On the structural level, linear elasticity means that, for example, the deflection of a beam is proportional to the load applied to it. The most fundamental material model is linear elasticity in which stresses are proportional to strains. Often, the relation also includes time derivatives (as in viscoelasticity) or a memory of previous strains (as in plasticity).įor each material, it is necessary to perform measurements and then fit these measurements to a suitable mathematical model. In a few cases, for elastic materials, this relation is unique. Mathematically, material models relate stresses to strains. The laws of thermodynamics, symmetry conditions, and similar arguments can at best provide some limitations on the allowed mathematical structure of the material models. As opposed to the two previous sets of equations, constitutive relations cannot be derived from first principles, but are purely empirical. Constitutive RelationsĪ constitutive relation, that is, a material model, forms the bridge from force to deformation or from stress to strain. Just as the equilibrium relations, these conditions are fundamental and do not contain any assumptions. These compatibility conditions, either on the structure level or the continuum level, are basically geometric relations. This provides the compatibility conditions for a continuum. The individual components of the strain tensor cannot have arbitrary spatial distributions, since they are derived from a displacement field. Where the individual elements are defined as derivatives of the displacements, In a general 3D setting, the strain is also represented by a symmetric tensor, ĭefinition of the engineering strain for pure extension. For a simple elongation of a bar, the engineering strain,, is a ratio of the displacement,, and the original length.
Inside the material, the local deformations are characterized by the strain that represents a relative deformation. For example, in a framework, the ends of all members joined at a point must move the same distance and in the same direction. Strain and Compatibility EquationsĬompatibility relations are requirements on the deformations. Where is a force per unit volume, is the mass density, and is the displacement vector. In terms of the stresses, Newton's second law can be formulated as From moment equilibrium considerations, the stress tensor is symmetric and contains six independent values. One index is the direction of the force component and the other index is the orientation of the normal to the surface on which the force acts. In three dimensions, the stresses in the material are represented by the stress tensor, which can be written asĪn element in the stress tensor represents a force component on a unit area in the material. The external forces on a bar are balanced by the internal stresses. These internal forces are called stresses. If you make a virtual cut through a material somewhere, there must be forces in the cut that are in balance with the external loads. The equilibrium equations are based on Newton's second law, stating that the sum of all forces acting on a body (including any inertial forces) sum up to zero, so that all parts of any structure must be in equilibrium. These equations can, however, come in different guises, depending on whether the analysis is at a continuum level or a large-scale structural level. The force distribution is influenced by the stiffness of each bar.ĭue to the static indeterminacy, almost all structural mechanics analyses rely on the same three types of equations, which express equilibrium, compatibility, and constitutive relations. The forces in three bars cannot be determined by only two force balance equations at the joint.
0 kommentar(er)
