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EN
In this research, a mathematical model is derived to enable analytical determination of effective ultimate forces in the process of plasticization of the surface layer of wood. The experimentally determined thermo-mechanical properties of the material subjected to the process of plasticization are used in defining the structure of the model. The analysis of plastic strain in the layer in consideration is based on a generalised model of an ideally rigid-plastic medium, including certain modifications. Considering the anisotropic properties of wood, the Azzi-Tsai-Hill (ATH) strength criterion is applied which takes into account variation in the response of the loaded material depending on the direction. The article presents also results of FEM analysis of the same process of hot rolling of wood.
EN
The paper concerns the formulation and analysis of an adhesive joint model, aimed at reinforcing or reconstructing weakened wooden elements. The joint is modeled as a plane stress problem of the theory of elasticity. It is assumed that wood is an orthotropic material. The reinforcement of an element is achieved by means of attaching a covering plate, while reconstruction is carried out by introducting an insert into the weakened (deteriorated) zone of an element. The influence of varying thickness of plates and inserts on the stress states in the adherends and adhesive is analyzed. The analyses are related to axially loaded elements.
EN
The purpose of this paper is to formulate an approximate numerical model for adhesive scarf joints in wooden beams under the assumption that the joint is deformable. This approximate model is founded on observations resulting from the general solution of the plane linear theory of elasticity. It is assumed that wood is orthotropic. The joint can be subjected to a complex load combination including an axial force, a shear force and a bending moment. Within this framework, analytical relations for stresses in the adhesive, as well as stresses and displacements of the adherends, were derived in a form useful for engineering practice. A numerical example proves the high accuracy of the approximate model when compared to the general model based on the plane linear theory of elasticity.
EN
The present paper is a continuation of the general formulation discussed in the paper: Mechanics of adhesive joints as a plane problem of the theory of elasticity. Part I: general formulation, Archives of Civil and Mechanical Engineering 10 (2) (2010). Adhesive joints between adherends of varying thickness, made of various orthotropic materials are considered. The adhesive surface can be curved and shapes of the adherends are arbitrary. The joints can be loaded with shear stress distributed arbitrarily on the adherend surfaces as well as by normal and shear stresses of any distribution along the adherend edges. The problem is formulated in terms of displacements in the form of four partial differential equations of the second order. The boundary conditions allow for existence of sharp edges of adherends. Notions of the obtuse sharp edge and the tangent sharp edge are introduced. The presented examples include complete solutions within the framework of the plane theory of elasticity for adhesive joints with adherends of constant thickness and adherends of varying thickness and curved adhesive surface. The influence of sharp edges on the reduction of stress concentration in the adhesive and the adherends is illustrated. The assumptions, notation and equations presented in the general formulation of Part I are used.
EN
The subject of this paper is the general formulation of a model for scarf adhesive joints in timber beams within the framework of plane linear elasticity. It is assumed that wood is orthotropic. The joint can be subjected to a complex loading state including an axial force, a bending moment and a shear force. The joint model is given in displacements by means of a set of four partial differential equations of the second order. Boundary conditions cater for sharp edges in the adherends. Complete solutions to theory of elasticity equations are presented and discussed. The manner in which the joint transmits the axial force, the bending moment and the shear force is presented. It is shown that the scarf joint does not feature stress concentrations and that there exists an approximate equivalence of displacements and stress states in scarf jointed and continuous elements.
6
Content available remote Mechanical properties of cervical dura mater
EN
The aim of the study was to determine experimentally the stress as strain function as well as the orthotropy and heterogeneity of porcine dura mater of the cervical spinal cord. Material was divided into groups based on the place of collection, considering the dorsal side and ventral side, specifying the number of cervical vertebra, and the direction of tension of the sample - longitudinal or circumferential. Experimental studies were conducted with the MTS Synergie 100 testing machine. The tensile test was performed for each sample at a speed of 2 mm/min until the sample's break. There were determined the characteristics of stress as a function of strain in particular samples. Distribution maps of the stress and strain values at the characteristic points were then drawn (the beginning and the end of the linear range of the stress-strain characteristic and the point corresponding to the complete sample damage) for each set of samples, taking account of their collection place and direction of tension. The results confirmed the orthotropy of mechanical properties of dura mater. Stress and strain differed also in the value at the height of each vertebra and exhibited diversification on the ventral side compared to dorsal one.
EN
A class of constitutive models of orthotropic non-homogeneous masonry materials in the framework of elasto-plasticity theory of small displacements with energetic hardening/softening is proposed. Analysed class of constitutive models reduces to the classical one for ideal elasto-plasticity with well-known Hoffman yield condition for orthotropic homogeneous materials. Proposed class of models is implemented in FEM system ABAQUS. Constitutive relationships are implemented in FORTRAN in user procedure UMAT. The numerical tests are proposed to check correctness of the implementation, and some boundary value problems are also solved.
PL
W artykule przedstawiono sformułowanie teorii sprężysto plastyczności niejednorodnych ortotropowych materiałów murowych z tzw. Energetycznym wzmocnieniem/osłabieniem. Analizowana klasa modeli redukuje się do znanych modeli idealnej sprężysto-plastyczności w warunkiem plastyczności Hoffmana. Zaproponowaną klasę modeli zaimplementowano w programie MES ABAQUS. Relacje konstytutywne zaprogramowano w języku FORTRAN w procedurze użytkownika UMAT. Przydatność zaimplementowanego modelu zaprezentowano na przykładzie zadania brzegowego ścinania ściany z otworem.
PL
W oparciu o informacje dostępne w literaturze, a także o wyniki badań własnych, przeanalizowano cechy sprężyste papierów opakowaniowych. Między innymi określono wpływ zmian zachodzących w strukturze włóknistej, poddanej działaniu naprężeń, na wartość modułu Younga, a także porównano rozkład modułu w płaszczyźnie papieru z rozkładem charakterystycznym dla ciał ortotropowych. Uzyskane wyniki wykazały dużą zgodność zachowań papierów opakowaniowych z zachowaniem ortotropowych materiałów sprężystych
EN
On the basis of available literature and results of author’s tests, the elastic properties of packaging paper were examined. Among other things, the author determined the impact of changes appearing in the fibrous structure under influence of the stress on the value of Young’s modulus as well as he compared modulus distribution in the paper plane with the distribution characteristic for orthotropic bodies. Obtained results demonstrated high similarity of packaging paper behaviour to orthotropic elastic materials.
9
Content available remote Experimental Determination of Anisotropy Coefficients for a Wood
EN
A special form of the yield condition of porous anisotropic materials, such as a wood containing the wood humidity, is a function of the modulus of elasticity, the modulus of humidity and anisotropy coefficients. It is necessary to determine anisotropy coefficients to develop this equation for engineers. A measuring laboratory and a methodology of experimental determination of anisotropy coefficients for a wood are given in the paper.
10
Content available remote Determination of the elasticity range of paper
EN
This research work presents a method for the determination of the limit of elasticity of paper considered as a variable dependent on stress and the time of its duration. For plane stress it was found that in the case of a low share of permanent strain in the total strain, we can assume that the highest value of its share appears on one of the principal axes of stress. Such an assumption allowed to use a one-dimensional rheological model to define the permanent range in a two dimensional state of stresses for paper.
PL
W ramach pracy zaproponowano sposób wyznaczania granicy sprężystości papieru, traktowanej jako zmienna zależna od wartości naprężeń i czasu ich działania. Dla płaskich stanów naprężeń, występujących w płaszczyźnie papieru wykazano, że w przypadku małego udziału odkształceń trwałych w całkowitych można przyjąć, że największa wartość udziału odkształceń trwałych w całkowitych występuje dla jednego z kierunków naprężeń głównych. Takie założenie, pozwoliło na wykorzystanie jednowymiarowego modelu reologicznego do określenia zakresu sprężystości w dwukierunkowym stanie naprężeń papieru.
EN
The study concerns the linear elastic and viscoelastic constitutive modelling of homogeneous orthotropic solid bodies. The considerations are based on well-known coupled standard/inverse constitutive equations of elasticity. The author has derived new uncoupled standard/inverse constitutive equations of elasticity, new uncoupled standard/inverse constitutive equations of viscoelasticity, and new coupled standard/inverse constitutive equations of viscoelasticity of orthotropic materials. A homogeneous orthotropic material is deseribed by 9 elastic and 18 viscoelastic constants, clearly interpreted physically. Simpler materials, i.e. monotropic and isotropic solid bodies, are also considered. In addition, the separation of shear and bulk strains in the uncoupled constitutive equations of elasticity has been examined numerically for exemplary materials.
PL
Praca dotyczy modelowania konstytutywnego jednorodnych ortotropowych ciał stałych w zakresie linowym, sprężystym i lepkosprężystym. Podstawą rozważań są znane sprzężone standardowe/odwrotne równania konstytutywne liniowej sprężystości tych materiałów. Wyznaczono niesprzężone standardowe/odwrotne równania konstytutywne liniowej sprężystości, sformułowano niesprzężone standardowe/odwrotne równania konstytutywne liniowej lepkosprężystości, a następnie wyznaczono sprzężone standardowe/odwrotne równania konstytutywne liniowej lepkosprężystości ortotropowych ciał stałych. Jednorodny materiał ortotropowy opisano za pomocą 9 stałych sprężystości i 18 stałych lepkosprężystości z podaniem przejrzystej interpretacji fizycznej tych stałych. Rozważono również przypadki szczególne materiału monotropowego i izotropowego. Dodatkowo, przetestowano numerycznie rozdzielenie odkształceń postaciowych i objętościowych w przypadku niesprzężonych równań konstytutywnych sprężystości materiałów ortotropowych i monotropowych.
PL
Liczne doświadczenia pokazują, że propagacja szczeliny w jednorodnym, izotropowym materiale, w warunkach złożonego stanu obciążenia, następuje często w innej płaszczyźnie niż pierwotna płaszczyzna szczeliny. Inaczej jest w przypadku materiału ortotropowego, dla którego kryterium pękania powinno uwzględniać fakt, że w materiale istnieją uprzywilejowane kierunki propagacji szczeliny (kierunki wzmocnienia), wzdłuż których najczęściej następuje pękanie. W związku z tym, odporność na pękanie materiału ortotropowego silnie zależy od układu propagacji szczeliny. Ortotropię drewna charakteryzują trzy osie symetrii, oznaczone odpowiednio literami: L (kierunek wzmocnienia, wzdłuż osi pnia drzewa), R (prostopadle do kierunku tzw. słojów, widocznych w przekroju pnia) i T (stycznie do kierunku tzw. słojów).
EN
Numerous fracture tests shows, that a crack propagation in a homogeneous, isotropic body under complex loading condition takes place in a direction, that don’t coincide with a crack direction. Contrary to that, it is known, that a crack oriented along the reinforcement direction in a orthotropic body propagates self–similarly, i.e. it don’t leave its original direction, even though the mode II load component may be significant. The fracture toughness of orthotropic material is highly dependent on both the crack propagation direction and the crack plane orientation. In other words, due to the large difference in fracture toughness between systems of principal material axes, cracks are often found in some of them, whereas they are extremely rare in other systems. The described above fracture phenomenon is well known in fracture mechanics of orthotropic materials such as wood or unidirectional reinforced fibrous composites. At a macroscopic level, wood is treated as a cylindrically orthotropic material with three main directions of the symmetry, namely longitudinal L, along the tree trunk, radial R, perpendicular to the year rings and tangential T, parallel to the year rings. The objective of the present work is to apply the non-local stress fracture criterion to mixed mode fracture in wood in the RL system for arbitrary oriented cracks. On the basis of a failure model taking into account the occurrence of microcracks and a assumption of the existence of the critical plane, the non-local stress fracture criterion of wood was derived for the RL extension system (the first letter alludes to the crack plane normal, the second one indicates the direction of crack propagation). In the most general terms, when the crack is notched with a inclination to the main orthotropy axis, the non–local stress fracture criterion in the system of critical stress intensity factors KI, KII is a rotated ellipse that is located in the center of the KI, KII system. Only when the crack is oriented along the main orthotropy axis, the non-local stress fracture criterion in the KI, KII system is a plain ellipse, whose semimajor and semiminor axes coincide with axes of the KI, KII system. In order to evaluate the derived non-local fracture criterion, a experimental investigation of the mixed mode fracture toughness of pine wood was made. Experiments have been carried out using specimens with single-edge crack. The mixed mode conditions were controlled by varying of the quotient between the stress intensity factors associated with mode I and mode II. This quotient depended both on the inclination of single-edge crack respect to main orthotropy axis and on the tensile and shear components of the loading force applied by machine. The mode I and II stress intensity factors used to characterize singular stress field near the crack tip have been evaluated using the finite element method and singular elements. Experimental data in terms of critical applied force were recalculated with respect to critical stress intensity factors associated with mode I and mode II and next they were compared with results obtained with the derived non-local fracture criterion. In this way, a high efficiency of the proposed fracture criterion of wood was demonstrated and proved.
EN
Softwood is a typical orthotropic and porous material. In a dry state the unit volume of wood consists of the so-called wood skeleton and of the air-filled pores. The variable humidity produces the internal forces within the loaded material bulk loading to destruction. In the paper the mathematical model of the transition of the softwood into the plastic state is formulated and discussed. The outset of yielding is examined for anisotropy with various ratios of humidity. The case of particular deformation is used as an example for which the material constants are identified and the limit load determined.
PL
W pracy przedstawiono matematyczny model uplastycznienia ortotropowego materiału • porowatego (drewna sosny) w warunkach zmiennej wilgotności. Wskazano na możliwość określenia nośności i wilgotności granicznej obciążonych elementów.
EN
The localization of failure initiation in a simple tension test applied to the infinite orthotropic disc with a circular hole is considered. The considerations are based on the Tsai and Wu model. According to the author's supposition, the angular coordinate of the failure at the hole boundary should depend on the form of the polynomial failure criterion. The computations based on accessible handbook data for the elastic and strength properties of chosen typical composite material have not confirmed the above supposition, giving evidence of surprising insensitivity of to changes in both elastic and strength characteristics of the material.
PL
Rozważania nad zniszczeniem kompozytów oparto na kryterium Tsai i Wu. Zbadano przydatność testu prostego rozciągania dla nieskończonej tarczy z otworem kołowym. Miejsce zniszczenia na obwodzie tarczy wyznacza kąt φ[0], który mógłby zależeć od postaci funkcji wytężenia F. Analizę przeprowadzono na podstawie danych literaturowych dla wybranego typowego kompozytu zbrojonego jedną rodziną włókien. Otrzymane wyniki pozwalają wnioskować, że kąt φ[0] zależy od właściwości badanego kompozytu dla pewnej klasy materiałów.
EN
An analysis of local buckling of thin-walled beam-columns, taking account global precritical bending within the first order approximation, is presented in the paper. The problem of interactive buckling of the structure is solved by means of Byskov and Hutchinson's (1977) or Koiter's (1976) approximation theory. Beam-columns made from orthotropic plates with the main directions of orthotropy parallel to the wall edges characterised by a widthwise varying orthotropy coefficient ni=Eyi/Exi are investigated. Beam-columns with open sections (i.e. channel sections), simply supported on the loaded edges, are analysed. The girders are subjected to loads which cause a uniform or linearly variable shortening of the edges.
PL
W pracy analizowano wyboczenie lokalne cienkościennych belek-słupów z uwzględnieniem globalnego dokrytycznego zginania w ramach przybliżenia pierwszego rzędu. Wyboczenie interakcyjne konstrukcji rozwiązano stosując aproksymacyjną teorię Byskova i Hutchinsona (1977) lub Koitera (1976). Badano belki-słupy zbudowane z płyt ortotropowych o głównych kierunkach ortotropii równoległych do krawędzi ścian charakteryzujących się zmiennym wzdłuż szerokości współczynniku ortotropii ni = Eyi / Exi. Analizowano belki-słupy oprzekro jach otwartych (ceowym i ceowym ze wzmocnieniem), przegubowo podparte na obciążonych brzegach. Dźwigary poddano obciążeniom ppowodującym równomierne i liniowo zmienne zbliżenie brzegów.
16
Content available remote Weight function for a crack in an orthotropic medium under normal impact loading
EN
The paper deals with the investigation of an elastodynamic response of an infinite orthotropic medium containing a central crack under normal impact loading. Laplace and Fourier integral transforms are employed to reduce the dimensional wave propagation problem to the solution of a pair of dual integral equations in the Laplace transform plane. These integral equations are then reduced to integral differential equations which have been solved in the low frequency domain by method of iteration. To determine time dependence of the parameters, these equations are inverted to yield the dynamic stress intensity factor (SIF) for normal point force loading. These results have been used to obtain the SIF at the crack tip which corresponds to the weight function for the crack under normal loading. Analytical expressions of the weight function are used to derive SIF for polynomial loading. Numerical results of normalized SIF for a large normalized time variable and for different concentrated point force loading at an arbitrary location of the crack surface have been calculated for different orthotropic materials. In the present paper, a numerical Laplace inversion technique is used to recover the time dependence of the solution. Finally, the results obtained are displayed graphically.
17
Content available remote Axi-symmetric ice sheet flow with evolving anisotropic fabric
EN
An axially symmetric, gravity driven, steady flow of a grounded polar ice sheet with a prescribed temperature field is considered. The ice is treated as an incompressible, non-linearly viscous, anisotropic fluid, the internal structure (fabric) of which evolves as ice descends from the free surface to depth in an ice sheet. The evolution of the ice fabric is described by an orthotropic constitutive law which relates the deviatoric stress to the strain-rate, strain, and three structure tensors based on the current (rotating) principal stretch axes. The solution of the problem is constructed as a leading-order approximation derived from asymptotic expansions in a small parameter that reflects the small ratio of stress and velocity gradients in the lateral direction of the ice sheet to those in the thickness direction. Numerical simulations of the flow problem have been carried out for various sets of rheological parameters defining the limit strength of the anisotropic fabric in ice. The results of calculations illustrate the influence of the ice anisotropy, basal melt conditions and temperature field in ice on the glacier thickness and lateral span, and on the depth profiles of the flow velocity.
18
Content available A method for solving rectangular beams and discs
EN
An analytic method for solving two-dimensional elasticity problems of orthotropic body has been worked out. A solution for rectangular elements of the beam or disc type has been developed. This solution satisfies the fundamental equations of the elasticity theory of two-dimensional body exactly and the boundary conditions with large accuracy.
PL
Opracowano metodę analityczną rozwiązywania dwuwymiarowych zagadnień teorii spreżystości ciała ortotropowego. Zaproponowano rozwiązanie dla elementów prostokatnych typu belka lub tarcza. Otrzymane rozwiazanie dokładnie spełnia wszystkie podstawowe równania teorii spreżystości ciała dwuwymiarowego i z dużą dokładnością spełnia warunki brzegowe.
19
Content available remote Formulas for the Rayleigh wave speed in orthotropic elastic solids
EN
Formulas for the speed of Rayleigh waves in orthotropic compressible elastic materials are obtained in explicit form by using the theory of cubic equations. Different formulas are obtained by using different forms of the (cubic) secular equation. Each formula is expressed as a continuous function of three dimensionless material parameters, which are the ratios of certain elastic constants. It is interesting to note that one of the formulas includes as a special case the formula obtained recently by Malischewsky for isotropic materials.
20
Content available remote Pure bending of the orthotropic elastic rectangle
EN
In this contribution, a new method for obtaining the solution to the linear elasticity problem is proposed. An idea of this method is based on certain special expansion of these displacement fields in finite trigonometric series with exponential coefficients. This approach leads to an equivalent problem which requires to derive the exact solution. The proposed method has a mixed analytic-numerical form and has been illustrated by the solution to the boundary value problem for a rectangular region subjected to bending by loads applied on the opposite sides of the region. The numerical results derived have been compared with solutions obtained in the framework of the Euler-Bernoulli beam theory.
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