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The purpose of this research is to evaluate the influence of through-the-thickness reinforcements on the fracture behavior of stitched sandwich composites and to develop predictive methodologies to aid in simulating their damage-tolerant capability. Sandwich composites are widely used for their high stiffness-to-weight ratio due to their unique material architecture, which is composed of two rigid, outer facesheets that are bonded to a light-weight internal core. However, sandwich composites are limited by their low interlaminar strengths and can develop core-to-facesheet separation when subjected to low out-of-plane loads. In this study, sandwich composites were manufactured with through-the-thickness reinforcements, or stitches, to act as crack-growth inhibitors and to improve interlaminar properties. Stitch processing parameters, such as the number of stitches per unit area (stitch density) and stitch diameter (linear thread density), have considerable influence on the in-plane and out-of-plane behavior of composite structures. A design of experiments (DoE) approach is used to investigate stitch processing parameters and their interaction on the fracture behavior of stitched sandwich composites. Single cantilevered beam (SCB) tests are performed to estimate the required energy to propagate crack growth, or Mode I fracture energy, during the separation of the facesheet from the core. Additionally, embedded optical fibers within the SCB test articles are used to determine the internal crack front variation. During testing, unique fracture morphologies are obtained and show dependency on stitch processing parameters. Furthermore, embedded optical fibers indicate that the internal crack front is approximately 10% greater than visual edge measurements, which is primarily attributed to Poisson’s effect. The DoE approach is then used to develop a statistically informed response surface model (RSM) to optimize stitch processing parameters based on a maximum predicted fracture energy. Novel analytical formulations are developed for estimating the mode I fracture energy using the J-integral approach. The DoE approach is then used to inform and validate finite element models that simulate the facesheet-to-core separation using a discrete cohesive zone modeling approach. The predicted load and crack growth response show good agreement to experimental measurements and highlights the capability of stitching to arrest delamination in stitched sandwich composites.

This paper includes some basic ideas about the computation of a function theta(G), the theta number of a graph G, which is known as the Lovasz number of G. theta(G^c) lies between two hard-to-compute graph numbers omega(G), the size of the largest lique in a graph G, and chi(G), the minimum number of colors need to properly color the vertices of G. Lovasz and Grotschel called this the "Sandwich Theorem". Donald E. Knuth gives four additional definitions of theta, theta_1, theta_2, theta_3, theta_4 and proves that they are all equal. First I am going to describe the proof of the equality of theta, theta_1 and theta_2 and then I will show the calculation of the theta function for some specific graphs: K_n, graphs related to K_n, and C_n. This will help us understand the theta function, an important function for graph theory. Some of the results are calculated in different ways. This will benefit students who have a basic knowledge of graph theory and want to learn more about the theta function.

Reducing the weight of the truck vehicle conveys more cargo to be carried by thetrailer. This has a significant impact on the efficiency of the transport lowering both the total cost of cargo moved and the total carbon dioxide emitted. Half of the body in-white weight of a truck is comprised out of panels made out of thin mild forming steel which cannot be made thinner to reduce weight due to the lowered stiffness it would entail. Sandwich materials have a high stiffness to weight ratio and would for the same panel thickness as regular forming steel have a comparable bending stiffness but lowered weight. This master thesis is intended to be a preliminary study for Scania CV AB on sandwich materials and its potential use as lightweight panels in their trucks. With the intention of investigating whether a commercial sandwich material is capable of filling the role as outer panels of a truck, comparative tests regarding significant matters such as forming and painting was made on identically manufactured demonstrators comparing a sandwich material and a regular forming steel material. The tests identified weaknesses in the current manufacturing process for parts of a sandwich material. Such limitations are problems with painting and joining due to isolated cover sheets, forming problems revealing sink marks likely due to different spring back of the material and hemming flaws due to inadequately optimized hemming technique and anisotropy. Now that more knowledge of sandwich materials has been gained, counter measures for these findings can be made in order to take another step towards lowering the weight of the truck and a more efficient way of transporting goods. / Genom att minska vikten på lastbilen frigörs mer last att bäras av släpvagnen. Detta har en betydande inverkan på effektiviteten hos transporten som sänker både den totala kostnaden för transporterad last och de totala koldioxidutsläppen. Hälften aven lastbils rena karossvikt består av paneler gjorda av tunt mjukt formningsstål vilke tinte kan bli tunnare för att minska vikten på grund av den sänkta styvheten som detskulle medföra. Sandwichmaterial har en hög styvhet till viktförhållande och skulle församma paneltjocklek som vanligt formningsstål ha en jämförbar böjstyvhet men sänkt vikt. Denna uppsats är avsedd att vara en preliminär studie för Scania CV AB om sandwichmaterial och dess potentiella användning av lättvitkspaneler i lastbilar.Med avsikt att undersöka huruvida ett kommersiellt sandwichmaterial kan fylla rollen som lastbilens ytterpaneler utfördes jämförande tester med avseende på signifikanta frågor såsom formning och målning på identiskt tillverkade demonstratorer som jämförde ett sandwichmaterial och ett vanligt formningsstål. Testerna identifierade svagheter med materialet samt hur processen behöver anpassas för att kunna använda sandwichmaterialet i rådande tillverkningsprocess. Identifierade problem var bland annat problem med målning och sammanfogning på grund av isolerade ytterskickt i sandwichmaterialet, problem med formning som gav upphov till limdragningar som troligen beror på materialets olika återfjädring samt falsningsfel på grund av otillräckligt optimerad falsteknik och anisotropi. Nu när mer kunskap om sandwichmaterial erhållits kan motåtgärder för de funna resultaten undersökas för att ta ytterligare ett steg mot att sänka lastbilens vikt och därmed få ett effektivare transportmedel.

Composite sandwich structures are prevalent in engineering applications where high strength to weight ratios are critical. A composite sandwich includes the addition of a core material between two composite face sheets to increase ultimate stress in compression and bending loading cases. The performance of many composite sandwich structure configurations is well understood in the undamaged case. This analysis examines a type of damage, low velocity indentation, and determines the effect on mechanical behavior. The scope of the analysis includes manufacturing sandwich composite structures, creating indentation in the composite, and testing the sandwich composite structure. The mechanical behavior of the composite sandwich structures is characterized through ASTM C364 test standard for compressive strength and ASTM C393 standard test standard for flexural properties of sandwich constructions. The experiment is conducted with varied indentation depth, core materials, composite sandwich thickness, and composite face sheet thickness. The findings are compared to control specimens and used to determine the effect of indentation depth and create a relationship for the mechanical performance of indented sandwich composites.

The use of composite sandwich structures is rapidly increasing in the aerospace industry because of their increased strength-to-weight and stiffness-to-weight characteristics. The effects of low velocity impacts on these structures, however, are the main weakness that hinders further use of them in the industry because the damages from these loadings can often be catastrophic. Impact behavior of composite materials in general is a crucial consideration for a designer but can be difficult to describe theoretically. Because of this, experimental analysis is typically used to attempt to describe the behavior of composite sandwiches under impact loads. Experimental testing can still be unpredictable, however, because low velocity impacts can cause undetectable damage within the composites that weaken their structural integrity. This is an important issue with composite sandwich structures because interlaminar damage within the composite facesheets is typical with composites but the addition of a core material results in added failure modes. Because the core is typically a weaker material than the surrounding facesheet material, the core is easily damaged by the impact loads. The adhesion between the composite facesheets and the core material can also be a major region of concern for sandwich structures. Delamination of the facesheet from the core is a major issue when these structures are subjected to impact loads. This study investigated, through experimental and numerical analysis, how varying the core and facesheet material combination affected the flexural strength of a composite sandwich subjected to low velocity impact. Carbon, hemp, aramid, and glass fiber materials as facesheets combined with honeycomb and foam as core materials were considered. Three layers of the same composite material were laid on the top and bottom of the core material to form each sandwich structure. This resulted in eight different sandwich designs. The carbon fiber/honeycomb sandwiches were then combined with the aramid fiber facesheets, keeping the same three layer facesheet design, to form two hybrid sandwich designs. This was done to attempt to improve the impact resistance and post-impact strength characteristics of the carbon fiber sandwiches. The two and one layer aramid fiber laminates on these hybrid sandwiches were always laid up on the outside of the structure. The sandwiches were cured using a composite press set to the recommended curing cycle for the composite facesheet material. The hybrid sandwiches were cured twice for the two different facesheet materials. The cured specimens were then cut into 3 inch by 10 inch sandwiches and 2/3 of them were subjected to an impact from a 7.56 lbf crosshead which was dropped from a height of 38.15 inches above the bottom of the specimen using a Dynatup 8250 drop weight machine. The impacted specimen and the control specimen (1/3 of the specimens not subjected to an impact) were loaded in a four-point bend test per ASTM D7250 to determine the non-impacted and post-impact flexural strengths of these structures. Each sandwich was tested under two four-point bend loading conditions which resulted in two different extension values at the same 100 lbf loading value. The span between the two supports on the bottom of the sandwich was always 8 inches but the span between the two loading pins on the top of the sandwich changed between the two loading conditions. The 2/3 of the sandwiches that were tested after being impacted were subjected to bending loads in two different ways. Half of the specimens were subjected to four-point bending loads with the impact damage on the top facesheet (compressive surface) in between the loading pins; the other half were subjected to bending loads with the damage on the bottom facesheet (tensile surface). Theoretical failure mode analysis was done for each sandwich to understand the comparisons between predicted and experimental failures. A numerical investigation was, also, completed using Abaqus to verify the results of the experimental tests. Non-impacted and impacted four-point bending models were constructed and mid-span deflection values were collected for comparison with the experimental testing results. Experimental and numerical results showed that carbon fiber sandwiches were the best sandwich design for overall composite sandwich bending strength; however, post-impact strengths could greatly improve. The hybrid sandwich designs improved post-impact behavior but more than three facesheet layers are necessary for significant improvement. Hemp facesheet sandwiches showed the best post-impact bending characteristics of any sandwich despite having the largest impact damage sizes. Glass and aramid fiber facesheet sandwiches resisted impact the best but this resulted in premature delamination failures that limited the potential of these structures. Honeycomb core materials outperformed foam in terms of ultimate bending loads but post-impact strengths were better for foam cores. Decent agreement between numerical and experimental results was found but poor material quality and high error in material properties testing results brought about larger disagreements for some sandwich designs.

In this study, Ritz method has been employed to analyze the following problems: free vibrations of plates with curvilinear stiffeners, the lowest 100 frequencies of thick isotropic plates, free vibrations of thick quadrilateral laminates and free vibrations and static deformations of rectangular laminates, and sandwich structures. Admissible functions in the Ritz method are chosen as a product of the classical Jacobi orthogonal polynomials and weight functions that exactly satisfy the prescribed essential boundary conditions while maintaining orthogonality of the admissible functions. For free vibrations of plates with curvilinear stiffeners, made possible by additive manufacturing, both plate and stiffeners are modeled using a first-order shear deformation theory. For the thick isotropic plates and laminates, a third-order shear and normal deformation theory is used. The accuracy and computational efficiency of formulations are shown through a range of numerical examples involving different boundary conditions and plate thicknesses. The above formulations assume the whole plate as an equivalent single layer. When the material properties of individual layers are close to each other or thickness of the plate is small compared to other dimensions, the equivalent single layer plate (ESL) theories provide accurate solutions for vibrations and static deformations of multilayered structures. If, however, sufficiently large differences in material properties of individual layers such as those in sandwich structure that consists of stiff outer face sheets (e.g., carbon fiber-reinforced epoxy composite) and soft core (e.g., foam) exist, multilayered structures may exhibit complex kinematic behaviors. Hence, in such case, C_z⁰ conditions, namely, piecewise continuity of displacements and the interlaminar continuity of transverse stresses must be taken into account. Here, Ritz formulations are extended for ESL and layerwise (LW) Nth-order shear and normal deformation theories to model sandwich structures with various face-to-core stiffness ratios. In the LW theory, the C⁰ continuity of displacements is satisfied. However, the continuity of transverse stresses is not satisfied in both ESL and LW theories leading to inaccurate transverse stresses. This shortcoming is remedied by using a one-step well-known stress recovery scheme (SRS). Furthermore, analytical solutions of three-dimensional linear elasticity theory for vibrations and static deformations of simply supported sandwich plates are developed and used to investigate the limitations and applicability of ESL and LW plate theories for various face-to-core stiffness ratios. In addition to natural frequency results obtained from ESL and LW theories, the solutions of the corresponding 3-dimensional linearly elastic problems obtained with the commercial finite element method (FEM) software, ABAQUS, are provided. It is found that LW and ESL (even though its higher-order) theories can produce accurate natural frequency results compared to FEM with a considerably lesser number of degrees of freedom. / Doctor of Philosophy / In everyday life, plate-like structures find applications such as boards displaying advertisements, signs on shops and panels on automobiles. These structures are typically nailed, welded, or glued to supports at one or more edges. When subjected to disturbances such as wind gusts, plate-like structures vibrate. The frequency (number of cycles per second) of a structure in the absence of an applied external load is called its natural frequency that depends upon plate's geometric dimensions, its material and how it is supported at the edges. If the frequency of an applied disturbance matches one of the natural frequencies of the plate, then it will vibrate violently. To avoid such situations in structural designs, it is important to know the natural frequencies of a plate under different support conditions. One would also expect the plate to be able to support the designed structural load without breaking; hence knowledge of plate's deformations and stresses developed in it is equally important. These require mathematical models that adequately characterize their static and dynamic behavior. Most mathematical models are based on plate theories. Although plates are three-dimensional (3D) objects, their thickness is small as compared to the in-plane dimensions. Thus, they are analyzed as 2D objects using assumptions on the displacement fields and using quantities averaged over the plate thickness. These provide many plate theories, each with its own computational efficiency and fidelity (the degree to which it reproduces behavior of the 3-D object). Hence, a plate theory can be developed to provide accurately a quantity of interest. Some issues are more challenging for low-fidelity plate theories than others. For example, the greater the plate thickness, the higher the fidelity of plate theories required for obtaining accurate natural frequencies and deformations. Another challenging issue arises when a sandwich structure consists of strong face-sheets (e.g., made of carbon fiber-reinforced epoxy composite) and a soft core (e.g., made of foam) embedded between them. Sandwich structures exhibit more complex behavior than monolithic plates. Thus, many widely used plate theories may not provide accurate results for them. Here, we have used different plate theories to solve problems including those for sandwich structures. The governing equations of the plate theories are solved numerically (i.e., they are approximately satisfied) using the Ritz method named after Walter Ritz and weighted Jacobi polynomials. It is shown that these provide accurate solutions and the corresponding numerical algorithms are computationally more economical than the commonly used finite element method. To evaluate the accuracy of a plate theory, we have analytically solved (i.e., the governing equations are satisfied at every point in the problem domain) equations of the 3D theory of linear elasticity. The results presented in this research should help structural designers.

Byggbranschens utsläpp av växthusgaser utgör en stor andel av Sveriges totala utsläpp. För att minska de byggrelaterade utsläppen är det på många fronter som byggbranschen behöver förändras och effektiviseras. De senaste åren har en succesiv ökning av byggandet i trä skett vilket är gynnsamt då trä alternativet är mer klimatvänligt än stål och betong. De tuffa klimatmålen vi nu står framför innebär dock att mer behöver göras än att endast öka andelen träbyggnader. Pappersmassaindustrin är lätt att bortse ifrån, då den hittills inte varit relevant för byggbranschen och för att återanvändning är relativt framträdande inom den branschen. Ifrån sågverken som sönderdelar trästockar till virke fraktas flis som blir över till pappersbruk. Av flisen görs sedan bland annat diverse pappförpackningar som går att återvinna. Problemet är att dessa förpackningar endast går att återvinna ett visst antal gånger innan fibrerna blir obrukbara och istället används som biobränsle. Om byggmaterial skulle gå att producera baserat på dessa fibrer, skulle detta innebära en mer långlivad användning av dem. Ett byggmaterial som uppfunnits, gjort på fibrer från pappersmassabruk är core-materialet ”Greenwood”. Eftersom materialet är nytt och egenskaperna till stor del är okända krävs det att diverse studier görs som undersöker materialets olika egenskaper som är relevanta för en eventuell tillämpning inom byggbranschen. Denna studie avser att undersöka skjuvstyvhet, böjstyvhet och bärförmåga hos sandwichelement uppbyggda av core-materialet Greenwood och ytskikt av papp. Detta genom att först  dynamiskt och statiskt testa de ingående materialens egenskaper, följt av böjprovning av nio sandwichbalkar med varierande tjocklek på ytskikten. Samtliga balkar testades även dynamiskt. Core-materialet Greenwood som ingick i sandwichelementen var endast den begränsande faktorn en gång av tio böjprov. När core-materialets skjuvstyvhet togs fram både dynamiskt och statiskt och när den omvandlades till en skjuvmodul visade det sig att Greenwood har en mer än dubbelt så stor styvhet som EPS-cellplast vid liknande densitet. Detta är intressant då denna cellplast ofta agerar som ett core-material i sandwichelement ute i byggbranschen. Testerna visar även på att balkarna har en relativt liten spridning vilket innebär att resultaten har god tillförlitlighet. Slutligen, kan det konstateras att dessa sandwichelement uppvisar sega egenskaper med en viss kvarvarande lastkapacitet även efter brott. Samtliga nämnda egenskaper ovan talar för en viss potential för tillämpning av dessa sandwichelement inom byggbranschen. Fortsatta studier av fukt- och krypegenskaper vid långtidsbelastning rekommenderas, vilket är viktigt för användning inom byggandet. De omfattande resultaten från föreliggande studie utgör dock ett bra underlag för fortsatta undersökningar och värdering av möjliga tillämpningar. / The construction industry`s greenhouse emissions, makes up for a large portion of Sweden’s total emissions. In order to reduce construction related emissions, a fair amount of fronts within the construction industry needs to be changed and streamlined. In the last couple of years, there has been a successive increase in the number of structures that are built from wood amongst other things, which is beneficial because the wood alternative is more climate friendly than steel and concrete. The current tough climate goals entails that more has to be done than just increasing the amount of wood constructions. The pulp industry is easy to write off because so far, it has not been relevant to the construction industry and because recycling is relatively prominent within that industry. From the sawmills that dismember wooden logs to lumber, leftover wood chips are transported to paper mills. Among other things, different cardboard packages that can be recycled are then made from those wood chips. The problem with these packages is that they can only be recycled a certain number of times before the fibers become unusable and instead, are used as biofuel. If building materials were to be able to be produced with these fibers, that would be a more long-lived use of them. A building material, recently invented, made of fiber from paper mills is the core-material “Greenwood”. Because the material is new and its properties for the most part are unknown, this requires that various studies are conducted that examines the different properties the material possesses that are relevant for a contingent enforcement within the construction industry. This study intends to examine the shear rigidity, flexural rigidity and maximum capacity for sandwich panels made from the core-material Greenwood and faces of paperboard. This was achieved by first dynamically and statically test the properties of the two different materials, followed by flexure testing nine sandwich beams with varying face thicknesses. Every beam was also tested dynamically. The core-material Greenwood which was a part of the sandwich panels, was only the limiting factor 1 time out of 10 flexure tests. When the shear rigidity of the core-material was calculated both statically and dynamically and when it was converted to a shear modulus it was shown that Greenwood has a rigidity of more than double that of EPS cellular plastic at similar density. This is interesting because this type of cellular plastic often acts as a core-material in sandwich structures found in the construction industry. The tests also show that the beams have a relatively small spread which means that the results have good reliability. Finally, it can be concluded that these  sandwich panels exhibit ductile properties with a certain lasting load capacity even after ultimate load has been reached. Every property mentioned above indicates that there is a certain potential for applicability of these sandwich panels within the construction industry. Continued studies of moisture properties and creep properties during long-term loading is recommended, which is important for a possible use within construction. The extensive results from this study constitutes a good basis for continued research and assessment of possible applications.