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弹塑性材料应力分析

创建于：2024年12月30日

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问题

一个由不可压缩理想弹塑性材料制成的薄壁圆筒，承受轴向拉力和扭矩作用。根据Mises条件：如果先使筒的剪应变 γ=σs3G\gamma = \frac{\sigma_s}{\sqrt{3G}}，让筒进入屈服状态，然后再拉伸使拉应变 ε=σsE\varepsilon = \frac{\sigma_s}{E}，要求求出此时筒中的应力 σ\sigma 和 τ\tau。
详细思考详细解答题目并详细解释用到的知识点

思考

Examining applied strains

I'm mapping out the solution for a thin-walled circular cylinder under axial tension and torque, made of incompressible ideal elasto-plastic material with Mises yield condition.

Examining stress invariants

I’m thinking through the Von Mises yield criterion for elasto-plastic materials, noting that yielding occurs at a critical deviatoric stress invariant value. Incompressible materials only undergo deviatoric stress changes.

Breaking down stress relations

Taking a closer look at expressing normal stress σ\sigma and shear stress τ\tau by linking them to yield stress σs\sigma_s, shear modulus GG, and Young's modulus EE.

Piecing together stress states

I'm pulling together how shear strain induces shear stress τ\tau and axial strain ϵ\epsilon influences σ\sigma, using stress-strain relations for incompressible materials to analyze stress states.

Linking stress and strain

Elastic stress-strain relations for incompressible, isotropic materials are connected through shear modulus GG and hydrostatic pressure pp. Plastic strain is incorporated into the Von Mises criterion.

Confirming shear stress

I'm noting that in pure shear, yielding happens when τ=σs3\tau = \frac{\sigma_s}{\sqrt{3}}. The initial shear strain is given by γ=τG\gamma = \frac{\tau}{G}.

Revisiting the formula

I'm focusing on the shear strain formula γ=σs3G\gamma = \frac{\sigma_s}{\sqrt{3} G} to ensure it's dimensionally accurate. This leads me to apply axial strain ϵ=σsE\epsilon = \frac{\sigma_s}{E}.

Connecting stress and strain

I'm thinking through how peak stress σs\sigma_s aligns with yield stress in tension, considering elastic-plastic behavior where additional strain post-yield doesn't elevate stress further.

Assessing stress impacts

I’m pulling together the stresses σ\sigma and τ\tau in a thin-walled cylinder, noting how elastic and plastic strains interplay, and leveraging Mises yield condition for combined loading effects.

Breaking down strain

I’m noting the decomposition of total strain into elastic and plastic components, setting aside hardening effects, and utilizing the Mises criterion for post-yield stress analysis.

Connecting stress and flow

I'm examining the relationship between axial and shear stress using the Mises criterion, focusing on the ideal elasto-plastic material's yield condition to manage combined loading effects.

Balancing stress states

I'm noticing that after shear yielding, additional axial strain causes plastic deformation to maintain the equivalent stress at σs\sigma_s, all while adhering to incompressibility and yield conditions.

Connecting stress and strain

I'm figuring out how axial stress σ\sigma and shear stress τ\tau align with the Mises equivalent stress equation, while also considering plastic strain adjustments to satisfy yield conditions.

Monitoring stress states

Piecing together the final stress state for axial stress σ\sigma and shear stress τ\tau to align with the Mises criterion, while considering the impact of applied strain and plastic deformation.

Connecting stress and strain

I'm validating the relationship between stress and strain, distinguishing between elastic and plastic components, and ensuring the final stress state adheres to the Mises criterion and additive strain principles.

Balancing stress states

Piecing together the stress states for shear τ\tau and axial σ\sigma stresses using the Mises criterion, while balancing linearily elastic and plastic behaviors under combined loading.

Balancing stress states

I’m mapping out the relationship between shear τ\tau and axial σ\sigma stress, while ensuring it satisfies the Mises criterion and incompressibility condition.

Formulating stress states

I’m working through the stress states to ensure they fit the Mises criterion and yield condition, factoring in elastic and plastic strains and maximizing clarity.

Mapping stress states

Analyzing the evolution of stress states in a thin-walled cylinder, I’m examining how shear and axial strains are managed post-yielding, and ensuring the stress state aligns with the Mises criterion.

Analyzing stress dynamics

I’m examining the final stress state for shear and axial stresses, ensuring they align with the Mises criterion, while understanding how plastic strains adjust to maintain null volumetric strain.

Considering proportionality

I might express stresses as proportional to strains, while ensuring the yield condition holds and stresses stay within physical limits.

Linking stress and strain

OK, let me see. By combining elastic and plastic strains, I’m ensuring the Mises criterion and yield condition are met, while assessing both normal and shear stress contributions.

Mapping out stress responses

OK, let me see. By examining how shear and axial strains affect irritability while keeping the final stress state in alignment with yield conditions, I’m focusing on incompressible material behaviors.

Balancing stress states

I’m pulling together how applied shear and axial strains impact stress, with the yield condition and incompressibility constraints, aiming to align with the Mises criterion.

Adjusting my approach

I’m piecing together how elastic and plastic strains interact, ensuring the yield condition is met, and establishing a consistent relationship between stress and strain.

Balancing axial and shear stress

Evaluating how axial strain ϵ\epsilon influences shear stress τ\tau within the yield conditions, ensuring that shear stress τ\tau aligns with the Mises criterion.

Revisiting stress states

I'm refining the relationship between shear stress τ\tau and axial stress σ\sigma, considering the interplay of elastic and plastic components, to ensure the final stress state satisfies the Mises criterion.

Balancing stress states

Considering the interplay between shear and axial stresses, ensuring the final state matches the Mises criterion, with elastic and plastic strain components thoughtfully balanced.

Balancing stress dynamics

OK, let me see. After yielding primarily from shear strain, axial strain leads to stress adjustment, maintaining yield conditions. The final stress state has axial stress at yield and shear stress at zero.

Ensuring stress balance

OK, let me see. The final stress state strikes a balance between axial and shear stresses, maintaining the Mises criterion. The she stress drops to zero as axial stress hits its peak.

Thought for 31 seconds

回答

要解决这个问题，我们需要分析由不可压缩的理想弹塑性材料制成的薄壁圆筒在轴向拉力和扭矩作用下的应力状态。我们将根据冯·米塞斯（Von Mises）屈服准则逐步分析。

已知条件

  1. 材料性质：

    • 不可压缩：体积应变为零，即各向的总应变之和为零。
    • 理想弹塑性：材料在达到屈服点后，发生塑性变形而应力不再增加。
    • 冯·米塞斯屈服准则。

  2. 加载步骤：

    • 第一步：施加剪应变 γ=σs3G\gamma = \frac{\sigma_s}{\sqrt{3} G}，使圆筒达到屈服状态。
    • 第二步：在已经屈服的状态下，再施加轴向拉应变 ε=σsE\varepsilon = \frac{\sigma_s}{E}。

分析步骤

第一步：施加剪应变使圆筒达到屈服

  • 剪应变 γ=σs3G\gamma = \frac{\sigma_s}{\sqrt{3} G} 导致的剪应力 τ\tau：

    τ=Gγ=G(σs3G)=σs3\tau = G \gamma = G \left(\frac{\sigma_s}{\sqrt{3} G}\right) = \frac{\sigma_s}{\sqrt{3}}

  • 冯·米塞斯屈服准则在纯剪应力下的形式为：

    σeq=3τ=σs\sigma_{\text{eq}} = \sqrt{3} \tau = \sigma_s

    这表明在施加剪应变后，材料刚好达到屈服点，剪应力 τ\tau 为 σs3\frac{\sigma_s}{\sqrt{3}}，而轴向应力 σ=0\sigma = 0。

第二步：施加轴向拉应变

  • 轴向拉应变 ε=σsE\varepsilon = \frac{\sigma_s}{E} 导致的轴向应力 σ\sigma：

    σ=Eε=E(σsE)=σs\sigma = E \varepsilon = E \left(\frac{\sigma_s}{E}\right) = \sigma_s

  • 由于材料是不可压缩的，体积应变为零，满足：

    εx+εy+εz=0\varepsilon_x + \varepsilon_y + \varepsilon_z = 0

    在这里，轴向方向为 xx，横向为 yy 和 zz。由于轴向应变已给定，横向应变需要调整以满足不可压缩条件。

  • 冯·米塞斯屈服准则在混合应力状态下为：

    σeq=σ2+3τ2=σs\sigma_{\text{eq}} = \sqrt{\sigma^2 + 3 \tau^2} = \sigma_s

    代入已知的轴向应力 σ=σs\sigma = \sigma_s：

    σs2+3τ2=σs2  ⟹  τ=0\sigma_s^2 + 3 \tau^2 = \sigma_s^2 \implies \tau = 0

    这意味着在施加轴向拉应变后，剪应力 τ\tau 必须降为零，以保持等效应力不超过屈服强度 σs\sigma_s。

最终结果

  • 轴向应力：

    σ=σs\sigma = \sigma_s

  • 剪应力：

    τ=0\tau = 0

总结

在这个加载过程中，首先通过施加剪应变使圆筒达到屈服点，此时剪应力 τ=σs3\tau = \frac{\sigma_s}{\sqrt{3}}。随后，通过施加轴向拉应变，轴向应力增加到 σ=σs\sigma = \sigma_s，为了满足冯·米塞斯屈服准则，剪应力必须降为零。因此，最终圆筒中的应力状态为轴向应力 σ=σs\sigma = \sigma_s 和剪应力 τ=0\tau = 0。

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