İleri Malzeme Mekaniği Lineer Gerilme - Genleme Bağıntıları 1 Origins Origins Origins Origins Origins Origins Origins Origins of of of of of of of of Elastisity Elastisity Elastisity Elastisity Elastisity Elastisity Elastisity Elastisity Lineer gerilme Lineer gerilme Lineer gerilme Lineer gerilme Lineer gerilme Lineer gerilme Lineer gerilme Lineer gerilme - - - - - - - - genleme genleme genleme genleme genleme genleme genleme genleme iliskileri iliskileri iliskileri iliskileri iliskileri iliskileri iliskileri iliskileri Dr. Dr. Nusret Nusret MEYDANLIK MEYDANLIK nm nm- -2011 2011 İleri Malzeme Mekaniği Dr. Nusret MEYDANLIK Ders Ders 4 4 İ İ LER LER İ İ MALZEME MEKAN MALZEME MEKAN İĞİ İĞİ M M K K 64 64 5 5 Kaynak Kaynakç ça: a: -Advanced Mechanics of Materials , 6 th Ed., Chapter 3 ARTHUR P. BORESI & RICHARD J. SCHMIDT What What is is elasticity elasticity and and why why should should we we care care? ? • • The The capability capability of a of a strained strained body body to to recover recover its its size size and and shape shape after after deformation deformation. . • • Most Most engineering engineering design design is done in is done in the the elastic elastic region region to to ensure ensure that that engineered engineered components components retain retain their their dimensions dimensions and and functionality functionality. . • • Generally Generally, , plastic plastic deformation deformation or or fracture fracture ? ? failure failure. . • • Sometimes Sometimes, , elastic elastic deformation deformation, , if if excessive excessive ? ? failure failure. . • • Want Want to to avoid avoid failure failure! ! • Microscopically, elastic behavior is inherently anisotropic for individual grains or single crystals. • Macroscopically, most polycrystalline materials are elastically isotropic. • Polycrystalline materials can be anisotropic if they exhibit strong crystallographic textures.3 What What is is the the origin origin of of elasticity elasticity? ? • Atomic bonding forces. • Originates from long range attractive forces which draw atoms together until short range repulsive forces become large enough to balance them out. • Elastik özellikler atomlararası bağ nedeniyle atomların tek tek deformasyonunun birleştirilmiş etkisidir. 4 Bulk Bulk Elastic Elastic Behavior Behavior • Applied force is transmitted by the network of bonds constituting the material.[*] • Thus, elastic behavior depends quantitatively on the magnitude of the interatomic forces. • Elastic properties do not depend on microstructure of the material. • Elastic properties do depend upon atomic structure of a material.5 Forces Forces between between atoms atoms • The minimum energy point corresponds to the equilibrium separation (i.e., the equilibrium bond length, r o ). • The force between atoms is simply the derivative (i.e., the slope) of the energy versus distance plot. • The bond stiffness is the derivative of the force versus distance plot. 6 • At the equilibrium bond length, r o , the variation of F with r is essentially linear, which means that the stiffness is essentially constant at small distances from r o . • Using this expression, the force to “stretch” n bonds in a solid is: where n is the number of bonds. • The applied stress is:7 • The strain can be expressed as: • Thus, stress becomes: • Hooke’s law: ? = E? • Thus, The derived expression shows that the modulus is only dependent upon bond strength. Elastic properties are “structure insensitive” (i.e., insensitive to microstructure) Elastic Elastic properties properties do do depend depend upon upon atomic atomic arrangement arrangement (i.e., (i.e., crystallography crystallography). ). 8 How How does does stress stress relate relate to to strain strain for for isotropic isotropic and and anisotropic anisotropic solids solids? ? • Hooke’s Law – Isotropic Solids (properties are the same in all directions) Stress-strain plot for a linearly elastic material – Anisotropic solids (properties are strain directional) Must Must define define differently differently! !9 What are the origins of anisotropy in materials? 10 What are the origins of anisotropy in materials?11 First First consider consider the the Poisson Poisson effect effect 1213 What What are are the the elastic elastic stress stress- -strain strain relations relations in 3D? in 3D? 14 • Bulk Modulus (K), also known as the volumetric elastic modulus: Another important elastic constant15 ?, µ = Lame constants Other Relations Between Isotropic Elastic Constants 16 Energy stored in a body due to deformation. ELASTIC STRAIN ENERGY ELASTIC STRAIN ENERGY • Work to deform a body elastically is stored as elastic strain energy. It is recovered when the applied forces are released. • Strain energy is proportional to the area under the load-deformation (stress-strain) curve.17 To relate the stress at a point in a material to the corresponding strain at that point, knowledge of material properties is required. These properties enter into the stress- strain-temperature relations as material coefficients. The theoretical basis for these relations is the first law of thermodynamics, but the material properties themselves must be determined experimentally. Stress-strain relations may be derived with the first law of thermodynamics, a precise statement of the law of conservation of energy. The total amount of internal energy in a system is generally indeterminate. Hence, only changes of internal energy are measurable. If electromagnetic effects are disregarded, this law is described as follows: The form of the stress-strain relations depends on material behavior. The derivation of load load- -stress stress and load load- -deflection deflection relations requires stress-strain relations. The form of the stress-strain relations depends on material behavior. We treat mainly materials that are isotropic; that is, at any point they have the same properties in all directions. Stress-strain relations for linearly elastic isotropic materials are well known for today. where ?W is the work performed on the system by external forces, ?H is the heat that flows into the system, ?U is the increase in internal energy, and ?K is the increase in kinetic energy. 18 For adiabatic conditions (no net heat flow into V,?H = 0) and static equilibrium (?K = 0), the first law of thermodynamics states that, during the displacement variations (?u, ?v, ?w), the variation in work of the external forces ?W is equal to the variation of internal energy ?U for each volume element. Hence, for V, we have • Consider an elemental cube that is subjected to only an elastic tensile stress along the x-axis. The elastic strain energy, U, is: ?W= ?U19 2021 ? = ? ? d U 0 ? = ? ? d U 0 ? = ? ? d C 0 Elastisite Elastisite de gerilme ve de gerilme ve genlemeleri genlemeleri elde etmek i elde etmek iç çin bu terimler kullan in bu terimler kullanı ılabilir. Ki labilir. Ki mekanikte bir mekanikte bir ç çok problemi ok problemi çö çözmek i zmek iç çin kullan in kullanı ılan lan Castigliano Castigliano’ ’s s teoremi teoremi, , virt virtü üel el i iş ş teoremi teoremi ve ve en k en küçü üçük i k iş ş teoremi teoremi bu terimleri kullanan tekniklerdir bu terimleri kullanan tekniklerdir 22 Relations between stress and strain for anisotropic crystals • Since interatomic forces, stresses, and strains are directional, elastic constants will also be directional. • We need to relate every stress component to every strain component. • Therefore, we must define two new elastic constants: – C ? stiffness – S ? compliance • farklı tip malzemeler için genel Hooke’s yasası : şeklinde ifade edilir.23 2425 rijitlik matrisi esneklik matrisi 2627 2829 Üç boyutlu bir kütle için, 1-2-3 ortogonal koordinat sistemindeki en genel şekil değiştirme-gerilme ilişkisi aşağıdaki gibidir. 3031 HOOKE'S LAW: ANISOTROPIC ELASTICITY HOOKE'S LAW: ANISOTROPIC ELASTICITY In the one-dimensional case, for a linear elastic material the stress a is proportional to the strain e; that is, ? = E? , where the proportionality factor E is called the modulus of elasticity. The modulus of elasticity is a property of the material. Thus, for the one- dimensional case, only one material property is required to relate stress and strain for linear elastic behavior. The relation ? = E? is known as Hooke's law. More generally, in the three-dimensional case, Hooke's law asserts that each of the stress components is a linear function of the components of the strain tensor; that is (with ? xy , ? xz , ? yz ), where the 36 coefficients, C 11 , ..., C 66 , are called elastic coefficients elastic coefficients. Materials that exhibit such stress-strain relations involving a number of independent elastic coefficients are said to be anisotropic anisotropic. 32 Slayt 19 ve 3.20 ifadelerinden elde edilir , bu katsay bu katsayı ılar aras lar arası ında simetri ve nda simetri ve C C ij ij = =C C ji ji ili iliş şki kisi si vard vardı ır r, O halde; lineer elastik lineer elastik anizotrop anizotrop malzemeler i malzemeler iç çin 21 farkl in 21 farklı ı C katsay C katsayı ıs sı ı vard vardı ır. r. In reality, Eq. 3.20 is not a law but merely an assumption that is reasonably accurate for many materials subjected to small strains. For a given temperature, time, and location in the body, the coefficients C ij are constants that are characteristics of the material. Bir noktada yirmibir adet bağımsız elastik sabite sahip olan malzemeye anizotropik malzeme denir. Bu sabitler bir kez özel bir nokta için bulunduğu zaman gerilme-şekil değiştirme ilişkisi o noktada geliştirilebilir. Eğer malzeme homojen değilse, bu sabitler noktadan noktaya değişiklik gösterebilirler. Malzeme homojen olsa bile (veya öyle olduğu farz edilsin) analitik olarak veya deneysel olarak, bu 21 adet elastik sabiti bulmak gerekir.33 • Birçok doğal ve sentetik malzeme, malzeme simetrisine sahiptir, yani elastik nitelikler simetri doğrultularında özdeştir. Bu simetri özelliği 6X6 rijitlik [C] ve 6X6 esneklik [S] matrislerindeki sabitlerin bazılarını ya sıfırlayarak yada birbirleriyle ilişkilendirerek bağımsız elastik sabitlerin sayısını düşürür. Bu durum, elastik simetrinin değişik türleri için Hooke kanunundaki ilişkileri basitleştirir. Örneğin , • Eğer malzemenin, bir tane malzeme simetri düzlemi varsa bu tip malzemelere monoklinik monoklinik malzemeler malzemeler denir. Simetri düzlemine dik olan doğrultu, “temel doğrultu” olarak adlandırılır. Bu tip malzemeler 13 adet ba 13 adet bağı ğıms msı ız elastik sabit z elastik sabite sahiptir. • Eğer malzeme, karşılıklı olarak birbirine dik üç adet malzeme simetri düzlemine sahipse bu tip malzemelere ortotropik ortotropik malzeme malzeme (ahşap, katmanlı plastikler, kompozit malzemeler, haddelenmiş ürünler gibi) denir. Bu tip malzemeler 9 adet 9 adet ba bağı ğıms msı ız elastik sabit z elastik sabite sahiptir. Ortotropik malzemeler için rijitlik ve esneklik matrisleri aşağıdaki gibidir. ( Daha detaylı bilgi için BORESİ Ch.3.5 i okuyunuz.) 34 HOOKE'S LAW: ISOTROPIC ELASTICITY HOOKE'S LAW: ISOTROPIC ELASTICITY .. that for isotropic isotropic linear elastic materials linear elastic materials, the stress-strain relations involve only two (3) elastic constants. What is these elastic constants? Gerilme –genleme ilişkileri ; Elastik izotrop mal. için rijitilik matrisi35 veya ; For the case of plane stress, ? zz = ? xz =? yz =0, For the case of plane strain, ? zz = ? xz =? yz =0, yazılabilir. 36 Isıl etkinin katılımı ile, Elastik genleme enerjisi,37 Çalışma soruları; (BORESİ , 6. Ed.) • 3.1 3.2 • 3.11-12 ve • 3.13 • 3.18 3.19 ? max =60.53 MPa ? 70 MPa 3839