1 第二讲 固体力学基础 张俊乾 近代力学基础

2 Contents 1.Stress and Kinetics 2.Strain and Kinematics 3.Constitutive Models for Materials 4.Material Failure 5.Boundary and Initial Value Problems

3 Stress vector at a point Stress and Kinetics Uniformly distributed Stress Dimension of stress: [force] / [length] 2 = N / m 2 (Pa) = MPa Non-uniformly distributed Stress Outward normal

4 Stress tensor at a point Stress and Kinetics Stress vectors on the plane perpendicular to x- axis, to y-axis, to y-axis, respectively : Stress tensor: 1.The first subscript indicates the direction of the plane normal upon the stress acts, the second subscript the direction of stress component. 2.Positive stress rule: The directions of stress component and of the plane are both positive, or both negative.

5 Cauchy’s formula Stress vectors on the plane with normal vector n Remark: Cauchy’s formula assures us that the nine components of stress are necessary and sufficient to define the traction on any surface element across a body. Hence the stress state in a body is characterized completely by the set of stress tensor σ Relationship: stress vector - stress tensor Stress tensor σ is symmetric Stress and Kinetics

6 Index notation and transformation of coordinates Coordinates: Subscript: (x, y, z) index (1, 2, 3) Stress vector: (t x, t y, t z ) Stress tensor: (t 1, t 2, t 3 ) Summation convention: The summation is implied by the repeated index, called dummy index. Use of any other index instead of i does not change the meaning. Stress and Kinetics

7 Index notation and transformation of coordinates Transformation of stress vector: Transformation of coordinates: Transformation of stress tensor: Stress and Kinetics

8 Equations of motion (equilibrium) 3-dimensional 2-dimensional x1x1 x2x2 O f1f1 f2f2 Symmetry: --- derived from the linear momentum balance or Newton’s second law of motion --- derived from the angular momentum balance Stress and Kinetics

9 Strain tensor (deformation measure) S y x z P Strain and Kinematics Displacement vector: 3 components: Strain-displacement relationship:, or Normal strain Shear strain The change of volume (volumetric strain):

x y O P A dx B dy u v Distortion of the right angle between two lines (Shear strain) ： Elongation of PA (Normal strain): Elongation of PB (Normal strain): Properties of Strain tensor Strain and Kinematics

11 Transformation of coordinates Transformation of coordinates: Transformation of strain tensor: Strain and Kinematics

Constitutive Model : Isotropic, linear elastic materials Uniaxial tension: Thermoelastic constitutive equations in multiaxial-stress state: Typical materials: polycrystalline metal, polymers and concrete etc. Young’s modulus Poisson’s ratio Coefficient of thermal expansion

Constitutive Model : Anisotropic linear elastic materials Stiffness matrix compliance matrix Coefficient of thermal expansion Strain energy density:

Constitutive Model : Linear elastic orthotropic materials 9 independent elastic constants; 3 CTE constants

Constitutive Model : Transversely isotropic materials 5 independent elastic constants; 2 CTE constants

Constitutive Model : Rate independent plasticity Features of the inelastic response of metals Decomposition of strain into elastic and plastic parts: Yield: If the stress exceeds a critical magnitude, the stress-strain curve ceases to be linear. Bauschinger effect: If the specimen is first deformed in compression, then loaded in tension, it will generally start to deform plastically at a lower tensile stress than an annealed specimen.

Constitutive Model : Rate independent plasticity Yield Criteria are the components of the `von Mises effective stress’ and `deviatoric stress tensor’ respectively. 1. A hydrostatic stress (all principal stresses equal) will never cause yield, no matter how large the stress; 2. Most polycrystalline metals are isotropic.

Constitutive Model : Rate independent plasticity Isotropic hardening model Perfectly plastic solid: Linear strain hardening solid Power-law hardening material

Constitutive Model : Rate independent plasticity Plastic flow law is the slope of the plastic stress-strain curve.

Complete incremental stress-strain relations Constitutive Model : Rate independent plasticity

MaterialYield Stress (MPa) MaterialYield Stress (MPa) Tungsten Carbide6000Mild steel220 Silicon Carbide10 000Copper60 Tungsten2000Titanium Alumina5000Silica glass7200 Titanium Carbide4000Aluminum & alloys Silicon Nitride8000Polyimides Nickel70Nylon Iron50PMMA Low alloy steels Polycarbonate55 Stainless steel PVC45-48 Constitutive Model : Rate independent plasticity Typical values for yield stress of some materials

Constitutive Model : Viscoplasticity Features of creep behavior (constant stress) Features of high-strain rate behavior 1.If a tensile specimen of a solid is subjected to a time independent stress, it will progressively increase in length. 2.The length-time plot has three stages 3.The rate of extension increases with stress 4.The rate of extension increases with temperature 1.The flow stress increases with strain rate 2.The flow stress rises slowly with strain rate up to a strain rate of about 10 6, and then begins to rise rapidly.

Constitutive Model : Viscoplasticity Flow potential for creep: Flow potential for High strain rate: Strain rate decomposition: Plastic flow rule:

Material Failure : Introduction The mechanisms involved in fracture or fatigue failure are complex, and are influenced by material and structural features that span 12 orders of magnitude in length scale, as illustrated in the picture below

Material Failure : Mechanisms Failure under monotonic loading Brittle 1.Very little plastic flow occurs in the specimen prior to failure 2.The two sides of the fracture surface fit together very well after failure 3. In many materials, fracture occurs along certain crystallographic planes. In other materials, fracture occurs along grain boundaries Ductile 1.Extensive plastic flow occurs in the material prior to fracture 2.There is usually evidence of considerable necking in the specimen 3.Fracture surfaces don’t fit together 4.The fracture surface has a dimpled appearance, you can see little holes, often with second phase particles inside them.

Material Failure : Mechanisms Failure under cyclic loading 1.S-N curve normally shows two different regimes of behavior, depending on stress amplitude 2.At high stress levels, the material deforms plastically and fails rapidly. In this regime the life of the specimen depends primarily on the plastic strain amplitude, rather than the stress amplitude. This is referred to as `low cycle fatigue’ behavior 3.At lower stress levels life has a power law dependence on stress, this is referred to as `high cycle’ fatigue behavior 4.In some materials, there is a clear fatigue limit, if the stress amplitude lies below a certain limit, the specimen remains intact forever. In other materials there is no clear fatigue threshold. In this case, the stress amplitude at which the material survives 10 8 cycles is taken as the endurance limit of the material. (The term `endurance’ appears to refer to the engineer doing the testing, rather than the material)

Material Failure : Stress and strain based failure criteria Failure criteria for isotropic materials: Tsai-Hill criterion for brittle fiber-reinforced composites and wood:. Ductile Fracture Criteria: Criteria for failure by low cycle fatigue : Criteria for failure by high cycle fatigue: