Applications of Matrix Theory in Stress-Strain Laws

SHANGHAI JIAO TONG UNIVERSITY, SHANGHAI CN Stress-Strain Laws Applications of Matrix Theory 12-12-2017 Submitted to: Dr. Fang Wang Submitted by: Muhammad Khizar Shafique 117050990004 School of Materials Science and Engineering Table of Contents Introduction .......................................................................................................................................... 2 Hooke’s Law .......................................................................................................................................... 2 Generalized Hooke’s Law ..................................................................................................................... 3 Generalized Hooke’s Law in Three Dimensions ................................................................................. 4 Strain-to-Stress Relation .................................................................................................................. 4 Stress-to-Strain Relation .................................................................................................................. 5 Thermal Effect Relation.................................................................................................................... 5 Generalized Hooke’s Law for Two Dimensions .................................................................................. 6 Plane Stress ....................................................................................................................................... 6 Conclusion ............................................................................................................................................. 7 1|Page Introduction Stress–strain analysis is an engineering discipline that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. Stress analysis is a primary task for material engineers involved in the design and material selection of structures of all sizes, such as tunnels, bridges and dams, aircraft and rocket bodies, mechanical parts, and even plastic cutlery and staples. Stress analysis is also used in the maintenance of such structures, and to investigate the causes of structural failures. In engineering, stress analysis is often a tool rather than a goal in itself; the ultimate goal being the design of structures and artifacts that can withstand a specified load, using the minimum amount of material or that satisfies some other optimality criterion. Stress analysis may be performed through classical mathematical techniques, analytic mathematical modelling or computational simulation, experimental testing, or a combination of methods. Here I will use matrices to explain the Stress-Strain calculation for materials. Hooke’s Law Hooke's law is a principle of physics that states that the force (F) needed to extend or compress a spring by some distance X scales linearly with respect to that distance. That is: F = kX, where k is a constant factor characteristic of the spring: its stiffness, and X is small compared to the total possible deformation of the spring. Hooke's law is only a first-order linear approximation to the real response of springs and other elastic bodies to applied forces. It must eventually fail once the forces exceed some limit, since no material can be compressed beyond a certain minimum size, or stretched beyond a maximum size, without some permanent deformation or change of state. Many materials will noticeably deviate from Hooke's law well before those elastic limits are reached. 2|Page The modern theory of elasticity generalizes Hooke's law to say that the strain (deformation) of an elastic object or material is proportional to the stress applied to it. However, since general stresses and strains may have multiple independent components, the "proportionality factor" may no longer be just a single real number, but rather a linear map (a tensor) that can be represented by a matrix of real numbers. Generalized Hooke’s Law The one-dimensional Hooke’s la relates 1D normal stress to 1D extensional strain through two material parameters introduced previously: the modulus of elasticity E, also called You g’s modulus a d Poisso ’s ratio ѵ. The modulus of elasticity connects axial stress σ to a ial strai Ɛ: � = �� ⇒ � = � � ⇒ �= � � Uniaxial tension test Poisso ’s ratio ѵ is defined as ratio of lateral strain to axial strain: Torsion Test ѵ = | ������� ����� ������� ����� |= − � ��� ����� � ��� ����� The − sig is i trodu ed for o e ie e so that ν o es out positive. For structural aterials ѵ lies i the ra ge . ≤ ѵ < 0.5. For most metals ѵ ≈ . 5–0.35. For concrete and era i s, ѵ ≈ . . For ork ѵ ≈ . For ru er, ѵ ≈ .5 to pla es. A aterial for hi h ѵ = 0.5 is called incompressible. 3|Page Generalized Hooke’s Law in Three Dimensions Strain-to-Stress Relation Pulli g the aterial Ne t, pull the appl i g σxx along x will produce normal strains: � aterial Fi all pull the aterial = � ;� � = � ;� � = � � = � ;� � = � ;� � = � � = � ;� � = � ;� � = � � σyy along y to get the strains: � σzz along z to get the strains: � In the general case the cube is subjected to combined normal stresses σxx, σyy and σzz. Since we assumed that the material is linearly elastic, the combined strains can be obtained by superposition of the foregoing results: ѵ� ѵ� � � � � = � = � = � +� +� +� +� +� +� − − = � − ѵ� − ѵ� � � � � � ѵ� ѵ� =− + − = −ѵ� +� − ѵ� � � � � ѵ� � ѵ� − + = −ѵ� − ѵ� + � =− � � � � = The shear strains and stresses are connected by the shear modulus as: � � � � � � � =� = = ; � =� = = ; � =� = = � � � � � � The above equations can be expressed in the following matrix: ѵ ѵ − − � � � ѵ ѵ − − � � � � � ѵ ѵ � � − − � = � � � � � � � � � [� ] [� ] [ � �] 4|Page Stress-to-Strain Relation To get stresses if the strains are given, the most expedient method is to invert the matrix equation � � � Here �̂ is a � � [� ] = effe ti e �̂ [ �̂ ѵ �̂ − ѵ �̂ ѵ −ѵ �̂ ѵ �̂ ѵ odulus odified � � Poisso ’s ratio: � − ѵ �̂ = Thermal Effect Relation �̂ ѵ �̂ ѵ �̂ − ѵ � � � � � � ] [� ] +ѵ To incorporate the effect of a temperature change ∆T with respect to a base or reference temperature, add α∆T to the three normal strains in the previous strain relations � = � = � = � � � (� − ѵ� (−ѵ� +� − ѵ� (−ѵ� − ѵ� ) + �∆T − ѵ� ) + �∆T + � ) + �∆T No change in the shear strain-stress relation is needed because if the material is linearly elastic and isotropic, a temperature change only produces normal strains. The stress-tostrain matrix relation expands as follows: � � � � � [� ] � ѵ − � ѵ − � = [ − ѵ � � ѵ − � ѵ � ѵ − � − � � � � � � � � [� �] ] + �∆ [ ] 5|Page Inverting this relation provides the stress-strain relations that account for a temperature change: � � � � � [� ] = As before �̂ is a �̂ [ −ѵ �̂ ѵ �̂ ѵ effe ti e �̂ ѵ �̂ − ѵ �̂ ѵ odulus �̂ = �̂ ѵ �̂ ѵ �̂ − ѵ odified � − ѵ � � � � � � � � ] [� ] − Poisso ’s ratio: ��∆ − ѵ [ ] +ѵ Note that if all mechanical normal strains � , � and � vanish, but ∆ ≠ , the normal stresses given above relation are nonzero. Those are called initial thermal stresses, and are important in engineering systems exposed to large temperature variations, such as rails, turbine engines, satellites or reentry vehicles1. Generalized Hooke’s Law for Two Dimensions Two specializations of the foregoing 3D equations to two dimensions are of interest in the applications: plane stress and plane strain. Plane stress is more important in Aerospace stru tures, hi h te d to e thi . I’ll o l e e plai i g these i short gi i g refere es fro the Hooke’s La for Three Di e sio s Plane Stress In this case all stress components with a z component are assumed to vanish. For a linearly elastic isotropi aterial. The σzz strain, often called the transverse strain or thickness strain in applications, in ge eral ill e o zero e ause of Poisso ’s ratio effe t. By considering temperature changes the matrix form for plane stress in 2D would become: ѵ − � � � � ѵ � − [� ] = [� ] + �∆ [ ] � � � ѵ ѵ � − − � � [ �] 1 University of Colorado (Online lectures from Department of Aerospace Engineering) 6|Page And also we can find the other expression � � �̂ �̂ ѵ ��∆ [� ] = [�̂ ѵ �̂ ] [� ] − [ ] − ѵ � � � Similarly we can also find the expressions for plane strain and later include the temperature change parameters in it to make a matrix for calculations of plane strain Conclusion A matrix stress and strain implementation is made that becomes simpler than tensor notation as usual. The effective stress is calculated using the indicated matrix notation, starting from a deviatoric stress state. The effective stress defines a functional flow condition which, besides the principle of normality, solves an equation for the incremental plastic deformation. The plastic work is determined, as usual, from the state of stresses and strains. Finally, the effective plastic strain is obtained as a function of the plastic working and the effective stress, both in the case of plastic deformation at constant volume and with variable volume model2. 2 Matrix stress-strain working method for determining the effective plastic strain, M. Sánchez et.al., 2015 7|Page