The processes of occurrence and propagation of
stresses and strains under loading of solid physical bodies take a certain
time. When testing building materials and elements of building structures, due
to the lack of technical ability to visually observe and describe a continuous
process, test methods have been adopted based on a discrete, phased application
of a test load with the necessary endurance after each stage of loading. Fixing
and recording the indicators of measuring equipment and instruments, as a rule,
are also carried out in stages.
If we proceed from the fact that the strain
propagation rates in solids are quite high, comparable with such a physical
property of solids as the speed of sound propagation in the material, then the
application rates of external test loads are relatively low, comparable to or
exceed thousandths of the strain propagation velocity. The time duration of the
application of the load is much greater than the time of occurrence,
propagation and attenuation of the resulting deformations in the physical body.
In turn, the deformations propagate at a very high, but with a finite speed, in
the volume of the test sample, the emergence and stabilization of a new
stress-strain state does not occur instantly.
The basis of the research hypothesis is the
assumption that in wooden structures, deformations and stresses propagate in
waves. The emergence, propagation and attenuation of strain waves,
respectively, and stresses, occur as physical processes of propagation of
longitudinal strain waves in the volume of an array of wooden structures, and
propagation of transverse strain waves on the surface of structures.
The authors, at the North-Eastern Federal
University named after M.K. Ammosov (Yakutsk), theoretical models of the
propagation of strain and stress waves in wood, solid wood and glued wood
elements of building structures were developed and implemented in several
numerical experiments.
As part of the specification of the main hypothesis
of wave propagation of strains and stresses, models of the propagation of
single transverse strain waves on the surface of bodies with boundary
conditions in the form of loading areas and a section of a sample were analyzed
when solving a plane problem [1–3]. Considered models of wave-like deformation
are the basis for the visual representation of the analyzed physical processes.
The problem of numerical modeling of an all-wood
sample with idealized anisotropy of elastic properties as a medium for the propagation
of elastic waves of deformations and stresses is solved. Calculations were
performed by the finite element method for spatial approximation and by the
finite difference method for discrediting in time. The defining conditions of
the problem were taken to be the physical properties of wood, such as elastic
characteristics, the velocity of propagation of the strain wave, and geometric
indicators, such as the length of the sample (finite or infinite), transverse dimensions
specified relative to the median plane. The elastic parameters of wood were
taken according to E.K. Ashkenazi [4], the orthotropic structure of the model
sample was specified in a cylindrical coordinate system.
For calculations in the
Cartesian coordinate system, the elastic parameters of wood were transformed
based on the corresponding transition matrix. The numerical experiment
demonstrated a correct qualitative visual picture of the wave propagation of
deformations, with wave manifestations and characteristic effects on the surface
of the sample, at axial and corner points. Visually, the numerical model showed
Rayleigh waves on the surface layer of the sample, depending on the ratio of
the external geometric dimensions of the sample model, with pronounced wave
interference on the outer shell. The visual manifestation of deformation on the
outer sides (faces) and the reflection of deformation waves from the outer
boundaries of the elastic medium of the sample in the form of Rayleigh waves
confirm the correctness of the general hypothesis and the implemented model.
Visualization of the process of emergence, propagation and attenuation of
deformation waves on the surface of the sample shows that in the quantitative
description of the deformation gradient, areas dangerous for the material can
be identified [5]. In subsequent tasks of the hypothesis, it is necessary to
establish the correct relationship between the visual qualitative picture of
deformation and the quantitative assessment of stresses in the considered wave
processes, based on the use of real attenuation coefficients of wave processes
obtained in full-scale vibration experiments.
In the development of this problem, a model of the
process of resistance of a wooden cross-glued panel (the so-called CLT-panel,
Cross-Laminated Timber) was calculated [6-8]. A breakdown into solid tetrahedra
is adopted, with the corresponding number of common points of their vertices,
according to the accepted breakdown.
The model provides for the application of a load
distributed along the panel, which excites a single wave with one pulse. The
load is distributed in the form of a triangle, the load value increases from
left to right, the maximum value is at the right edge of the panel. The load
simulates a transient load applied vertically from the bottom up. The panel
support along the upper face is continuous. The described computational
numerical model is implemented in a certain range of physical and mechanical
characteristics and geometric dimensions of wooden CLT panels (Fig. 1).
Fig. 1. Visualization of deformation
of the sample model at t = 0,50 milliseconds ( ms ).
The following numerical experiment also describes
the resistance process of the CLT-panel when a shear force is applied. Panel
dimensions are accepted: length 2000 mm, width (height) 1000 mm, thickness 175
mm. The panel consists of 7 layers of pine wood boards, 25 mm thick, glued
crosswise. The volume of the panel is divided into 293360 vertices and 1516873
elements - individual tetrahedra with common adjacent faces.
The short-term load (sudden) is applied to a
platform 175 mm wide, in the corner of the panel, and is directed parallel to
the long side of the panel. The load increases linearly for 1 ms, after which
it is released. The support is organized in a similar way, that is, the panel
rests diagonally to the site of application of the external load. The support
also has a width of 175 mm. The design scheme is shown in Fig. 2.
Fig. 2. Loading the CLT-panel with a shear force.
The shear force, within the considered time period
from zero to 200 milliseconds, causes a process of continuous deformation,
which is visualized with minimal time steps specified by the experimental
conditions. The time steps are set, as a rule, tolerably small to optimize the
total time for solving the problem. For visual comparison and analysis of an
array of images, the researcher has the ability to select visual images based
on the need to obtain a sufficient visual difference between images of the
position in space of the locus of tangential strain tensors.
The calculations were carried out using finite
element methods for spatial approximation and finite difference methods for
time discretization. The numerical implementation was performed on a freely
distributed open source computing platform FeniCS [9-11].
FEniCS is a free and open-source computing
platform in the C++ and Python programming languages for the numerical solution
of partial differential equations using the finite element method. The problem
statement for FEniCS is initialized personally by the user in the form of a
variational formulation of partial differential equations describing the
phenomenon, while the construction of a finite element basis and translation
into systems of linear algebraic equations (SLAE) is carried out automatically.
To solve the resulting SLAE, FEniCS has a fairly large selection of
preconditioners and solvers.
When working in C++, a convenient FFC compiler (FEniCS
form compiler), which creates header files for integration into the main program
from a variational formulation from ufl code. The convenience lies in writing
the equation in a form similar to the analytical one. When working in Python,
equations are initialized in the main code. The main program for the numerical
implementation of finite elements in FEniCS in the C++ programming language
consists of the following parts: initialization of the finished mesh, setting
function spaces, boundary conditions, linear and bilinear forms, solving the
SLAE and saving the results.
The movement of an elastic wave
along a wooden sample is described by a non-stationary elasticity equation,
which consists in the condition of conservation of linear momentum:
|
|
(1)
|
where
is
the displacement field vector,
is
the acceleration,
is
the density of the material,
is
the given body force, and
is
the stress tensor, which is related to the displacement by means of the
generalized Hooke's law for an anisotropic body:
where
is
the linearized strain tensor,
is the
fourth-rank elasticity tensor, which in our timber approximation can be
described in terms of the Voigt notation for an orthotropic body in the form:
where the coefficients are given as
Due to the symmetry of the problem (the wooden sample is
symmetrical, with idealized anisotropy), the solution was carried out on a
quarter of the sample. Equation (1) is supplemented with boundary conditions
responsible for the absence of transverse deformations on the symmetry planes.
The initial deformed state is to assume the initial conditional deformation in
Δx
at time
t =
0.
To obtain a variational formulation, we use
integration by parts using the test function
,
where
is the
space of test functions that satisfies the boundary conditions for
displacements:
.
The previous equation can be rewritten as:
.
where
is the
symmetric bilinear form associated with the mass matrix, and
is the form
associated with the stiffness matrix.
We also add dissipation to the variational form,
as a term proportional to
:
.
The dissipative term can be considered bilinear and symmetric.
Post-processing was done with ParaView [12, 13].
Based on the vtk library [14, 15], ParaView is a software with a clear
graphical interface and a powerful toolkit for building the necessary graphs,
animated video files and presenting the distribution of the solution on the
computational domain.
Paraview is an open source multi-platform data
analysis and visualization application. Paraview uses the VTK (Visualization
Toolkit) as a tool for rendering images and for data processing. Paraview's
user interface is written using the Qt environment [16].
Qt is a framework for creating programs based on
the C++ programming language, Python. Among the advantages of Qt are convenient
libraries and an application programming interface.
Visualization Toolkit (VTK) is a free and open
source software system for creating 3D computer graphics, modeling, image
processing, volumetric rendering, 2D plotting and most importantly, scientific
visualization. Based on VTK, you can create your own visualizer of calculation
results with the ability to process images using various useful filters. In our
case, filters were used to visualize isosurfaces.
When developing a problem hypothesis, the
conditions are:
-
the magnitude and nature of the applied external load;
-
the duration of the application of the load (impulse), from the
minimum possible to the maximum allowable, determined by the speed of wave
propagation and the size of the sample under consideration (time, and period -
wavelength);
-
for a numerical experiment, if it is necessary to compare with a
full-scale experiment, restrictions may be dictated by the technical
capabilities of the test bench, power or measuring equipment, climatic or other
conditions;
-
certain restrictions can be introduced by software,
characteristics of computers or computer programs, computer time resource (for
example, breaking down the simulated structures into too small finite elements
dramatically increases their number and, in turn, increases the time for
solving problems).
The color scale for visual assessment of stress
strains is adopted in the range from blue to red, which corresponds to values
from 0.3 to 3 MPa for stresses, and 0 to 5 mm for strains (Fig. 3).
Next, the task was formulated to represent a solid
continuous body simulating the structure of a wooden glued CLT-panel, which is
also loaded according to the design scheme in Fig. 2.
Visualization of the results of a numerical
experiment made it possible to see a detailed picture of the stress
distribution according to a given scale (Fig. 4-6). As can be seen from the
figures below, part of the physical volume of the model is not identified
(there are "voids" and other discontinuities of the body). This
circumstance is explained by the fact that in places of "emptiness"
stresses have an intermediate value between the levels of a given breakdown of
the numerical value of the scale. In turn, high stresses exceeding the maximum
numerical value of the scale specified by the boundary condition (red color)
turned out to be less than the actual maximum stresses according to the
numerical experiment, and “emptiness” is also represented at these points of
the model volume. In a numerical experiment, researchers have the opportunity
to observe the emergence and propagation of stress waves, reflection from the
edge of a given space, complex processes of interaction of reflected stress
waves as a single continuous physical process. A qualitative and quantitative
continuous wave pattern of stresses shows areas of the model volume where
stress and strain waves arise and move. Stress gradients are visualized;
visualization of strain gradients is possible.
In general, such an approach to a numerical
experiment makes it possible, by varying the ratio of the initial and final
values of the specified stress scale in a wide range, to obtain various visual
and quantitative representations of wave processes in structural models. It is
possible to visualize and quantify stress waves in almost any narrow or wide
range of values. It is possible to divide the stress values expected in a
numerical experiment into arbitrarily narrow intervals and separately track these
processes in numerical models.
Fig. 3. CLT - panel. Axonometric image
of surfaces with the same values of the strain vector, at t=2 milliseconds. The
color scale of deformations visualizes gradual distribution of deformations in
the volume of the sample.
Fig. 4. CLT panel. Visualization of stresses at
the beginning of shear, t=1,2 ms.
Fig. 5. CLT panel. Visualization of
shear stresses. The formation of the reflected wave, the beginning of the
backward movement of the reflected wave. t=2 ms.
Fig. 6. CLT panel. Visualization of
shear stresses. t=5 ms .
Fig. 7. CLT panel. Visualization of
shear stresses. Attenuation of stress waves, t = 20 ms.
The article gives a consistent description of the modeling of the
elastic properties of the material - solid and glued wood. As a result of
numerical experiments, the main research hypothesis was confirmed - that in
wooden structures, deformations and stresses propagate in waves. The emergence,
propagation and attenuation of strain waves, respectively, and stresses, occur
as physical processes of propagation of longitudinal strain waves in the volume
of an array of wooden structures, and propagation of transverse strain waves on
the surface of structures. For the first time, wave processes of propagation of
strains and stresses have been identified and visualized, and qualitative
patterns of changes in the volume of the studied wooden surface samples with
the same values of strain vectors and stresses are especially indicative.
We noted that wave manifestations of deformations, as a result of
the resistance of the sample material, after the application of an external
test load, in the future, it is possible to fix using appropriate equipment
[5-8]. High-speed special video recording, combined with the visualization of
the numerical description of wave deformations on the surface of the samples,
will allow a deeper understanding of the physical picture of the deformation of
a complex system, such as glued wood. Naturally, the most characteristic
picture of the visualization of these wave manifestations throughout the
entirety and spectrum of wave characteristics will be presented on the surfaces
of large-sized planar structures, such as CLT panels.
The obtained visualization results are the correct physical basis
for further improvement of the research hypothesis in order to obtain correct
quantitative results, with the refinement of the initial data and boundary
conditions by natural experiments.
The work is supported by Russian Science Foundation (no. 23-41-00037).
1.
Chakhov D.K., Romanov
P.G., Syrovatsky V.S., Samsonov A.M. Structural strength of coniferous wood
growing in Yakutia (n.-t. report) RSTP "Problems of the construction
complex of Yakutia". Block 1. Section 3.15. 4. Propagation of transverse
waves of plastic deformations on the surface of wood. Rationale for the
development of a theoretical model. Yakutsk. 1994. 69-85 p.
2.
Chakhov D.K., Romanov
P.G., Syrovatsky V.S., Samsonov A.M. Structural strength of coniferous wood
growing in Yakutia (n.-t. report) RSTP "Problems of the construction
complex of Yakutia". Block 1. Section 3.4-55. 3. Propagation of transverse
waves of plastic deformations on the surface of wooden samples. Yakutsk. 1995.
54-69 p.
3.
Romanov P.G. To the
formulation of the problem of wave manifestations of resistance of large-sized
wooden unidirectionally and cross-glued panel structures. III All-Russian
scientific and practical conference "Modern problems of construction and
life support: safety, quality, energy and resource saving". Yakutsk,
North-Eastern Federal University named after M.K. Ammosova, March 3-4, 2014,
pp. 171-174.
4.
Ashkenazi E.K. Anisotropy
of wood and wood materials. – M.: Timber industry, 1978. – 223 p.
5.
Romanov P.G., Sivtsev P.V.
Numerical simulation of wave manifestations of the resistance of a wooden
sample with an idealized anisotropy of elastic parameters. Construction
Mechanics And calculation structures. 2020, No. 1, p. 37-43.
6.
Romanov PG, Sivtsev PV
Features of the formation of cross-glued wooden panel structures for northern
construction. 5th International Conference "Arctic: History and
Modernity".18-19 March 2020, Saint-Petersburg, Russia. IOP Publishing. IOP
Conf. Series: Earth and Environmental Science 625 (2021) 012018. doi:10.1088/1755-1315/625/1/012018.
7.
Romanov PG, Sivtsev PV
Features of modeling of stress and strain waves in an anisotropic medium on the
example of a wooden element. 6th International Conference "Arctic: History
and Modernity".14-15 April 2021, Saint-Petersburg, Russia. IOP Publishing.
IOP Conf. Series: Earth and Environmental Science 625 (2021). 032089 doi :
10.1088/1742-6596/2131/3/032089.
8.
Romanov P.G. Sivtsev P.V.
Numerical experiment on shear tests of a cross-glued wooden element. Structural
mechanics and calculation of structures. 2023, No. 1. With. 56-64. doi:
10.37538/0039-2383.2023.1.56.64.
9.
A. Logg, K.-A. Mardal, G.
N. Wells et al. Automated Solution of Differential Equations by the Finite
Element Method, Springer (2012).
10.
M. S. Alnaes, J. Blechta,
J. Hake, A. Johansson, B. Kehlet, A. Logg, C. Richardson, J. Ring, M. E. Rognes
and G. N. Wells. The FEniCS Project Version 1.5, Archive of Numerical Software
3 (2015).
11.
Software package
FEniCS. ––
https://fenicsproject.org/.
12.
James Ahrens, Berk Geveci,
and Charles Law. “ParaView: An End-User Tool for Large Data Visualization.” In
Visualization Handbook. Elsevier, 2005. ISBN 978-0123875822.
13.
Software package
ParaView. –– https://www.paraview.org/.
14.
Schroeder W., Martin K.,
Lorensen B. The visualization toolkit, 4th edn. Kitware, 2006.
15.
Software VTK. ––
https://vtk.org/.
16.
Software Qt.
–– https://www.qt.io/.
