to intenal varaition --
In order to develop a better understanding of how growth strain variation around the surface of stems effects testing results both at an individual and population wide level a mathematical model was developed where surface strain patterns were veried.
Simulating an individual sample
In order to investigate the roll differing surface stress profiles play on the reliability of both the rapid splitting test procedure and 'point' based procedures such as using strain gauges or CIRAD an orthotropic elastic mathematical model of a typical very early selection stem sample was developed. This generic sample was assumed to be a truncated cone with a length of 400 mm, a small end diameter of 34.8 mm and a big end diameter of 39.55 mm. The material of the sample was amused to be orthotropic with longitudinal stiffness coming from Eucalyptus argo---see previous chapter?--- --- and the remainder being taken from, or derived as a ratio from the EA longitudinal value, I think, Goncalves 2014----. --ref--- solmone --- was used to create a mesh of xxxx nodes, xxx verticies and xxxx elements to approximate the sample, and a slit from the big end, though the pith with a width of 0.9 mm and a length of 300 mm was added to simulate the splitting test, as can be seen in Figure xxx. Further the slit was rotated 90 degrees about the pith and a second mesh created to provide multiple splitting test measurements from each modeled sample (Figure xxx).
Material properties derived from experiments such as in ----Goncalves 2014---- exist in their native radial coordinate system and hence to be used in a Cartesian coordinate system as was required for some functions of modeling, a transformation between the two was needed. Voigt (engineering) notation was used to convert the stiffness matrix (Equation xxx) from radial to Cartesian coordinates at any point in the domain, Equations xxx to xxx follow logically to provide this transformation.
--- describe what the equations are doing---
In the sense of the stiffness matrix, it was assumed that no taper (ie that longitudinal stiffness exists parallel to the vertical axis regardless of the coordinate system), no spiral grain, knots, grain wobble etc. exist and that there is no change in material properties within the volume (i.e. the pith has the same 9 material constants that the periphery has). Further it was assumed that no external forces such as gravity were activating significantly on the simulated samples, the only forcing was the internal stress field.
Traditionally the growth stress field is assumed to be axis-symmetric and follow a curve similar to that presented by --- gillis and hsu 1979 --- as can be seen in Figure ---- Here the stress field existing in a longitudinally ordinated plane from the pith to the periphery can be described by in the same way by Equations xxxx and xxx. However at every point the value of the surface strain changes, ie the stress field is not axis-symmetric, and is instead governed by Equations xxx to xxx. --- describe how these equations work ---- Further while the peaks and troughs of the surface strain are 90 degrees apart, their orientation with the splitting test is random and only by chance will peaks/troughs intersect with a cut. This was done as in real world experiments on straight stems it is not known where high or low surface stress is located and hence it can be reasonably assumed that the cut orientation will be randomly aligned with the surface stress pattern. Figure xxx shows some examples of surface strain values around the circumference of some theoretical samples.
The individual sample simulation is solved --- see page 62 of masters ---
From the resulting deformed coordinate positions, the average displacement of the two halves at the inner edge on the big end of the cut can be calculated (the digital equivalent of the opening measurement in the experimental version of the rapid splitting test --ref previous chapter ---). A cut perpendicular (the second mesh, Figure xxx) to the first was made, with the same stress field remained identical to that in the first instance. The two openings provide theoretical results of multiple testing of the same individual, without one test influencing another.
A simulated population here refers to a set of 1000 simulated individual samples which have a (simulated) rapid splitting test mean of xxx+-xxx and a standard deviation of \(630\pm5\ \mu\epsilon\).
In order for the the theoretical sample described in Section xxx to be created (which is needed to provide the individuals of the populations), the four input values xxx stress_x_xy xxx in Equations xxx to xxx need to be defined. For each sample they are calculated from a multivariate normal distribution (Equation xxx). The generation of the normal distribution takes the mean matrix (Equation xxx) which is constant for all samples regardless of their population, and the Covarience matrix (Equation xxx) which is made up of a population specific input variance xxx and two correlations, CAdjoining and COpposing, describe how related each of the four evenly spaced stress points on the circumference of the sample are at the populations level (see Figure xxx for a visual representation). The input variance xxxx is manipulated to give the output population a standard deviation of \(630\pm5\ \mu\epsilon\) (Note that the output population means are all between xxx and xxx).
By systematically varying CAdjoining and COpposing along with xx input var xx (see Equations xxx to xxx for values) populations with the same descriptive statistics (mean and standard deviation) are produced but consist of very different individuals. Note that some populations are not producible statistically, for example CAdjoining and COpposing can not both be equal to negative one as no such statistical distribution can exist. For each individual the surface stress profile, true mean stress, and two rapid splitting test values are now known, allowing for comparisons of how well different tests predict each other and the true mean value. Further more, some populations can be removed as unlikely, and from previous experimental work some can be removed not fitting previous experimental evidence.
As the results produced from the model above
results - note these assume no measurment error as there is neglagable error in the computer